Final Results of the MAJORANA DEMONSTRATOR’s Search for Double-Beta Decay of 76Ge to Excited States of 76Se (2024)

Majorana Collaboration

I.J.ArnquistPacific Northwest National Laboratory, Richland, WA 99354, USA F.T.AvignoneIIIDepartment of Physics and Astronomy, University of South Carolina, Columbia, SC 29208, USAOak Ridge National Laboratory, Oak Ridge, TN 37830, USA A.S.Barabash \orcidlink0000-0002-5130-0922National Research Center “Kurchatov Institute”, Kurchatov Complex of Theoretical and Experimental Physics, Moscow, 117218 Russia E.BlalockDepartment of Physics, North Carolina State University, Raleigh, NC 27695, USATriangle Universities Nuclear Laboratory, Durham, NC 27708, USA B.BosDepartment of Physics and Astronomy, University of North Carolina, Chapel Hill, NC 27514, USATriangle Universities Nuclear Laboratory, Durham, NC 27708, USA M.BuschDepartment of Physics, Duke University, Durham, NC 27708, USATriangle Universities Nuclear Laboratory, Durham, NC 27708, USA Y.-D.ChanNuclear Science Division, Lawrence Berkeley National Laboratory, Berkeley, CA 94720, USA J.R.Chapman \orcidlink0009-0004-9815-2981Department of Physics and Astronomy, University of North Carolina, Chapel Hill, NC 27514, USATriangle Universities Nuclear Laboratory, Durham, NC 27708, USA C.D.Christofferson \orcidlink0009-0005-1842-9352South Dakota Mines, Rapid City, SD 57701, USA P.-H.Chu \orcidlink0000-0003-1372-2910Los Alamos National Laboratory, Los Alamos, NM 87545, USA C.Cuesta \orcidlink0000-0003-1190-7233Centro de Investigaciones Energéticas, Medioambientales y Tecnológicas, CIEMAT 28040, Madrid, Spain J.A.Detwiler \orcidlink0000-0002-9050-4610Center for Experimental Nuclear Physics and Astrophysics, and Department of Physics, University of Washington, Seattle, WA 98195, USA Yu.EfremenkoDepartment of Physics and Astronomy, University of Tennessee, Knoxville, TN 37916, USAOak Ridge National Laboratory, Oak Ridge, TN 37830, USA H.EjiriResearch Center for Nuclear Physics, Osaka University, Ibaraki, Osaka 567-0047, Japan S.R.Elliott \orcidlink0000-0001-9361-9870Los Alamos National Laboratory, Los Alamos, NM 87545, USA N.Fuad \orcidlink0000-0002-5445-2534Center for Exploration of Energy and Matter, and Department of Physics, Indiana University, Bloomington, IN 47405, USA G.K.GiovanettiPhysics Department, Williams College, Williamstown, MA 01267, USA M.P.Green \orcidlink0000-0002-1958-8030Department of Physics, North Carolina State University, Raleigh, NC 27695, USATriangle Universities Nuclear Laboratory, Durham, NC 27708, USAOak Ridge National Laboratory, Oak Ridge, TN 37830, USA J.Gruszko \orcidlink0000-0002-3777-2237Department of Physics and Astronomy, University of North Carolina, Chapel Hill, NC 27514, USATriangle Universities Nuclear Laboratory, Durham, NC 27708, USA I.S.Guinn \orcidlink0000-0002-2424-3272Oak Ridge National Laboratory, Oak Ridge, TN 37830, USA V.E.Guiseppe \orcidlink0000-0002-0078-7101Oak Ridge National Laboratory, Oak Ridge, TN 37830, USA C.R.HaufeDepartment of Physics and Astronomy, University of North Carolina, Chapel Hill, NC 27514, USATriangle Universities Nuclear Laboratory, Durham, NC 27708, USA R.Henning \orcidlink0000-0001-8651-2960Department of Physics and Astronomy, University of North Carolina, Chapel Hill, NC 27514, USATriangle Universities Nuclear Laboratory, Durham, NC 27708, USA D.HervasAguilar \orcidlink0000-0002-9686-0659Present address: Technical University of Munich, 85748 Garching, GermanyDepartment of Physics and Astronomy, University of North Carolina, Chapel Hill, NC 27514, USATriangle Universities Nuclear Laboratory, Durham, NC 27708, USA E.W.Hoppe \orcidlink0000-0002-8171-7323Pacific Northwest National Laboratory, Richland, WA 99354, USA I.Kim \orcidlink0000-0002-8394-6613Present address: Lawrence Livermore National Laboratory, Livermore, CA 94550, USALos Alamos National Laboratory, Los Alamos, NM 87545, USA R.T.Kouzes \orcidlink0000-0002-6639-4140Pacific Northwest National Laboratory, Richland, WA 99354, USA T.E.LannenVDepartment of Physics and Astronomy, University of South Carolina, Columbia, SC 29208, USA A.Li \orcidlink0000-0002-4844-9339Halıcıoğlu Data Science Institute, Department of Physics, University of California San Diego, CA 92093, USA R.Massarczyk \orcidlink0000-0001-8001-9235Los Alamos National Laboratory, Los Alamos, NM 87545, USA S.J.Meijer \orcidlink0000-0002-1366-0361Los Alamos National Laboratory, Los Alamos, NM 87545, USA T.K.Oli \orcidlink0000-0001-8857-3716Present address: Argonne National Laboratory, Lemont, IL 60439, USADepartment of Physics, University of South Dakota, Vermillion, SD 57069, USA L.S.Paudel \orcidlink0000-0003-3100-4074Department of Physics, University of South Dakota, Vermillion, SD 57069, USA W.Pettus \orcidlink0000-0003-4947-7400Center for Exploration of Energy and Matter, and Department of Physics, Indiana University, Bloomington, IN 47405, USA A.W.P.Poon \orcidlink0000-0003-2684-6402Nuclear Science Division, Lawrence Berkeley National Laboratory, Berkeley, CA 94720, USA D.C.RadfordOak Ridge National Laboratory, Oak Ridge, TN 37830, USA A.L.Reine \orcidlink0000-0002-5900-8299Center for Exploration of Energy and Matter, and Department of Physics, Indiana University, Bloomington, IN 47405, USA K.Rielage \orcidlink0000-0002-7392-7152Los Alamos National Laboratory, Los Alamos, NM 87545, USA D.C.Schaper \orcidlink0000-0002-6219-650XPresent address: Indiana Universty, Bloomington, IN 47405, USALos Alamos National Laboratory, Los Alamos, NM 87545, USA S.J.Schleich \orcidlink0000-0003-1878-9102Center for Exploration of Energy and Matter, and Department of Physics, Indiana University, Bloomington, IN 47405, USA D.TedeschiDepartment of Physics and Astronomy, University of South Carolina, Columbia, SC 29208, USA R.L.Varner \orcidlink0000-0002-0477-7488Oak Ridge National Laboratory, Oak Ridge, TN 37830, USA S.VasilyevJoint Institute for Nuclear Research, Dubna, 141980 Russia S.L.Watkins \orcidlink0000-0003-0649-1923Present address: Pacific Northwest National LaboratoryLos Alamos National Laboratory, Los Alamos, NM 87545, USA J.F.Wilkerson \orcidlink0000-0002-0342-0217Department of Physics and Astronomy, University of North Carolina, Chapel Hill, NC 27514, USATriangle Universities Nuclear Laboratory, Durham, NC 27708, USAOak Ridge National Laboratory, Oak Ridge, TN 37830, USA C.Wiseman \orcidlink0000-0002-4232-1326Center for Experimental Nuclear Physics and Astrophysics, and Department of Physics, University of Washington, Seattle, WA 98195, USA C.-H.Yu \orcidlink0000-0002-9849-842XOak Ridge National Laboratory, Oak Ridge, TN 37830, USA B.X.ZhuPresent address: Jet Propulsion Laboratory, California Institute of Technology, Pasadena, CA 91109, USALos Alamos National Laboratory, Los Alamos, NM 87545, USA

(October 11, 2024)

Abstract

⁷⁶Ge can $\beta\beta$ decay into three possible excited states of ⁷⁶Se, with the emission of two or, if the neutrino is Majorana, zero neutrinos.None of these six transitions have yet been observed.The Majorana Demonstratorwas designed to study $\beta\beta$ decay of ⁷⁶Ge using a low background array of high purity germanium detectors.With 98.2kg-y of isotopic exposure, the Demonstratorsets the strongest half-life limits to date for all six transition modes.For $2\nu\beta\beta$ to the $0^{+}_{1}$ state of ⁷⁶Se, this search has begun to probe for the first time half-life values predicted using modern many-body nuclear theory techniques, setting a limit of $T_{1/2}>1.5\times 10^{24}$ y (90% CL).

pacs:

23.40-s, 23.40.Bw, 14.60.Pq, 27.50.+j

Double-beta ( $\beta\beta$ ) decay is a rare second-order weak nuclear process in which two neutrons simultaneously decay to two protons and emit two electrons. $\beta\beta$ decay was predicted by Goeppert-Mayer to occur in even-even nuclei in which a single $\beta$ decay is forbidden[1].Furthermore, if the neutrino is a Majorana fermion, meaning it is its own antiparticle[2], then it is possible for neutrinoless double-beta decay ( $0\nu\beta\beta$ ) to occur[3]. $\beta\beta$ decay with the emission of two neutrinos ( $2\nu\beta\beta$ ) has been directly measured in 11isotopes, with half-lives in a range of $10^{18}-10^{22}$ yr[4]. $0\nu\beta\beta$ has not been observed, but its discovery would prove that the neutrino is a Majorana fermion[5], provide an example of lepton number violation in nature, and might provide a mechanism for the generation of the observed matter-antimatter asymmetry in the universe[6, 7].As a result, a robust experimental program has risen to search for $0\nu\beta\beta$ in a variety of isotopes[8, 9, 10, 11].

$\beta\beta$ decay can cause a transition of parent nuclei to daughters in either the ground state (G.S.) or an energetically allowed excited state (E.S.)[12].Decays to E.S.s have a lower Q-value than decays to the G.S., and they include the prompt emission of one or more $\gamma$ rays.The branching ratios are suppressed for E.S.decays relative to G.S.decays, due to the smaller phase space of the decay.To date, only $\beta\beta$ transitions to the first $0^{+}$ E.S.of two isotopes have been observed, in ¹⁰⁰Mo ( $T_{1/2}=6.7^{+0.5}_{-0.4}\times 10^{20}$ y)[13, 14, 15, 16, 17, 18, 19, 20] and ¹⁵⁰Nd ( $T_{1/2}=1.18^{+0.23}_{-0.20}\times 10^{20}$ y)[21, 22, 23, 24, 25].

Applying Fermi’s golden rule and the closure approximation, we can express the half-life for $2\nu\beta\beta$ as:

T_{1/2}^{-1}=G^{2\nu}_{d.s}\cdot(g_{A}^{eff,2\nu})^{4}\cdot|M^{2\nu}_{d.s}|^{2}

(1)

where $G^{2\nu}_{d.s}$ is the phase space factor (PSF), which depends on the daughter nuclear state, $(g_{A}^{eff,2\nu})^{2}$ is the axial vector coupling constant with an empirical quenching term applied, and $|M^{2\nu}_{d.s}|$ is the nuclear matrix element.The PSF can be accurately calculated[26, 27, 28], but there is large uncertainty on $(g_{A}^{eff,2\nu})^{4}\cdot|M^{2\nu}_{d.s}|^{2}$ [29]; this means that half-life measurements of $2\nu\beta\beta$ to both the G.S.and E.S.sserve as useful tests of nuclear many-body models used to compute the nuclear matrix element.In addition, the nuclear matrix element for $2\nu\beta\beta$ transitions to $2^{+}$ states is sensitive to a bosonic component of the neutrino wave function[30, 31].

For $0\nu\beta\beta$ when dominated by light neutrino exchange, the half-life can be expressed as:

T_{1/2}^{-1}=G^{0\nu}\cdot(g^{eff,0\nu}_{A})^{4}|M^{0\nu}|^{2}\langle m_{\beta%\beta}\rangle^{2}

(2)

where $m_{\beta\beta}$ is the effective Majorana mass of the electron neutrino, a coherent average of the neutrino masses.Given an accurate calculation of $(g^{eff,0\nu}_{A})^{4}|M^{0\nu}|^{2}$ , a measurement of the $0\nu\beta\beta$ half-life would provide information about the neutrino mass and Majorana CP-phases[32].Furthermore, the branching ratio for $\beta\beta$ decay modes to E.S.s can vary depending on the physics mechanisms; this means that a measurement of $0\nu\beta\beta$ to an E.S.of the daughter nucleus could help inform how to extend the Standard Model in order to accommodate Majorana neutrinos[33].

⁷⁶Ge is a promising isotope with an active experimental program for measuring $\beta\beta$ decay[34].Arrays of high purity germanium (HPGe) detectors manufactured from germanium that has been isotopically enriched in ⁷⁶Ge are capable of achieving high detection efficiency, an ultra-low background rate, and excellent energy resolution.The Majorana Demonstrator[35], which operated HPGe detectors in vacuum, and GERDA[36], which operated HPGe detectors submerged in liquid argon instrumented to act as an active veto, both recently completed their data-taking campaigns and achieved the two lowest background indices and best energy resolutions in their searches for $0\nu\beta\beta$ out of any experiments performed to date.

⁷⁶Ge can decay into three E.S.s of ⁷⁶Se with a decay structure shown in Fig.1; these transitions have never been observed before.Searches for E.S.decay modes were performed by both the Demonstrator[37] and GERDA[38] by searching for peaks produced when the deexcitation $\gamma$ rays escape the detector of origin and are fully absorbed in a second HPGe detector.

The Majorana Demonstratorsearched for $0\nu\beta\beta$ and $\beta\beta$ decay to E.S.using an array of HPGe detectors.The experiment consisted of two modules, each of which consisted of an array of HPGe detectors operated in a separate vacuum cryostat.The modules were constructed from ultra-low background materials[39, 40] and placed in a low-background passive shield, surrounded by a muon veto with nearly $4\pi$ -coverage[41, 42].For each module, a ²²⁸Th line source was stored outside of this shield and deployed once per week along a helical track surrounding the cryostat to calibrate the detectors[43].The experiment was located at the 4850ft level (4300 m.w.e.) in the Davis campus of the Sanford Underground Research Facility, in Lead, SD[44].

The Demonstratorutilized three types of p-type HPGe detector geometries, each of which had a $p^{+}$ -type point-like electrode on one face, and an $n^{+}$ -type electrode on the other surfaces.These detector geometries were chosen for their excellent energy resolution and for their ability to discriminate single- and multi-site events using pulse-shape discrimination (PSD) techniques.Most of the detectors (up to 35detectors totalling 29.7kg) were p-type point-contact (PPC) detectors[45], with an isotopic fraction of $87.4\%~{}^{76}$ Ge.In addition, up to 23 BEGe™detectors[46] totalling 14.4kg were used, with a natural isotopic abundance of 7.8% ⁷⁶Ge.Near the end of the Demonstrator’s operation for $0\nu\beta\beta$ , 4 inverted coaxial point contact (ICPC) detectors[47] were installed, totalling 6.7kg, with an isotopic fraction of $88\%~{}^{76}$ Ge.

Module1 began operation in its low background configuration in December 2015, and Module2 began in August 2016.For this analysis, we divide data into five datasets, listed in Tab.2, based on major hardware changes; these are combinations of the 13datasets described in Ref.[35].We exclude a period from Oct.2016 to Jan.2017 with higher electronic noise due to sub-optimal grounding.During most of its operation, blinding was applied via data parsing, with cycles of 31h of open data followed by 93h of blind data.

Signals from the HPGe detectors were digitized[48] and analyzed on-disk.Each detector was read out using two channels with differing gains; the high-gain channels had a dynamic range up to ${\sim}3$ MeV and the low-gain channels had a dynamic range up to ${\sim}10$ MeV.Each digitizer channel was triggered independently using an internal trapezoidal filter, typically with energy thresholds of $<1$ keV for the high-gain channels.Waveform energies are corrected for digitizer non-linearity[49] and charge-trapping[50], and were calibrated once per week using Th-chain gamma peaks[51].

The detection signature used to identify $\beta\beta$ to E.S.s is to search for energy peaks created by the full absorption of a gamma in a detector different from the site of the decay.This signature directly takes advantage of the Demonstrator’s strength in peak searches, which derives from its excellent energy resolution and operation in vacuum cryostats.Because the site of the decay will be inside of a detector, we can expect a typical event following this signature to involve multiple detector hits in coincidence; thus, we reject events with a detector multiplicity of 1. $\beta\beta$ to E.S.events with multiplicity1 will produce a broad spectral signal.Since the spectral signal from $2\nu\beta\beta$ to the G.S.will be a large background for this feature we would gain little sensitivity from this signal compared to our selected signature; in addition, analysis of broad spectral features is subject to greater systematic uncertainties from the modelling of backgrounds to which this peak search analysis is immune.

Detector hits that fall within a rolling $4~{}\mu$ s window are combined into events, with event multiplicity defined as the number of HPGe detector hits in a single event.Most hits have data from both the high and low gain channels; the signal from the high-gain channels were preferentially used for the analysis, unless the energy surpassed the dynamic range or the high gain channel did not have a trigger that passed all data cleaning cuts while the low gain did; if both channels fail data cleaning cuts the entire event is rejected.In addition, we reject events that occur during periods of high microphonic noise during fills of the LN dewars used to cool the cryostats, and within 20ms before and 1s after a muon event, removing $<1\%$ of events[42, 52].Additional digital signal processing algorithms are used to remove non-physical events, keeping $>99.9\%$ of physics events.High multiplicity events also contain valuable information from multiple detectors which we can use to achieve a large reduction in backgrounds.Similar techniques were used for each E.S.transition, and any differences will be noted.This analysis technique was also used in Ref.[37]; since then, multiple refinements have been made to further improve sensitivity, which will be noted.The background rejection cuts were developed and optimized using open data.

We use simulations to estimate the detection efficiency of these peaks including the effect of background cuts, and to optimize the tradeoff between signal sacrifice and background reduction in order to boost our sensitivity.MaGe[53] is a Geant4[54] based software library that implements the full as-built geometry for each experimental configuration of the Majorana Demonstratorand produces Monte-Carlo simulations of a variety of physical processes.To generate $\beta\beta$ decays to E.S.s, the DECAY0[55] library was used, with several modifications.DECAY0 was modified to include angular correlations in the deexcitation $\gamma$ rays from the $2^{+}_{2}$ E.S.of ⁷⁶Se (the angular correlation for the $0^{+}_{1}$ E.S.was already included), and the precision of the energy values used to generate $\gamma$ rays was increased, from 559 to 559.101keV, from 563 to 563.178keV, and from 1216 to 1216.104keV[56].For other radioactive backgrounds and calibration source simulations, the standard radioactive decay module built into Geant4 was used to generate events.

Step data produced by Geant4 is post-processed to simulate the observables produced by HPGe detectors.This stage simulates the effect of dead time from detectors that are disabled or unstable, from data cleaning cuts, and from hardware retriggering by randomly rejecting detector hits in proportion with the time spent in that configuration.The effect of transition dead layers is simulated by reducing the energy for steps in the transition layers caused by low charge collection efficiency within $\sim 1$ mm of the $n^{+}$ detector surfaces.One hundred separate sets of post-processed simulations are produced for each dataset and E.S.decay mode, varying the transition dead layer parameters to study the systematic effect from uncertainty in the dead layer thickness.In addition, a background model simulation is produced by sampling from about 100 post-processed simulations of a variety of isotopes in different hardware components, in proportion with the fitted activities from Ref.[57].

Final Results of the MAJORANA DEMONSTRATOR’s Search for Double-Beta Decay of 76Ge to Excited States of 76Se (1)

We apply a sequence of background reduction cuts, determined based on our simulations to improve the sensitivity of the experiment.The “Enriched Source Detector Cut” rejects hits that are not in coincidence with an enriched detector.The “Hot Detector Cut” rejects events that include one of two detectors closest to the Module1 crossarm; this cut was not included in Ref.[37]These detectors have significantly elevated background rates consistent with ²³²Th progeny in a cavity in the interface between the Module1 cryostat cold plate and crossarm[58, 57].For $\beta\beta$ to the $2^{+}_{1}$ E.S.and $2^{+}_{2}$ E.S.with the emision of a 1216keV $\gamma$ ray, we additionally require that an event has a multiplicity of exactly 2, since these modes only emit a single $\gamma$ ray.

Because $\gamma$ rays that are fully absorbed inside a detector typically Compton scatter at least once, we use the $AvsE$ PSD parameter to select for multi-site events[59]. $AvsE$ uses a comparison between the current amplitude ( $A$ ) and energy ( $E$ ) of a pulse to discriminate between single- and multi-site events; multi-site events usually have a lower $A$ for a given $E$ in point-contact HPGe detectors relative to single-site events. $AvsE$ is calibrated to measure 90% single-site events in the ²⁰⁸Tl double-escape peak (DEP), and is corrected for correlations with drift time and energy[35].Because of the difficulty of simulating the effect of $AvsE$ for a wide variety of energies and event topologies, we instead calculate this cut efficiency for each peak using ²²⁸Th calibration data.We measure the multi-site acceptance of 16 full energy peaks (FEPs) in the range of 400keV to 1700keV, and model dependence on energy ( $E_{FEP}$ ) as

\varepsilon_{AvsE}(E_{FEP},p_{0},p_{1})=p_{0}-\frac{p_{1}}{E_{FEP}}

(3)

We perform best fit of the parameters $p_{0}\simeq 0.92$ and $p_{1}\simeq 3.4\times 10^{4}$ keV over all detectors and calibration runs for each dataset, and the fit performs well with a $\chi^{2}$ value ranging from 10-20 for 14 d.o.f.We additionally check for a systematic effect based on the incidence angle of $\gamma$ rays using the acceptance of the 511keV annihilation peak in coincidence with a DEP or single-escape peak (SEP) event, which originates inside of detectors instead of from the calibration track; this is consistent with the above model.We also check for variance over time in the DEP, SEP, and Compton continuum and in variance between the acceptance in many detectors of the 583keV FEP; these are used to calculate systematic uncertainty terms, with the dominant uncertainty arising from variance over time.Based on this, we measure an acceptance of $80.9\pm 0.2\%$ for the 559and 563keV $\gamma$ rays from the $0^{+}_{1}$ E.S.of ⁷⁶Se.This cut was not used in Ref.[37]

The last set of cuts we apply are the Coincident- and Sum-Energy cuts, which reject events where either the sum over all hits or any hit in a coincident detector fall within a set of energy ranges.These cuts are designed to reject multi-detector events from $\gamma$ ray cascades and Compton-scattered $\gamma$ rays from common backgrounds, respectively.The energy ranges were determined algorithmically to optimize the discovery sensitivity of the experiment, based on the signal and background acceptance efficiencies determined using simulations of the E.S.modes and of the background model.The events were binned both by sum- and coincident hit energies, and bins were added to the cut if doing so improved the sensitivity.To avoid statistical biases towards cutting statistical fluctuations in the simulations, a new energy range was only introduced to the cut if we estimated it to have a $>97\%$ chance of improving sensitivity.For the $0\nu\beta\beta$ decay to the $2^{+}_{1}$ and the single- $\gamma$ branch of the $2^{+}_{2}$ mode, we do not apply this algorithm; instead we only apply a coincident energy cut around the $Q_{\beta\beta}$ -value of the decays since the coincident energy spectrum is strongly peaked.

Cut Description	$\varepsilon_{signal}$	$\varepsilon_{background}$
Gamma FEP Efficiency	5.4%	–
Multiplicity $\geq 2$	71.5%	8.0%
Enriched Source Detector Cut	97.5%	63.6%
Hot Detector Cut	97.4%	88.9%
Multi-site Gamma	80.9%	56.2%
Coincident Energy Cut	82.6%	54.8%
Sum Energy Cut	88.8%	56.9%
Total	2.2%	0.8%

The effect of the cuts for $\beta\beta$ to the $0^{+}_{1}$ E.S.is shown in Tab.1 and Fig.2, and the final detection efficiency for each decay mode can be seen in Tab.3.These were determined by measuring the efficiency in simulations and multiplying the $AvsE$ efficiency determined for each energy peak.One source of systematic uncertainty is derived from the variance introduced by changing the simulation post-processing parameters, measured to be $0.06\%$ ; the dominant source of uncertainty arises from the thickness of the dead layer.In addition, uncertainty in the spectral shape from DECAY0 was estimated by performing a Kolmogorov-Smirnov test comparing the G.S.spectrum to a more precise determination from Ref.[26]; this was found to be $0.01\%$ .Finally, because of the reliance on simulations to calculate the detection efficiency, we validate the simulations using DEPs in coincidence with full absorption of a 511keV annihilation $\gamma$ as proxies for $\beta\beta$ to E.S.s, since these events originate from inside of detectors.To do this, we use data collected using a ⁵⁶Co line source that was inserted into each calibration track for a period of one week; this source emits many high-energy $\gamma$ rays, and we used 6 DEPs and 7 SEPs.To validate the simulation, we compared the measured ratio of the peak amplitudes for events in coincidence with a 511keV hit and for multiplicity1 hits.We found an average disagreement in this ratio of 2.2%, with a significant difference of 8.4% in DEPs in Module1.The source of this disagreement cannot be explained by effects such as dead layers, so we include it as a systematic uncertainty term, which scales to $0.23\%$ ; this is our dominant uncertainty.

The measured half-life is calculated using

T_{1/2}=\frac{\mathrm{ln}2N_{A}\varepsilon M_{iso}T_{live}}{m_{76}\langle s\rangle}

(4)

where $N_{A}$ is avogadro’s number, $m_{76}=75.9$ g is the molar mass of ⁷⁶Ge, and $\langle s\rangle$ is the estimated combined amplitude of the signal peaks.The isotopic exposure $M_{iso}T_{live}=98.2\pm 0.5$ kg-y is the product of the total mass of ⁷⁶Ge in a module times the operating time of the module summed over datasets.Note that this differs from the active exposure defined in Ref.[35] which includes all Ge in active regions of the detector and subtracts dead time; instead, these effects are included as reductions in detection efficiency, and variation in the number of active detectors is the primary driver for the variation in efficiency between datasets seen in Tab.2.

Dataset	Time	Efficiency	Exposure	BG Index
		Period	(kg-y)	(cts/keV-kg-y)
DS I M1	7/15-10/15	$2.04(24)\%$	$1.92(1)$	$0.056(23)$
DS II M1	12/15-8/16	$2.12(24)\%$	$5.02(3)$	$0.021(9)$
DS III M1	8/16-11/19	$2.83(25)\%$	$39.68(22)$	$0.028(4)$
DS III M2	8/16-11/19	$0.99(22)\%$	$30.90(16)$	$0.012(3)$
DS IV M1	11/19-8/20	$2.07(25)\%$	$7.36(4)$	$0.015(6)$
DS V M1	8/20-3/21	$2.30(24)\%$	$7.14(4)$	$0.023(8)$
DS V M2	8/20-3/21	$3.62(27)\%$	$6.14(4)$	$0.018(7)$

We use a profile likelihood analysis to construct Neyman confidence intervals for the half-life of each E.S.decay mode.The data are modelled using one or more peaks, using the measured peakshape function, on a flat background; we also include nuisance parameters for uncertainty in the detection efficiency, drift in the peak position, and uncertainty in the peak width.The model is applied for events in an energy range of $520-575$ keV for the 559-and 563-keV peaks, $620-690$ keV for the 657keV peak, omitting $660-670$ keV to remove a U-chain peak, and $1180-1300$ keV for the 1216keV peak, omitting $1230-1245$ keV to remove another U-chain peak.For each dataset, we independently calculate the exposure and detection efficiency, and we float an independent background index.This is done due to the large variations based on which detectors were enabled; most noticeably, the efficiency and backgrounds both increased significantly in Module2 after the upgrade resulted in almost all detectors being active.We use an extended unbinned likelihood function, implemented with iminuit[60], to calculate our confidence intervals.For the $2^{+}_{2}$ decay modes, we simultaneously profile over all three peaks.Wilks’ theorem is applied to calculate p-values for all modes except for the $0\nu\beta\beta$ to $2^{+}_{1}$ , which had zero counts in the background window; in this case, p-values were calculated through Monte-Carlo sampling.For all $\beta\beta$ to E.S.decay modes, we measure a null result, with detailed results shown in Tab.3.

Final Results of the MAJORANA DEMONSTRATOR’s Search for Double-Beta Decay of 76Ge to Excited States of 76Se (2)

Decay Mode	Peak Energies (keV)	Peak FWHM (keV)	Efficiency	$\langle s\rangle$ BF	$\langle s\rangle$ Limit	$T_{1/2}$ Limit	$T_{1/2}$ Sensitivity
$0^{+}_{g.s.}\xrightarrow{2\nu\beta\beta}0^{+}_{1}$	559.1, 563.2	1.12, 1.13	$2.15(24)\%$	1.3	8.0	$1.5\times 10^{24}$ y	$2.2\times 10^{24}$ y
$0^{+}_{g.s.}\xrightarrow{2\nu\beta\beta}2^{+}_{1}$	559.1	1.12	$1.15(26)\%$	0.0	2.1	$3.0\times 10^{24}$ y	$2.1\times 10^{24}$ y
$0^{+}_{g.s.}\xrightarrow{2\nu\beta\beta}2^{+}_{2}$	559.1, 657.0, 1216.1	1.12, 1.22, 1.73	$1.76(29)\%$	2.1	10.9	$0.88\times 10^{24}$ y	$1.5\times 10^{24}$ y
$0^{+}_{g.s.}\xrightarrow{0\nu\beta\beta}0^{+}_{1}$	559.1, 563.2	1.12, 1.13	$2.83(32)\%$	0.0	2.0	$7.6\times 10^{24}$ y	$5.9\times 10^{24}$ y
$0^{+}_{g.s.}\xrightarrow{0\nu\beta\beta}2^{+}_{1}$	559.1	1.12	$1.58(35)\%$	0.0	2.5	$3.5\times 10^{24}$ y	$3.5\times 10^{24}$ y
$0^{+}_{g.s.}\xrightarrow{0\nu\beta\beta}2^{+}_{2}$	559.1, 657.0, 1216.1	1.12, 1.22, 1.73	$2.16(32)\%$	0.0	1.7	$6.6\times 10^{24}$ y	$4.3\times 10^{24}$ y

Combined with the measured half-life for $2\nu\beta\beta$ to the G.S.of ⁷⁶Se of $T_{1/2}=2.05^{+0.04}_{-0.05~{}syst}\pm 0.01_{stat}\times 10^{21}$ yr[57], the limit for the $0^{+}_{1}$ E.S.corresponds to a branching ratio of $\mathrm{BR}<0.0014$ (sensitivity of $\mathrm{BR}<0.0009$ ).We compare this BR limit to predictions using theoretical calculations of the PSFs[28] and nuclear matrix elements, under the assumption that the same value for $g^{eff,2\nu}_{A}$ applies to each daughter state, using $\mathrm{BR}=\frac{G^{2\nu}_{E.S.}|M^{2\nu}_{E.S.}|^{2}}{G^{2\nu}_{G.S.}|M^{2%\nu}_{G.S.}|^{2}}$ .Several variants of Quasiparticle Random Phase Approximation (QRPA) have been applied[61, 62, 63], with the minimum predicted BR of 0.0045 strongly disfavored with a P-value of $1\times 10^{-5}$ .An Effective Theory (ET) predicted a BR of 0.0011-0.0012[64], which is disfavored with a p-value of 0.23.The Nuclear Shell Model (NSM) predicted a BR of 0.00068-0.00076[38] and the Interacting Boson Model (IBM2) predicted 0.00025-0.00027[65], both beyond the sensitivity of this search.For $2\nu\beta\beta$ to the $2^{+}_{1}$ E.S., BR predictions range from $4.2\times 10^{-7}-2.1\times 10^{-6}$ [66, 64, 67], well beyond the sensitivity of this search.

The Majorana Demonstratorhas set the most stringent limits to date for all E.S.decay modes in ⁷⁶Ge. We achieved sensitivity to half-life values for $2\nu\beta\beta$ to the $0^{+}_{1}$ state of ⁷⁶Se in a range predicted by recent calculations.We have benefited from the excellent energy resolution of the experiment and from operating the detectors in vacuum.Future experimental efforts from the LEGEND collaboration[34] will use detectors operated in a liquid argon active veto, which will increase shielding between detectors and introduce backgrounds from ⁴²K; thus, LEGEND will likely require new analysis techniques to significantly improve on this result.

This material is based upon work supported by the U.S.Department of Energy, Office of Science, Office of Nuclear Physics under contract / award numbers DE-AC02-05CH11231, DE-AC05-00OR22725, DE-AC05-76RL0130, DE-FG02-97ER41020, DE-FG02-97ER41033, DE-FG02-97ER41041, DE-SC0012612, DE-SC0014445, DE-SC0017594, DE-SC0018060, DE-SC0022339, and LANLEM77/LANLEM78. We acknowledge support from the Particle Astrophysics Program and Nuclear Physics Program of the National Science Foundation through grant numbers MRI-0923142, PHY-1003399, PHY-1102292, PHY-1206314, PHY-1614611, PHY-13407204, PHY-1812409, PHY-1812356, PHY-2111140, and PHY-2209530. We gratefully acknowledge the support of the Laboratory Directed Research & Development (LDRD) program at Lawrence Berkeley National Laboratory for this work. We gratefully acknowledge the support of the U.S.Department of Energy through the Los Alamos National Laboratory LDRD Program, the Oak Ridge National Laboratory LDRD Program, and the Pacific Northwest National Laboratory LDRD Program for this work. We gratefully acknowledge the support of the South Dakota Board of Regents Competitive Research Grant.We acknowledge the support of the Natural Sciences and Engineering Research Council of Canada, funding reference number SAPIN-2017-00023, and from the Canada Foundation for Innovation John R.Evans Leaders Fund.We acknowledge support from the 2020/2021 L’Oréal-UNESCO for Women in Science Programme.This research used resources provided by the Oak Ridge Leadership Computing Facility at Oak Ridge National Laboratory and by the National Energy Research Scientific Computing Center, a U.S.Department of Energy Office of Science User Facility. We thank our hosts and colleagues at the Sanford Underground Research Facility for their support.

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