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 ββ𝛽𝛽\beta\betaitalic_β italic_βdecay 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 ββ𝛽𝛽\beta\betaitalic_β italic_βdecay 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 2νββ2𝜈𝛽𝛽2\nu\beta\beta2 italic_ν italic_β italic_βto the 01+subscriptsuperscript010^{+}_{1}0 start_POSTSUPERSCRIPT + end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT 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 T1/2>1.5×1024subscript𝑇121.5superscript1024T_{1/2}>1.5\times 10^{24}italic_T start_POSTSUBSCRIPT 1 / 2 end_POSTSUBSCRIPT > 1.5 × 10 start_POSTSUPERSCRIPT 24 end_POSTSUPERSCRIPTy (90% CL).

pacs:

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

Double-beta (ββ𝛽𝛽\beta\betaitalic_β italic_β) decay is a rare second-order weak nuclear process in which two neutrons simultaneously decay to two protons and emit two electrons.ββ𝛽𝛽\beta\betaitalic_β italic_βdecay was predicted by Goeppert-Mayer to occur in even-even nuclei in which a single β𝛽\betaitalic_βdecay 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 (0νββ0𝜈𝛽𝛽0\nu\beta\beta0 italic_ν italic_β italic_β) to occur[3].ββ𝛽𝛽\beta\betaitalic_β italic_βdecay with the emission of two neutrinos (2νββ2𝜈𝛽𝛽2\nu\beta\beta2 italic_ν italic_β italic_β) has been directly measured in 11isotopes, with half-lives in a range of 10181022superscript1018superscript102210^{18}-10^{22}10 start_POSTSUPERSCRIPT 18 end_POSTSUPERSCRIPT - 10 start_POSTSUPERSCRIPT 22 end_POSTSUPERSCRIPTyr[4].0νββ0𝜈𝛽𝛽0\nu\beta\beta0 italic_ν italic_β italic_βhas 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 0νββ0𝜈𝛽𝛽0\nu\beta\beta0 italic_ν italic_β italic_βin a variety of isotopes[8, 9, 10, 11].

ββ𝛽𝛽\beta\betaitalic_β italic_βdecay 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 γ𝛾\gammaitalic_γrays.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 ββ𝛽𝛽\beta\betaitalic_β italic_β transitions to the first 0+superscript00^{+}0 start_POSTSUPERSCRIPT + end_POSTSUPERSCRIPT E.S.of two isotopes have been observed, in 100Mo (T1/2=6.70.4+0.5×1020subscript𝑇12subscriptsuperscript6.70.50.4superscript1020T_{1/2}=6.7^{+0.5}_{-0.4}\times 10^{20}italic_T start_POSTSUBSCRIPT 1 / 2 end_POSTSUBSCRIPT = 6.7 start_POSTSUPERSCRIPT + 0.5 end_POSTSUPERSCRIPT start_POSTSUBSCRIPT - 0.4 end_POSTSUBSCRIPT × 10 start_POSTSUPERSCRIPT 20 end_POSTSUPERSCRIPTy)[13, 14, 15, 16, 17, 18, 19, 20] and 150Nd (T1/2=1.180.20+0.23×1020subscript𝑇12subscriptsuperscript1.180.230.20superscript1020T_{1/2}=1.18^{+0.23}_{-0.20}\times 10^{20}italic_T start_POSTSUBSCRIPT 1 / 2 end_POSTSUBSCRIPT = 1.18 start_POSTSUPERSCRIPT + 0.23 end_POSTSUPERSCRIPT start_POSTSUBSCRIPT - 0.20 end_POSTSUBSCRIPT × 10 start_POSTSUPERSCRIPT 20 end_POSTSUPERSCRIPTy)[21, 22, 23, 24, 25].

Applying Fermi’s golden rule and the closure approximation, we can express the half-life for 2νββ2𝜈𝛽𝛽2\nu\beta\beta2 italic_ν italic_β italic_βas:

T1/21=Gd.s2ν(gAeff,2ν)4|Md.s2ν|2superscriptsubscript𝑇121subscriptsuperscript𝐺2𝜈formulae-sequence𝑑𝑠superscriptsuperscriptsubscript𝑔𝐴𝑒𝑓𝑓2𝜈4superscriptsubscriptsuperscript𝑀2𝜈formulae-sequence𝑑𝑠2T_{1/2}^{-1}=G^{2\nu}_{d.s}\cdot(g_{A}^{eff,2\nu})^{4}\cdot|M^{2\nu}_{d.s}|^{2}italic_T start_POSTSUBSCRIPT 1 / 2 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT = italic_G start_POSTSUPERSCRIPT 2 italic_ν end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_d . italic_s end_POSTSUBSCRIPT ⋅ ( italic_g start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_e italic_f italic_f , 2 italic_ν end_POSTSUPERSCRIPT ) start_POSTSUPERSCRIPT 4 end_POSTSUPERSCRIPT ⋅ | italic_M start_POSTSUPERSCRIPT 2 italic_ν end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_d . italic_s end_POSTSUBSCRIPT | start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT(1)

where Gd.s2νsubscriptsuperscript𝐺2𝜈formulae-sequence𝑑𝑠G^{2\nu}_{d.s}italic_G start_POSTSUPERSCRIPT 2 italic_ν end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_d . italic_s end_POSTSUBSCRIPT is the phase space factor (PSF), which depends on the daughter nuclear state, (gAeff,2ν)2superscriptsuperscriptsubscript𝑔𝐴𝑒𝑓𝑓2𝜈2(g_{A}^{eff,2\nu})^{2}( italic_g start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_e italic_f italic_f , 2 italic_ν end_POSTSUPERSCRIPT ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT is the axial vector coupling constant with an empirical quenching term applied, and |Md.s2ν|subscriptsuperscript𝑀2𝜈formulae-sequence𝑑𝑠|M^{2\nu}_{d.s}|| italic_M start_POSTSUPERSCRIPT 2 italic_ν end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_d . italic_s end_POSTSUBSCRIPT | is the nuclear matrix element.The PSF can be accurately calculated[26, 27, 28], but there is large uncertainty on (gAeff,2ν)4|Md.s2ν|2superscriptsuperscriptsubscript𝑔𝐴𝑒𝑓𝑓2𝜈4superscriptsubscriptsuperscript𝑀2𝜈formulae-sequence𝑑𝑠2(g_{A}^{eff,2\nu})^{4}\cdot|M^{2\nu}_{d.s}|^{2}( italic_g start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_e italic_f italic_f , 2 italic_ν end_POSTSUPERSCRIPT ) start_POSTSUPERSCRIPT 4 end_POSTSUPERSCRIPT ⋅ | italic_M start_POSTSUPERSCRIPT 2 italic_ν end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_d . italic_s end_POSTSUBSCRIPT | start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT[29]; this means that half-life measurements of 2νββ2𝜈𝛽𝛽2\nu\beta\beta2 italic_ν italic_β italic_βto 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 2νββ2𝜈𝛽𝛽2\nu\beta\beta2 italic_ν italic_β italic_βtransitions to 2+superscript22^{+}2 start_POSTSUPERSCRIPT + end_POSTSUPERSCRIPT states is sensitive to a bosonic component of the neutrino wave function[30, 31].

For 0νββ0𝜈𝛽𝛽0\nu\beta\beta0 italic_ν italic_β italic_β  when dominated by light neutrino exchange, the half-life can be expressed as:

T1/21=G0ν(gAeff,0ν)4|M0ν|2mββ2superscriptsubscript𝑇121superscript𝐺0𝜈superscriptsubscriptsuperscript𝑔𝑒𝑓𝑓0𝜈𝐴4superscriptsuperscript𝑀0𝜈2superscriptdelimited-⟨⟩subscript𝑚𝛽𝛽2T_{1/2}^{-1}=G^{0\nu}\cdot(g^{eff,0\nu}_{A})^{4}|M^{0\nu}|^{2}\langle m_{\beta%\beta}\rangle^{2}italic_T start_POSTSUBSCRIPT 1 / 2 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT = italic_G start_POSTSUPERSCRIPT 0 italic_ν end_POSTSUPERSCRIPT ⋅ ( italic_g start_POSTSUPERSCRIPT italic_e italic_f italic_f , 0 italic_ν end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT ) start_POSTSUPERSCRIPT 4 end_POSTSUPERSCRIPT | italic_M start_POSTSUPERSCRIPT 0 italic_ν end_POSTSUPERSCRIPT | start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ⟨ italic_m start_POSTSUBSCRIPT italic_β italic_β end_POSTSUBSCRIPT ⟩ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT(2)

where mββsubscript𝑚𝛽𝛽m_{\beta\beta}italic_m start_POSTSUBSCRIPT italic_β italic_β end_POSTSUBSCRIPT is the effective Majorana mass of the electron neutrino, a coherent average of the neutrino masses.Given an accurate calculation of (gAeff,0ν)4|M0ν|2superscriptsubscriptsuperscript𝑔𝑒𝑓𝑓0𝜈𝐴4superscriptsuperscript𝑀0𝜈2(g^{eff,0\nu}_{A})^{4}|M^{0\nu}|^{2}( italic_g start_POSTSUPERSCRIPT italic_e italic_f italic_f , 0 italic_ν end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT ) start_POSTSUPERSCRIPT 4 end_POSTSUPERSCRIPT | italic_M start_POSTSUPERSCRIPT 0 italic_ν end_POSTSUPERSCRIPT | start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT, a measurement of the 0νββ0𝜈𝛽𝛽0\nu\beta\beta0 italic_ν italic_β italic_βhalf-life would provide information about the neutrino mass and Majorana CP-phases[32].Furthermore, the branching ratio for ββ𝛽𝛽\beta\betaitalic_β italic_βdecay modes to E.S.s can vary depending on the physics mechanisms; this means that a measurement of 0νββ0𝜈𝛽𝛽0\nu\beta\beta0 italic_ν italic_β italic_βto 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 ββ𝛽𝛽\beta\betaitalic_β italic_βdecay[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 0νββ0𝜈𝛽𝛽0\nu\beta\beta0 italic_ν italic_β italic_βout 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 γ𝛾\gammaitalic_γ rays escape the detector of origin and are fully absorbed in a second HPGe detector.


The Majorana Demonstratorsearched for 0νββ0𝜈𝛽𝛽0\nu\beta\beta0 italic_ν italic_β italic_βand ββ𝛽𝛽\beta\betaitalic_β italic_βdecay 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 4π4𝜋4\pi4 italic_π-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+superscript𝑝p^{+}italic_p start_POSTSUPERSCRIPT + end_POSTSUPERSCRIPT-type point-like electrode on one face, and an n+superscript𝑛n^{+}italic_n start_POSTSUPERSCRIPT + end_POSTSUPERSCRIPT-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%7687.4\%~{}^{76}87.4 % start_FLOATSUPERSCRIPT 76 end_FLOATSUPERSCRIPTGe.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 0νββ0𝜈𝛽𝛽0\nu\beta\beta0 italic_ν italic_β italic_β, 4 inverted coaxial point contact (ICPC) detectors[47] were installed, totalling 6.7kg, with an isotopic fraction of 88%7688\%~{}^{76}88 % start_FLOATSUPERSCRIPT 76 end_FLOATSUPERSCRIPTGe.

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 3similar-toabsent3{\sim}3∼ 3MeV and the low-gain channels had a dynamic range up to 10similar-toabsent10{\sim}10∼ 10MeV.Each digitizer channel was triggered independently using an internal trapezoidal filter, typically with energy thresholds of <1absent1<1< 1keV 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 ββ𝛽𝛽\beta\betaitalic_β italic_βto 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.ββ𝛽𝛽\beta\betaitalic_β italic_βto E.S.events with multiplicity1 will produce a broad spectral signal.Since the spectral signal from 2νββ2𝜈𝛽𝛽2\nu\beta\beta2 italic_ν italic_β italic_βto 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 4μ4𝜇4~{}\mu4 italic_μs 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%absentpercent1<1\%< 1 % of events[42, 52].Additional digital signal processing algorithms are used to remove non-physical events, keeping >99.9%absentpercent99.9>99.9\%> 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 ββ𝛽𝛽\beta\betaitalic_β italic_βdecays to E.S.s, the DECAY0[55] library was used, with several modifications.DECAY0 was modified to include angular correlations in the deexcitation γ𝛾\gammaitalic_γ rays from the 22+subscriptsuperscript222^{+}_{2}2 start_POSTSUPERSCRIPT + end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPTE.S.of 76Se (the angular correlation for the 01+subscriptsuperscript010^{+}_{1}0 start_POSTSUPERSCRIPT + end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPTE.S.was already included), and the precision of the energy values used to generate γ𝛾\gammaitalic_γrays 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 1similar-toabsent1\sim 1∼ 1mm of the n+superscript𝑛n^{+}italic_n start_POSTSUPERSCRIPT + end_POSTSUPERSCRIPT 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 ββ𝛽𝛽\beta\betaitalic_β italic_βto the 21+subscriptsuperscript212^{+}_{1}2 start_POSTSUPERSCRIPT + end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT E.S.and 22+subscriptsuperscript222^{+}_{2}2 start_POSTSUPERSCRIPT + end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT E.S.with the emision of a 1216keV γ𝛾\gammaitalic_γray, we additionally require that an event has a multiplicity of exactly 2, since these modes only emit a single γ𝛾\gammaitalic_γray.

Because γ𝛾\gammaitalic_γrays that are fully absorbed inside a detector typically Compton scatter at least once, we use the AvsE𝐴𝑣𝑠𝐸AvsEitalic_A italic_v italic_s italic_E PSD parameter to select for multi-site events[59].AvsE𝐴𝑣𝑠𝐸AvsEitalic_A italic_v italic_s italic_E uses a comparison between the current amplitude (A𝐴Aitalic_A) and energy (E𝐸Eitalic_E) of a pulse to discriminate between single- and multi-site events; multi-site events usually have a lower A𝐴Aitalic_A for a given E𝐸Eitalic_E in point-contact HPGe detectors relative to single-site events.AvsE𝐴𝑣𝑠𝐸AvsEitalic_A italic_v italic_s italic_E 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 AvsE𝐴𝑣𝑠𝐸AvsEitalic_A italic_v italic_s italic_E 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 (EFEPsubscript𝐸𝐹𝐸𝑃E_{FEP}italic_E start_POSTSUBSCRIPT italic_F italic_E italic_P end_POSTSUBSCRIPT) as

εAvsE(EFEP,p0,p1)=p0p1EFEPsubscript𝜀𝐴𝑣𝑠𝐸subscript𝐸𝐹𝐸𝑃subscript𝑝0subscript𝑝1subscript𝑝0subscript𝑝1subscript𝐸𝐹𝐸𝑃\varepsilon_{AvsE}(E_{FEP},p_{0},p_{1})=p_{0}-\frac{p_{1}}{E_{FEP}}italic_ε start_POSTSUBSCRIPT italic_A italic_v italic_s italic_E end_POSTSUBSCRIPT ( italic_E start_POSTSUBSCRIPT italic_F italic_E italic_P end_POSTSUBSCRIPT , italic_p start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , italic_p start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ) = italic_p start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT - divide start_ARG italic_p start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_ARG start_ARG italic_E start_POSTSUBSCRIPT italic_F italic_E italic_P end_POSTSUBSCRIPT end_ARG(3)

We perform best fit of the parameters p00.92similar-to-or-equalssubscript𝑝00.92p_{0}\simeq 0.92italic_p start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ≃ 0.92 and p13.4×104similar-to-or-equalssubscript𝑝13.4superscript104p_{1}\simeq 3.4\times 10^{4}italic_p start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ≃ 3.4 × 10 start_POSTSUPERSCRIPT 4 end_POSTSUPERSCRIPTkeV over all detectors and calibration runs for each dataset, and the fit performs well with a χ2superscript𝜒2\chi^{2}italic_χ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT value ranging from 10-20 for 14 d.o.f.We additionally check for a systematic effect based on the incidence angle of γ𝛾\gammaitalic_γ 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±0.2%plus-or-minus80.9percent0.280.9\pm 0.2\%80.9 ± 0.2 % for the 559and 563keV γ𝛾\gammaitalic_γrays from the 01+subscriptsuperscript010^{+}_{1}0 start_POSTSUPERSCRIPT + end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT 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 γ𝛾\gammaitalic_γray cascades and Compton-scattered γ𝛾\gammaitalic_γ 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%absentpercent97>97\%> 97 % chance of improving sensitivity.For the 0νββ0𝜈𝛽𝛽0\nu\beta\beta0 italic_ν italic_β italic_βdecay to the 21+subscriptsuperscript212^{+}_{1}2 start_POSTSUPERSCRIPT + end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT and the single-γ𝛾\gammaitalic_γ branch of the 22+subscriptsuperscript222^{+}_{2}2 start_POSTSUPERSCRIPT + end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT mode, we do not apply this algorithm; instead we only apply a coincident energy cut around the Qββsubscript𝑄𝛽𝛽Q_{\beta\beta}italic_Q start_POSTSUBSCRIPT italic_β italic_β end_POSTSUBSCRIPT-value of the decays since the coincident energy spectrum is strongly peaked.

Cut Descriptionεsignalsubscript𝜀𝑠𝑖𝑔𝑛𝑎𝑙\varepsilon_{signal}italic_ε start_POSTSUBSCRIPT italic_s italic_i italic_g italic_n italic_a italic_l end_POSTSUBSCRIPTεbackgroundsubscript𝜀𝑏𝑎𝑐𝑘𝑔𝑟𝑜𝑢𝑛𝑑\varepsilon_{background}italic_ε start_POSTSUBSCRIPT italic_b italic_a italic_c italic_k italic_g italic_r italic_o italic_u italic_n italic_d end_POSTSUBSCRIPT
Gamma FEP Efficiency5.4%
Multiplicity 2absent2\geq 2≥ 271.5%8.0%
Enriched Source Detector Cut97.5%63.6%
Hot Detector Cut97.4%88.9%
Multi-site Gamma80.9%56.2%
Coincident Energy Cut82.6%54.8%
Sum Energy Cut88.8%56.9%
Total2.2%0.8%

The effect of the cuts for ββ𝛽𝛽\beta\betaitalic_β italic_βto the 01+subscriptsuperscript010^{+}_{1}0 start_POSTSUPERSCRIPT + end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT 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 AvsE𝐴𝑣𝑠𝐸AvsEitalic_A italic_v italic_s italic_E 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%percent0.060.06\%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%percent0.010.01\%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 γ𝛾\gammaitalic_γ as proxies for ββ𝛽𝛽\beta\betaitalic_β italic_βto 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 γ𝛾\gammaitalic_γ 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%percent0.230.23\%0.23 %; this is our dominant uncertainty.

The measured half-life is calculated using

T1/2=ln2NAεMisoTlivem76ssubscript𝑇12ln2subscript𝑁𝐴𝜀subscript𝑀𝑖𝑠𝑜subscript𝑇𝑙𝑖𝑣𝑒subscript𝑚76delimited-⟨⟩𝑠T_{1/2}=\frac{\mathrm{ln}2N_{A}\varepsilon M_{iso}T_{live}}{m_{76}\langle s\rangle}italic_T start_POSTSUBSCRIPT 1 / 2 end_POSTSUBSCRIPT = divide start_ARG ln2 italic_N start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT italic_ε italic_M start_POSTSUBSCRIPT italic_i italic_s italic_o end_POSTSUBSCRIPT italic_T start_POSTSUBSCRIPT italic_l italic_i italic_v italic_e end_POSTSUBSCRIPT end_ARG start_ARG italic_m start_POSTSUBSCRIPT 76 end_POSTSUBSCRIPT ⟨ italic_s ⟩ end_ARG(4)

where NAsubscript𝑁𝐴N_{A}italic_N start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT is avogadro’s number, m76=75.9subscript𝑚7675.9m_{76}=75.9italic_m start_POSTSUBSCRIPT 76 end_POSTSUBSCRIPT = 75.9g is the molar mass of 76Ge, and sdelimited-⟨⟩𝑠\langle s\rangle⟨ italic_s ⟩ is the estimated combined amplitude of the signal peaks.The isotopic exposure MisoTlive=98.2±0.5subscript𝑀𝑖𝑠𝑜subscript𝑇𝑙𝑖𝑣𝑒plus-or-minus98.20.5M_{iso}T_{live}=98.2\pm 0.5italic_M start_POSTSUBSCRIPT italic_i italic_s italic_o end_POSTSUBSCRIPT italic_T start_POSTSUBSCRIPT italic_l italic_i italic_v italic_e end_POSTSUBSCRIPT = 98.2 ± 0.5kg-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.

DatasetTimeEfficiencyExposureBG Index
Period(kg-y)(cts/keV-kg-y)
DS I M17/15-10/152.04(24)%2.04percent242.04(24)\%2.04 ( 24 ) %1.92(1)1.9211.92(1)1.92 ( 1 )0.056(23)0.056230.056(23)0.056 ( 23 )
DS II M112/15-8/162.12(24)%2.12percent242.12(24)\%2.12 ( 24 ) %5.02(3)5.0235.02(3)5.02 ( 3 )0.021(9)0.02190.021(9)0.021 ( 9 )
DS III M18/16-11/192.83(25)%2.83percent252.83(25)\%2.83 ( 25 ) %39.68(22)39.682239.68(22)39.68 ( 22 )0.028(4)0.02840.028(4)0.028 ( 4 )
DS III M28/16-11/190.99(22)%0.99percent220.99(22)\%0.99 ( 22 ) %30.90(16)30.901630.90(16)30.90 ( 16 )0.012(3)0.01230.012(3)0.012 ( 3 )
DS IV M111/19-8/202.07(25)%2.07percent252.07(25)\%2.07 ( 25 ) %7.36(4)7.3647.36(4)7.36 ( 4 )0.015(6)0.01560.015(6)0.015 ( 6 )
DS V M18/20-3/212.30(24)%2.30percent242.30(24)\%2.30 ( 24 ) %7.14(4)7.1447.14(4)7.14 ( 4 )0.023(8)0.02380.023(8)0.023 ( 8 )
DS V M28/20-3/213.62(27)%3.62percent273.62(27)\%3.62 ( 27 ) %6.14(4)6.1446.14(4)6.14 ( 4 )0.018(7)0.01870.018(7)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 520575520575520-575520 - 575keV for the 559-and 563-keV peaks, 620690620690620-690620 - 690keV for the 657keV peak, omitting 660670660670660-670660 - 670keV to remove a U-chain peak, and 11801300118013001180-13001180 - 1300keV for the 1216keV peak, omitting 12301245123012451230-12451230 - 1245keV 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 22+subscriptsuperscript222^{+}_{2}2 start_POSTSUPERSCRIPT + end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT decay modes, we simultaneously profile over all three peaks.Wilks’ theorem is applied to calculate p-values for all modes except for the 0νββ0𝜈𝛽𝛽0\nu\beta\beta0 italic_ν italic_β italic_βto 21+subscriptsuperscript212^{+}_{1}2 start_POSTSUPERSCRIPT + end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT, which had zero counts in the background window; in this case, p-values were calculated through Monte-Carlo sampling.For all ββ𝛽𝛽\beta\betaitalic_β italic_βto 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 ModePeak Energies (keV)Peak FWHM (keV)Efficiencysdelimited-⟨⟩𝑠\langle s\rangle⟨ italic_s ⟩ BFsdelimited-⟨⟩𝑠\langle s\rangle⟨ italic_s ⟩ LimitT1/2subscript𝑇12T_{1/2}italic_T start_POSTSUBSCRIPT 1 / 2 end_POSTSUBSCRIPT LimitT1/2subscript𝑇12T_{1/2}italic_T start_POSTSUBSCRIPT 1 / 2 end_POSTSUBSCRIPT Sensitivity
0g.s.+2νββ01+2𝜈𝛽𝛽subscriptsuperscript0formulae-sequence𝑔𝑠subscriptsuperscript010^{+}_{g.s.}\xrightarrow{2\nu\beta\beta}0^{+}_{1}0 start_POSTSUPERSCRIPT + end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_g . italic_s . end_POSTSUBSCRIPT start_ARROW start_OVERACCENT 2 italic_ν italic_β italic_β end_OVERACCENT → end_ARROW 0 start_POSTSUPERSCRIPT + end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT559.1, 563.21.12, 1.132.15(24)%2.15percent242.15(24)\%2.15 ( 24 ) %1.38.01.5×10241.5superscript10241.5\times 10^{24}1.5 × 10 start_POSTSUPERSCRIPT 24 end_POSTSUPERSCRIPTy2.2×10242.2superscript10242.2\times 10^{24}2.2 × 10 start_POSTSUPERSCRIPT 24 end_POSTSUPERSCRIPTy
0g.s.+2νββ21+2𝜈𝛽𝛽subscriptsuperscript0formulae-sequence𝑔𝑠subscriptsuperscript210^{+}_{g.s.}\xrightarrow{2\nu\beta\beta}2^{+}_{1}0 start_POSTSUPERSCRIPT + end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_g . italic_s . end_POSTSUBSCRIPT start_ARROW start_OVERACCENT 2 italic_ν italic_β italic_β end_OVERACCENT → end_ARROW 2 start_POSTSUPERSCRIPT + end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT559.11.121.15(26)%1.15percent261.15(26)\%1.15 ( 26 ) %0.02.13.0×10243.0superscript10243.0\times 10^{24}3.0 × 10 start_POSTSUPERSCRIPT 24 end_POSTSUPERSCRIPTy2.1×10242.1superscript10242.1\times 10^{24}2.1 × 10 start_POSTSUPERSCRIPT 24 end_POSTSUPERSCRIPTy
0g.s.+2νββ22+2𝜈𝛽𝛽subscriptsuperscript0formulae-sequence𝑔𝑠subscriptsuperscript220^{+}_{g.s.}\xrightarrow{2\nu\beta\beta}2^{+}_{2}0 start_POSTSUPERSCRIPT + end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_g . italic_s . end_POSTSUBSCRIPT start_ARROW start_OVERACCENT 2 italic_ν italic_β italic_β end_OVERACCENT → end_ARROW 2 start_POSTSUPERSCRIPT + end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT559.1, 657.0, 1216.11.12, 1.22, 1.731.76(29)%1.76percent291.76(29)\%1.76 ( 29 ) %2.110.90.88×10240.88superscript10240.88\times 10^{24}0.88 × 10 start_POSTSUPERSCRIPT 24 end_POSTSUPERSCRIPTy1.5×10241.5superscript10241.5\times 10^{24}1.5 × 10 start_POSTSUPERSCRIPT 24 end_POSTSUPERSCRIPTy
0g.s.+0νββ01+0𝜈𝛽𝛽subscriptsuperscript0formulae-sequence𝑔𝑠subscriptsuperscript010^{+}_{g.s.}\xrightarrow{0\nu\beta\beta}0^{+}_{1}0 start_POSTSUPERSCRIPT + end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_g . italic_s . end_POSTSUBSCRIPT start_ARROW start_OVERACCENT 0 italic_ν italic_β italic_β end_OVERACCENT → end_ARROW 0 start_POSTSUPERSCRIPT + end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT559.1, 563.21.12, 1.132.83(32)%2.83percent322.83(32)\%2.83 ( 32 ) %0.02.07.6×10247.6superscript10247.6\times 10^{24}7.6 × 10 start_POSTSUPERSCRIPT 24 end_POSTSUPERSCRIPTy5.9×10245.9superscript10245.9\times 10^{24}5.9 × 10 start_POSTSUPERSCRIPT 24 end_POSTSUPERSCRIPTy
0g.s.+0νββ21+0𝜈𝛽𝛽subscriptsuperscript0formulae-sequence𝑔𝑠subscriptsuperscript210^{+}_{g.s.}\xrightarrow{0\nu\beta\beta}2^{+}_{1}0 start_POSTSUPERSCRIPT + end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_g . italic_s . end_POSTSUBSCRIPT start_ARROW start_OVERACCENT 0 italic_ν italic_β italic_β end_OVERACCENT → end_ARROW 2 start_POSTSUPERSCRIPT + end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT559.11.121.58(35)%1.58percent351.58(35)\%1.58 ( 35 ) %0.02.53.5×10243.5superscript10243.5\times 10^{24}3.5 × 10 start_POSTSUPERSCRIPT 24 end_POSTSUPERSCRIPTy3.5×10243.5superscript10243.5\times 10^{24}3.5 × 10 start_POSTSUPERSCRIPT 24 end_POSTSUPERSCRIPTy
0g.s.+0νββ22+0𝜈𝛽𝛽subscriptsuperscript0formulae-sequence𝑔𝑠subscriptsuperscript220^{+}_{g.s.}\xrightarrow{0\nu\beta\beta}2^{+}_{2}0 start_POSTSUPERSCRIPT + end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_g . italic_s . end_POSTSUBSCRIPT start_ARROW start_OVERACCENT 0 italic_ν italic_β italic_β end_OVERACCENT → end_ARROW 2 start_POSTSUPERSCRIPT + end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT559.1, 657.0, 1216.11.12, 1.22, 1.732.16(32)%2.16percent322.16(32)\%2.16 ( 32 ) %0.01.76.6×10246.6superscript10246.6\times 10^{24}6.6 × 10 start_POSTSUPERSCRIPT 24 end_POSTSUPERSCRIPTy4.3×10244.3superscript10244.3\times 10^{24}4.3 × 10 start_POSTSUPERSCRIPT 24 end_POSTSUPERSCRIPTy

Combined with the measured half-life for 2νββ2𝜈𝛽𝛽2\nu\beta\beta2 italic_ν italic_β italic_βto the G.S.of 76Se of T1/2=2.050.05syst+0.04±0.01stat×1021subscript𝑇12plus-or-minussubscriptsuperscript2.050.040.05𝑠𝑦𝑠𝑡subscript0.01𝑠𝑡𝑎𝑡superscript1021T_{1/2}=2.05^{+0.04}_{-0.05~{}syst}\pm 0.01_{stat}\times 10^{21}italic_T start_POSTSUBSCRIPT 1 / 2 end_POSTSUBSCRIPT = 2.05 start_POSTSUPERSCRIPT + 0.04 end_POSTSUPERSCRIPT start_POSTSUBSCRIPT - 0.05 italic_s italic_y italic_s italic_t end_POSTSUBSCRIPT ± 0.01 start_POSTSUBSCRIPT italic_s italic_t italic_a italic_t end_POSTSUBSCRIPT × 10 start_POSTSUPERSCRIPT 21 end_POSTSUPERSCRIPTyr[57], the limit for the 01+subscriptsuperscript010^{+}_{1}0 start_POSTSUPERSCRIPT + end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT E.S.corresponds to a branching ratio of BR<0.0014BR0.0014\mathrm{BR}<0.0014roman_BR < 0.0014 (sensitivity of BR<0.0009BR0.0009\mathrm{BR}<0.0009roman_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 gAeff,2νsubscriptsuperscript𝑔𝑒𝑓𝑓2𝜈𝐴g^{eff,2\nu}_{A}italic_g start_POSTSUPERSCRIPT italic_e italic_f italic_f , 2 italic_ν end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT applies to each daughter state, using BR=GE.S.2ν|ME.S.2ν|2GG.S.2ν|MG.S.2ν|2BRsubscriptsuperscript𝐺2𝜈formulae-sequence𝐸𝑆superscriptsubscriptsuperscript𝑀2𝜈formulae-sequence𝐸𝑆2subscriptsuperscript𝐺2𝜈formulae-sequence𝐺𝑆superscriptsubscriptsuperscript𝑀2𝜈formulae-sequence𝐺𝑆2\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}}roman_BR = divide start_ARG italic_G start_POSTSUPERSCRIPT 2 italic_ν end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_E . italic_S . end_POSTSUBSCRIPT | italic_M start_POSTSUPERSCRIPT 2 italic_ν end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_E . italic_S . end_POSTSUBSCRIPT | start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG start_ARG italic_G start_POSTSUPERSCRIPT 2 italic_ν end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_G . italic_S . end_POSTSUBSCRIPT | italic_M start_POSTSUPERSCRIPT 2 italic_ν end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_G . italic_S . end_POSTSUBSCRIPT | start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG.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 1×1051superscript1051\times 10^{-5}1 × 10 start_POSTSUPERSCRIPT - 5 end_POSTSUPERSCRIPT.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 2νββ2𝜈𝛽𝛽2\nu\beta\beta2 italic_ν italic_β italic_βto the 21+subscriptsuperscript212^{+}_{1}2 start_POSTSUPERSCRIPT + end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT E.S., BR predictions range from 4.2×1072.1×1064.2superscript1072.1superscript1064.2\times 10^{-7}-2.1\times 10^{-6}4.2 × 10 start_POSTSUPERSCRIPT - 7 end_POSTSUPERSCRIPT - 2.1 × 10 start_POSTSUPERSCRIPT - 6 end_POSTSUPERSCRIPT[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 2νββ2𝜈𝛽𝛽2\nu\beta\beta2 italic_ν italic_β italic_βto the 01+subscriptsuperscript010^{+}_{1}0 start_POSTSUPERSCRIPT + end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT 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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