Results on $\beta\beta$ decay with emission of two neutrinos or Majorons in $^{76}$Ge from GERDA Phase I

0.48 MB·English

by  M. Agostini

Eur. Phys. J. C manuscript No.
(will be inserted by the editor)
Results on ββ decay with emission of two neutrinos or
Majorons in 76Ge from GERDA Phase I
M. Agostini15, M. Allardt4, A.M. Bakalyarov13, M. Balata1, I. Barabanov11,
N. Barros4, L. Baudis19, C. Bauer7, N. Becerici-Schmidt14, E. Bellotti8,9,
S. Belogurov12,11, S.T. Belyaev13, G. Benato19, A. Bettini16,17,
L. Bezrukov11, T. Bode15, D. Borowicz3,5, V. Brudanin5, R. Brugnera16,17,
D. Budj´aˇs15, A. Caldwell14, C. Cattadori9, A. Chernogorov12,
V. D’Andrea1, E.V. Demidova12, A. di Vacri1, A. Domula4,
5 E. Doroshkevich11, V. Egorov5, R. Falkenstein18, O. Fedorova11,
1 K. Freund18, N. Frodyma3, A. Gangapshev11,7, A. Garfagnini16,17,
0
P. Grabmayr18, V. Gurentsov11, K. Gusev13,5,15, A. Hegai18, M. Heisel7,
2
S. Hemmer16,17, G. Heusser7, W. Hofmann7, M. Hult6, L.V. Inzhechik11,a,
n
J. Janicsk´o Cs´athy15, J. Jochum18, M. Junker1, V. Kazalov11, T. Kihm7,
a
J I.V. Kirpichnikov12, A. Kirsch7, A. Klimenko7,5,b, K.T. Kn¨opfle7,
0 O. Kochetov5, V.N. Kornoukhov12,11, V.V. Kuzminov11, M. Laubenstein1,
1 A. Lazzaro15, V.I. Lebedev13, B. Lehnert4, H.Y. Liao14, M. Lindner7,
I. Lippi17, A. Lubashevskiy7,5, B. Lubsandorzhiev11, G. Lutter6,
]
x C. Macolino1, B. Majorovits14, W. Maneschg7, E. Medinaceli16,17,
e M. Misiaszek3, P. Moseev11, I. Nemchenok5, D. Palioselitis14, K. Panas3,
-
l L. Pandola2, K. Pelczar3, A. Pullia10, S. Riboldi10, N. Rumyantseva5,
c
C. Sada16,17, M. Salathe7, C. Schmitt18, J. Schreiner7, O. Schulz14,
u
n B. Schwingenheuer7, S. Sch¨onert15, O. Selivanenko11, M. Shirchenko13,5,
[ H. Simgen7, A. Smolnikov7, L. Stanco17, M. Stepaniuk7, C.A. Ur17,

L. Vanhoefer14, A.A. Vasenko12, A. Veresnikova11, K. von Sturm16,17,
1
v V. Wagner7, M. Walter19, A. Wegmann7, T. Wester4, H. Wilsenach4,
5 M. Wojcik3, E. Yanovich11, P. Zavarise1, I. Zhitnikov5, S.V. Zhukov13,
4 D. Zinatulina5, K. Zuber4, G. Zuzel3
3
2 1INFNLaboratoriNazionalidelGranSassoand GranSassoScienceInstitute,Assergi, Italy
0 2INFNLaboratoriNazionalidelSud,Catania, Italy
1. 3InstituteofPhysics,JagiellonianUniversity,Cracow, Poland
0 4Institutfu¨rKern-undTeilchenphysik,TechnischeUniversit¨atDresden,Dresden, Germany
5 5JointInstituteforNuclearResearch,Dubna,Russia
1 6InstituteforReference MaterialsandMeasurements,Geel,Belgium
: 7MaxPlanckInstitutfu¨rKernphysik,Heidelberg,Germany
v 8DipartimentodiFisica,Universita` MilanoBicocca, Milano,Italy
i 9INFNMilano Bicocca,Milano, Italy
X
10DipartimentodiFisica,Universita` degliStudidiMilanoe INFNMilano,Milano,Italy
r 11InstituteforNuclear ResearchoftheRussian AcademyofSciences,Moscow,Russia
a
12InstituteforTheoreticaland ExperimentalPhysics, Moscow,Russia
13NationalResearchCentre“KurchatovInstitute”, Moscow,Russia
14Max-Planck-Institut fu¨rPhysik,Mu¨nchen,Germany
15Physik DepartmentandExcellenceClusterUniverse, TechnischeUniversit¨atMu¨nchen,Mu¨nchen, Germany
16DipartimentodiFisicaeAstronomiadell‘Universit`adiPadova,Padova,Italy
17INFNPadova,Padova,Italy
18PhysikalischesInstitut,EberhardKarlsUniversita¨t Tu¨bingen,Tu¨bingen,Germany
19Physik InstitutderUniversita¨tZu¨rich,Zu¨rich,Switzerland
Received:date/Accepted:date
2
Abstract Asearchforneutrinolessββ decayprocesses where K is the summed energy of the two electrons, G
accompaniedwithMajoronemissionhasbeenperformed isthephasespace,Q istheQvalueofthe0νββdecay
ββ
usingdatacollectedduringPhaseIoftheGERmanium andnisthespectralindexofthemodel.SingleMajoron
DetectorArray(Gerda)experimentattheLaboratori emitting ββ decays can be roughly divided into three
Nazionali del Gran Sasso of INFN (Italy). Processes classes, n = 1, n = 2, and n = 3. Double Majoron
with spectral indices n = 1, 2, 3, 7 were searched for. emitting decays can have either n = 3 or n = 7. Their
No signals were found and lower limits of the order of characteristic spectral shapes differ from that of two-
1023 yr on their half-lives were derived, yielding sub- neutrinoββ decay(2νββ),forwhichn=5.Thisallows
stantially improved results compared to previous ex- for discrimination between the processes.
periments with 76Ge. A new result for the half-life of Experimental searches for ββ decay mediated by
the neutrino-accompanied ββ decay of 76Ge with sig- emission of one or two Majorons (0νββχ) have been
nificantly reduced uncertainties is also given, resulting performedbytheHeidelberg-Moscowexperiment(HdM)
in T2ν =(1.926±0.095)·1021 yr. for76Ge[8,9];byNemo-2andNemo-3for100Mo,116Cd,
1/2
82Se,96Zr,130Te [10,11];byELEGANTVfor100Mo[12];
Keywords double beta decay · Majoron emission ·
by DAMA [13] and by KAMland-Zen for 136Xe [14].
enriched 76Ge
Noneoftheseexperimentshaveseenanexcessofevents
PACS 23.40.-s β decay; double β decay; electron that could be interpreted as a Majoron signal; they re-
and muon capture · 14.80.Va majorons · 21.10.Tg ported lower limits on the half-lives of the processes
Lifetimes, widths · 27.50.+e mass 59 ≤ A ≤ 89 that involve Majoron emission.
The 2νββ decay process conserves lepton number
andisindependentofthenatureoftheneutrino.Ithas
1 Introduction been detected for eleven nuclides so far, with measured
half-lives(T2ν )intherangeof7×1018−2×1024yr[15,
1/2
Neutrinoless double beta (0νββ) decay is regarded as
16,17]. The knowledge of T2ν allows for extraction of
the gold-plated process for probing the fundamental 1/2
the nuclear matrix element, M2ν, which can provide
characterofneutrinos.Observationofthisprocesswould
some constraints on that of 0νββ decay, M0ν, if the
imply total lepton number violation by two units and
evaluations of M for the two processes are performed
that neutrinos have a Majorana mass component. Al-
within the same model [18,19].
thoughthemainfocusofexperimentaleffortsliesonthe
This paper reports on the search for neutrinoless
detection of 0νββ decay mediated by light Majorana
doublebetadecayof76GewithMajoronemission(0νββχ)
neutrino exchange, there are also many other proposed
and a new analysis of the half-life of the 2νββ decay of
mechanisms which are being searched for. Some exotic
76Ge using data collected by the Gerda experiment
models predict 0νββ decays proceeding through the
duringitsPhaseI.2νββ decayisawellestablishedand
emissionofamasslessGoldstoneboson,calledMajoron.
previouslyobservedprocess,while0νββχdecayisahy-
Predictions of different models depend on its transfor-
potheticalone.Inthefirstcasethehalf-lifeisextracted,
mation properties under weak isospin, singlet [1], dou-
while for the second one a limit is set. This leads to
blet [2] and triplet [3]. Precise measurements of the in-
slightly different approaches in the analyses leading to
visiblewidthoftheZbosonatLEP[4]greatlydisfavour
different data sets and background components being
triplet and pure doublet models. Several new Majoron
used.
models have been developed subsequently in which the
Majoron carries leptonic charge and cannot be a Gold-
stone boson [5,6] or in which the 0νββ decay proceeds
2 The GERDA experiment
through the emission of two Majorons [7].
All these models predict different shapes of the two
The main aim of the Gerda experiment [20] at the
emitted electrons’ summed energy spectrum. The pre-
Laboratori Nazionali del Gran Sasso (LNGS) of INFN
dicted spectral shapes are essentially defined by the
inItalyistosearchfor0νββ decayof76Ge.Thecoreof
phase space of the emitted particles:
thesetupisanarrayofhigh-puritygermanium(HPGe)
dN detectorsmadefromisotopicallymodifiedmaterialwith
∼G∼(Q −K)n (1)
dK ββ 76Ge enriched to ∼86% (enrGe), mounted in low-mass
copper supports (holders) and immersed in a 64 m3
aalso at: Moscow Inst.ofPhysics andTechnology,Russia
cryostat filled with liquid argon (LAr). The LAr serves
balso at: Int. Univ. for Nature, Society and Man “Dubna”,
Dubna,Russia as cooling medium and shield against external back-
cCorrespondence,email: [email protected] grounds. The shielding is complemented by water in a
3
tank of 10 m in diameter which is instrumented with 3 Data taking and data selection
photomultiplierstodetectCherenkovlightgeneratedin
muon-induced showers [20].
Phase I data taking lasted from November 9, 2011, to
The array of HPGe detectors is arranged in strings. May 21, 2013. The total exposure collected comprises
Eachstringisenclosedwithacylinder,madefrom60µm 19.2 kg·yr for the coaxial detectors and 2.4 kg·yr for
thick Cu foil, called mini-shroud, to mitigate the back- the BEGe detectors. In this paper, the entire expo-
ground coming from the decay of 42Ar present in the sure collected by the BEGe detectors (BEGe data set)
LAr.Moreover,inordertopreventcontaminationfrom and 17.9 kg·yr from the coaxial detectors (golden data
radonwithinthecryostat,acylinder,madefrom30µm set) are used [25,26]. For the coaxial detectors, a data
thickCufoil,calledradon-shroud,separatesthecentral set collected for 1.3 kg·yr exposure during a restricted
part of the cryostat, where the detectors are located, timeperiodaroundthedeploymentoftheBEGedetec-
from the rest. The HPGe detector signals are read out torsisdiscardedduetoahigherbackgroundlevel.Also
with custom-made charge sensitive preamplifiers opti- one of the coaxial detectors, RG2, is not considered for
mizedforlowradioactivity,whichareoperatedcloseto the data analysis starting from March 2013, as its high
the detectors in the LAr. The analog signals are digi- voltage had to be reduced below depletion voltage due
tizedwith100MHzFlashADCs(FADC)andanalyzed to increased leakage current. The energy calibration of
offline. If one of the detectors has an energy deposition thedetectorswasperformedusingtheinformationfrom
above the trigger threshold (40-100 keV), all channels dedicated calibration runs. For these calibration runs,
arereadout.Reprocessedp-typecoaxialdetectorsfrom three 228Th sources were lowered to the vicinity of the
theHdM[21]andIgex[22]experimentswereoperated detectors. The stability of the energy scale was moni-
together with Broad Energy Germanium (BEGe) type tored by performing such calibration runs every one or
detectors manufactured by Canberra [23,24]. two weeks. Moreover, the stability of the system was
continuously monitored by injecting charge pulses into
Asexplainedinsection5,somebackgroundcompo-
the test input of the preamplifiers. Using physics data,
nents have different effects on the two detector types
the interpolated FWHM values at Q averaged with
due to their peculiar geometry. A schematic drawing ββ
the exposure are (4.8 ± 0.2) keV for the coaxial detec-
of a coaxial detector type is shown in the top part of
tors and (3.2 ± 0.2) keV for the BEGe detectors.
Fig. 1, while the lower part depicts that for a BEGe
type detector. AllstepsoftheofflineprocessingoftheGerdadata
were performed within the software framework Gela-
tio [27]. The energy deposited in each detector was
60-80 mm extracted from the respective charge pulse by applying
aapproximateGaussianfilter[28].Non-physicalevents,
such as discharges, cross-talk and pick-up noise events,
wererejectedbyqualitycutsbasedonthetimeposition
m C of the rising edge, the information from the Gaussian
0 m oax filter, the rise time and the charge pulse height, which
0-11 ial mustnotexceedthedynamicrangeoftheFADCs.Pile-
7 up and accidental coincidences were removed from the
p-type data set using cuts based on the baseline slope, the
Ge
number of triggers and the position of the rising edge.
The rate of pile-up and accidental coincidence events
~2 mm is negligible in the Gerda data due to the extremely
p+ electrode n+ electrode
(read-out) 0 V 3-4 kV loweventrate.Thelossduetomis-classificationbythe
≲1 mm
quality cuts was <0.1% for events with energies above
m 1 MeV. All events that come within 8 µs of a signal
0 m p-Gtyepe BEG from the muon veto were rejected. Finally, only events
25-5 e thatsurvivethedetectoranti-coincidencecutwerecon-
sidered. This means, that all events with an energy de-
65-80 mm
position >50 keV in more than one detector in the ar-
Fig. 1 Schematic sketch of a coaxial HPGe detector (top)
raywerenottakenintoaccount.Since2νββ and0νββχ
and a BEGe detector (bottom) with their different sur-
faces and dead layers (drawings not to scale), adapted from eventsreleasetheirenergywithinasmallvolumeinside
Ref.[25]. the detectors, almost no signal events were lost by this
4
cut, while a part of the γ-induced background events ismotivatedbytheβ-decayof39Ar,whichgivesalarge
were rejected. contribution up to its Q -value of 565 keV.
β
Thefollowingbackgroundcomponentswereusedfor
theextractionoftheT2ν (minimummodelinRefs.[25,
1/2
4 Analysis Strategy 29]):(1)76Ge2νββ decay,(2)214Bi,228Ac,228Th,60Co
and 40K decays in the close vicinity of the detectors
The two analyses described in this paper are different (<2 cm, represented by decays in the detector holders
in the sense that for 2νββ decay a parameter is ex- in the MC simulation), (3) decays of 60Co inside the
tracted for a well established and known process, while detectors, constrained by the maximum expected ac-
in the case of the search for 0νββχ decay limits for tivity from their cosmogenic activation history,(4) 42K
a hypothetical process are set. In order to minimize decays in LAr assuming a uniform distribution, (5) α-
the systematic uncertainties for the extraction of the modelthataccountsforαdecaysoriginatingfrom210Po
T2ν it is favorable to use a well defined and controlled and 226Ra contaminations on the p+ surface of the de-
1/2
subset of the data and to use only well identified back- tectors as well as from 222Rn in the LAr, and finally
groundprocesses.For0νββχlimitsettingitisfavorable (6) 214Bi decays on the p+ surface, constrained by the
to maximize the exposure and to take into account all estimated 226Ra activity from the α-model.
known possible background processes that can not be The parameters of all components besides the con-
unambiguouslydetectedbutcouldmimic0νββχdecay. strained ones were given a flat prior probability dis-
FortheT2ν analysisthegoldendataset(17.9kg·yr) tribution.Therearenostrongcorrelationsbetweenthe
1/2
with the coaxial detectors is used in order to have a modelparameterssinceallconsideredbackgroundcom-
large data sample obtained in well controlled experi- ponents have characteristic features such as γ-ray lines
mental conditions. The Majoron analysis uses both the or peak-like structures at different energies. The ratios
goldendatasetandtheBEGedatasetforatotalexpo- of the γ-ray line intensities from the individual consid-
sure of 20.3 kg·yr in order to maximize the sensitivity. ered background sources suggest contaminations dom-
The background model for the T2ν analysis uses a inantly in locations close to the detectors. Hence, the
1/2
minimal number of components, assuming all sources minimum model takes into account only the close-by
neartothedetectors[25,29].FortheMajoronanalysis, source locations. Nevertheless, the screening measure-
anexpandedmodelisused[30],takingintoaccountalso ments indicate contaminations of materials in farther
additionalmediumandfardistantpositionsforsomeof locations as well. An additional contribution can come
thesources.Thisbecomesnecessarywhensearchingfor from 42K decays at or near the detector n+ surfaces
rareprocessessuchasMajoronemission,whereallpos- (seeFig.1)withaspecificactivityhigherthanthatfor
sible sources of background which could simulate the the uniform distribution assumption. This component
exotic process have to be considered. Therefore, even is the dominating one for the BEGe data set, as the
theslightdifferencesresulting,forexamplefromavari- thinnerdeadlayerthicknessofBEGesofroughly1mm
ation of the source position, have to be evaluated. allowspenetrationoftheelectronsemittedinthedecay
In both analyses, the experimental spectra of the of 42K to the active volume, while for coaxial detectors
coaxialandBEGedetectorsareanalyzedusingtheBaye- the dead layer thickness of ∼2 mm efficiently shields
sian Analysis Toolkit (Bat) [31]. this background component.
The spectral shapes of the contributions from the
backgroundsourceswithoutsignificantmultipleγpeaks
5 The background model atdifferentsourcelocationsdifferonlymarginally.This
makes it impossible to pinpoint the exact source loca-
The background sources considered in the models were tionsgiventheavailablestatisticsofthemeasuredspec-
identified by their prominent structures in the energy tra.Therefore,variationsofthesourcelocationsforthe
spectra and were also expected on the basis of ma- considered decays were taken into account when evalu-
terial screening measurements. The spectral shapes of ating the systematic uncertainty on T2ν .
1/2
individual background contributions were obtained by FortheMajoronanalysisadditionalbackgroundcom-
using a detailed implementation of the experimental ponents were used [30], including also medium and far
setup in the Monte Carlo (MC) simulation framework distant contributions. For the coaxial detectors 42K on
MaGe [32]. A Bayesian spectral fit of the measured the n+ and on the p+ contacts was added to the list of
energy spectrum with the simulated spectra was per- theclosesourcesofthepreviousbackgroundmodel.For
formed in an energy range from 570 keV up to the end medium distances, i.e. between 2 cm and 50 cm from
ofthedynamicrangeat7500keV.Thelowenergylimit the detectors, contributions from the following sources
5
were added: 214Bi, 228Th and 228Ac. A 228Th contam- Table 1 Parameters for the coaxial detectors (upper part)
ination was chosen as a representative for far distant and for the BEGe detectors (lower part): live time, t, total
mass, M, the fraction of 76Ge atoms, f , and the active
sources (above 50 cm). Whenever possible, screening 76
volumefraction,f .Forthecoaxialdetectors,thefirstun-
AV
measurementswereusedtoconstrainthelowerlimitof
certaintyonf istheuncorrelatedpart,thesecondonethe
act
the expected background events. correlated contribution. The values for M, f and f are
76 AV
IntheMajoronanalysis,alsothedatacollectedwith taken fromRef.[25].
the BEGe diodes were used in order to maximize the
exposure. Consequently, the background model devel- detectors t M f76 fAV
[days] [kg] [%] [%]
oped for these detectors was used [25,30]. The same
enrichedcoaxialdetectors
close,mediumandfardistantsourcesasforthecoaxial
detectorswereused.68Gewasaddedasinternalsource. ANG2 485.5 2.833 86.6±2.5 87.1±4.3±2.8
ANG3 485.5 2.391 88.3±2.6 86.6±4.9±2.8
This was necessary in order to take into account the
ANG4 485.5 2.372 86.3±1.3 90.1±4.9±2.9
cosmic activation of the germanium due to the recent ANG5 485.5 2.746 85.6±1.3 83.1±4.0±2.7
production of these diodes. RG1 485.5 2.110 85.5±1.5 90.4±5.2±2.9
RG2 384.8 2.166 85.5±1.5 83.1±4.6±2.7
enrichedBEGedetectors
6 Determination of the half-life of 2νββ decay GD32B 280.0 0.717 87.7±1.3 89.0±2.7
GD32C 304.6 0.743 87.7±1.3 91.1±3.0
GD32D 282.7 0.723 87.7±1.3 92.3±2.6
6.1 Analysis
GD35B 301.2 0.812 87.7±1.3 91.4±2.9
TheT2ν of2νββ decayof76Gewasdeterminedconsid-
1/2
ering the golden data set of Phase I, amounting to an The detection efficiencies, on average εfit =0.667 and
AV
exposureof17.9kg·yr,andusingthebackgroundmodel εfit =0.011, are obtained through dedicated MC sim-
DL
predictionforthecontributionofthe2νββ spectrumto
ulations.Thestatisticaluncertaintyduetothenumber
the overall energy spectrum. Details of the background
of simulated events is on the order of 0.1%.
analysis can be found in Ref. [29].
Thebackgroundmodelresultedinascalingparame-
Theglobalfitforthebackgroundmodelingwasper- terofNfit =25690+310forthe2νββspectrum,whichis
2ν −330
formed on the summed energy spectrum of the coaxial
the best fit parameter. The uncertainty is given by the
detectors using a bin width of 30 keV. Thus, the scal-
smallest 68% probability interval of the marginalized
ingparameterofthe2νββ spectruminthemodel,Nfit,
2ν posteriorprobabilitydistribution.Usingthisresult,the
givesthenumberofeventsinthe2νββ spectruminthe
half-life derived according to Eq. 2 is
fitwindowof570–7500keVforalldetectors.Usingthis
result for the number of measured 2νββ events, the T2ν =(1.926+0.025)·1021 yr . (3)
1/2 −0.022
half-life is calculated as
T12/ν2 = m(ln2)NNfiAt N(cid:88)detMitif76,i(cid:2)fAV,i εfiAtV,i 6.2 Systematic Uncertainties
enr 2ν i=1
+(1−f ) εfit (cid:3), (2) The systematic uncertainties affecting the results for
AV,i DL,i
T2ν weregroupedintothethreecategories(i) detector
1/2
parameters and fit model, (ii) MC simulation, and (iii)
where N is Avogadro’s constant and m =75.6 g is
A enr data acquisition and selection.Thecontributionstothe
the molar mass of the enriched material. The summa-
totalsystematicuncertaintyonT2ν aresummarizedin
tionrunsoverallthedetectors(N )consideredinthe 1/2
det Table 2.
data set. All detector related parameters like the de-
(i) detector parameters and fit model
tector mass (M ), the time of the data taking for each
i
detector (t ), the fraction of 76Ge atoms (f ), the ac- – Thesystematicuncertaintyontheactive76Geexpo-
i 76,i
tive volume fraction (f ), and the detection efficien- sure(E )wasdeterminedusingaMCapproach.
AV,i AV,76
cies in the active volume (εfit ) and in the dead layer E is defined as
AV,i AV,76
(εfit ) are taken into account separately for the indi-
DL,i
vidual detectors. All values are listed in Table 1. The N(cid:88)det
E = M t f f . (4)
efficiency εfit (εfit ) corresponds to the probability AV,76 i i AV,i 76,i
AV,i DL,i
i=1
that a 2νββ decay taking place in the active volume
(dead layer) of the detector deposits detectable energy Forevaluatingitsuncertainty,theparametersofthe
inthefitwindowconsideredforthebackgroundmodel. individual detectors were randomly sampled from
6
Table2 ContributionstothesystematicuncertaintyonT2ν takenintoaccountinthiswork.Thetotalsystematicuncertainty
1/2
isobtainedbycombiningtheindividualcontributionsinquadrature.
Item UncertaintyonT2ν
1/2
[%]
Active76Geexposure ±4
Backgroundmodelcomponents +1.4
−1.2
Binning ±0.5
Shape ofthe2νββ spectrum <0.1
Subtotalfitmodel ±4.3
Precisionofthe MonteCarlogeometrymodel ±1
AccuracyoftheMonteCarlotracking ±2
SubtotalMonteCarlosimulation ±2.2
Dataacquisitionandhandling <0.1
Total ±4.8
Gaussian distributions with mean values and stan- 214Bi and 60Co decays in the holders, respectively.
dard deviations according to the corresponding val- Consequently, the source locations are moved fur-
ues listed in Table 1. The correlated terms for f ther out with respect to the reference model. Con-
AV
were also taken into account. The uncertainty on sistently, the models that include additional con-
the live time t is 0.3%, whereas the total detec- tributions from close source locations yield a de-
tor masses are known with good accuracy (uncer- crease in the T2ν value, e.g. including 214Bi in LAr
1/2
tainty smaller than 0.1%). The calculation yields close to the p+ surface (-1.0%) or 42K on the n+
EAV,76 = (13.45±0.54) kg·yr. The uncertainty of (-1.2%) and p+ (-0.6%) surfaces. Comparing alter-
4% is driven by the uncertainties on fAV and f76, native background models to the reference one, the
which mainly affect the number of 76Ge nuclei in deviations in the T2ν result range between -1.2%
1/2
the active volume of the detectors, with a relatively and +1.4%.
smaller impact on the detection efficiency for the – Forthestandardfit,abinwidthof30keVwasused
background sources. forthedataandMCenergyspectra.Inordertotake
– Thereferencebackgroundmodelusedfordetermin- into account the systematic uncertainty related to
ingT2ν accountsonlyforthedominantsourceloca- binning effects, the fit was repeated twice using bin
1/2
tions in the setup. The systematic uncertainty due widths of 10 and 50 keV. The bin width of 10 keV
to the choice of the background model components waschoseninordertominimizeasmuchaspossible
was evaluated by repeating the global fit with al- the bin size taking into account the energy resolu-
ternativemodels,whichaccountfordifferentsource tion of ≈4.5 keV of the coaxial detectors and the
locations for all the background sources considered necessitytohaveenoughstatisticsinallbins.Above
inthereferencemodel.Themodelthataccountsfor 50 keV, peak structures are washed out, leading to
228Th and 228Ac contributions also in the radon- adeteriorationofthefit.ThedeviationsintheT2ν
1/2
shroud instead of only in the holders results in a resultrangebetween-0.5%and+0.5%withrespect
1.4% longer T2ν . The same increase occurs if 40K to that using the standard bin width.
1/2
in the radon-shroud is added to the model compo- – The primary spectrum of the two electrons emitted
nents. The model including the contribution from in the 2νββ decay of 76Ge, which was then fed into
214Biintheradon-shroudinadditiontothep+ sur- the MC simulation, was sampled according to the
faceandholdersyieldsa0.7%longerT2ν .Inallthe distribution given in Ref. [33] implemented in De-
1/2
casesmentionedabove,thecontributionfromback- cay0[34].Thesystematicuncertaintyduetotheas-
groundinthe2νββ spectrumregionincreases,since sumed 2νββ spectral shape was evaluated by com-
the peak-to-Compton ratio of the γ-rays decreases paring the spectrum generated by Decay0 to the
for farther source locations leading to longer T2ν onegiveninRef.[35].Consideringtheanalysiswin-
1/2
estimates. Excluding contributions from very close dow used for background modeling, the maximum
source locations, like 214Bi on the p+ surface and deviation is 0.2% and the total deviation of the in-
60Coonthegermanium,resultsinasmallerincrease tegral in the analysis window is 0.1%. When the fit
of the best T2ν estimate. In this case, the contri- with the background model is repeated using the
1/2
butionsfromthesecomponentsarecompensatedby
7
spectrum of Ref. [35], the difference from the refer- (4%), which can only be reduced by performing new
ence T2ν result is less than 0.1%. and more precise measurements of the active masses
1/2
– Apossibleeffectofatransitionlayer,whereitisas- ofthecoaxialdetectors.Othersignificantcontributions
sumed that the n+ dead layer on the detector sur- are related to the Monte Carlo simulations (2.2%) and
faces is partially active, has been investigated [36, tothebackgroundmodelassumptions(+1.4%).Thelat-
−1.2%
37]. The dead layer thickness for individual detec- terhavebeensignificantlyreducedinthisanalysiscom-
tors assumed in MC simulations were given accord- paredtotheanalysisofthefirst5kg·yrofPhaseIdata
ing to the listed values in Ref. [25]. The transition reported in Ref. [41], where the systematic uncertainty
layer is modeled using two different assumptions: a due to the background model was +5.7%. The new re-
−2.1%
linearly and an exponentially increasing charge col- sult is in good agreement with that mentioned above.
lection efficiency in the dead layer. The systematic Adding further identified components to the reference
uncertainty on T2ν due to the 2νββ spectrum sim- backgroundmodelresultsinaslightincreaseofthebest
1/2
ulated with the transition layer is found to be neg- T2ν estimate.
1/2
ligible. The background level achieved in Gerda Phase I
is about one order of magnitude lower with respect
(ii) MC simulation
to predecessor 76Ge experiments, and has allowed the
The uncertainty related to the MC simulation arises
measurement of T2ν with an unprecedented signal-to-
from the precision of the experimental geometry model 1/2
background ratio of 3:1 in the 570–2039 keV interval.
implemented in MaGe (1%) and from the accuracy of
The ratio amounts to 4:1 for the smaller interval of
particle tracking (2%) performed by Geant4 [38,39].
600–1800 keV.
The total MC simulation uncertainty was estimated to
be2.2%bysumminginquadraturetheaforementioned
contributions. 7 Limits on Majoron-emitting double β-decays
of 76Ge
(iii) Data acquisition and selection
The trigger and reconstruction efficiencies for physical 7.1 Analysis
events are practically 100% above 100keV in Gerda.
The performance of the quality cuts applied in Phase I The search for 0νββχ was performed using the golden
data has been investigated through a visual analysis. and BEGe data sets, amounting to a total exposure
The total uncertainty related to data acquisition and of 20.3 kg·yr. The analysis employed the background
selection was estimated to be less than 0.1%. modeldescribedinsection5.Theinformationfromthe
Summinginquadraturetheuncertaintiesofthethree two data sets was combined in one fit, while keeping
groups gives a total systematic uncertainty of ±4.8%. their energy spectra distinct. A separate fit was per-
formed for each spectral index, containing the back-
ground contributions, the contributions from 2νββ de-
6.3 Results and Discussion cay, and also the Majoron component under study. A
single parameter, T0νχ, is considered common for the
Fig. 2 shows the experimental data together with the 1/2
twodatasets.Itisdefinedasthehalf-lifeoftherespec-
best fit model for the golden data set. The different
tive Majoron accompanied mode.
componentsoftheminimumbackgroundmodelarealso
In order to improve the detection efficiency for the
reported. The model is able to reproduce the experi-
Majoron processes with low n (n = 1,2), a slightly
mental data well, as shown in the lower panel of the
different event selection was used with respect to the
figure by the residuals. T2ν analysis.Ifaneventoccurswithenergydeposition
The best estimate of the T2ν of the 2νββ decay of 1/2
1/2 in two detectors and the energy deposit in the detec-
76Ge is: tor where the decay took place is below the threshold
T2ν =(cid:0)1.926+0.025 +0.092 (cid:1)·1021yr for the anti-coincidence cut, the event contributes to
1/2 −0.022stat−0.092syst
the energy spectrum of the other detector. Therefore,
=(1.926±0.095)·1021yr, (5)
when determining the total energy spectrum resulting
from decays in one of the detectors, the energy spectra
with the latter combining in quadrature the statistical from all detectors in the array have to be taken into
(fit)andsystematicuncertainties.Thetotaluncertainty account. Such a selection has no impact on the detec-
of 4.9% is dominated by the systematic uncertainties. tion efficiency for the Majoron process with n=3 and
Thelargestcontributiontothesystematicuncertainties 7 and 2νββ decay. The content of the i-th bin in the
comesfromtheuncertaintyontheactive76Geexposure combined energy spectrum of all N detectors in the
det
8
2n b b alpha
data
K42 LAr Co60 H
model K40 H Co60 Ge
Ac228 H Bi214 H
Th228 H Bi214 p+
)
keV golden data set (17.9 kg(cid:215)yr) A 14-12
D
30 103 GER
(
/
s
t
n
e
v
e 102
10
1
o 1000 1500 2000
ti data/model
ra 2.0 68% energy (keV)

el 95%
d
1.5 99.9%
o
m
/
a
1.0
t
a
d
1000 1500 2000
energy (keV)
Fig. 2 Upper panel: experimental data (markers) and the best fit model (black histogram) for the golden data set. The
contribution from 2νββ (green) and from the single background components are also shown. Lower panel: ratio between
experimental data and the prediction of the best fit model. The green, yellow and red regions are the smallest intervals
containing 68%,95%and99% probabilityfortheratioassumingthebest fitparameters,respectively[40].
array, for decays taking place in the active and dead spectrum:
part of detector α, becomes:
N(cid:88)det
λ0νχ = λα,0νχ . (7)
 i i
λαi,0νχ = m(ln2)TN0νAχMαf76,α·fAV,α N(cid:88)det tjεαAV,jΦαA,V0,νiχ,j For all fαo=u1r Majoron modes (n = 1,2,3,7) only lower
enr 1/2 j=1
limitsonthehalf-lifecanbegiven.Theywereobtained

N(cid:88)det from the 90% quantiles of the marginalized posterior
+(1−fAV,α) tjεαDL,jΦαD,L0ν,iχ,j (6) distributions. These lower limits for T0νχ, not taking
1/2
j=1 into account the systematic uncertainties, are in units
of 1023 yr: >4.4, >1.9, >0.9, and >0.4 for n = 1, 2, 3,
withΦα,0νχ (Φα,0νχ)givingthecontentofthei-thbin and7,respectively.Therespectivehalf-lifeofthe2νββ
AV,i,j DL,i,j
ofthenormalizedenergydistributionrecordedwithde- processderivedfromthisanalysisamountstoinunitsof
tector j for 0νββχ taking place in the active (dead) 1021yr:1.96±0.03 ,1.97±0.03 ,1.98±0.03 ,and
stat stat stat
volume of detector α. Summing up the simulations of 1.99±0.03 . Within the uncertainties coming from
stat
decays in all N detectors results in the final model the different background models and the different data
det
9
sets of the two analyses, the derived T2ν values are in tors plays a role when converting the number of
1/2
agreement (<1σ) with that discussed in section 6.3. events attributed to 0νββχ into T0νχ. The prob-
1/2
ability distribution of f for each detector is given
76
by a Gaussian function with mean values and stan-
7.2 Systematic Uncertainties
dard deviations as listed in Table 1.
– The data does not allow the resolution of the ambi-
Thesystematicuncertaintiesweredividedintothethree
guity regarding the exact positions of the near and
categories(i)detectorparametersandfitmodel,(ii)MC
medium distant sources. The 214Bi decays serves as
simulation, and (iii) data acquisition and selection.
a representative in order to estimate the impact of
(i) detector parameters and fit model this uncertainty. Their near position is represented
Uncertainties from the fitting procedure were folded by decays in the holders, in the mini-shroud or on
into the posterior distribution of T0νχ with a MC ap- then+ surfaceofthedetectors,eachhavingaprob-
1/2
proach. Each source of uncertainty is described by a ability of 1/3 in the sampling process. The medium
probability distribution. The fitting procedure was re- distantpositionisrepresentedbydecaysintheradon-
peated1000times,eachtimedrawingarandomnumber shroud or in the LAr, having a probability 1/2 in
for each source of uncertainty according to its proba- contrast.
bility distribution: – ExtensivestudiesofthecharacteristicsoftheBEGe
diodes suggest the presence of a transition layer be-
– Material screening measurement results were used
tweentheregionwherethedetectorisfullyefficient
toconstraintheminimumnumberofeventsexpected
andtheexternaldeadregion[36,37].Anuncertainty
from close and medium distant sources of the 214Bi
as high as ±0.5% on the lower limits of T0νχ is es-
and228Thdecays.Gaussiandistributionsdescribing 1/2
timated for this effect in the case of the BEGe de-
these lower limits used in the fit were derived from
tectors. This uncertainty was folded into the total
the mean and standard deviations of the screening
marginalizedposteriordistributionaposteriori.The
measurements. For details refer to Ref. [25].
corresponding uncertainty for the coaxial detectors
– As for the T2ν analysis, the standard fit uses a bin
1/2 is estimated to be negligible.
widthof30keVforthedataandMCenergyspectra.
Inordertodeterminethesystematicuncertaintyre- The marginalized posterior distributions for T0νχ
1/2
lated to binning effects the bin width was sampled derived from each of the 1000 individual fits were sum-
uniformly from 10 keV to 50 keV. med up. The resulting total marginalized posterior dis-
– Uncertainties on the active volume fractions enter tribution accounts for the statistical as well as for the
the model in several ways. On the one hand, the listed systematic uncertainties related to the fit model.
MC energy spectra for all internal sources, that is AsfortheT2ν analysis,theuncertaintiesontheac-
1/2
for 2νββ, 0νββχ, 60Co, and 68Ga decays, are af- tive volume fractions and on the enrichment fractions
fected, as the fraction of decays taking place in the are major contributions to the total uncertainty on the
active and dead part of the detectors changes with limits for T0νχ. However, the largest source of uncer-
1/2
changing f . On the other hand, the uncertainty tainty is the composition of the fit model and the indi-
AV
on the active volume fraction also plays a role for vidual background contributions. In the case of n = 1,
theshapeoftheenergyspectrumdueto42Kdecays a fit with a bin width of 50 keV weakens the limit by
on the n+ surface. Larger f means thinner n+ ≈ 16% compared to the standard fit, while the result
AV
dead layer and thus the possibility of an increased for T2ν is not affected at all. The stability of the T2ν
1/2 1/2
contributionfromtheelectronstothespectrum.For results shows the validity of the fit. The use of the al-
smaller f and thicker n+ dead layer, their con- ternativecloseandmediumdistantsourcepositionsfor
AV
tributions are expected to be reduced. The active 214Bi decays leads to maximal variations of +8.3 % of
−12.6
volumefractionforeachdetectorwassampledfrom the limit on T0νχ.
1/2
aGaussiandistributionwithmeanandstandardde-
(ii) MC simulation
viation according to Table 1. For the coaxial detec-
As in the case of the T2ν measurement, a total MC
tors,thepartialcorrelationsoftheuncertaintywere 1/2
simulation uncertainty of 2.2% has to be taken into
takenintoaccount.Thesimulatedspectraofthein-
account for effects related to the geometry implemen-
ternal sources as well as of the 42K decays on the
tation and particle tracking. It is folded into the total
n+ surface are composed according to the sampled
marginalized posterior distributions. No effect on the
active volume fractions.
lower limits is observed for any of the spectral modes.
– The uncertainty on the fraction of enrichment in
76Ge of the germanium that constitutes the detec- (iii) Data acquisition and selection
10
Table 3 Experimental results for the limits on T0νχ of 76Ge for the Majoron models given in Refs. [7,42,43,44]. The first
1/2
sectionconsidersleptonnumberviolatingmodels(I)allowing0νββdecay,whileinthesecondsectionleptonnumberconserving
models(II)arelisted,where0νββ decayisnotallowed.Thefirstcolumngivesthemodelname,thesecondthespectralindex,
n, the third the information on whether one Majoron, χ, or two Majorons, χχ, is emitted, the fourth if the Majoron is a
Goldstone boson, the fifth provides its lepton number, L, the sixth the experimental limit on T0νχ of 76Ge obtained in this
1/2
analysis. The nuclear matrix elements, M0νχ, the phase space factor, G0νχ, and the resulting effective coupling constants,
(cid:104)g(cid:105),aregivenintheseventh,eighthandninthcolumns,respectively.ThelimitsonT0νχ of76GefortheMajoronmodelsand
1/2
(cid:104)g(cid:105) correspond to the 90% quantiles of the marginalized posterior probability distribution. For the case of n=1, the nuclear
matrix element, M0νχ, from Refs. [45,46,47,48,49,50,51] and the phase space factor, G0νχ, from Ref. [52] are used for the
calculation of (cid:104)g(cid:105). The given range covers the variations of M0νχ in these works. For n = 3 and 7, (cid:104)g(cid:105) is determined using
the matrix elements and phase space factors from Ref. [42]. The results for 0νββχ (n=3, 7) account for the uncertainty on
M0νχ.Forn=2,onlythe experimentalupper limitisgiven.
Model n Mode Goldstone L T0νχ M0νχ G0νχ (cid:104)g(cid:105)
1/2
boson [1023yr] [yr−1]
IB 1 χ no 0 >4.2 (2.30−5.82) 5.86·10−17 <(3.4−8.7)·10−5
IC 1 χ yes 0 >4.2 (2.30−5.82) 5.86·10−17 <(3.4−8.7)·10−5
ID 3 χχ no 0 >0.8 10−3±1 6.32·10−19 <2.1+4.5
−1.4
IE 3 χχ yes 0 >0.8 10−3±1 6.32·10−19 <2.1+4.5
−1.4
IF 2 χ bulkfield 0 >1.8 – – –
IIB 1 χ no -2 >4.2 (2.30−5.82) 5.86·10−17 <(3.4−8.7)·10−5
IIC 3 χ yes -2 >0.8 0.16 2.07·10−19 <4.7·10−2
IID 3 χχ no -1 >0.8 10−3±1 6.32·10−19 <2.1+4.5
−1.4
IIE 7 χχ yes -1 >0.3 10−3±1 1.21·10−18 <2.2+4.9
−1.4
IIF 3 χ gaugeboson -2 >0.8 0.16 2.07·10−19 <4.7·10−2
The uncertainty from data acquisition and selection is fit model, are (in units of 1023 yr): >4.2, >1.8, >0.8
estimated to be below 0.1% and does not alter the de- and >0.3 for n=1,2,3 and 7, respectively. The results
rived limits on T0νχ. are summarized in Table 3 for the different Majoron
1/2
models.
The limits on T0νχ presented here are the most
1/2
7.3 Results and Discussion stringent limits obtained to date for 76Ge. The limits
for n=1 and n=3 are improved by more than a fac-
Fig.3showstheglobalmodelforthecaseofspectralin- tor six [9], the limit for n = 7 is improved by a factor
dexn=1togetherwiththeenergyspectraforboththe five [8] compared to previous measurements. The limit
coaxialandtheBEGedatasets.Thecontributionsfrom for the mode with n = 2 is reported here for the first
the background contaminations, from the 2νββ decay time.
only, and the combined spectra from the background From the lower limits on T0νχ, upper limits on the
1/2
contaminations and 2νββ decay are drawn separately. effective neutrino-Majoron coupling constants (cid:104)g(cid:105) for
The 35868 events in the data spectrum of the golden the models with n=1,3 and 7 can be calculated using
data set were matched with 35834 events in the best- the following equations:
fit model for n = 1. Of those events, in the best fit,
54.5 are attributed to 0νββχ. For the BEGe data set, 1/T10/ν2χ =|(cid:104)g(cid:105)|2·G0νχ(Qββ,Z)·|M0νχ|2 (8)
the best-fit model contains 5081.4 counts for the 5035
and
measured events. In this fit, 7.8 events are attributed
to 0νββχ decay. The limit of T10/ν2χ at 90% C.I. derived 1/T0νχ =|(cid:104)g(cid:105)|4·G0νχχ(Q ,Z)·|M0νχχ|2 (9)
from the fit is also drawn (green histogram). The up- 1/2 ββ
per limits at 90% C.I. for the remaining three modes for single and double Majoron emission, respectively.
arereportedforillustrativepurpose(bluehistogramfor The matrix element for the models with n = 1 (IB,
n=2, orange for n=3 and red for n=7). The maxi- IC and IIB) are taken from Refs. [45,46,47,48,49,50,
mumofthecorrespondingdistributionsshiftstohigher 51], whereas the phase space factor is that of Ref. [52].
energy with the diminishing of the spectral index n. The matrix elements for the models with n = 3 (ID,
The resulting lower limits on T0νχ, determined as the IE, IIC, IID, IIF) and with n = 7 (IIE) as well as
1/2
90%quantilesoftheposteriorprobabilitydistributions the corresponding phase space factors are taken from
and taking into account all uncertainties related to the Ref.[42].Theresultsfortheupperlimitson(cid:104)g(cid:105)arealso