# Constraints on partial half-lives of 136Ce and 138Ce double electron
captures

B. Lehnert<EMAIL_ADDRESS>Nuclear Science Division, Lawrence Berkeley
National Laboratory, Berkeley, CA 94720, U.S.A.  M. Hult  European Commission,
Joint Research Centre, 2440 Geel, Belgium  J. Kotila  Finnish Institute for
Educational Research, University of Jyväskylä, P.O. Box 35, FI-40014
Jyväskylä, Finland  Center for Theoretical Physics, Sloane Physics Laboratory,
Yale University, New Haven, Connecticut 06520-8120, USA  G. Lutter  European
Commission, Joint Research Centre, 2440 Geel, Belgium  G. Marissens  European
Commission, Joint Research Centre, 2440 Geel, Belgium  A. Oberstedt  Extreme
Light Infrastructure - Nuclear Physics (ELI-NP) / Horia Hulubei National
Institute for Physics and Nuclear Engineering (IFIN-HH), 077125 Bucharest-
Magurele, Romania  S. Oberstedt  European Commission, Joint Research Centre,
2440 Geel, Belgium  H. Stroh  European Commission, Joint Research Centre, 2440
Geel, Belgium  K. Zuber  Institute for Nuclear and Particle Physics, TU
Dresden, 01069 Dresden, Germany

###### Abstract

The $\gamma$-ray emissions from a radiopure cerium-bromide crystal with a mass
of 4381 g were measured for a total of 497.4 d by means of high-resolution
$\gamma$-ray spectrometry in the HADES underground laboratory at a depth of
500 m.w.e. A search for $\rm 0/2\nu\epsilon\epsilon$ and $\rm
0/2\nu\epsilon\beta^{+}$ double beta decay transitions of 136Ce and 138Ce was
performed using Bayesian analysis techniques. No signals were observed for a
total of 35 investigated decay modes. 90% credibility limits were set in the
order of $10^{18-19}$ a. Existing constraints from a cerium oxide powder
measurement were tested with a different cerium compound and half-life limits
could be improved for most of the decay modes. The most likely accessible
decay mode of the 136Ce $\rm 2\nu\epsilon\epsilon$ transition into the
$0^{+}_{1}$ state of ${}^{136}\rm Ba$ results in a new best 90% credibility
limit of $5.0\times 10^{18}$ a.

Suggested keywords

###### pacs:

Valid PACS appear here

††preprint: AIP/123-QED

##  I Introduction

Neutrinoless double beta ($\rm 0\nu\beta\beta$) decay is a second-order weak
nuclear decay process requiring physics beyond the Standard Model (SM) of
particle physics. The observation of $0\nu\beta\beta$ decay would establish
lepton number violation (LNV) and neutrinos as Majorana particles. It would
also open up possibilities to explain the matter-antimatter asymmetry in the
Univserse converting Leptogenesis into Baryogenesis (see e.g. Ref. dep18 ).
The inverse of the $0\nu\beta\beta$ decay half-life $T^{0\nu}_{1/2}$ in a
given isotope is conventionally expressed as

$\left[T^{0\nu}_{1/2}\right]^{-1}=\left|f(m_{i},U_{ei})\right|^{2}\cdot
G_{0\nu}\cdot\left|M_{0\nu}\right|^{2},$ (1)

with the phase space factor $G$ and the nuclear matrix element $M$. The LNV
mechanism $f(m_{i},U_{ei})$ can have many origins. Under the assumption of
light Majorana neutrino exchange, it is connected with $m^{eff}_{\nu}$ and
allows to constrain the absolute neutrino mass scale. While $\rm
0\nu\beta\beta$ decay is one of the most promising avenues to search for new
physics, the similar neutrino-accompanied double beta ($\rm 2\nu\beta\beta$)
decay is allowed within the SM:

$2\nu\beta^{-}\beta^{-}:(Z,A)\longrightarrow(Z+2,A)+2\,e^{-}+2\,\overline{\nu}_{e}\
,$ (2)

with half-life

$\left[T^{2\nu}_{1/2}\right]^{-1}=G_{2\nu}\cdot\left|M_{2\nu}\right|^{2}.$ (3)

It was detected directly in 11 isotopes so far with half-lives between 1018
and 1021 a tre02 ; sak13  and provides valuable experimental information to
better understand the complex nuclear physics of double beta decay processes
and isotopes. Equivalent processes based on the right side of the mass
parabola are decays such as (1) double electron capture $\epsilon\epsilon$,
(2) electron capture with positron emission $\epsilon\beta^{+}$ and (3) double
positron emission $\beta^{+}\beta^{+}$:

$\displaystyle 2\nu{\rm\epsilon\epsilon}:2\,e^{-}+(Z,A)$
$\displaystyle\longrightarrow$ $\displaystyle(Z-2,A)+2\,\nu_{e}$ (4)
$\displaystyle 2\nu{\rm\epsilon}\beta^{+}:\,e^{-}+(Z,A)$
$\displaystyle\longrightarrow$ $\displaystyle(Z-2,A)+e^{+}+2\,\nu_{e}$ (5)
$\displaystyle 2\nu\beta^{+}\beta^{+}:(Z,A)$ $\displaystyle\longrightarrow$
$\displaystyle(Z-2,A)+2\,e^{+}+2\,\nu_{e}$ (6)

The experimental sensitivity for those processes is considerably lower
compared to $2\nu\beta^{-}\beta^{-}$ decay and, except for two cases, only
lower limits for the half-lives in the range of $10^{18-21}$ a could be
obtained for a few isotopes tre02 ; tre95 ; gav06 ; bel09 ; ruk10 . In Ref.
mesh01  the total weak decay ($\beta^{+}\beta^{+}$ \+ $\epsilon\epsilon$ \+
$\epsilon\beta^{+}$) half-life for 130Ba is reported as T1/2 = (2.2 $\pm$ 0.5)
$\times$ 1021 a (68% C.L.) based on a geochemical analysis of natural barite
(BaSO4). Recently a first direct observation of $\rm 2\nu\epsilon\epsilon$
could be achieved in ${}^{124}\rm Xe$ with a half-life of $1.8\times 10^{22}$
a apr19 .

$\rm 2\nu\beta\beta$ decays into excited states, the focus of this work, are
so far only observed in two nuclides, ${}^{150}\rm Nd$ and ${}^{100}\rm Mo$,
with average half-lives of $1.33^{+0.45}_{-0.26}\times 10^{20}$ a and
$5.9^{+0.8}_{-0.6}\times 10^{20}$ a, respectively ESAverage . The first
observations of these decay modes were performed with samples on HPGe
detectors in the “source$\neq$detector” approach that is also employed here.

Neutrinoless double-beta decay has not been observed, yet. Although much
effort is being made to improve the sensitivity by using the
“source$=$detector” approach, increasing the target mass, isotopic fraction of
the nuclide under consideration and improving the radio-purity of the sample,
only lower limits of the decay half-lives are reported. Leading half-life
limits and sensitivities recently exceeded $10^{26}$ a by the KamLAND-ZEN
KLZ0nbb  and GERDA Gerda0nbb  experiments for ${}^{136}\rm Xe$ and ${}^{76}\rm
Ge$, respectively.

The recent development of production of large cerium-bromide detectors put
cerium isotopes in the focus of double beta decay searches. Cerium has three
isotopes, which are candidates for double beta decay: 136,138Ce and 142Ce.
Moreover, being used as $\gamma$-ray detectors offers the possibility to
exploit the “source$=$detector” approach for cerium bel03 ; ber97 ; bel11 .

In this work, we measure double beta decays in a cylindrical CeBr3 crystal of
size 10.2 cm $\times$ 10.2 cm (diameter $\times$ length) with 4381 g mass with
an ultra-low-background HPGe detector in the “source$\neq$detector” approach.
The large size of the crystal contains a significantly greater number of
cerium atoms than were available in previous searches. Thorough
characterization of the material itself and other crystals produced by the
same company revealed a very high radio-purity lutter13 ; bil11 . This
provides excellent background conditions for investigating rare decay
processes and the possibility for future “source$=$detector” measurements if
additional background from the light readout is kept low.

##  II Double beta decays in cerium

Natural cerium consists of 4 isotopes out of which only ${}^{140}\rm Ce$
(88.45% abundance) is expected to be stable. The long lived isotopes
${}^{142}\rm Ce$, ${}^{138}\rm Ce$, and ${}^{136}\rm Ce$ are candidates for
double beta decay:

Figure 1: Decay scheme of ${}^{136}\rm Ce$ to ${}^{136}\rm Ba$. Each excited
state of ${}^{136}\rm Ba$ shows the de-excitation cascade including
$\gamma$-ray emission probabilities. Decay branches not considered in the
analysis are shown in gray. All other $\gamma$-rays are included in the
combined fits. The radiative $\rm 0\nu\epsilon\epsilon$ decay is illustrated
on the right for the single bremsstrahlung $\gamma$-ray emission case. $\rm
0\nu\epsilon\epsilon$ to a final 2+ state can occur with captures from any
combination of the K and L shell, where each shell electron reduces the
available energy for the bremsstrahlung $\gamma$-ray. For $\rm
0\nu\epsilon\epsilon$ to a final 0+ state only KL and LL captures are allowed
due to spin constraints. The highlighted example in blue shows the $\rm
0\nu\epsilon\epsilon$ decay modes into the first excited 0+ and its subsequent
$\gamma$-ray de-excitation. $\rm 0/2\nu\epsilon\beta^{+}$ and $\rm
0/2\nu\beta^{+}\beta^{+}$ decay can only occur into the ground and first
excited state. All other excited states are populated by $\rm
0/2\nu\epsilon\epsilon$.

  * •

${}^{142}\rm Ce$ has a high natural abundance of 11.11% and is a candidate for
the $\rm 2\nu\beta^{-}\beta^{-}$ decay with an energy release of $Q=(1416.8\pm
2.9)$ keV Qvalues . The first excited state is at 1575.8 keV which is higher
than the Q-value. Hence, no excited state of its daughter 142Nd can be
populated and the decay cannot be measured by $\gamma$-ray spectrometry.

  * •

${}^{138}\rm Ce$ has a natural abundance of 0.251% and is a candidate for the
$\rm 2\nu\epsilon\epsilon$ decay with $Q=(690.6\pm 5.0)$ keV Qvalues . Also in
this case no excited state of ${}^{138}\rm Ba$ can be populated and the
experimental signature for the $2\nu$ mode are only barium X-rays with
energies below 40 keV. Due to the high absorption of the low energy X-rays,
this decay channel will not be studied. In case of the $0\nu$ mode, the excess
of energy due to the non-emission of the 2 neutrinos can be released through a
bremsstrahlung photon in addition to the X-rays. The energy of the emitted
photon, E0ν, is equal to:

$\displaystyle E_{0\nu}=Q-E_{\beta 1}-E_{\beta 2}$ (7)

where Eβ1 and Eβ2 are the binding energies of the K, L1, L2, L3 shells. In
case of barium these are EK = 37.4 keV, EL1 = 6.0 keV, EL2 = 5.6 keV and EL3 =
5.2 keV. Due to the energy resolution of the $\gamma$-ray spectrometer the EL
energies cannot be distinguished. Therefore, the mean value of EL, 5.6 keV, is
taken for the analysis.

For the $0\nu$LK, and $0\nu$LL decays of 138Ce we expect the emission of a
photon at (647.6 $\pm$ 5) keV and (679.4 $\pm$ 5) keV. Note that due to
angular momentum constraints the $0\nu$KK decay to a $0^{+}$ state is not
possible doi93 .

  * •

${}^{136}\rm Ce$ has a natural abundance of 0.185% and is a candidate for $\rm
2\nu\epsilon\epsilon$, $\rm 2\nu\epsilon\beta^{+}$, and $\rm
2\nu\beta^{+}\beta^{+}$ decay with $Q=(2378.53\pm 0.27)$ keV QValueCe136 .
Double beta decays of this isotope are the main focus of this work and the
large number of possible decay modes are illustrated in Fig. 1. The $\rm
2\nu\beta^{+}\beta^{+}$ mode can only populate the ground state of
${}^{136}\rm Ba$ since each $\beta^{+}$ reduces the Q-value twice by 511 keV.
Consequently, the $\rm 2\nu\epsilon\beta^{+}$ mode can populate the ground
state and first excited $2^{+}_{1}$ state at 818.5 keV. The $\rm
2\nu\epsilon\epsilon$ decay mode can populate a total of 8 excited states in
${}^{136}\rm Ba$ as shown in Fig. 1. The most likely mode among the excited
state transitions investigated here is the first excited $0^{+}_{1}$ state at
1579.0 keV. The $0\nu$ modes are possible into the ground and excited states
with additional emission of a radiative bremsstrahlung photon to release the
excess energy as shown in Eq. 7. This is illustrated in Fig. 1 on the right.
For each final state, the bremsstrahlung energy depends on the shell
combination from which the two electrons are captured. For the $2^{+}$ final
states, KK, KL, and LL combinations are considered, each differing in the
energy $E_{0\nu}$ but with an otherwise identical ${}^{136}\rm Ba$
$\gamma$-ray de-excitation cascade. For the $0^{+}$ states only the KL and LL
shell captures are considered. Figure 1 shows the example of bremsstrahlung
energies for the $\rm 0\nu\epsilon\epsilon$ $0^{+}_{1}$ state transitions in
blue.

In ${}^{136}\rm Ce$ a potential resonant double-electron capture to the
excited levels in ${}^{136}\rm Ba$ with energies of 2392.1 keV and 2399.9 keV
could reduce the half-life by several orders of magnitude to the order of
$10^{24}$ a (e.g. bern83 ). However, a later measurement determined the
reaction Q-value of this process to Q = (2378.53 $\pm$ 0.27) keV QValueCe136
ruling out such a resonance enhancement in the ${}^{136}\rm Ce$ \-
${}^{136}\rm Ba$ system.

Table 1 summarizes the fundamentally different decay modes for each cerium
isotope of interest. Each of these modes has a number of possible “sub-modes”
either going to different excited states or starting from captures on
different electronic shell combinations (see Fig. 1). The number of sub-modes
investigated in this work are listed in the second column. In this manuscript,
the $\rm 2\nu\epsilon\epsilon$ transition into the $0^{+}_{1}$ with the 818.5
keV and 760.5 keV $\gamma$-ray is used to exemplarily illustrate the analysis
for all other decay modes.

A clear signature of the 511 keV annihilation peak in combination with the
510.8 keV $\gamma$-line from ${}^{208}\rm Tl$ is present in the background
data (as shown later in Fig. 2). Hence, the search for decay modes including a
$\beta^{+}$ cannot reliably use this signature and we only constrain the $\rm
0/2\nu\epsilon\beta^{+}$ mode into the $2^{+}_{1}$ state with the de-
excitation $\gamma$-line at 818.5 keV. Due to the possible coincidences of the
de-excitation $\gamma$-ray with the annihilation $\gamma$-rays, this mode has
a slightly reduced detection efficiency compared to the pure de-excitation
$\gamma$-ray search of the $\rm 0/2\nu\epsilon\epsilon$ into the same final
state. Also note that the signature for the $\rm 0\nu\epsilon\beta^{+}$ and
$\rm 2\nu\epsilon\beta^{+}$ modes are identical since the $\beta^{+}$ can
carry the remaining decay energy and no radiative bremsstrahlung emission is
expected for the $0\nu$ case with a $\beta^{+}$ emission.

The right column in Tab. 1 lists corresponding theoretical half-life estimates
when available. The theoretical half-lives are given for the ground state
(g.s.) and first excited $0^{+}_{1}$, as well as first and second $2^{+}$
state transitions when applicable. For completeness, Tab. 1 also lists
theoretical results for ${}^{142}\rm Ce$, which are however not investigated
in this work.

Table 1: Summary of the double beta decay isotopes in cerium together with their theoretical half-life estimates. Listed are the main $2\nu$ and $0\nu$ decay modes in the first column and the number of sub-modes available due to various excited final states or due to different starting energies based on captures from different shell combinations in the second column. The number of sub-modes investigated in this work are shown in parenthesis. Half-life predictions in the literature are shown in the third column for ground state transitions, as well as excited state transitions where available. For the $0\nu$ modes, the values are given for $m_{\nu}^{\rm eff}=1$ eV. The theoretical expectations marked as IBM-2 are calculated with matrix elements taken from/calculated as described in Ref. CeNME  and phase space factors taken from/calculated as described in Refs. Kot12 ; CePSF .  decay mode | sub modes | theoretical T1/2
---|---|---
| all (investigated) | [a]
136Ce |  |
$\rm 2\nu\epsilon\epsilon$ | 9 (8) |  g.s: $1.2\times 10^{21}$ [IBM-2], $1.7\times 10^{22}$ hir94 , $9.6\times 10^{21}$ rum98 ,
|  |  $0.3-6.4\times 10^{19}$ suh93 , $3.2-5.1\times 10^{21}$ civ98 , $3.7\times 10^{23}$ aba84 ,
|  |  $1.6-5.9\times 10^{22}$ pir15
|  |  $2^{+}_{1}$: $2.3-8.5\times 10^{29}$ pir15
|  |  $0^{+}_{1}$: $5.7\times 10^{25}$ [IBM-2], $7.7-28\times 10^{29}$ pir15
|  |  $2^{+}_{2}$: $7.3-27\times 10^{33}$ pir15
$\rm 2\nu\epsilon\beta^{+}$ | 2 (1) |  g.s: $7.8\times 10^{22}$ [IBM-2], $9.2\times 10^{23}$ hir94 , $6.0\times 10^{23}$ rum98 ,
|  |  $2.8\times 10^{24}$ aba84 , $1.0-3.7\times 10^{24}$ pir15
|  |  $2^{+}_{1}$: $7.8-29\times 10^{30}$ pir15
$\rm 2\nu\beta^{+}\beta^{+}$ | 1 (0) |  g.s: $5.2\times 10^{30}$ [IBM-2], $5.2\times 10^{31}$ hir94 , $9.6\times 10^{31}$ aba84 ,
|  |  $6.8-25\times 10^{31}$ pir15
$\rm 0\nu\epsilon\epsilon$ | 23 (23) | -
$\rm 0\nu\epsilon\beta^{+}$ | 2 (1) |  g.s: $3.3\times 10^{26}$ [IBM-2], $1.8\times 10^{26}$ suh03 , $6.4-110\times 10^{25}$ suh98
$\rm 0\nu\beta^{+}\beta^{+}$ | 1 (0) |  g.s: $2.0\times 10^{29}$ [IBM-2], $3.8\times 10^{30}$ suh03 , $5.6-7.3\times 10^{29}$ suh98
138Ce |  |
$\rm 2\nu\epsilon\epsilon$ | 2 (0) |  g.s: $1.5\times 10^{24}$ [IBM-2],$2.1\times 10^{26}$ aba84
$\rm 0\nu\epsilon\epsilon$ | 2 (2) | -
142Ce |  |
$\rm 2\nu\beta^{-}\beta^{-}$ | 1 (0) |  g.s: $2.5\times 10^{22}$ [IBM-2], $1.6\times 10^{21}$ sta90 , $0.16-23\times 10^{22}$ bob04 ,
|  |  $3.8\times 10^{22}$ del17
$\rm 0\nu\beta^{-}\beta^{-}$ | 1 (0) |  g.s: $2.1\times 10^{25}$ [IBM-2], $2.8\times 10^{24}$ sta90

For this work, the half-lives labeled IBM-2 (microscopic Interacting Boson
Model) in Tab. 1 were obtained by dedicated calculations of nuclear matrix
elements as described in Ref. CeNME  and phase space factors (PSF) as
described in Ref. Kot12  and CePSF . In the calculation of PSFs, the same
binding energies for K and L shells were used as in the following analysis,
i.e, EK = 37.4 keV and the mean value, 5.6 keV for EL. In these IBM-2
calculations a bare value of $g_{A}=1.269$ was used and the values shown give
thus a lower estimate for the half-lives. The quenching of $g_{A}$ is
intensely discussed in literature and values from 1.2694 (the free nucleon
value) to values much less than 1 have been suggested (for a review see Ref.
suh17 ). For example in Ref. pir15  quenched values from $0.6-0.8$ have been
used leading to the longest half-life predictions for the mode in question
kos21 .

Decays into the ground states are estimated to have half-lives in the range of
$10^{18}-10^{26}$ a and $10^{22}-10^{24}$ a for decay modes $\rm
2\nu\epsilon\epsilon$ and $\rm 2\nu\epsilon\beta^{+}$, respectively. The
predictions of $\rm 2\nu\epsilon\epsilon$ transition to the first excited
$0^{+}_{1}$ state vary in the range of $10^{25}-10^{29}$ a. This large range
is partly due to the fact that the lower estimate is calculated using the bare
$g_{A}$ value and the higher estimate is using quenched values. For $\rm
0\nu\epsilon\beta^{+}$ the estimates are $10^{25}-10^{27}$ a assuming
$m_{\nu}^{\rm eff}=1$ eV. All other listed theoretical calculations for $\rm
2\nu\beta^{+}\beta^{+}$ and $\rm 0\nu\beta^{+}\beta^{+}$ modes, as well as,
for the transitions to $2^{+}$ states suggest half-lives longer than $10^{29}$
a.

Previous investigations of double beta decays in ${}^{136}\rm Ce$ and
${}^{138}\rm Ce$ have been performed in Refs. bel09 ; ber97 ; bel11 ; dan01 ;
bel14  and were recently updated in Ref. bel17 . The main differences in this
work is the use of a CeBr3 crystal compared to a cerium oxide powder sample
used in Ref. bel17 , as well as the use of a fully Bayesian statistical
analysis using all signature $\gamma$-rays in combined fits. A list of
previous limits is compiled further below together with results from this
work.

##  III Experimental setup and data taking

Table 2: Radioactivity in the Al-bag#1 and in bags from the same batch as Al-bag#2. Decision thresholds are given at 90% confidence level following the ISO11929:2010 standard. | Al-bag#1 | Al-bag#2
---|---|---
radionuclide | massic activity | activity per bag | activity | activity per bag
| [Bq/kg] | [mBq] | [Bq/kg] | [mBq]
${}^{238}\rm U$ | $6.7\pm 0.9$ | $190\pm 26$ | $5.8\pm 0.6$ | $165\pm 17$
${}^{226}\rm Ra$ | $<0.04$ | $<1.2$ | $<0.04$ | $<0.6$
${}^{210}\rm Pb$ | $4.2\pm 0.7$ | $119\pm 20$ | $2.4\pm 0.8$ | $68\pm 23$
${}^{228}\rm Th$ | $0.4\pm 0.1$ | $11\pm 3$ | $0.28\pm 0.05$ | $8.0\pm 1.5$
${}^{228}\rm Ra$ | $<0.1$ | $<3$ | $<0.04$ | $<1.2$
${}^{40}\rm K$ | $<0.3$ | $<9$ | $<0.1$ | $<3$

The measurements of the CeBr3 crystal were carried out in the 225 m (500 m
water equivalent) deep underground research facility HADES in Belgium hades ;
hades2 . The muon flux is reduced by about a factor 5000 compared to above
ground. An HPGe-detector named “Ge-4” was used for all measurements of the
CeBr3 crystal. It is a coaxial detector with submicron top deadlayer
(Canberra, XtRa-type) and has a relative efficiency of 106% geDet . The
detector has a good long-term stability which is checked by measuring a
quality control source with ${}^{60}\rm Co$, ${}^{137}\rm Cs$ and ${}^{241}\rm
Am$ at regular intervals. The shift in peak centroid for the 661.7 keV
$\gamma$-line was less than 0.12 keV during the measurement campaigns. The
full width at half maximum of the detector is at present 1.86 keV at 661.7 keV
and 2.22 keV at 1332.5 keV. An EGSnrc egs-nrc  computer model of detector Ge-4
was established when it was installed in 2000. It has since then been refined
and validated in several proficiency tests and generally produces results on
an absolute scale with an accuracy of around 3%. This computer model was used
in combination with the Decay0 event generator decay0  for all calculations of
all full energy peak efficiencies in this study. The detector was connected to
a digital signal analyzer (Mirion’s LYNX) for high voltage and data
acquisition. Spectra are collected on average every two days.

A CeBr3 crystal produced in March 2018 by the company Schott in Jena, Germany,
was procured from the company Scionix, Netherlands scionix . Its height and
diameter are 102.55 mm and the mass is ($4380.7\pm 2.0$) g, which corresponds
to $6.9456\times 10^{24}$ cerium atoms. Due to the hygroscopic nature of
CeBr3, the crystal was delivered to JRC-Geel in three layers of protective
wrapping. The innermost protection is 0.24 mm of Teflon, enveloped by a
plastic bag, and followed by an aluminized plastic bag on the outside. The
aluminized bag, hereafter referred to as Al-bag, is filled with nitrogen gas.
The CeBr3 crystal was measured as it was delivered with these three layers of
protection. After the first measurement campaign, the Al-bag used there (Al-
bag#1) was changed by Scionix and replaced by another Al-bag (Al-bag#2).

The Al-bag#1 and several Al-bags from the same batch as Al-bag#2 were measured
using $\gamma$-ray spectrometry on a low-background detector in HADES and the
detected radio-impurities are reported in Tab. 2. A similar Al-bag was used in
a previous study of a smaller $1.5^{\prime\prime}\times 1.5^{\prime\prime}$
CeBr3 crystal lutter13 .

For practical reasons, the measurements of the CeBr3 crystal were carried out
in five campaigns starting May 5th 2018 and ending February 23rd 2021.
Background and quality control sources were measured before and after each
campaign. In total, data from 497.4 days passed the quality controls. In the
first campaign of 96.8 d the crystal was placed 25 mm above the endcap of the
detector. In the later campaigns the distance was reduced to 4 mm in order to
increase the $\gamma$-ray detection efficiencies. For the analysis the data is
split in two datasets (M1 and M2) as indicated in Tab. 3 showing the
resolution and detection efficiencies for both datasets.

The full spectrum for measurements M1, M2 as well as a 104 d background
measurement without sample is shown in Fig. 2. Prominent background
$\gamma$-lines are labeled. Some of the $\gamma$-lines are only observed in
the sample measurements, notably the ${}^{138}\rm La$ $\gamma$-lines at 788.7
keV and 1435.8 keV. The contribution of natural decay chain background is
higher in the sample measurements. This is especially visible for the low
energy $\gamma$-lines of ${}^{210}\rm Pb$ and ${}^{234}\rm Th$ which are
better shielded by the detector setup than high energy $\gamma$-lines and are
thus not as prominent in the background spectrum.

Table 3: Key parameters for the two measurement campaigns: measurement time and crystal distance to endcap. The detection efficiency ($\epsilon$) and resolution in $\sigma_{E}$ are shown exemplary for the emission of the 818.5 keV $\gamma$-ray in the $0^{+}_{1}$ decay mode of ${}^{136}\rm Ce$. meas. | time | distance | det. efficiency | resolution $\sigma_{E}$
---|---|---|---|---
M1 | 96.8 d | 25 mm |  $(0.44\pm 0.04)$% |  $(0.83\pm 0.04)$ keV
M2 | 400.6 d | 4 mm |  $(0.65\pm 0.07)$% |  $(0.83\pm 0.04)$ keV

Figure 2: (color online) Recorded $\gamma$-ray spectrum of the CeBr3 crystal
(+Al-bag) taken in the 25 mm distance configuration (red) and in the 4 mm
distance configuration (blue). Both spectra are compared to a 104 d background
spectrum (black). Prominent background peaks are highlighted.

The radioactive impurities in the CeBr3 crystal itself were quantified by
subtracting the activity in the Al-bag (Tab. 2) using Monte Carlo simulations.
They are presented in Tab. 4. The most noteworthy difference between the
present crystal and the crystal used in the previous measurement is the
absence of ${}^{227}\rm Ac$, whose contribution was relatively high in the
previous one. Also, the massic activity of ${}^{138}\rm La$ is reduced by
almost a factor of 4, likely indicating a change in the production process. A
peak at 810.8 keV could be observed in the data from the first measurement
campaign M1 (May 2018). The peak can be explained by the activation of copper
from the shield, which is located just next to the endcap of Ge4. This copper
piece had been brought above ground, end of January 2018, for a few days to be
slightly modified. Refs. Co58Argument1 ; Co58Argument2  shows that production
of ${}^{58}\rm Co$ by cosmic activation is quite fast. In addition, the
measured half-life ($120\pm 55$) d is compatible with the ${}^{58}\rm Co$
half-life (70.9 d). We also note two unidentified peaks around 1821 keV and
1825 keV in a subset of data from the M2 campaign with 4 mm detector-to-sample
distance. In background data, data from the M1 campaign, and other subsets of
the M2 campaign, these peaks are not observed. An extensive search of nuclear
databases and the spurious nature of the signal leads us to assume that the
origin of the peaks is likely not in the CeBr3 crystal. In either case, these
peaks do not interfere with the regions of interest for the double beta decay
search in this work.

Table 4: Radioimpurities in the CeBr3-crystal from both the present study and from a previous study lutter13  on a smaller crystal. | 4” crystal 4381 g | 1.5” crystal 222 g
---|---|---
radionuclide | massic activity | activity | massic activity | activity
| [mBq/kg] | [mBq] | [mBq/kg] | [mBq]
${}^{238}\rm U$ | $<43$ | $<186$ | $<135$ | $<30$
${}^{226}\rm Ra$ | $<1.1$ | $<4.9$ | $<0.5$ | $<0.12$
${}^{210}\rm Pb$ | $<400$ | $<1700$ | $<600$ | $<134$
${}^{235}\rm U$ | $<5.4$ | $<24$ | $<1.5$ | $<0.32$
${}^{227}\rm Ac$ | $<2.3$ | $<10$ | $300\pm 20$ | $64\pm 4$
${}^{228}\rm Th$ | $<0.6$ | $<2.6$ | $<2$ | $<0.44$
${}^{228}\rm Ra$ | $<0.48$ | $<2.1$ | $<0.7$ | $<0.15$
${}^{40}\rm K$ | $<0.96$ | $<4.2$ | $<1.9$ | $<0.43$
${}^{138}\rm La$ | $2.0\pm 0.2$ | $8.8\pm 0.9$ | $7.4\pm 1.0$ | $1.66\pm 0.19$
${}^{139}\rm Ce$ | $11.7\pm 1.5$ | $51\pm 7$ | $4.3\pm 0.3$ | $0.96\pm 0.08$
${}^{82}\rm Br$ | $5.0\pm 1.0$ | $21\pm 5$ | $18\pm 4$ | $3.9\pm 0.9$
${}^{60}\rm Co$ | $0.091\pm 0.022$ | $0.4\pm 0.1$ | $1.4\pm 0.4$ | $0.20\pm 0.12$

##  IV Data analysis

The analysis is performed independently for each of the 35 considered decay
modes using combined fits to the two datasets $d$, M1 and M2. Each de-
excitation $\gamma$-line $k$ in a given decay mode has its own fit region,
typically $\pm 10$ keV around the $\gamma$-line of interest111Note that in
some cases the fit range is adjusted to include or exclude background
$\gamma$-lines on the region borders. Another exception is the $0\nu$KL
$0^{+}_{1}$ decay mode where the 760.5 keV $\gamma$-line from the de-
excitation cascade and the 756.6 keV bremsstrahlung $\gamma$-line are fitted
in a single wider region.. The signal count expectation $s_{d,k}$ of each
$\gamma$-line in each dataset depends on the single half-life $T_{1/2}$
parameter of the decay mode as

$\displaystyle s_{d,k}=\ln{2}\cdot\frac{1}{T_{1/2}}\cdot\epsilon_{d,k}\cdot
N_{A}\cdot t_{d}\cdot m\cdot f_{\rm iso}\cdot\frac{1}{M_{\rm Ce}}\ .$ (8)

Here, $\epsilon_{d,k}$ is the full energy detection efficiency of $\gamma$-ray
$k$ in dataset $d$, $N_{A}$ is Avogadro’s constant, $t_{d}$ is the live-time
of the dataset, $m$ is the mass of cerium in the CeBr3 crystal (1616.0 g),
$M_{\mathrm{Ce}}$ ist the molar mass of natural cerium (140.1) and $f_{\rm
iso}$ is the natural isotopic abundance of ${}^{136}\rm Ce$ (0.186%) and
${}^{138}\rm Ce$ (0.251%), respectively. The data is binned in 0.5 keV steps
for both datasets. The fit is performed combining all datasets and
$\gamma$-lines for a given decay mode. The Bayesian Analysis Toolkit (BAT)
Caldwell:2009kh  is used to obtain full posterior probability distributions
for all parameters using Markov Chain Monte Carlo and a binned likelihood.
Each free parameter in the fit has an associated prior which is either non-
informative e.g. for the half-life, or informed by systematic uncertainties
e.g. in case of the energy resolution. The likelihood $\mathcal{L}$ is defined
as the product of the Poisson probabilities of each bin $i$ in the fit region
for $\gamma$-line $k$ in every dataset $d$

$\displaystyle\mathcal{L}(\mathbf{p}|\mathbf{n})=\prod\limits_{d}\prod\limits_{k}\prod\limits_{i}\frac{\lambda_{d,k,i}(\mathbf{p})^{n_{d,k,i}}}{n_{d,k,i}!}e^{-\lambda_{d,k,i}(\mathbf{p})}\
,$ (9)

where n denotes the data and p the set of floating parameters. $n_{d,k,i}$ is
the measured number of counts in bin $i$. $\lambda_{d,k,i}$ is the expected
number of counts taken as the integral of the model $P_{d,k}$ in this bin. The
model is composed of three components: (1) A linear background, (2) the
Gaussian signal peak, and (3) a number of Gaussian background peaks. The
number and type of background peaks and consequently also the number of fit
parameters depend on the fit region. The full expression of $P_{d,k}$ is
written as:

$\displaystyle P_{d,k}(E|\mathbf{p})$
$\displaystyle=B_{d,k}+C_{d,k}\left(E-E_{0}\right)$
$\displaystyle+\frac{s_{d,k}}{\sqrt{2\pi}\sigma_{d,k}}\cdot\exp{\left(-\frac{(E-E_{k})^{2}}{2\sigma_{d,k}^{2}}\right)}$
$\displaystyle+\sum\limits_{l_{k}}\left[\frac{b_{d,l_{k}}}{\sqrt{2\pi}\sigma_{d,k}}\cdot\exp{\left(-\frac{(E-E_{l_{k}})^{2}}{2\sigma_{d,k}^{2}}\right)}\right].$

The first row is describing the linear background with the two parameters
$B_{d,k}$ and $C_{d,k}$. The second row is describing the signal peak with the
energy resolution $\sigma_{d,k}$ and the $\gamma$-line energy $E_{k}$. The
third row is describing the $l_{k}$ background peaks in the fit region of
$\gamma$-line $k$ with the strength of the peak $b_{d,l_{k}}$ and the peak
position $E_{l_{k}}$. The same probability density function with different
parameter values is used for both datasets. Hence, the same number of
background peaks is used in each dataset even if not all background peaks are
prominent in both datasets.

The free parameters $\mathbf{p}$ in the fit and their associated priors are:

  * •

1 inverse half-life $(T_{1/2})^{-1}$ with flat prior

  * •

2 x $2$ x $k$ linear background parameters $B_{d,k}$ and $C_{d,k}$ with flat
priors

  * •

$2$ x $k$ energy resolutions $\sigma_{d,k}$ with Gaussian priors

  * •

$2$ x $k$ detection efficiencies $\epsilon_{d,k}$ with Gaussian priors

  * •

$2$ x $l_{k}$ x $k$ background peak strength $b_{d,l_{k}}$ with flat priors

Depending on the decay mode, the contributing $\gamma$-lines (see Fig. 1) and
the background peaks in the vicinity, this amounts to 10 fit parameters in the
easiest case for the $\rm 0\nu\epsilon\epsilon$ g.s. KL and LL transitions and
to 79 fit parameters in the most complex case for the $\rm
0\nu\epsilon\epsilon$ $2_{5}^{+}$ KK, KL, and LL transitions. Note that the
large number of parameters is necessary to fully describe the two datasets and
multiple regions of interests within each dataset.

The energy resolution for each $\gamma$-line of interest, $\sigma_{d,k}$, is
obtained from calibrations and are included in the analysis with a Gaussian
prior. The mean is centered around the best fit value and the width is set to
an estimated systematic uncertainty of 5%. The detection efficiencies
$\epsilon_{d,k}$ are determined with MC simulations for each dataset and decay
mode and are also included with a Gaussian prior. The uncertainty is estimated
at 10%, which includes comparably negligible uncertainties from the sample
mass (1%) and the isotopic abundance (1%). Additional uncertainties for the
peak positions of the signal $E_{k}$ are typically small and neglected here. A
special case is the radiative $\gamma$-rays from $\rm 0\nu\epsilon\epsilon$ in
${}^{138}\rm Ce$ with $\pm 5$ keV that is discussed below.

Figure 3: Data and fit function for the ${}^{136}\rm Ce$ $\rm
2\nu\epsilon\epsilon$ transition into the $0^{+}_{1}$ state. Two regions of
interest for the 818.5 keV and 760.5 keV $\gamma$-lines are shown for both
measurement campaigns M1 and M2. The blue curve shows the best fit value
whereas the red curve shows the signal strength set according to the half-life
limit excluded with 90% credibility. Background $\gamma$-lines are included in
the fit as described in the text.

The full posterior probability obtained by BAT is used for parameter
estimation. The global mode of the posterior space is the best fit. The
posterior space is marginalized for $(T_{1/2})^{-1}$ and the 90% quantile of
this distribution is used for limit setting. The values quoted are 90%
credibility intervals (C.I.) on the half-life222Note that in most cases
Frequentist confidence levels (C.L.) and Bayesian credibility intervals (C.I.)
are numerically similar. A distinction is made due to different definitions of
probability in the two concepts. In practice, numerical differences can occur
in special cases when reducing multi-dimensional parameter spaces into one
dimension for the parameter of interest e.g. $(T_{1/2})^{-1}$. This is often
done by profiling in the Frequentist case and marginalization in the Bayesian
case.. Systematic uncertainties are naturally included via the width of
Gaussian priors but typically have a small effect in the case of limit
setting, where the fit is dominated by statistical uncertainties. Fixing the
free parameters with Gaussian priors to their mean value results in about 1%
better half-life limits.

Fig. 3 exemplarily shows the combined fit for the ${}^{136}\rm Ce$ $\rm
2\nu\epsilon\epsilon$ transition into the $0^{+}_{1}$ state for the region of
interest of the 818.5 keV and 760.5 keV $\gamma$-lines, for both datasets,
respectively. This fit contains 43 free parameters. The best fit function is
shown in blue and the signal strength, set to the 90% credibility limit, is
shown in red. The 818.5 keV fit region includes the 810.8 keV $\gamma$-line
from ${}^{58}\rm Co$ at 99.5% emission probability as well as 821.2 keV
(0.16%, ${}^{214}\rm Bi$), and 826.5 keV (0.12%, ${}^{214}\rm Bi$) background
$\gamma$-lines in the fit. The 760.5 keV region includes the 752.9 keV (0.13%,
${}^{214}\rm Bi$), 755.3 keV (1.0%, ${}^{228}\rm Ac$), 763.1 keV (0.64%,
${}^{208}\rm Tl$), 766.4 keV (0.32%, ${}^{234}\rm Pa$), and 768.4 keV (4.9%,
${}^{214}\rm Bi$) background $\gamma$-lines in the fit.

Some decay modes are discussed in more detail in the following. For $\rm
0\nu\epsilon\epsilon$ radiative $\gamma$-rays the available energy decreases
with increasing excitation level. $\gamma$-rays below 150 keV are not used in
this analysis due to small detection efficiencies. Thus, these $\gamma$-lines
are removed from searches for the highest excited states of the $\rm
0\nu\epsilon\epsilon$ $0^{+}_{3}$ and $2^{+}_{5}$ transitions. The radiative
$\gamma$-ray for the $0\nu\rm LL$ $2^{+}_{4}$ mode at 238.6 keV has the
background $\gamma$-line of ${}^{212}\rm Pb$ (43.6%) overlapping at the same
energy and is thus ignored. Equally, the 287.4 keV $\gamma$-line for the
$0\nu\rm LL$ $2^{+}_{3}$ mode is ignored due to a background $\gamma$-line at
288.2 keV from ${}^{214}\rm Bi$ (0.34%) and the 752.8 keV $\gamma$-line in the
$0\nu\rm KK$ $2^{+}_{2}$ mode due to the 752.9 keV $\gamma$-line from
${}^{214}\rm Bi$ (0.13%)

For the $\rm 0\nu\epsilon\epsilon$ modes in ${}^{138}\rm Ce$ the analysis
becomes more complex since the Q-value and thus the signal peak energy are
only known within $\pm 5$ keV. This is significantly larger than the energy
resolution of the detector system and special care is taken to include the
look elsewhere effect and correctly estimate the half-life probability within
a large energy window. In the Bayesian framework the prior probability for the
signal peak position $E_{k}$ (see Eq. IV) is thus included as (647.6 $\pm$ 5)
keV and (679.4 $\pm$ 5) keV for the KL and LL mode which is shown in Fig. 4 in
the left and right top panels, respectively. The input prior probability for
$E_{k}$ is shown in black. The resulting posterior probability is shown in red
and shows significant structures. Any background upward fluctuation in the
spectrum (shown in the bottom panels) will result in a larger or smaller
probability for the signal peak at the position of the fluctuation. The global
best fit value for $E_{k}$ is shown as a blue diamond marker and also in the
blue best-fit functions in the bottom panels. It appears as if a non-zero
signal has been found at the indicated locations of the marker.

However, considering the full probability for a signal anywhere in the fit
region, the most probable signal value is zero. This is shown in Fig. 5. The
histograms show the marginalized posterior for $T_{1/2}^{-1}$ with the 90%
quantile highlighted in red and the global best fit value of $T_{1/2}^{-1}$
marked as blue diamond. In other words, if one were to search for any peak in
the wide search window, one would find one at the marked location with rather
large significance as indicated by the best fit in Fig. 5. However, when
including the look elsewhere effect and marginalizing over all possible peak
positions, the evidence for a signal peak does not exist.

Figure 4: Fit results of the ${}^{138}\rm Ce$ KL shell capture mode (left) and
LL shell capture mode (right). The top panels show the marginalized posterior
probability of the signal peak energy $E_{0}$ in red and its input prior
probability of $\pm 5$ keV in black. Also shown is the global best fit value
of $E_{0}$ as blue diamond marker. The bottom panels show the spectra for the
M1 dataset in black and the M2 dataset in green. Also shown are the fit
functions for the best fit parameters in blue and the signal strength set to
the 90% credibility limit in red.

Figure 5: Marginalized posterior distribution for $T_{1/2}^{-1}$ for the
${}^{138}\rm Ce$ KL shell capture mode (left) and LL shell capture mode
(right). The red shaded area shows the 90% quantile used for limit setting.
Also shown is the best fit as blue diamond marker. The global best fit is
significantly different from the most likely $T_{1/2}^{-1}$ value due to the
large uncertainty in the peak energy and the look elsewhere effect.

##  V Results

All fits are consistent with zero signal count and hence no double beta decay
signal has been found. For 27 of the 35 considered decay modes, the
$(T_{1/2})^{-1}$ posterior distributions peaks at zero. For the other eight
channels a positive value is found; however zero is included within 1.1 sigma
even for the most extreme case. For consistency, the 90% quantile of the
$(T_{1/2})^{-1}$ distribution is used for limit setting in all cases. The
lower $T_{1/2}$ limits set in this work are shown in Tab. 6 together with
previous results. The obtained half-life limits are at a similar order of
magnitude as previous results in bel17 , but improve the global constraints
for most decay modes.

The most likely decay, $\rm 2\nu\epsilon\epsilon$ of ${}^{136}\rm Ce$ to the
$0^{+}_{1}$ (1579.0 keV) state in ${}^{136}\rm Ba$, could be constrained to
$>5.0\times 10^{18}$ a (90% C.I.).

Table 5:  Fit results for all accessible ${}^{136}\rm Ce$ and ${}^{138}\rm Ce$ decay modes. Previous results from Ref. bel17  are given as Frequentist 90% confidence levels (C.L.) while results from this work are given as Bayesian 90% credibility intervals (C.I.). decay mode | final state |  previous result bel17  | this work
---|---|---|---
| [keV] | 90% C.L. [a] | 90% C.I. [a]
${}^{136}\rm Ce$ |  |  |
$\rm 2\nu\epsilon\epsilon$ |  $2^{+}_{1}$ (818.5) | $2.9\times 10^{18}$ | $3.6\times 10^{18}$
$\rm 2\nu\epsilon\epsilon$ |  $2^{+}_{2}$ (1551.0) | $3.4\times 10^{18}$ | $2.5\times 10^{18}$
$\rm 2\nu\epsilon\epsilon$ |  $0^{+}_{1}$ (1579.0) | $2.5\times 10^{18}$ | $5.0\times 10^{18}$
$\rm 2\nu\epsilon\epsilon$ |  $2^{+}_{3}$ (2080.0) | $2.8\times 10^{18}$ | $2.9\times 10^{18}$
$\rm 2\nu\epsilon\epsilon$ |  $2^{+}_{4}$ (2128.8) | $1.4\times 10^{18}$ | $2.6\times 10^{18}$
$\rm 2\nu\epsilon\epsilon$ |  $0^{+}_{2}$ (2141.3) | $4.4\times 10^{18}$ | $5.9\times 10^{18}$
$\rm 2\nu\epsilon\epsilon$ |  $(2)^{+}_{5}$ (2222.7) | $2.0\times 10^{18}$ | $3.6\times 10^{18}$
$\rm 2\nu\epsilon\epsilon$ |  $0^{+}_{3}$ (2315.3) | $2.5\times 10^{18}$ | $4.4\times 10^{18}$
$\rm 0\nu KL$ |  $0^{+}_{0}$ (g.s.) | $3.4\times 10^{18}$ | $7.3\times 10^{18}$
$\rm 0\nu LL$ |  $0^{+}_{0}$ (g.s.) | $8.4\times 10^{18}$ | $4.1\times 10^{18}$
$\rm 0\nu KK$ |  $2^{+}_{1}$ (818.5) | $3.0\times 10^{18}$ | $7.7\times 10^{18}$
$\rm 0\nu KL$ |  $2^{+}_{1}$ (818.5) | $3.0\times 10^{18}$ | $5.6\times 10^{18}$
$\rm 0\nu LL$ |  $2^{+}_{1}$ (818.5) | $3.0\times 10^{18}$ | $3.5\times 10^{18}$
$\rm 0\nu KK$ |  $2^{+}_{2}$ (1551.0) | $2.9\times 10^{18}$ | $2.4\times 10^{18}$
$\rm 0\nu KL$ |  $2^{+}_{2}$ (1551.0) | $2.9\times 10^{18}$ | $2.2\times 10^{18}$
$\rm 0\nu LL$ |  $2^{+}_{2}$ (1551.0) | $2.9\times 10^{18}$ | $2.0\times 10^{18}$
$\rm 0\nu KL$ |  $0^{+}_{1}$ (1579.0) | $2.2\times 10^{18}$ | $4.6\times 10^{18}$
$\rm 0\nu LL$ |  $0^{+}_{1}$ (1579.0) | $2.2\times 10^{18}$ | $4.4\times 10^{18}$
$\rm 0\nu KK$ |  $2^{+}_{3}$ (2080.0) | $2.6\times 10^{18}$ | $2.5\times 10^{18}$
$\rm 0\nu KL$ |  $2^{+}_{3}$ (2080.0) | $2.6\times 10^{18}$ | $1.8\times 10^{18}$
$\rm 0\nu LL$ |  $2^{+}_{3}$ (2080.0) | $2.6\times 10^{18}$ | $2.7\times 10^{18}$
$\rm 0\nu KK$ |  $2^{+}_{4}$ (2128.8) | $1.6\times 10^{18}$ | $3.1\times 10^{18}$
$\rm 0\nu KL$ |  $2^{+}_{4}$ (2128.8) | $1.6\times 10^{18}$ | $2.2\times 10^{18}$
$\rm 0\nu LL$ |  $2^{+}_{4}$ (2128.8) | $1.6\times 10^{18}$ | $2.5\times 10^{18}$
$\rm 0\nu KL$ |  $0^{+}_{2}$ (2141.3) | $4.2\times 10^{18}$ | $6.3\times 10^{18}$
$\rm 0\nu LL$ |  $0^{+}_{2}$ (2141.3) | $4.2\times 10^{18}$ | $6.2\times 10^{18}$
$\rm 0\nu KK$ |  $(2)^{+}_{5}$ (2222.7) | $2.0\times 10^{18}$ | $3.6\times 10^{18}$
$\rm 0\nu KL$ |  $(2)^{+}_{5}$ (2222.7) | $2.0\times 10^{18}$ | $3.5\times 10^{18}$
$\rm 0\nu LL$ |  $(2)^{+}_{5}$ (2222.7) | $2.0\times 10^{18}$ | $3.5\times 10^{18}$
$\rm 0\nu KL$ |  $0^{+}_{3}$ (2315.3) | $2.5\times 10^{18}$ | $4.7\times 10^{18}$
$\rm 0\nu LL$ |  $0^{+}_{3}$ (2315.3) | $2.5\times 10^{18}$ | $4.6\times 10^{18}$
$\rm 0\nu\epsilon\beta^{+}$ |  $2^{+}_{1}$ (818.5) | $2.3\times 10^{18}$ | $3.0\times 10^{18}$
$\rm 2\nu\epsilon\beta^{+}$ |  $2^{+}_{1}$ (818.5) | $2.4\times 10^{18}$ | $3.0\times 10^{18}$
${}^{138}\rm Ce$ |  |  |
$\rm 0\nu KL$ |  $0^{+}_{0}$ (g.s.) | $8.3\times 10^{17}$ | $1.3\times 10^{18}$
$\rm 0\nu LL$ |  $0^{+}_{0}$ (g.s.) | $4.2\times 10^{18}$ | $3.1\times 10^{18}$

This analysis considers many decay modes - some with a complex decay scheme
and many de-excitation $\gamma$-rays. The order of magnitude of the half-life
limits is mainly determined by the exposure and background of the experiment.
Smaller variations are due to different detection efficiencies of
$\gamma$-rays in the different decay modes. The use of information from all
prominent $\gamma$-lines for a given decay mode in a combined fit largely
mitigates strong differences as would be seen when e.g. only a single
$\gamma$-line is used for limit setting. Another source of variations are
statistical fluctuations of the background. Also here, the use of a combined
fit to multiple $\gamma$-lines reduces the probability of strong background
fluctuation compared to using a single $\gamma$-line fit. It is also worth
noting that the approach of combined fits reduces the selection bias. This
would occur, if one selects the limit from the $\gamma$-line which happens to
result in the highest limit. Especially for decay modes with multiple
$\gamma$-rays at similar detection efficiency, this would systematically
select favorable background fluctuations.

Each decay mode is analyzed independently. However, the value of the limits is
not independent, since many decay modes share the same $\gamma$-line. This is
especially true for the 818.5 keV $\gamma$-line from the ${}^{136}\rm Ce$
$2^{+}_{1}$ state which is the final $\gamma$-ray in the de-excitation
cascade.

##  VI Discussion and Conclusion

A 4381 g CeBr3 crystal was measured with $\gamma$-ray spectrometry over the
last three years at the HADES underground lab. An extensive search for double
electron capture transitions in cerium isotopes has been performed on 479.4
days of data. No signals have been observed and 90% credibility limits have
been set using a Bayesian analysis for all accessible decay modes in
${}^{136}\rm Ce$ and ${}^{138}\rm Ce$. Previously existing limits could be
improved by up to a factor of two. Special care has been taken to avoid
selection biases and look elsewhere effects using all available spectral
information in the analysis as well as implementing knowledge of systematic
uncertainties using Bayesian priors.

The measurement was performed in a state-of-the-art low-background
$\gamma$-ray spectrometry setup for an extended time. Further reducing the
radioactive background or increasing the measurement time / exposure is
limited in feasibility and practicality. Instead, a much stronger improvement
on the half-life sensitivity may be achieved by instrumenting the CeBr3
crystal as a scintillation detector - its intended purpose. This allows
analysis-driven background rejection by coincidence or anti-coincidence
requirements between the CeBr3 and the HPGe detectors. In addition, turning
the crystal into a detector would enable searches in the source$=$detector
configuration with a two order of magnitude increase in detection efficiency
close to 100%. Furthermore, additional decay modes become accessible, e.g.
those which only emit X-rays, which do not easily escape the crystal, or
${}^{142}\rm Ce$ $\rm 0/2\nu\beta\beta$ modes with continuous double beta
spectrum. With enough resources, isotopic enrichment of the low mass side of
cerium isotopes can increase the half-life sensitivity by multiple orders of
magnitude. Natural abundances of ${}^{138}\rm Ce$ and ${}^{136}\rm Ce$ are
only 0.25% and 0.19%, respectively, with strong potential for enrichment.

The newly achieved experimental constraints on the half-lives of the order of
$10^{18-19}$ a are still considerably far away from theoretical predictions.
The shortest half-lives are expected for the $\rm 2\nu\epsilon\epsilon$ ground
state transitions which were not accessible in this search. However, they are
predicted as low as $3\times 10^{18}$ a for ${}^{136}\rm Ce$ and come into
reach with an instrumented CeBr3 crystal. $\rm 2\nu\epsilon\beta^{+}$ decay
modes to the ground state have slightly longer half-life predictions but an
enhanced experimental signature with two annihilation photons. This signature
could not be exploited in this setup but becomes a powerful discriminator
using CeBr3-HPGe coincidences. The shortest decay with $\gamma$-ray emission
is the ${}^{136}\rm Ce$ $0^{+}_{1}$ mode, predicted at $5.7\times 10^{25}$ a.
Reaching this sensitivity requires significant improvement and investment.
However, the only two observed double beta decay excited state transitions in
${}^{100}\rm Mo$ and ${}^{150}\rm Nd$ were measured at a significant lower
half-life than predicted ESReview . Surprises are possible for the complicated
nuclear physics of double beta decay and future experimental searches are well
motivated.

##  VII Acknowledgements

This work received support from the EC-JRC open access scheme EUFRAT under
Horizon-2020, project number 35375-1. EIG EURIDICE and the staff of HADES are
gratefully acknowledged for their work. One of the authors (A. O.)
acknowledges the support from the Extreme Light Infrastructure Nuclear Physics
(ELI-NP) Phase II, a project co-financed by the Romanian Government and the
European Union through the European Regional Development Fund, the
Competitiveness Operational Programme (1/07.07.2016, COP, ID 1334). Another
authors (J. K.) acknowledges the support from the Academy of Finland, Grant
Nos. 314733 and 320062.

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