Detection prospects for the second-order weak decays of 124 Xe in multi-tonne xenon time projection chambers
Christian Wittweg, Brian Lenardo, Alexander Fieguth, Christian Weinheimer
DDetection prospects for the second-order weak decays of
Xe in multi-tonne xenontime projection chambers
Christian Wittweg, ∗ Brian Lenardo, Alexander Fieguth, and Christian Weinheimer Institut f¨ur Kernphysik, Westf¨alische Wilhelms-Universit¨at M¨unster, 48149 M¨unster, Germany Physics Department, Stanford University, 382 Via Pueblo Mall, Stanford, CA 94305, USA (Dated: February 12, 2020)We investigate the detection prospects for two-neutrino and neutrinoless second order weak de-cays of
Xe – double electron capture (0 / ν ECEC), electron capture with positron emission(0 / ν EC β + ) and double-positron emission (0 / νβ + β + ) – in multi-tonne xenon time projectionchambers. We simulate the decays in a liquid xenon medium and develop a reconstruction algo-rithm which uses the multi-particle coincidence in these decays to separate signal from background.This is used to compute the expected detection efficiencies as a function of position resolution andenergy threshold for planned experiments. In addition, we consider an exhaustive list of possiblebackground sources and find that they are either negligible in rate or can be greatly reduced usingour topological reconstruction criteria. In particular, we draw two conclusions: First, with a half-lifeof T ν EC β + / = (1 . ± . · yr, the 2 ν EC β + decay of Xe will likely be detected in upcomingDark Matter experiments (e.g. XENONnT or LZ), and their major background will be from gammarays in the detector construction materials. Second, searches for the 0 ν EC β + decay mode are likelyto be background-free, and new parameter space may be within the reach. To this end we investigatethree different scenarios of existing experimental constraints on the effective neutrino mass. Thenecessary 500 kg-year exposure of Xe could be achieved by the baseline design of the DARWINobservatory, or by extracting and using the
Xe from the tailings of the nEXO experiment. Wedemonstrate how a combination of
Xe results with those from 0 νβ − β − searches in Xe couldhelp to identify the neutrinoless decay mechanism.
Keywords: Dark matter, double beta decay, double electron capture, direct detection, liquid xenon, dual-phase time projection chamber, nuclear recoils, nuclear matrix element, low-energy calibration, neutronelastic scattering
I. INTRODUCTION
The origin of the matter-antimatter asymmetry in theUniverse and the mechanism generating the neutrinomasses are among the great unsolved questions of mod-ern particle physics. Neutrinoless second-order weak de-cays are one of the experimental channels available toaddress these questions by testing the Majorana natureof neutrinos [1–4]. Most experimental effort to date hasfocused on searching for the neutrinoless double beta de-cay (0 νβ − β − ) of neutron-rich candidate isotopes [5–9],due to their relatively high natural abundance comparedto proton-rich candidates. However, proton-rich isotopesoffer unique decay topologies that make them of consid-erable experimental interest as well. In particular, thosewith Q-values greater than 2044 keV (4 m e c ) can de-cay in three possible modes – double-electron capture(0 / ν ECEC), double-positron emission (0 / νβ + β + ) andsingle electron-capture with coincident positron emission(0 / ν EC β + ) [10] – which each produce a different ex-perimental signature. In detectors with high-fidelity po-sition reconstruction, tagging the specific combinationsof emitted particles would be a powerful tool for discrim-inating signal events from backgrounds, potentially pro-viding an extremely low-background or background-free ∗ [email protected] experiment.While searches for the neutrinoless decays can com-plement 0 νβ − β − searches [2, 3] positronic, second-orderdecays with neutrino emission are theoretically well-established [11]. Here, new measurements can be usedas a benchmark for nuclear matrix element calculationsat the long half-life extreme.The isotope Xe is of particular interest as its Q-value of (2856 . ± .
12) keV [12] energetically allows allthree two-neutrino and neutrinoless decay modes. Itsdouble-K-electron capture (2 ν ECEC) has recently beenmeasured with the XENON1T Dark Matter detector [13].At T ν KK1 / = (1 . ± . stat ± . sys ) × yr the measure-ment agrees well with recent theoretical predictions [14–16]. In this decay, the measurable signal is constituted bythe atomic deexcitation cascade of X-rays and Auger elec-trons that occurs when the vacancies of the captured elec-trons are refilled. In the XENON1T measurement thiscascade was resolved as a single signal at 64.3 keV. Anobservation of the KL-capture and LL-capture [10] couldbe within reach in future experiments if background lev-els can be controlled, which would allow the decouplingof the nuclear matrix element from phase-space factors.Furthermore, the discovery potential for the positron-emitting modes (2 ν EC β + or 2 νβ + β + ) in future, longer-exposure experiments could be enhanced by their dis-tinct experimental signatures [17]. Position-sensitive de-tectors could tag the γ -rays emitted by the annihilatingpositron, providing a tool for rejecting γ -ray and β -decay a r X i v : . [ nu c l - e x ] F e b backgrounds which arise from natural radioactivity. Inbeyond-the-Standard-Model neutrinoless decays, the en-tire energy must be emitted in the form of charged par-ticles or photons, favoring the positron-emitting decaychannel 0 ν EC β + [18–23]. As in the two-neutrino case,the coincidence signature of the atomic relaxation, themono-energetic positron and the two subsequent back-to-back γ -rays could be used to reject background. Xemay also allow a resonant enhancement in 0 ν ECEC toan excited state of
Te [12], which would be needed toprovide accessible experimentally half-lives [24]. The ex-perimental signature contains multiple γ -rays emitted ina cascade, so coincidence techniques can be used to in-crease experimental sensitivity by suppressing the back-ground substantially.Liquid xenon time projection chambers (TPCs) areideally suited to search for Xe decays, due to theirlarge relatively target masses with 1 kg of
Xe per tonneof natural xenon, low backgrounds, O (1 %) energy reso-lution at Q = 2 . ν EC β + , 2 νβ + β + ,0 ν ECEC, 0 ν EC β + and 0 νβ + β + in multi-tonne xenonTPCs such as the next-generation Dark-Matter detectorsLZ [25], PandaX-4t [26] and XENONnT [27], as well asthe future nEXO [28] double- β decay experiment, and theDARWIN [29] Dark Matter detector. We simulate the ex-perimental signatures of the second-order Xe decaysin such detectors, compute the expected signal detectionefficiencies, assess background sources, and calculate theexperimental sensitivity as a function of the
Xe expo-sure. We close with a brief discussion on the physics casefor pursuing these efforts.The signal modeling and estimated half-lives of
Xeare discussed in II. Relevant details of liquid xenon TPCsare described in III. The detection efficiencies for the dif-ferent decay channels will be affected by a given detec-tor’s energy resolution, spatial resolution, energy thresh-old and exposure. We outline the analysis of these effectsand give the resulting efficiencies in section IV. Potentialbackgrounds and their impact are discussed in section V.The experimental sensitivities are then given in VI andfollowed by the discussion in section VII.
II. SIGNALS FROM
XE DECAY
The decay modes under investigation provide distinctsignatures that can be measured by the coincidence andmagnitude of energy depositions (Table I) in a detector.We group the decay modes by the number of emittedpositrons. Each emitted positron will lead to the emis-sion of at least two γ -rays and reduce the energy thatis initially available for the positrons and neutrinos bytwice the positron mass. Each of the 0 ν decays will ex-hibit a monoenergetic total energy deposition while the2 ν decays have continuous spectra due to the neutrinosleaving the detector without further interaction. We only consider decays to the ground state of thedaughter nucleus for the positronic decay modes. A spe-cial treatment is required for 0 ν ECEC, as only decayswhich resonantly populate an excited state of
Te maybe experimentally accessible.
A. Signal models of decay modes / ν EC β + The electron capture with coincident positron emissioncan be written as
Xe + e − → Te + e + (+2 ν e ) + X k , (1)where the Standard Model decay features the emissionof two electron-neutrinos ( ν e ) in addition to the positron(e + ). We assume the most-likely case of an electron cap-ture from the K-shell. This will produce a cascade ofX-rays and Auger electrons (X k ) with a total energy of(31 . ± . + and the two ν e is then given by E e (+ E ν ) = Q − m e c − E k = (2856 . ± .
12) keV − .
00 keV − .
81 keV= (1802 . ± .
12) keV , (2)where one has a monoenergetic positron for the neutri-noless decay and a β -like spectrum for the two-neutrinodecay. Upon thermalization the e + annihilates with anatomic electron resulting in two back-to-back 511 keV γ -rays . / νβ + β + The reaction equation for the β + β + -decay to theground state is Xe → Te + 2e + (+2 ν e ) . (3)The energy available for the two e + and the two ν e isgiven by E (+ E ν ) = Q − m e c = (2856 . ± .
12) keV − .
99 keV= (812 . ± .
12) keV , (4)where one has a continuous spectrum for the energiesof the two positrons for the two-neutrino decay and a The electron mass uncertainty of 44 ppb and the uncertainty onthe K-shell X-ray energy in xenon are neglected in our calcula-tions, as they will not affect the results. Moreover, we note thatthe 2 γ -annihilation is by far the most likely case for positronium,but more γ -rays are possible. TABLE I: Signatures of the different decay modes of
Xe.
Decay mode Emitted quanta Coincidence ν EC β + X-rays/e
Auger , e + , 2 ν X-rays/e
Auger + e + and 2 γ from (e + +e − )0 ν EC β + X-rays/e
Auger , e + X-rays/e
Auger + e + and 2 γ from (e + +e − )2 νβ + β + + , 2 ν + and 4 γ from (2e + +2e − )0 νβ + β + + + and 4 γ from (2e + +2e − )0 ν ECEC X-rays/e
Auger , 2 − γ X-rays/e
Auger and 3 γ peak for the neutrinoless decay. Upon thermalizationthe positrons annihilate to at least four 511 keV γ -raysemitted as back-to-back pairs. We do not model the an-gular correlation of the positrons, as their thermalizationrange is smaller than the spatial resolution in existingand planned experiments.
3. Resonant ν ECEC
In contrast to the former decay modes, the energy re-leased in the 0 ν ECEC decay has to be transferred to amatching excited nuclear state Te ∗ of the daughterisotope, since no initial quanta are emitted from the nu-cleus. For a double-K capture one only has the atomicdeexcitation cascade (X ): Xe + 2e − → Te ∗ + X , Te ∗ → Te + multiple γ . (5)The corresponding energy match has to be exact withinuncertainties to avoid a violation of energy and mo-mentum conservation. Therefore, the excitation energy E exc,res of the state Te ∗ has to fulfill the resonancecondition E exc,res = Q − E = (2856 . ± .
12) keV − (64 . ± . . ± .
13) keV . (6)Here, E = (64 . ± . . E exc,res = (2790 . ± .
09) keV and a correspondingdeviation of (1 . ± .
15) keV [12, 31]. The angular mo-mentum of this state is not precisely known, but 0 + to 4 + The authors of [12] recommend to perform at least one more in-dependent measurement of the Xe → Te Q-value in orderto resolve discrepancies between existing measurements. In ad-dition a determination of J P of the (2790 . ± .
09) keV excitedstate would be helpful in order to further assess the feasibility ofthis decay mode. + + + + + —4 + Te + Xe ν ECEC 2 ν ECEC2 ν EC β + νβ + β + ν EC β + νβ + β + FIG. 1: Decay scheme of
Xe. While 0 / ν EC β + ,0 / νβ + β + and 2 ν ECEC most likely occur to theground state of
Te, 0 ν ECEC resonantly populates anexcited state at (2790 . ± .
09) keV. There are fivedifferent known γ -cascades along three differentintermediate states. The energy level and J P are givenfor each state and the γ -intensities I γ , i for thetransitions have been normalized, such that (cid:80) i I γ , i = 100 % [31].are possible J P configurations. The level scheme relevantto the decay is shown in shown in Fig. 1. There are fivedifferent γ -cascades that are either ≥ + → + → + or ≥ + → + → + → + for two- and three- γ tran-sitions, respectively. As a considerable decay rate isonly expected to 0 + and 1 + states [12], we assume thatthe resonantly populated state is 0 + and focus on the0 + → + → + → + transition that occurs in 57 .
42 %of all decays.
B. Half-life calculations
1. Two-neutrino decays
The half-life predictions for the two-neutrino decaymodes are constructed from( T ν / ) − = G ν | M ν | , (7)where G ν is a phase-space factor (PSF) and | M ν | isthe nuclear matrix element (NME). While the PSF isdifferent among the decay modes [10, 18, 32], the NMEdiffers only slightly between 2 ν ECEC and 2 ν EC β + andis about a factor of two smaller for 2 νβ + β + [14, 15]. Forsimplicity, we assume M ν ECEC = M ν EC β + = 2 × M νβ + β + (8)and use the existing 2 ν ECEC measurement to constrain M ν ECEC . This is justified by the relatively large uncer-tainty from the measured half-life [13] which outweighsthe expected NME differences.As only the value for the double K-capture has beenreported, the half-life has to be scaled by the fraction ofdouble-K decays f ν KK = 0 .
767 [20]. One obtains a totalhalf-life of T ν ECEC1 / = (1 . ± . × yr . (9)Using Eq. (7) with the measured half-life and calculatedPSFs one has T ν EC β + / = G ν ECEC G ν EC β + × T ν ECEC1 / ,T νβ + β + / = 4 × G ν ECEC G νβ + β + × T ν ECEC1 / . (10)The resulting expected half-lives for 2 ν EC β + and2 νβ + β + are given in Tab. II. Due to the smaller availablephase-spaces, the 2 ν EC β + half-life is about one orderof magnitude longer than the one for 2 ν ECEC, whereasthe 2 νβ + β + half-life is about six orders of magnitudelonger. This makes 2 ν EC β + a promising target for next-generation experiments such as LZ or XENONnT whilethe double-positronic mode will be challenging to mea-sure.
2. Neutrinoless decays
In case of the neutrinoless decays the equation relatingPSF and NME to the half-life changes to( T ν / ) − = G ν | M ν | | f ( m i , U ei ) | . (11)Note that the PSF ( G ν ) and NME ( | M ν | ) are dif-ferent from those used previously due to the absence ofneutrino emission. The additional factor f ( m i , U ei ) con-tains physics beyond the Standard Model. Typically thedecay is assumed to proceed via light neutrino exchange,for which we have f ( m i , U ei ) = (cid:104) m ν (cid:105) m e = (cid:80) k=light ( U m k ) m e . (12)Here the effective neutrino mass (cid:104) m ν (cid:105) is a linear combi-nation of neutrino masses m i and elements of the PMNS mixing matrix U ei [24, 33]. For 0 ν ECEC a resonancefactor R has to be added to Eq. (11):( T ν ECEC1 / ) − = G ν | M ν | | f ( m i , U ei ) | · R . (13)The mismatch ∆ = | Q − E − E exc | = (1 . ± .
15) keVbetween the available energy and the energy level of thedaughter nucleus in the excited state E exc [12] definesthe resonance factor R , which – with the two-hole widthΓ = 0 . R = m e c Γ∆ + Γ / = 2 . ± . . (14)We take the PSF values again from the review [32]which summarizes work by the reviewers and from[20, 33]. In order to calculate half-life expectations forneutrinoless decays of Xe, we also need estimates forthe NME, and the effective neutrino mass (cid:104) m ν (cid:105) . TheNMEs have never been measured for the neutrinolesscase. Only for the case with two neutrinos a few half-liveshave been determined experimentally. Unfortunately theNMEs for the 2 ν and the 0 ν cases are not strongly con-nected. Moreover, the effective neutrino mass has neverbeen measured, and we must choose among different ex-perimental constraints accordingly. To account for thesetwo sources of unknowns we use the following two ap-proaches to get lower limits for the expected half-lives ofneutrinoless double-weak decays of Xe.
Method 1:
In the first approach, to constrain theeffective neutrino mass we take the newest result fromthe neutrino mass experiment KATRIN which set themost stringent direct, model-independent limit on m ν < . (cid:104) m ν (cid:105) < . − . /c (90% C.L.) . (15)For the NMEs in our first approach, we take three avail-able sets of calculations into account. The first set isbased on the quasi-random phase approximation (QRPA)and was calculated in [14]. The second comes from theinteracting boson model (IBM) [24, 36]. The third setis based on nuclear shell model (NSM) calculations asperformed for the two-neutrino case [16] and is limitedby lower and upper values of the full shell model similarto normal neutrinoless double- β decay as shown in [37]and [38]. Both the QRPA and NSM calculations pro-vided good predictions of T / for 2 ν ECEC while therewere no 2 ν -predictions for IBM. We summarize the rele-vant PSF- and NME-values and the corresponding lowerhalf-life limits in Tab. III. Method 2:
In our second approach, we use a similaridea as for the prediction of the half-lives in the 2 ν case.Instead of one measured half-life value we take the half-life limits obtained in the search for 0 νβ − β − decay ofTABLE II: The different 2 ν decay modes of Xe with the corresponding phase-space factors (PSF), theassumptions of the corresponding matrix elements according to Eq. (8), and the measured or predicted half-livesaccording to Eq. (9) and Eq. (10), respectively. The PSF values were taken from the review [32] which summarizeswork by the reviewers and from [10, 33]. Therefore, we give a range of PSF values. For predicting the half-lives ofthe decay modes 2 ν EC β + and 2 νβ + β + we use the central value of this range as the most probable PSF value andhalf of this range as the uncertainty. Decay mode G ν [yr − ] M ν (8) Half-life [yr]Measured (9) Predicted (10) ν ECEC (1 . − . · − M ν ECEC (1 . ± . · ν EC β + (1 . − . · − M ν ECEC (1 . ± . · νβ + β + (4 . − . · −
26 12 · M ν ECEC (2 . ± . · Xe [5, 6]. The most stringent lower limit on the half-life comes from the KamLAND-Zen experiment [5] with T νβ − β − , / > . · yr (90% C.L.) . (16)Unlike the case for the various 2 ν -decays in Eq. (10), theNMEs of 0 ν -decays of Xe are different from the NMEsof the 0 νβ − β − -decay of Xe and do not cancel. Butfor this comparison of the half-life limits of neutron-poorand neutron-rich isotopes of the same element xenon, thesubstantial uncertainties connected to these calculationsdrop out to a large extent if the NMEs are calculatedwithin the same framework and if the main uncertaintiesstem from the unknown quenching q of the axial couplingconstant g A , which can be factorized out of the NME M :˜ M := Mq · g (17)This is of advantage, as the uncertainties of the NME areoften summarized in the g A -quenching, and the quench-ing factors q can be assumed to be similar for neutron-poor and neutron-rich nuclei of the same element [39].We perform the calculations using the NMEs from the in-teracting boson model (IBM) [24, 36] which possess theirmain uncertainties in the quenching of the axial couplingconstant. If we solve equation (11) or (13) according tothe factor f ( m i , U ei ) and equate for two decays of differ-ent xenon istopes, we obtain: T νβ + β + , / = T νβ − β − , / · G νβ − β − G νβ + β + · | ˜ M νβ − β − | | ˜ M νβ + β + | ,T ν EC β + , / = T νβ − β − , / · G νβ − β − G ν EC β + · | ˜ M νβ − β − | | ˜ M ν EC β + | ,T ν ECEC , / = T νβ − β − , / R · G νβ − β − G ν ECEC · | ˜ M νβ − β − | | ˜ M ν ECEC | . (18)In Eq. (18) the effective neutrino mass and its uncertaintydrop out of the equation. We would like to underline the validity of this kind of comparison by mentioning thata similar approach has been used in Ref. [40] (Fig. 6therein) to relate effective neutrino mass limits for twodifferent isotopes within the same theoretical frameworkof NME calculations. Again, we summarize the resultsin Tab. IV.Among the neutrinoless decays, the 0 ν EC β + mode isexpected to have the shortest – and thus most experi-mentally accessible – half-life. The other decay modes,0 νβ + β + and resonant 0 ν ECEC, exhibit considerablylonger half-lives owed to unfavourable phase-space anda lack of resonance enhancement R , respectively. Wealso note that the half-life limits calculated with the firstmethod are systematically lower than the ones for thesecond. This is in line with the smaller predicted upperlimits on the effective neutrino mass (cid:104) m ν (cid:105) < . − .
165 eV /c (19)given by KamLAND Zen [5].The ability to observe any of the described decay chan-nels is given not only by the theoretical prediction ontheir half-lives, but also by the detection efficiencies ina given experiment. In the following sections, we dis-cuss the detection prospects of the neutrinoless decaymodes in future experiments which could have signifi-cantly larger samples of Xe than current detectors,such as DARWIN or nEXO.
III. DETECTOR RESPONSE
In a liquid xenon TPC, the energy and position of anenergy deposit is reconstructed using two observed sig-nals: scintillation light and ionization charge [43]. Theformer is typically detected directly using UV-sensitivephotodetectors, producing a prompt signal referred to asS1. The latter is detected by applying an electric fieldacross the liquid xenon volume and drifting the charges toa collection plane. The charge can be either detected di-rectly using charge-sensitive amplifiers, or extracted intoa gas-phase region and accelerated, producing propor-tional electroluminescence light that is detected in theTABLE III: Predicted lower limits on the half-life of the 0 ν decay modes of Xe according to Eq. (11) as well asEq. (13),
Method 1:
Limits are given for the range of (cid:104) m ν (cid:105) from Eq. (15) [34, 35]. The PSFs ( G ν ) were takenfrom [24], and the review [32] which summarizes work by the reviewers and from [18, 20]. We use the central valueof the PSF-range as the most probable value and half of this range as the uncertainty. The same is done for theNMEs ( M ν ) in all cases were a range of values is given in the original publication. For 0 ν ECEC the NMEs valuesfrom the quasi-random phase approximation (QRPA) [14] and the interacting boson model (IBM) [24] were used.The NME for IBM is obtained by taking the single value given in the publication and assuming g A = 1 . g A = 1 .
25 under the assumption ofdifferent bases and short-range correlations. For the 0 ν EC β + and the 0 νβ + β + QRPA [14], NSM (calculated [41] asin [16]), and IBM [36] NMEs were considered. The range of NMEs for QRPA and the value for IBM are obtained asabove. However, for the latter an uncertainty is given in the publication instead of a value range. For the NSM theNME-range is given by different model configurations and the most probable value and uncertainty are derived inthe same fashion as for QRPA. All uncertainties are propagated by drawing 10 independent samples from theparameter distributions and multiplying with the upper limit on (cid:104) m ν (cid:105) . Then the 90 % C.L. upper limit on T − / isdetermined from the resulting distribution and inverted to obtain the corresponding lower half-life limit. Decay mode G ν [yr − ] M ν Model Predicted lower T / limit [yr] (90% C.L.) (cid:104) m ν (cid:105) < . / c (cid:104) m ν (cid:105) < . / c ν ECEC 2 . · − . − .
298 QRPA 1 . · . · .
478 IBM 1 . · . · ν EC β + (1 . − . · − . − .
617 QRPA 8 . · . · . .
23) IBM 4 . · . · . − .
77 NSM 1 . · . · νβ + β + (1 . − . · − . − .
617 QRPA 1 . · . · . .
23) IBM 8 . · . · . − .
77 NSM 2 . · . · photodetectors. The delayed secondary signal producedby the drifted charge is referred to as S2. The combi-nation of the two signals allows one to reconstruct the3D-position of the interaction inside the detector: the S2hit pattern on the collection plane gives the x-y coordi-nate and the S1-S2 time delay gives the depth z. Thedeposited energy is reconstructed using the magnitudeof the S1 and S2 signals. A linear combination of bothsignals has been shown to greatly improve the energyresolution compared to either signal individually, due torecombination of electron-ion pairs producing anticorre-lated fluctuations in the energy partitioning between lightand charge [44, 45].For events with multiple energy deposits, the promptS1 signals for each vertex are typically merged, resultingin a single scintillation pulse for the entire event. How-ever, individual vertices can be resolved as individual S2signals arriving at different positions and times on thecharge collection plane. A schematic of the signature ex-pected from a typical 0 / ν EC β + decay of Xe is shownin Fig. 2. In this example, there are five different S2signals produced from the positron, X-ray cascade, andeach of the annihilation γ -rays (one of which undergoesCompton scattering before being absorbed). With suf-ficient position and energy resolution, one can use this information to classify events and perform particle iden-tification, providing a tool for separating backgroundsfrom the signal of interest.The capability for a detector to resolve each vertex de-pends on the time resolution in the charge channel, thewidth of the S2 signals, and the x-y resolution of thecharge collection plane. These properties are highly de-pendent on the specific readout techniques employed ineach experiment. In addition, the detection of each en-ergy deposit requires its individual S2 signal to be abovethe detector’s charge energy threshold, a property thatis again specific to each experiment.In this work, we compute the detection efficiency ( (cid:15) )for the various modes of Xe decay as a function of thex-y- and z-position resolution and energy threshold, toprovide estimates that apply across the possible range ofexisting and future experiments.
IV. ANALYSISA. Simulation
We generate the emitted quanta and their initial mo-mentum vectors for each decay channel with the eventTABLE IV: Predicted lower limits on the half-life of the 0 ν decay modes of Xe according to Eq. (18),
Method 2:
Limits are given for T νβ − β − , / > . · yr as measured by KamLAND Zen [5]. The PSFs for Xe ( G Xe0 ν )were taken from [24], and the review [32] which summarizes work by the reviewers and from [18, 20]. Those for Xe ( G Xe0 ν ) were also taken from [32] and [42], cited therein. We use the central value of the PSF-range as themost probable value and half of this range as the uncertainty. For the NMEs of Xe ( ˜ M Xe ν ) and Xe ( ˜ M Xe ν )we only consider the interacting boson model (IBM) and do not multiply the NMEs with g . We consider [36] for˜ M Xe ν . For 0 ν ECEC the NME was taken from [24]. For 0 ν EC β + and 0 νβ + β + the NMEs were taken from [36].An uncertainty on the NME is only given in this publication. All uncertainties are propagated by drawing 10 independent samples from the parameter distributions and multiplying with the lower limit on T νβ − β − , / . Thenthe 90 % C.L. lower limit of the half-life is determined from the resulting distribution. Decay mode G
Xe0 ν [yr − ] ˜M Xe0 ν G Xe0 ν [yr − ] ˜M Xe0 ν Predicted lower T / limit [yr] (90% C.L.) ν ECEC 2 . · − .
297 (14 . − . · − . · ν EC β + (1 . − . · − . .
76) (14 . − . · − . · νβ + β + (1 . − . · − . .
76) (14 . − . · − . · generator DECAY0 [46]. The version used here has beenmodified previously for the simulation of the positronic Xe decay modes [17]. In the scope of this work, we ver-ified the implementation, added the resonant 0 ν ECECdecay mode, and implemented the angular correlationsfor the γ -cascades under the assumption of J P = 0 + forthe resonantly populated state [47, 48]. In order to inves-tigate the efficiency, at least 10 events per decay channelhave been used.The particles generated for each decay are propagatedthrough simplified models of the detectors under inves-tigation using the XeSim package [49], based on Geant4[50]. These detector models consist of a cylindrical liq-uid xenon volume in which we uniformly generate Xedecay events. This volume is surrounded by a thin shellof copper which is used for modeling the impact of ex-ternal γ -backgrounds. We simulate two different sizes ofcylinders in this work, characteristic of two classes of fu-ture experiments. The “Generation 2” (G2) experimentsare defined as experiments which have height/diameterdimensions of between one and two meters. This classincludes the LZ [25] and XENONnT Dark Matter ex-periments, which will use dual-phase TPCs filled withnatural xenon. It also includes the future nEXO neutri-noless double- β decay experiment, which will use a single-phase liquid TPC filled with xenon enriched to 90% in Xe. For simplicity, we model all G2 experiments asa right-cylinder of liquid xenon with a height and diam-eter of 120 cm each. We also simulate a “Generation3” (G3) experiment, which is intended to model the pro-posed DARWIN Dark Matter experiment [29]. This de- LZ and XENONnT are designed slightly larger than this, butwe show below that this assumption has a minimal effect on thecalculated efficiency. tector is modeled as a right-cylinder of liquid xenon witha height and diameter of 250 cm each.For experiments using nat
Xe targets, there will be ap-proximately 1 kg of
Xe per tonne of target material.The G2 Dark Matter experiments would therefore be ableto reach
Xe-exposures of ∼ −
100 kg-year in 10 yearsof run time. By scaling the target mass up to 50 tonnes,the G3 experiment DARWIN will amass an exposure of ∼
500 kg-year. For nEXO, the enrichment of the targetin
Xe will remove all of the
Xe; however, here weconsider the possibility of extracting the
Xe from thedepleted xenon and mixing it back into the target. Therewill be approximately 50 kg of
Xe in the nEXO tail-ings, meaning a 10 year experiment could amass an ex-posure of ∼
500 kg-year, competitive with a G3 naturalxenon experiment.
B. Energy resolution model
Within this study all simulated detectors use the en-ergy dependence of the resolution on the combined signalas reported in [13], which is modeled as σ E E = a √ E + b. (20)Here E is the energy and a = 31 keV / and b = 0 .
37 areconstants extracted from a fit to calibration data from41 −
511 keV. The model predicts a resolution of ∼ Xe decay. For the neutrino-less modes, this corresponds to a narrow energy windowaround the Q-value.
Te e + X-ray/e
Auger 𝜸 𝜸 𝜸 z/t 511 keV31.8 keV 511 keV ≤ keV 𝜸 e + 𝜸 𝜸 X-ray/e
Auger xy 𝜸 𝜸 𝜸 e + X-ray/e
Auger
FIG. 2: Schematic of a 0 / ν EC β + -decay signatureinside a xenon time projection chamber. As shown inthe bottom panel, an initial positron and atomicexcitation quanta are emitted and deposit their energyclose to the nucleus. Two secondary γ s are emittedafter the annihilation of the positron. One of those isdirectly absorbed and the other Compton-scattersbefore photoabsorption. On the dt /z-axis the ionizationsignals of γ , γ , and γ , can be distinguished fromone another. The positron and atomic deexcitationsignals are merged with γ , in this example. The toppanel shows the corresponding hit-pattern of theionization signal. In x-y-coordinates the individualscatters of γ can be clearly distinguished from γ ,while the discrimination of the atomic deexcitationquanta and the positron from γ is not trivial. Thescintillation signal is merged for all energy depositionsand is not shown in the figure. The sizes of thescintillation signals in x-y and z-coordinates roughlycorrespond to the magnitudes of the energy depositions.For the filtering of single energy depositions within anevent, the reconstruction can only be based on the S2.To model the broadening of the charge-only energy res-olution due to recombination fluctuations, we scale b inthe above formula to a value of 4.4. This gives a charge-only resolution of about 6 % at ∼
500 keV, consistent withmeasurements reported in the literature [52, 53].
C. Event reconstruction and efficiency calculation
This analysis utilizes the information on the variousenergy depositions at a given spatial position in all threedimensions. In order to reconstruct and validate the effi-ciency for detecting the unique event topologies, severalfiltering and clustering steps have to be performed .First the events are filtered by the total energy de-posited in the detector, in order to account for eventswhere decay products leave the detector. This criterionis a fixed value, only broadened by the energy resolutionfor the neutrinoless modes, but a broad range with amaximum cut-off at the Q-value and a decay-dependentthreshold for the two-neutrino decays.For any remaining event the vertices are sorted by theiraxial position in the detector (z-coordinate) and thesevertices are grouped within a spatial range determinedby the assumed position resolution of the detector in theaxial direction. For detector configurations where a sep-aration in the radial direction (x-y-coordinate) is alsopossible, the grouping algorithm also takes separationsin x-y into account – according to the assumed positionresolution. The energies of all vertices within each groupare summed and provide the individual S2 signals that adetector with the chosen properties would see.From this point the further filtering targets the recon-struction of the vertices of the annihilation products ofthe positrons. The procedure is analogously applied tothe de-excitation γ s in the case of the 0 ν ECEC. It isdepicted together with an illustration of the spatial clus-tering in Figure 2. All clustered energy depositions ofa given event are permuted for each possible interactioncombination and the total sum of the energy is comparedagainst the expected value, which is e.g. 511 keV for each γ produced in the positron’s annihilation. The combina-tion with the smallest difference between the summedenergy and the expected value is then removed from thelist of energy depositions if it lies within the energy reso-lution around the expected value. This raises the counterof measurable signatures by one. Afterwards, this pro-cedure is repeated until all desired signatures have beenfound and the counter matches the expectation (e.g. 4in 2 νβ + β + ). For any left-over energy it is then checkedif it fulfills the requirement for the point-like depositionexpected from the positron and/or the electron capturesignal. In case of 0 ν EC β + /2 ν EC β + a single mergedenergy deposition of the positron and atomic relaxationprocesses is expected. While this requirement is a fixedmaximum value for a single signature in case of the neu-trinoless mode, it is again a continuous distribution rang-ing from zero or the 31 .
81 keV K-shell hole energy to a We note that the application of machine learning techniques,such as boosted decision trees, in these filtering and clusteringsteps would likely enhance the signal efficiencies and backgroundrejection. However, their application exceeds the scope of thiswork, so the conventional analysis outlined here can be regardedas a baseline scenario.
Threshold [keV] E ff i c i e n c y [ % ] ν ECEC0 ν ECβ + ν ECβ + FIG. 3: Dependence of the charge-only energy thresholdon the detection efficiency for selected decay channels.The efficiency for the 0 ν EC β + (blue) and 2 ν EC β + (black) show a decrease with energy up to about250 keV with efficiencies ranging from about 41 % to10 %. More striking is the behavior of the 0 ν ECEC(red), which has a sharp cut-off as soon as the doubleelectron capture energy (64.3 keV) is below thethreshold. Since this signature is required within thisanalysis in order to provide a clear evidence andnecessary background suppression, this willautomatically drop the efficiency to zero. For thisexample a position resolution of 10 mm has been used inboth directions.cut-off depending on the Q-value. The requirement re-moves energy signatures which are merged by the detec-tor due to the aforementioned limited spatial and timeresolution. If not all signatures have been found or if theremaining energy is not a single deposition, the event isdiscarded. The ratio of all events which survive the fil-tering algorithm and the original generated number ofevents corresponds to the desired efficiency (cid:15) . D. Influence of thresholds, detector size andposition resolution
We first investigate the effect of the S2 energy thresh-old on the detection efficiency. While for Dark Matterdetectors the threshold usually is only a few keV thanksto amplification via electroluminescence, the situation isdifferent for an experiment like nEXO, which will mea-sure charge directly. In this case, the electronics noise inthe readout circuit introduces a larger energy thresholdand thus influences the efficiency for the example decaymodes as shown in Figure 3. In this work, the threshold isimplemented in the simulation simplified as a sharp cut-off for any given energy signature, assuming a positionresolution of 1 cm in radial and axial direction. It is evi-dent that the efficiency depends on this energy threshold.Therefore, an improvement from O (100 keV) as achieved in EXO-200 [54] would be beneficial for nEXO as thishas a direct impact on the sensitivity for any given de-cay channel. Especially, in order to look for a possiblesmoking gun evidence of the 0 ν ECEC decay, a thresholdbelow the energy of twice the K-shell electron energy isnecessary. In the following we assume that a sufficientlylow threshold is achieved that it can be considered neg-ligible.Next, we investigate the effect of a detector’s positionresolution on the detection efficiency. Our results areshown in Figure 4 where we emphasize the importanceof x-y resolution. For any detector with an axial positionresolution (z-coordinate) of a few mm, which is funda-mentally limited by electron diffusion, an additional res-olution of event topologies in the radial direction is highlybeneficial. Already at an achieved 10 mm separation inthe axial direction, an x-y resolution of also 10 mm canimprove the efficiency by more than a factor of two. Fora nEXO-type detector this resolution is mostly a func-tion of the pitch of the charge readout strips [55], andtherefore can become as small as a few mm. The situ-ation is less clear for dual-phase detectors used in DarkMatter searches; no detector dedicated for Dark Mattersearch has reported its x-y resolution for multiple en-ergy depositions arriving at the charge detection planesimultaneously. In principle this should be achievable bypattern recognition in the top array of the detector, andis a good candidate for future work in better matchingalgorithms and machine learning techniques.Finally, an interesting comparison arises between anEXO like detector and a G3 Dark Matter experiment,as both could have the same amount of
Xe withindifferent-sized detector volumes. The influence of thedetector size on the efficiency for the decay mode of0 ν EC β + is shown in Fig. 4. It is evident that an in-creased detector size only increases the efficiency by afew %. This is due to the ratio of events leaving thedetector in comparison to the events confined in the fullvolume. Therefore, the findings for a G3 detector thatare summarized in Table V are approximately also validfor a nEXO-like detector. V. BACKGROUNDS
From the above analysis, it is clear that the most exper-imentally accessible decay channels are the 0 ν / ν EC β + .As described, the key feature in a search for β + -emittingdecay modes is the ability to reject backgrounds usingthe distinct event topology. We consider possible sourcesof backgrounds below and estimate the expected rates ofevents passing the topological selection criteria describedin Section IV.As comparison points, we compute the expected num-ber of Xe decays per tonne-year exposure of nat
Xe(corresponding to 0.95 kg-year of
Xe) using the half-lives estimated in Table II, Table III and Table IV. Af-ter including the respective efficiencies for a G2 exper-0TABLE V: Efficiencies for all evaluated decay channels in a G2 and a G3 experiment assuming three different radialresolutions and an axial position resolution of 10 mm. Threshold effects are considered to be negligible.
Decay channel Only z [%] 30 mm x-y [%] 10 mm x-y [%]G2 G3 G2 G3 G2 G3 ν EC β +
22 24 27 31 42 472 νβ + β + ν EC ν EC a ν EC β +
19 21 23 27 37 410 νβ + β + a Here we considered the most probable branch (57.42%) with a three-fold γ -signature. An analysis using the two-fold signatures wouldyield higher efficiency but can add coincidental γ -backgrounds, which would weaken the sensitivity of a given search.
10 20 30 40
Z separation [mm] E ff i c i e n c y [ % ] G3 3mm x - y G2 3mm x - y G3 10mm x - y G2 10mm x - y G3 30mm x - y G2 30mm x - y G3 no x - y G2 no x - y FIG. 4: Comparison of efficiencies for the 0 ν EC β + decay, for different x-y resolutions and detector sizes asa function of the axial resolution. The black (red) linesshow the efficiencies for a G2 (G3) detector with 3 mm(solid), 10 mm (dashed), 30 mm (dotted) and no(dashdotted) x-y resolution.iment with 10 mm resolution in x-y-z and assuming anatural xenon ( nat Xe) target, we expect 8 . ± . ν EC β + . Under the assumption oflight-neutrino exchange and given the most optimisticassumptions described above, we expect a rate of lessthan 2 . · − decays per tonne-year for 0 ν EC β + . A. Radiogenic backgrounds from detectormaterials
Gamma rays from radioactivity in the laboratory en-vironment and detector construction materials are a pri-mary background in rare event searches. There are twomain concerns for the analysis presented here: first, thata γ -ray Compton-scatters multiple times and producesthe expected event signature. Second, that a γ -ray of sufficient energy creates a positron by pair production.In the latter case, the positron will annihilate and pro-duce a background event which, by design, passes ourevent topology cuts.We investigate the sources for falsely identified eventsfrom the U and
Th decay chains, the most com-mon sources of radiogenic backgrounds in most 0 νββ searches. For each decay step within the chain, 10 events have been uniformly generated in a copper shellof 1 cm thickness surrounding the liquid xenon volume ofa G2-sized detector using Geant4. Afterwards, the eventswhich interacted in the active volume were run throughthe respective event search algorithms for 2 ν EC β + and0 ν EC β + . We find that the only relevant decays are β -decays into excited daughters, as only these produce γ sof sufficient energies.For the neutrinoless case there are two particularlyproblematic transitions. The first is the β -decay of Biin the
U-chain, which has a small branching to the2880 keV state of
Po. If this γ -ray interacts via pairproduction, it creates an event identical to our signaldirectly in the ROI. We find that 1 . · − events per Bi primary decay pass the selection criteria. The sec-ond problematic transition is the decay of
Tl to
Pbin the
Th-chain, for which there are various transitionsin which different γ -rays are detected in coincidence withthe one from the 2614 keV state. Such events can depositenough energy to create events in the ROI, and may sim-ilarly produce a sequence of energy depositions whichpass our topological criteria. We find that 1 . · − events pass our cuts per Tl primary, but the 35.9 %branching fraction for creating
Tl in the first place re-duces its impact in a real detector to 4 . · − events per Th primary decay. Both sources of background canbe reduced by a subselection of an inner volume in the By simulating the same number of events for each decay we im-plicitly assume decay chain equilibrium. This is not necessarilyrealized in actual detector construction materials. γ -rays with energiesbelow 300 keV are paired with a high energy γ in the Tl decay signature, fiducializing is especially effectiveagainst these events; we find that cutting away the outer10 cm of LXe reduces its background contribution by al-most an order of magnitude. For a 20 cm cut, no eventout of the 10 simulated for any isotope passes the selec-tion criteria. We conclude that these backgrounds cantherefore be eliminated in a real experiment (dependingon the actual U/ Th contamination) by selecting anappropriate fiducial volume.Radiogenic backgrounds have a greater impact on2 ν EC β + searches, as the larger energy window allowsmore events to pass the selection criteria. We find threeisotopes in the U chain producing events which passour selection criteria, with decays of
Bi into differ-ent excited states of
Po being the major backgroundcomponent ( >
99 %). The surviving fraction for the totalchain is 6 . · − events per U primary decay withouta fiducial volume selection. This is reduced to 1 . · − and 1 . · − decays per primary with the 10 cm and20 cm cuts respectively. For the Th-chain, the Pb γ -rays following Tl β -decay are again the main con-tributor ( ∼ γ -rays from Th after the β -decay of Ac also contribute ( ∼ ∼ Pbfollowing the β -branch of the Bi decay. The surviv-ing fractions for the whole chain are 7 . · − , 1 . · − and 1 . · − events per primary Th decay with nofiducial volume cut, a 10 cm cut and a 20 cm cut, re-spectively. Due to the less-stringent energy selection thefiducial volume cuts are less efficient for the
Tl eventsin the two-neutrino case, but still significantly reduce thebackground contribution.In conclusion, two factors play a role for the exactevaluation of a given experimental setting: the fiducialvolume cut and the actual amount of contaminants sur-rounding the TPC. While this study cannot provide ananswer for all given experimental settings – this wouldneed a dedicated Monte Carlo study following a materialradioassay – we use reported contamination levels andexperimental details projected for the nEXO experiment(reported in Ref. [56]) to benchmark our calculations.Our approximate evaluation of a nEXO-like experimentis provided in Fig. 5. The nEXO experiment identi-fies the main source of external γ -ray backgrounds asthe copper cryostat, for which the collaboration reports U and
Th concentrations of 0 .
26 ppt and 0 .
13 ppt,respectively. This corresponds to 2 . · primary de-cays per year as indicated in Figure 5 by the dotted grayline. Accordingly, it would only require a mild 10 −
20 cmfiducial volume cut to achieve a favorable signal to back-ground ratio . Dark matter experiments, on the other This assumes the ∼
650 kg copper TPC vessel, the main contrib-utor according to [56], as the sole background source.
FIG. 5: Expected events detected falsely as signal for agiven number of primary decays per year for the
Uand
Th decay chains. A reduction from theexpectation in the full detector volume (solid black) isachieved by cutting the fiducial volume in alldimensions by 10 cm (red), 20 cm (blue) or 35 cm (gold).For reference the number of expected 2 ν EC β + signalevents is shown for a 50 kg-year Xe-exposure (dashedblack) and for the respective fiducial volumes (dashedred, blue and gold) with correspondingly reducedexposures. The expected number of
Th and
Uprimary decays per year for the nEXO cryostat isindicated as the dotted gray line.hand, are optimized for the low-energy regime, and typi-cally have higher background levels in the ∼ MeV regime,and may therefore require more aggressive fiducial cutsto achieve a similar signal-to-background ratio.We emphasize that these results are only approximate,and that in a full likelihood analysis the modeling of theevents’ spatial components and energy distributions willimprove the signal to background ratio beyond what hasbeen discussed above. More precise estimates of the im-pact of these backgrounds are beyond the scope of thiswork, but will be necessary to understand the true im-pact of externally-produced γ -ray backgrounds in realexperiments. B. Rn Rn may dissolve into the active LXe volume and cre-ate backgrounds via β -decays that emit γ -rays ( α -decayevents can be easily rejected by the ratio of ionized chargeto scintillation light [57]). There are only two β -decaysin the Rn chain with enough energy to create back-grounds in this analysis. The first,
Bi, is accompa-nied by the subsequent α decay of Po, which occurswith T / = 164 µ s. Thus, we assume that it can be re-jected via a coincidence analysis. The second, Bi, hasa Q-value of 1.2 MeV – just at the low-energy end of ourregion of interest for the 2 ν decays, but well below the2ROI for 0 ν signals – and decays with no accompanying γ . Therefore, it almost always is a single-scatter signaland does not pass our cuts. C. Charged-current scattering of (anti)neutrinos
Charged-current (CC) scattering of neutrinos and an-tineutrinos, while rare, may produce positrons which canexactly mimic our signal of interest.The CC scattering of low-energy antineutrinos pro-duces a fast positron in the final state. Here we con-sider two sources of antineutrinos: nuclear reactors andradioactive decay in the earth (geoneutrinos). Both ofthese are sources of electron antineutrinos in the few-MeV range. The threshold for the charged-current re-action is set by the mass difference between the xenonisotopes and their iodine isobars. The cross-sections as afunction of energy were computed in Ref. [58], and wereobtained in tabular form from the authors.We calculate the expected rates for geoneutrinos us-ing the two xenon isotopes with the lowest CC reactionthresholds:
Xe and
Xe, which have thresholds of1.2 MeV and 2.0 MeV, respectively. Convolving the en-ergy spectra and flux with the cross section, we find thatthe rates for
Xe and
Xe are 5.0 × − and 4.9 × − events per tonne-year of nat Xe exposure, respectively. Ina G3 detector filled with nat
Xe, there will, therefore, beless than 0.01 events in a 10-year exposure, renderingthis background negligible. An experiment using xenonenriched in the heaviest isotopes (
Xe and
Xe) willbe completely insensitive to geoneutrinos due to the highthreshold for CC reactions. The flux of geoneutrinos isexpected to vary by a factor of ∼ . × − and 3 . × − events per tonne-year for scat-tering on Xe and
Xe, respectively. The expectedrates at the other candidate locations are smaller by atleast an order of magnitude.In contrast to antineutrinos, CC scattering of low-energy neutrinos does not directly create positrons. There are, however, two possible backgrounds that mayarise from this reaction: the emission of a fast elec-tron and a daughter nucleus in an excited state (whichcan de-excite and create additional energy deposits thatmay mimic the signal event topology), and the creationof a daughter radioisotope which later decays via β + -emission.The primary source of neutrinos incident on deep un-derground detectors is the sun. For our purposes, themost important are those produced by the decay of B,which have energies of ∼ −
10 MeV. These are the onlysolar neutrinos with enough energy to react above thresh-old and populate an excited state in the daughter nu-cleus, for all xenon isotopes. The energy-averaged cross-section for these reactions is tabulated in Ref. [58], andis of O (cid:0) − − − (cid:1) cm . This may produce 10’s ofevents per tonne-year for each isotope in a nat Xe detector.However, scattering into low-lying excited states is sup-pressed, with partial cross-sections an order of magnitudesmaller than the total. Therefore, most of the events willdeposit too much energy in the detector and will be re-jected. We find that the low probability of the remainingevents passing our topological selection criteria rendersthese backgrounds negligible for both the 2 ν EC β + andthe 0 ν EC β + decay modes.Neutrino CC scattering on xenon may create radioac-tive isotopes of caesium in the liquid target. Of particularconcern are Cs and
Cs, which each have half-livesof < β + -emission with Q-valuesof 3.9 MeV and 2.9 MeV, respectively, exactly mimick-ing our expected event signature. Again using the B-averaged cross sections from Ref. [58], we calculate aproduction rate of 0.02 nuclei of
Cs and 0.07 nucleiof
Cs per tonne-year of nat
Xe exposure. The resulting β + decays are distributed across a broad spectrum, andour simulations indicate that they will be a small back-ground for the 2 ν EC β + process, with expected rates anorder of magnitude lower than the expected signal rate.The narrow ROI for 0 ν searches will render these back-grounds negligible. There are also two isotopes of xenonwith CC reaction thresholds low enough to react withCNO, Be, and pp neutrinos: Xe and
Xe. How-ever, the relevant Cs daughter isotopes have half-lives of O (10) days. Next-generation experiments plan to recir-culate and purify the liquid xenon with a turnover timeof ∼ D. Neutron-induced backgrounds
A final possibility for backgrounds are those from neu-tron scattering or capture. In neutron capture, thedaughter nucleus is generally left in a highly-excitedstate, and relaxes to the ground state via the emissionof several γ -rays. As the sum total of the energy lost inthis process is well above the Q-value for Xe decay, we3expect these events will be easy to reject and we neglectthis as a background source.For neutron scattering, it is of particular interest to es-timate the activation rate of Xe-radioisotopes that maydecay via β + emission in the region of interest. Weidentify the fast neutron scattering Xe( n, n ) Xereaction as the only one of significance. It has aneutron-energy threshold of 10.5 MeV and the cross-section reaches ∼ ∼
20 MeV [61]. The high threshold prevents radiogenicneutrons (which come from ( α ,n) reactions in the lab-oratory environment) from producing this background,but muon-induced neutrons, which can extend in energyup to the GeV scale, are of concern. We use an esti-mate of the muon-induced neutron flux at Gran Sasso of10 − n/cm /s and multiply by a factor of 10 − to accountfor the expected reduction due to shielding typically em-ployed in these experiments [62]. We find an expectedactivation rate of ∼ − atoms per kg ( Xe) per year,each of which we assume will produce a background eventin the TPC. However, this decay has a small branchingratio for β + decay, and even then always proceeds to anexcited state of I. Accordingly, it is unlikely to passour selection criteria, and we consider it negligible.
E. Summary
After considering an exhaustive list of backgroundsources, for 2 ν EC β + we conclude that the only signif-icant background originates from external γ -rays. Withstrong fiducial volume cuts, a likelihood-analysis utiliz-ing energy information and γ -background suppression,near-future G2 Dark Matter experiments have a strongchance of measuring this decay mode. For 0 ν EC β + , weconclude that the searches in G2 and G3 experimentswill basically be “background-free,” and the sensitivitywill only be limited by the detection efficiencies and theattainable Xe exposure in each experiment.
VI. SENSITIVITY
The half-life measured by a detector configuration withno expected background for a number of N observed de-cay events is given by T / = ln(2) N A × (cid:15) × m × tN × M Xe . (21)Here, N A is Avogadro’s constant, (cid:15) is the detection effi-ciency, and M Xe corresponds to the molar mass of Xe.The available mass of
Xe, m , and the measurementtime, t , depend on the detector configuration. If noevents are observed and if a Poissonian process withoutbackground is assumed, a 90 % C.L. lower limit on T / can be calculated by inserting N = 2 . Xe; the only difference is the ∼
10% decrease in detec-tion efficiency due to the increased probability of energybeing deposited outside the sensitive volume of the de-tector. The sensitivities are compared to the range oftheoretical predictions from Table II for 2 ν -decays, andTables III and IV for 0 ν -decays.Regarding the two-neutrino decays 2 ν EC β + will likelybe detected by a G3 experiment, but are already be ac-cessible to a G2 detector with a nat Xe target if the γ -background is properly addressed. However, due to anunfavourable phase-space 2 νβ + β + will likely be out ofreach of even a G3 detector. On the neutrinoless side0 νβ + β + is also pushed to experimentally inaccessiblehalf-lives by the unfavourable phase-space. An eventualdetection of 0 ν ECEC relies on the presence of a suffi-cient resonance enhancement that could boost the decayrate approximately four orders of magnitude. However,given current measurements of decay energies and
Teenergy levels this is not present [12, 31]. A final inde-pendent measurement is recommended by the authors of[12] would be needed for a final verdict on the detectionprospects of this decay. Thus, it is evident that the mostpromising neutrinoless decay is 0 ν EC β + .For this decay we compare the experimental sensitiv-ity derived in this study with three possible theoreticalscenarios. Scenarios one and two are based on the directcalculation (Method 1, Eq. 11 and Eq. 13) using the effec-tive neutrino mass range from Eq. 15. Scenario three isbased on the comparison of Xe-NMEs with the NMEfor 0 νβ − β − of Xe using the KamLAND Zen half-lifelimit (Method 2, Eq. 18). The results are shown as afunction of exposure in Fig. 6. Within a 500 kg-year-exposure, a background-free experiment would cover asignificant portion of the parameter space given by theKATRIN limit translated to (cid:104) m ν (cid:105) . Once this value is re-duced, e.g. by phase cancellations in the PMNS-matrix,the lower limits on the half-life are an order of magni-tude above the experimental sensitivity. Assuming thesame decay mechanism for Xe and
Xe – here light-neutrino exchange – the expected half-lives are two or-ders of magnitude above the experimental sensitivity tak-ing into account the current limits placed by KamLANDZen. Exposures larger than 10 kg-year would be neededto probe this parameter space. VII. DISCUSSION
This work has summarized the possible decay modes of
Xe and investigated possible efficiencies of future liq-uid xenon detectors to the respective channels. For a G2Dark Matter detector a detection of 2 ν EC β + is feasiblegiven a proper treatment of potential γ -backgrounds. An4TABLE VI: Results and theoretical predictions for the various decay channels of Xe. The experimental sensitivityis calculated for a 500 kg-year exposure assuming a G3-experiment with 10 mm position resolution in all threedimensions, a negligible threshold, and no backgrounds. The range of theoretical predictions for neutrinoless decaysis given between the weakest limit from the direct calculation with (cid:104) m ν (cid:105) < . Decay Exp. Sensitivity [10 yr] Experiment/Theory ν EC β + . ± . · νβ + β + . ± . · − ν ECEC 2.4 1 . · − − . · − ν EC β + . − . · − νβ + β + . − . · − Exposure [kg × yr] T ν E C β + / [ y r ]
90% C.L. lower limit on T ν ECβ + , G3 10 mm x - y - z Method 1: ⟨ m ν ⟩<1.1eV, 90% C.L.Method 1: ⟨ m ν ⟩<0.3eV, 90% C.L.Method 2: T νβ − β − ,1361/2 >1.07⋅10 yr, 90% C.L. FIG. 6: Projected 90 %C.L. lower limit on T ν EC β + / fora background-free experiment with 10 mm resolution inx-y-z, as a function of the exposure (red). Thiscalculation assumes the G3 geometry; the sensitivitycurve decreases by ∼
10% for a G2-sized detector at allexposures. Three ranges of lower limits on the0 ν EC β + -decay half-life are shown: the directcalculation (Table III) with (cid:104) m ν (cid:105) < . , eV (light blue)and with (cid:104) m ν (cid:105) < . Xe0 νβ − β − half-life limit (dark blue). For the directmethod the lower bound is given by the weakest limitamong the three NMEs for each (cid:104) m ν (cid:105) .experiment with the expected background of a double- β decay detector like nEXO, would be able to clearlydetect the decay and could study it with precision, if Xe would be added to the xenon inventory. A G3Dark Matter experiment like DARWIN would have thesignal strength to detect this decay with a few thousandsignals, but would need to optimize its fiducial volume inorder to reduce the γ -background.For a possible neutrinoless mode of this decay, achiev-ing a background-free experiment is a realistic prospect owed to the decay signatures. However, we have shownthat in this case a detection is only within reach of a G3or an enriched nEXO-like detector for the most conser-vative half-life predictions. It has to be emphasized thatsuch a scenario would require a mechanism that leads toa difference in the decay of proton-rich nuclei comparedto their neutron-rich counterparts. Otherwise it wouldbe excluded by KamLAND Zen.As mentioned previously, such a mechanism would bean exciting prospect in searches for neutrinoless decays ofproton-rich isotopes. If detected, it would provide com-plementary information on the physical mechanism medi-ating the decay process. One example for this possibilitywas studied in detail in Ref. [21] in the context of left-right symmetric models, in which one assumes that thereis a right-handed weak sector in addition to left-handedneutrinos, which can mediate neutrinoless double- β de-cays. Detectors with the capability of measuring bothisotopes simultaneously may therefore be attractive forboth the discovery of the neutrinoless process and sub-sequent study of the underlying physics. Here we brieflyreexamine the analysis of left-right symmetric models us-ing the projected sensitivities described in this work. Byadding right-handed terms to the Standard Model La-grangian, one derives a new expression for the half-life ofneutrinoless second-order weak decays:[ T α / (0 + i → + f )] − = C αmm (cid:18) (cid:104) m ν (cid:105) m e (cid:19) + C αηη (cid:104) η (cid:105) + C αλλ (cid:104) λ (cid:105) + C αmη (cid:104) m ν (cid:105) m e (cid:104) η (cid:105) + C αmλ (cid:104) m ν (cid:105) m e (cid:104) λ (cid:105) + C αηλ (cid:104) η (cid:105)(cid:104) λ (cid:105) , (22)where α represents the decay mode (0 νβ − β − , 0 ν EC β + ,etc.), (cid:104) m ν (cid:105) is the effective light neutrino mass definedabove, and (cid:104) η (cid:105) and (cid:104) λ (cid:105) are the effective coupling pa-rameters for the new interaction terms containing right-handed currents. The coefficients C αij are combinationsof nuclear matrix elements and phase space factors, anddiffer between the decay modes. In particular, it was5 −1.0 −0.5 0.0 0.5 1.0 ⟨ m ββ ⟩ −0.6−0.4−0.20.00.20.40.6 ⟨ λ ⟩ × 10 T ν ECβ + ,1241/2 > 2.9 ⋅ 10 yr T νβ − β − ,1361/2 > 1.07 ⋅ 10 yr FIG. 7: Comparison of exclusion limits at 90 %C.L. forleft-right symmetric models, in the (cid:104) m ν (cid:105) vs. (cid:104) λ (cid:105) plane.Parameter space outside the colored regions is excluded.Here we assume (cid:104) η (cid:105) = 0. The exclusion limits comparethe present limits on the 0 νβ − β − -decay of Xe [5]with the possible limits on 0 ν EC β + derived in thiswork. We assume the full 500 kg-year exposure for the Xe search – comparable to the 504 kg-year exposureused for the
Xe measurements. The dashed linerepresents the boundary of the excluded zone afterarbitrarily scaling the NMEs for
Xe by a factor oftwo, to mimic uncertainties in NME calculations.pointed out in Ref. [21] that the λ terms are significantlyenhanced in the case of the mixed-mode decays, meaningthe shape of the parameter space explored by 0 ν EC β + searches differs from that explored by the more common0 νβ − β − experiments. We illustrate this in Figure 7,where we compare the possible limits for 0 ν EC β + de-rived in this work with the current limits for the 0 νβ − β − of Xe decay from the Kamland-Zen experiment [5].We see that the sensitivity of the mixed-mode
Xedecay to the effective neutrino mass is significantlyweaker; this is due to the reduced phase space in thepositron-emitting decay mode. However, the sensitivityof the mixed-mode decay is within a factor of two for theright-handed coupling (cid:104) λ (cid:105) , which is within the uncertain-ties typically assumed for nuclear matrix element calcula- tions (usually a factor of ∼ ν decay mode in either iso-tope. It must be acknowledged that future experimentsexpect to reach sensitivities considerably larger than theexisting limits. Unless the 0 νβ − β − decay of Xe is justbeyond the reach of present experiments, we show thatthe
Xe mixed-mode decays will not be competitivein constraining left-right symmetric models with a G3experiment’s exposure. However, exploring proton-richisotopes may still provide complementary information indetermining the mechanism of lepton number violation;for example, an (unexpected) discovery of neutrinolessdecays in either only
Xe or in both
Xe and
Xecould prove that neither the light neutrino exchange norright-handed currents mediate the decay processes, andcould point towards alternative new physics. Therefore,we emphasize that future xenon-based TPC experimentsshould explore this decay channel, as the striking mul-tiple coincidence structure is straightforward to look forand distinguish from backgrounds. Also the consider-ation to expand an existing program like nEXO, whichwould require an additional enrichment on the light massside after the initial enrichment, could be thought of, inorder to gain further insight into the neutrinoless decaymodes – especially once it has been found in
Xe.
ACKNOWLEDGMENTS
We thank Javier Men´endez for calculating the nu-clear shell model matrix elements as well as for consulta-tion. We thank Lutz Althueser for providing the XeSimsoftware package, Pekka Pirinen for providing the crosssection data used to compute CC solar neutrino back-grounds, and Michael Jewell for useful discussions.This work was supported, in part, by DOE-NP grantDE-SC0017970. B.L. acknowledges the support of a KarlVan Bibber fellowship from Stanford University. C. Wi.acknowledges support by DFG through the ResearchTraining Group 2149 ”Strong and weak interactions –from hadrons to dark matter”. [1] E. Majorana, Il Nuovo Cimento (1924-1942) , 171(1937), ISSN 1827-6121, URL https://doi.org/10.1007/BF02961314 .[2] J. Bernabeu, A. De Rujula, and C. Jarlskog, Nucl. Phys. B223 , 15 (1983).[3] Z. Sujkowski and S. Wycech, Phys. Rev.
C70 , 052501(2004), hep-ph/0312040.[4] W. Buchm¨uller, R. Peccei, and T. Yanagida, Annual Re-view of Nuclear and Particle Science , 311 (2005).[5] A. Gando, Y. Gando, T. Hachiya, A. Hayashi,S. Hayashida, H. Ikeda, K. Inoue, K. Ishidoshiro, Y. Karino, M. Koga, et al. (KamLAND-Zen Collab-oration), Phys. Rev. Lett. , 082503 (2016), URL https://link.aps.org/doi/10.1103/PhysRevLett.117.082503 .[6] G. Anton, I. Badhrees, P. S. Barbeau, D. Beck, V. Belov,T. Bhatta, M. Breidenbach, T. Brunner, G. F. Cao,W. R. Cen, et al. (EXO-200 Collaboration), Phys. Rev.Lett. , 161802 (2019), URL https://link.aps.org/doi/10.1103/PhysRevLett.123.161802 .[7] S. I. Alvis, I. J. Arnquist, F. T. Avignone, A. S. Barabash,C. J. Barton, V. Basu, F. E. Bertrand, B. Bos, M. Busch, M. Buuck, et al. (Majorana Collaboration), Phys. Rev. C , 025501 (2019), URL https://link.aps.org/doi/10.1103/PhysRevC.100.025501 .[8] M. Agostini, A. M. Bakalyarov, M. Balata, I. Barabanov,L. Baudis, C. Bauer, E. Bellotti, S. Belogurov, A. Bettini,L. Bezrukov, et al. (GERDA Collaboration), Phys. Rev.Lett. , 132503 (2018), URL https://link.aps.org/doi/10.1103/PhysRevLett.120.132503 .[9] C. Alduino, F. Alessandria, K. Alfonso, E. Andreotti,C. Arnaboldi, F. T. Avignone, O. Azzolini, M. Balata,I. Bandac, T. I. Banks, et al. (CUORE Collaboration),Phys. Rev. Lett. , 132501 (2018), URL https://link.aps.org/doi/10.1103/PhysRevLett.120.132501 .[10] M. Doi and T. Kotani, Prog. Theor. Phys. , 1207(1992).[11] R. G. Winter, Phys. Rev. , 142 (1955).[12] D. A. Nesterenko et al., Phys. Rev. C86 , 044313 (2012).[13] E. Aprile et al. (XENON), Nature , 532 (2019),1904.11002.[14] J. Suhonen, J. Phys.
G40 , 075102 (2013).[15] P. Pirinen and J. Suhonen, Phys. Rev. C ,054309 (2015), URL https://link.aps.org/doi/10.1103/PhysRevC.91.054309 .[16] Coello P´erez, E. A. and Men´endez, J. and Schwenk,A., Physics Letters B , 134885 (2019), ISSN 0370-2693, 1809.04443, URL http://dx.doi.org/10.1016/j.physletb.2019.134885 .[17] N. Barros, J. Thurn, and K. Zuber, J. Phys. G41 , 115105(2014), 1409.8308.[18] J. Barea, J. Kotila, and F. Iachello, Phys. Rev. C , 057301 (2013), URL https://link.aps.org/doi/10.1103/PhysRevC.87.057301 .[19] C. W. Kim and K. Kubodera, Phys. Rev. D , 2765 (1983), URL https://link.aps.org/doi/10.1103/PhysRevD.27.2765 .[20] M. Doi and T. Kotani, Progress of Theo-retical Physics , 139 (1993), ISSN 0033-068X, http://oup.prod.sis.lan/ptp/article-pdf/89/1/139/5207768/89-1-139.pdf, URL https://dx.doi.org/10.1143/ptp/89.1.139 .[21] M. Hirsch, K. Muto, T. Oda, and H. V. Klapdor-Kleingrothaus, Zeitschrift f¨ur Physik A Hadrons andNuclei , 151 (1994), ISSN 0939-7922, URL https://doi.org/10.1007/BF01292371 .[22] J. Suhonen and M. Aunola, Nuclear PhysicsA , 271 (2003), ISSN 0375-9474, URL .[23] P. K. Rath, R. Chandra, K. Chaturvedi, P. K. Raina,and J. G. Hirsch, Phys. Rev. C , 044303 (2009),URL https://link.aps.org/doi/10.1103/PhysRevC.80.044303 .[24] J. Kotila, J. Barea, and F. Iachello, Phys. Rev. C , 064319 (2014), URL https://link.aps.org/doi/10.1103/PhysRevC.89.064319 .[25] D. S. Akerib et al. (LZ) (2019), 1910.09124.[26] L. Zhao and J. Liu, Mod. Phys. Lett. A33 , 1830013(2018).[27] E. Aprile et al. (XENON), JCAP , 027 (2016).[28] J. B. Albert et al. (nEXO), Phys. Rev.
C97 , 065503(2018), 1710.05075.[29] J. Aalbers et al. (DARWIN), JCAP , 017 (2016),1606.07001.[30] R. D. Deslattes, E. G. Kessler, P. Indelicato, L. de Billy, E. Lindroth, and J. Anton, Rev. Mod. Phys. ,35 (2003), URL https://link.aps.org/doi/10.1103/RevModPhys.75.35 .[31] J. Katakura and Z. Wu, Nuclear Data Sheets , 1655 (2008), ISSN 0090-3752, URL .[32] S. Stoica and M. Mirea, Front.in Phys. , 12 (2019).[33] J. Kotila and F. Iachello, Phys. Rev. C ,024313 (2013), URL https://link.aps.org/doi/10.1103/PhysRevC.87.024313 .[34] M. Aker et al. (KATRIN), Phys. Rev. Lett. , 221802(2019), 1909.06048.[35] I. Esteban, M. C. Gonzalez-Garcia, A. Hernandez-Cabezudo, M. Maltoni, and T. Schwetz, JHEP , 106(2019), 1811.05487.[36] J. Barea, J. Kotila, and F. Iachello, Phys. Rev. C , 034304 (2015), URL https://link.aps.org/doi/10.1103/PhysRevC.91.034304 .[37] E. Caurier, G. Mart´ınez-Pinedo, F. Nowacki, A. Poves,and A. P. Zuker, Rev. Mod. Phys. , 427 (2005), URL https://link.aps.org/doi/10.1103/RevModPhys.77.427 .[38] E. Caurier, J. Men´endez, F. Nowacki, and A. Poves,Phys. Rev. Lett. , 052503 (2008), URL https://link.aps.org/doi/10.1103/PhysRevLett.100.052503 .[39] A. Schwenk, private communication.[40] M. Auger et al. (EXO-200), Phys. Rev. Lett. , 032505(2012), 1205.5608.[41] J. Men´endez, private communication.[42] J. Kotila and F. Iachello, Phys. Rev. C ,034316 (2012), URL https://link.aps.org/doi/10.1103/PhysRevC.85.034316 .[43] E. Aprile and T. Doke, Rev. Mod. Phys. ,2053 (2010), URL https://link.aps.org/doi/10.1103/RevModPhys.82.2053 .[44] E. Conti et al. (EXO-200), Phys. Rev. B68 , 054201(2003), hep-ex/0303008.[45] E. Aprile, K. L. Giboni, P. Majewski, K. Ni, and M. Ya-mashita, Phys. Rev. B , 014115 (2007), URL http://link.aps.org/doi/10.1103/PhysRevB.76.014115 .[46] O. A. Ponkratenko, V. I. Tretyak, and Y. G. Zdesenko,Physics of Atomic Nuclei , 1282 (2000), ISSN 1562-692X, URL https://doi.org/10.1134/1.855784 .[47] T. Yamazaki, Nuclear Data Sheets. Section A , 1 (1967), ISSN 0550-306X, URL .[48] J. Smith, A. MacLean, W. Ashfield, A. Chester, A. Gar-nsworthy, and C. Svensson, Nuclear Instruments andMethods in Physics Research Section A: Accelerators,Spectrometers, Detectors and Associated Equipment , 4763 (2019), ISSN 0168-9002, URL http://dx.doi.org/10.1016/j.nima.2018.10.097 .[49] L. Althueser, l-althueser/xesim: v0.1.0 (2019), URL https://doi.org/10.5281/zenodo.3541115 .[50] S. Agostinelli, J. Allison, K. Amako, J. Apostolakis,H. Araujo, P. Arce, M. Asai, D. Axen, S. Banerjee,G. Barrand, et al., Nuclear Instruments and Methods inPhysics Research Section A: Accelerators, Spectrometers,Detectors and Associated Equipment , 250 (2003),ISSN 0168-9002, URL .[51] J. B. Albert, G. Anton, I. Badhrees, P. S. Barbeau, R. Bayerlein, D. Beck, V. Belov, M. Breidenbach,T. Brunner, G. F. Cao, et al. (EXO-200 Collaboration),Phys. Rev. Lett. , 072701 (2018), URL https://link.aps.org/doi/10.1103/PhysRevLett.120.072701 .[52] E. Aprile et al. (XENON100), Astropart. Phys. , 573(2012), 1107.2155.[53] M. Jewell, A. Schubert, W. Cen, J. Dalmasson,R. DeVoe, L. Fabris, G. Gratta, A. Jamil, G. Li,A. Odian, et al., Journal of Instrumentation , P01006(2018), URL https://doi.org/10.1088%2F1748-0221%2F13%2F01%2Fp01006 .[54] J. B. Albert et al. (EXO-200), Phys. Rev. C89 , 015502(2014), 1306.6106.[55] S. A. Kharusi et al. (nEXO) (2018), 1805.11142.[56] J. B. Albert, G. Anton, I. J. Arnquist, I. Badhrees,P. Barbeau, D. Beck, V. Belov, F. Bourque, J. P. Brod-sky, E. Brown, et al. (nEXO Collaboration), Phys. Rev.C , 065503 (2018), URL https://link.aps.org/doi/10.1103/PhysRevC.97.065503 .[57] J. B. Albert, D. J. Auty, P. S. Barbeau, D. Beck,V. Belov, M. Breidenbach, T. Brunner, A. Burenkov,G. F. Cao, C. Chambers, et al. (EXO-200 Collabora- tion), Phys. Rev. C , 045504 (2015), URL https://link.aps.org/doi/10.1103/PhysRevC.92.045504 .[58] P. Pirinen, J. Suhonen, and E. Ydrefors, Phys. Rev. C , 014320 (2019), URL https://link.aps.org/doi/10.1103/PhysRevC.99.014320 .[59] S. Usman, G. Jocher, S. Dye, W. McDonough, andJ. Learned, Nature Scientific Reports , 13945 (2015),URL https://doi.org/10.1038/srep13945 .[60] P. Huber, Phys. Rev. C , 024617 (2011), URL https://link.aps.org/doi/10.1103/PhysRevC.84.024617 .[61] D. Brown, M. Chadwick, R. Capote, A. Kahler, A. Trkov,M. Herman, A. Sonzogni, Y. Danon, A. Carlson,M. Dunn, et al., Nuclear Data Sheets (2018),ISSN 0090-3752, URL https://doi.org/10.1016/j.nds.2018.02.001 .[62] E. Aprile, F. Agostini, M. Alfonsi, K. Arisaka, F. Ar-neodo, M. Auger, C. Balan, P. Barrow, L. Baudis,B. Bauermeister, et al., Journal of Instrumentation , P11006 (2014), URL https://doi.org/10.1088%2F1748-0221%2F9%2F11%2Fp11006https://doi.org/10.1088%2F1748-0221%2F9%2F11%2Fp11006