# Nuclear and Detector Sensitivities for Neutrinoless Double Beta-Decay Experiments

Authors: Hiroyasu Ejiri
Journal: Advances in High Energy Physics
 (2021)

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## Abstract

Neutrinoless double beta-decay (DBD) is of current interest in high-sensitivity frontiers of particle physics. The decay is very sensitive to Majorana neutrino masses, neutrino CP phases, right-handed weak interactions, and others, which are beyond the standard electroweak model. DBDs are actually ultrarare events, and thus, DBD experiments with ultrahigh sensitivity are required. Critical discussions are presented on nuclear and detector sensitivities for high-sensitivity DBD experiments to study the neutrino masses in the normal and inverted mass hierarchies.


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## Body

## 1. Introduction

Neutrinoless double beta-decay (DBD) is of current interest in sensitivity frontiers of particle physics. The decay violates the lepton number conservation law, and thus, DBD is beyond the standard electroweak model. DBD is very sensitive to the Majorana nature (neutrino=antineutrino) of neutrino, the absolute neutrino mass scale and the neutrino mass hierarchy, the neutrino CP phases, the possible right-handed weak interactions, and others, which are beyond the standard model. Accordingly, it has been used to study these fundamental questions on the neutrino and the weak interaction for decades. Recent experimental and theoretical DBD works are discussed in the review articles and references therein [1–5].In fact, neutrinoless DBDs are due to various modes such as the light neutrino mass mode, the right-handed weak current mode, and the SUSY particle exchange mode, which are all beyond the standard model [1, 4]. In the present work, we discuss mainly the neutrinoless DBD due to the light neutrino mass mode, which is of current interest.DBD, however, is a very low-energy and ultrarare decay. The energy is of the order ofEββ≈10−3 GeV, and the decay rate is in the range of Tr0ν≈10−27-10−30 per year (y), depending on the neutrino mass and the nuclear matrix element (NME). The neutrino mass to be studied is around m=3 meV and 30 meV in cases of the normal mass hierarchy (NH) and the inverted mass hierarchy, respectively. Then, low-background high-sensitivity DBD detectors are required to search for such low-energy and ultrarare DBD signals [1, 4].Many experimental groups are working hard for high-sensitivity DBD experiments to search for the small Majorana neutrino mass and others beyond the standard model. Current lower limits on the DBD half-lives are of the order of1026 y for both 76Ge and 136Xe. The neutrino mass limits are, respectively, 600 meV/M76 and 230 meV/M136 with M76 and M136 being the nuclear matrix elements (NMEs) for 76Ge and 136Xe, respectively. So, they are in the range of 300-100 meV if NMEs are around 2 (see Section 4 for discussions on DBD experiments). Then, one needs to improve the sensitivity by factors 5 and 50 to access the IH and NH mass regions, respectively. In fact, the DBD sensitivity to the neutrino mass depends much on NME.DBD with a light neutrino mass mode is considered a neutrino exchange process, where a light Majorana neutrino is exchanged between two neutrons in the DBD nucleus. Here, the nucleus acts as a high-luminosity microcollider of two neutrons, as shown in Figure1. The neutrino exchange cross-section for the IH neutrino is as small as σ≈10−83 cm2 because of the second-order weak process and the small neutrino mass [4, 6, 7]. The luminosity of one nucleus is as high as L≈1048cm−2s−1 because of the very small (a few fm) distance between the two neutrons in the nucleus. The summed luminosity for 1 ton DBD nuclei is L≈1076cm−2s−1. Thus, one gets 2-3 DBD signals per year for the IH neutrino exchange by using a large DBD detector with 1 ton DBD isotopes.Figure 1
Schematic views of neutrinoless doubleβ-decay with Majorana neutrino exchange (a) and nuclear collider (b).
(a)(b)The DBD neutrino mass sensitivity to search for the small neutrino mass is defined as the minimum neutrino massmm to be measured by the DBD experiment. It is given by a product of a nuclear sensitivity and a detector sensitivity. The nuclear sensitivity depends on nuclear parameters such as the nuclear phase space and the NME, while the detector sensitivity depends on the detector parameters such as the total number of the DBD isotopes, the enrichment, the exposure time, the detection efficiency, and the backgrounds. The minimum neutrino mass to be measured and the nuclear and detector sensitivities are discussed in the review articles [1, 4, 7].The present report is aimed at critically discussing the nuclear and detector sensitivities to search for the ultrarare DBD events associated with very small IH and NH masses. Critical discussions on DBD NMEs are presented elsewhere [8].


## 2. Neutrino Mass Sensitivity for DBD Experiment

The neutrinoless DBD transition rate is expressed in terms of the axial-vector weak couplinggA=1.27 in units of the vector coupling of gV as [1, 4]
(1)Tr0ν=gA4G0νM0νfν2,where G0ν is the phase space volume, M0ν is the NME, and fν stands for the effective neutrino mass and the other neutrino and weak interaction parameters beyond the standard model. These three quantities are associated with the kinematic factor, the nuclear physics factor, and the particle physics factor, respectively.In case of the light neutrino mass process,fν is given by the effective neutrino mass of m=∑‍Ui2mieαi with mi, Ui, and αi being the ith neutrino mass, the mixing coefficient, and the phase, respectively. In case of the right-handed weak-boson process, fν includes the term kMWL/MWR2 with MWL and MWR being the left-handed and right-handed weak-boson masses, respectively [1, 4].The DBD NME for the light neutrino mass mode is expressed in terms of the Gamow-Teller (GT), the Fermi (F), and the tensor (T) NMEs as [1, 4, 7]
(2)M0ν=gAeffgA2MGT+gVeffgAeff2MF+MT,where gAeff and gVeff are, respectively, the effective axial-vector and vector couplings and gA and gV are the axial-vector and vector couplings for a free nucleon. MGT, MF, and MT are, respectively, the model NMEs for Gamow-Teller, Fermi, and tensor transition operators.The GT and T NMEs are very sensitive to nucleonic and nonnucleonic correlations and nuclear medium effects, and thus, the ratiosgAeff/gA and gVeff/gV stand for the renormalization coefficients due to such correlations and nuclear medium effects that are not explicitly included in the nuclear model. Thus, they depend on the models and the transition operators. Since the axial-vector NMEs are strongly modified by the strong isobar (quark spin-isospin excitation) and other nonnucleonic correlations, we usually consider explicitly the renormalization coefficient gAeff, but the renormalization coefficient gVeff for the vector coupling should be well considered unless the models include all correlations and the medium effect [6–8]. The DBD NMEs are discussed in review articles on neutrino nuclear responses [6–8]. Theoretical works on DBD NMEs are discussed in reviews [9–11].The DBD rateTr0ν per year (y) and per ton (t) of the DBD isotope-mass is expressed by using a neutrino mass unit m0 as [1, 4, 7, 8]
(3)Tr0ν=mm02,m0=m0′M0ν,m0′=7.8meVA1/2gA2G0ν1/2,whereA is the mass number of the DBD nucleus, G0ν is the phase space factor in a unit of 10−14/y, and m0 is the neutrino mass that gives the DBD rate Tr0ν=1 per t y. We define m0 as the nuclear sensitivity of the DBD nucleus, being inversely proportional to G0ν1/2M0ν. Here, high sensitivity to the neutrino mass means high capability of measuring a small neutrino mass. m0′ is the neutrino mass m0 in case of M0ν=1.The valuesm0′ for typical DBD nuclei of 82Se, 100Mo, 116Cd, and 130Te, which are of current interest, are all around 40 meV, which are close to the upper bound of the IH neutrino mass. The values for 76Ge and 150Nd are, respectively, larger and smaller by a factor 2 than the value of 40 meV because of the smaller and larger phase space volumes.In order to study the neutrino mass, the number of the DBD signals is required to be larger than the fluctuationδ of the number of the background signals. Then, it is expressed as [1, 7, 8]
(4)Tr0νηεNT≥δ,δ=δ0×BNT1/2,where N is the total DBD isotope-mass in units of t, T is the exposure in units of y, δ0 is around 2, B is the number of the backgrounds per N=1 ton per T=1 year, η is the enrichment coefficient of the DBD isotope, and ε is the DBD signal detection efficiency. Note that the actual DBD isotope-mass is ηN ton, the signal yield is Tr0νηεNT, and the BG yield is BNT.Then, the minimum effective neutrino mass to be measured with a 90% confidence level is expressed in terms of the nuclear sensitivitym0 and the detector sensitivity d as
(5)mm=m0×d,d=d0×η−1/2ε−1/2NTB−1/4,where d0 is around 1.4. The neutrino mass to be measured is mm=2m0′/M0ν in a typical case of the DBD detector exposure of NT=1ty, the BG rate of B=1/ty, ε=0.55, and η=0.9.On the basis of the simple expression of the mass sensitivitymm as given in equation (5), key points for high-sensitivity DBD experiments are as follows.
(1)
The neutrino mass to be measured is proportional to 1/M0ν and B/NT1/4. Therefore, the DBD nucleus with larger M0ν by a factor 2 is equivalent to the DBD isotope-mass (N) larger by a factor 16. Then, the detector volume gets more than an order of magnitude larger if M0ν gets smaller by 40%(2)
The minimum mass to be measured depends on the ratio ofB/N1/4. Then, the neutrino mass sensitivity is improved by a factor 2 by increasing the DBD isotope-mass (N) by a factor 16, or by decreasing the BG rate by a factor 16, or by increasing the DBD isotope-mass by a factor 4 and decreasing the BG rate by a factor 4(3)
The effective neutrino mass depends on the neutrino mixing phases and is in the regions ofm≈17-45 meV and m≈1.5-4 meV, respectively, in cases of IH and NH. Assuming a typical NME of M0ν=2, one needs the DBD detectors with the mass sensitivities around the nuclear sensitivity of m0≈20 meV and one-tenth of that, respectively, in cases of IH and NH. Then, one needs DBD detectors with N≈1 t and B≈1/t y and N≈100 t and B≈0.01/t y, respectively, to measure the IH and NH neutrino masses(4)
DBD isotope enrichment is very effective. Enrichment by a factor 3 is equivalent to the increase of DBD isotope-mass by a factor 10. Accordingly, it is effective to use ton scale 80-90% enriched DBD isotopes


## 3. DBD Nuclear and Detector Sensitivities

In this section, we discuss DBD nuclear and detector sensitivities as given in equation (5) for DBD experiments to access the IH and NH neutrino masses.


### 3.1. Nuclear Sensitivity

The nuclear sensitivitym0 is a ratio of the unit mass m0′ and the NME M0ν. In fact, M0ν is so sensitive to the details of the nuclear structures that accurate evaluation for M0ν is very hard. So, we discuss in the present work mainly m0′ and assume that M0ν is in the region of 1-3 (see the recent review on DBD NMEs [7] and references therein for detailed discussions on M0ν).The unit massm′0 is proportional to A1/2G0ν−1/2. The phase space factor G0ν increases with the increase of the DBD Q value. Thus, double β− nuclei with the large Q around 3 MeV are used. The large Q is also effective to reduce much BGs from 208Tl and 214Bi, which are two major BGs. The mass number A dependence reflects the number of the DBD nuclei per ton of the total DBD isotope-mass. The mass sensitivities for a typical DBD nucleus with m0′=40 meV are plotted as a function of the exposure NT in Figure 2. In order to access the IH mass of 20 meV, one needs a DBD exposure NT of around 16 t y, i.e., N≈5 ton DBD isotopes and T≈3 y measurement in case of a typical NME of M0ν=2 and BG rate of B=1/t y. On the other hand, one needs NT of around 3 t y, i.e., N≈1 ton DBD isotopes and T≈3 y measurement in case of M0ν=3, and NT of around 250 t y, i.e., N≈60 ton DBD isotopes and T≈4 y measurement, in case of M0ν=1. So, it is very crucial to know the NME even for designing the DBD detector.Figure 2
Neutrino mass sensitivitiesmm as a function of the exposure NT. M and B are the NME and the BG rate per t y. d0=1.4, ε=0.55, and η=0.9 (see equation (5)).The mass sensitivities for the enrichment factors ofη=10, 50, and 100% are plotted as a function of the total DBD isotope-mass N in Figure 3.Figure 3
Neutrino mass sensitivitiesmm for η=1.0, 0.5, and 0.1 as a function of the total isotope-mass N. The exposure time is T=5 y, and the nuclear sensitivity of m0=20meV, ε=0.55, d0=1.4, and B=1/t y.In order to access the IH mass of 20 meV, one needsN=3.4 t and 4.1 t of the total DBD isotopes, respectively, in case of η=100% and 90%, while it is N=14 t in case of η=50% (ηN=7 t), and N=340 t in case of η=10% (ηN=34 t). Therefore, enrichment around η≥80% is very effective. The 1-ton DBD total isotope-mass with η≈90% is equivalent to the N≈10 ton of the total DBD isotope-mass with η≈30%.


### 3.2. DBD Detector Sensitivity and Backgrounds

The DBD experiment requires an ultra-low-BG detector since the DBD signal is very low in energy and the event rate is very rare. The neutrino mass sensitivity for a typical exposure time ofT=5 y and η=0.9, ε=0.55, and d0=1.4 is rewritten in terms of the ratio of the DBD isotope-mass N and the BG rate B as
(6)mm≈1.35m0′M0νN/B1/4.The mass sensitivities for them0′=40 meV with typical NMEs of M0ν=2 and M0ν=3 are simply given as mm≈27 meV/N/B1/4 and mm≈18 meV/N/B1/4. They are shown as a function of N/B in Figure 4. So, in the case of M0ν≈2, DBD detectors to search for the IH mass of 20-30 meV and the NH mass of 2-3 meV are required to be large isotope-mass and low-BG detectors with N/B≈1 and N/B≈104, respectively.Figure 4
Neutrino mass sensitivitiesmm for η=0.9, ε=0.55, and d0=1.4 as a function of the ratio of the total isotope-mass N to the BG rate B. The exposure time is T=5 y, and the unit mass m0′=40 meV.Let us see how the mass sensitivity changes if one gets a BG-free (B=0) detector. In this case, the mass sensitivity is derived by requiring the signal yield≥2.3 as [4]
(7)mm=1.5×m0ε−1/2η−1/2NT−1/2.The mass sensitivities for three BG cases ofB=1, 0.01, and 0 are compared in Figure 5. Then, the DBD isotope-mass N required to access the IH mass of 20 meV is 1 t, while the DBD isotope-mass to access the NH mass of 2 meV is 100 t. They are a factor 3 smaller than those for the detectors with B=1 and 0.01. Thus, one needs 1-ton scale and 100-ton scale DBD isotope-masses to access the IH and NH neutrino masses even by using the BG-free detectors.Figure 5
Neutrino mass sensitivitiesmm as a function of the DBD isotope-mass N in cases of B=1, 0.01, and 0. The exposure time is T=5 y. The unit mass m0′=40 meV and M0ν=2, ε=0.55, and η=0.9.The DBD signal is expected to appear as a sharp peak at the energy ofE=Q in the energy spectrum. The peak width is around the FWHM (full width half maximum) of the detector. Thus, one usually sets the energy window of ΔE=2 FWHM as the region of interest (ROI).BGs in ROI are mostly due toβ-rays and Compton-scattered γ-rays, which are continuum spectra. Backgrounds due to the solar neutrino interaction, which gets serious in the NH mass search, are also continuum [12, 13]. Then, the BG rate at the energy window of ROI is proportional to FWHM. Since the mass sensitivity is proportional to B/N1/4, improvement of the energy resolution by an order of magnitude is equivalent to an increase of the DBD isotope-mass by the same order of magnitude.The mass sensitivity depends on the detection efficiency asmm=kε−1/2. The efficiency includes all efficiencies associated with the energy and PSA windows/cuts and analyses to select the DBD signals and to reject BG ones. It is around ε≈0.5-0.6. Severe cuts to reject BGs reduce both the efficiency ε and the BG rate B. The decrease of ε by 20% is compensated if B gets smaller by 40%.


## 3.1. Nuclear Sensitivity

The nuclear sensitivitym0 is a ratio of the unit mass m0′ and the NME M0ν. In fact, M0ν is so sensitive to the details of the nuclear structures that accurate evaluation for M0ν is very hard. So, we discuss in the present work mainly m0′ and assume that M0ν is in the region of 1-3 (see the recent review on DBD NMEs [7] and references therein for detailed discussions on M0ν).The unit massm′0 is proportional to A1/2G0ν−1/2. The phase space factor G0ν increases with the increase of the DBD Q value. Thus, double β− nuclei with the large Q around 3 MeV are used. The large Q is also effective to reduce much BGs from 208Tl and 214Bi, which are two major BGs. The mass number A dependence reflects the number of the DBD nuclei per ton of the total DBD isotope-mass. The mass sensitivities for a typical DBD nucleus with m0′=40 meV are plotted as a function of the exposure NT in Figure 2. In order to access the IH mass of 20 meV, one needs a DBD exposure NT of around 16 t y, i.e., N≈5 ton DBD isotopes and T≈3 y measurement in case of a typical NME of M0ν=2 and BG rate of B=1/t y. On the other hand, one needs NT of around 3 t y, i.e., N≈1 ton DBD isotopes and T≈3 y measurement in case of M0ν=3, and NT of around 250 t y, i.e., N≈60 ton DBD isotopes and T≈4 y measurement, in case of M0ν=1. So, it is very crucial to know the NME even for designing the DBD detector.Figure 2
Neutrino mass sensitivitiesmm as a function of the exposure NT. M and B are the NME and the BG rate per t y. d0=1.4, ε=0.55, and η=0.9 (see equation (5)).The mass sensitivities for the enrichment factors ofη=10, 50, and 100% are plotted as a function of the total DBD isotope-mass N in Figure 3.Figure 3
Neutrino mass sensitivitiesmm for η=1.0, 0.5, and 0.1 as a function of the total isotope-mass N. The exposure time is T=5 y, and the nuclear sensitivity of m0=20meV, ε=0.55, d0=1.4, and B=1/t y.In order to access the IH mass of 20 meV, one needsN=3.4 t and 4.1 t of the total DBD isotopes, respectively, in case of η=100% and 90%, while it is N=14 t in case of η=50% (ηN=7 t), and N=340 t in case of η=10% (ηN=34 t). Therefore, enrichment around η≥80% is very effective. The 1-ton DBD total isotope-mass with η≈90% is equivalent to the N≈10 ton of the total DBD isotope-mass with η≈30%.


## 3.2. DBD Detector Sensitivity and Backgrounds

The DBD experiment requires an ultra-low-BG detector since the DBD signal is very low in energy and the event rate is very rare. The neutrino mass sensitivity for a typical exposure time ofT=5 y and η=0.9, ε=0.55, and d0=1.4 is rewritten in terms of the ratio of the DBD isotope-mass N and the BG rate B as
(6)mm≈1.35m0′M0νN/B1/4.The mass sensitivities for them0′=40 meV with typical NMEs of M0ν=2 and M0ν=3 are simply given as mm≈27 meV/N/B1/4 and mm≈18 meV/N/B1/4. They are shown as a function of N/B in Figure 4. So, in the case of M0ν≈2, DBD detectors to search for the IH mass of 20-30 meV and the NH mass of 2-3 meV are required to be large isotope-mass and low-BG detectors with N/B≈1 and N/B≈104, respectively.Figure 4
Neutrino mass sensitivitiesmm for η=0.9, ε=0.55, and d0=1.4 as a function of the ratio of the total isotope-mass N to the BG rate B. The exposure time is T=5 y, and the unit mass m0′=40 meV.Let us see how the mass sensitivity changes if one gets a BG-free (B=0) detector. In this case, the mass sensitivity is derived by requiring the signal yield≥2.3 as [4]
(7)mm=1.5×m0ε−1/2η−1/2NT−1/2.The mass sensitivities for three BG cases ofB=1, 0.01, and 0 are compared in Figure 5. Then, the DBD isotope-mass N required to access the IH mass of 20 meV is 1 t, while the DBD isotope-mass to access the NH mass of 2 meV is 100 t. They are a factor 3 smaller than those for the detectors with B=1 and 0.01. Thus, one needs 1-ton scale and 100-ton scale DBD isotope-masses to access the IH and NH neutrino masses even by using the BG-free detectors.Figure 5
Neutrino mass sensitivitiesmm as a function of the DBD isotope-mass N in cases of B=1, 0.01, and 0. The exposure time is T=5 y. The unit mass m0′=40 meV and M0ν=2, ε=0.55, and η=0.9.The DBD signal is expected to appear as a sharp peak at the energy ofE=Q in the energy spectrum. The peak width is around the FWHM (full width half maximum) of the detector. Thus, one usually sets the energy window of ΔE=2 FWHM as the region of interest (ROI).BGs in ROI are mostly due toβ-rays and Compton-scattered γ-rays, which are continuum spectra. Backgrounds due to the solar neutrino interaction, which gets serious in the NH mass search, are also continuum [12, 13]. Then, the BG rate at the energy window of ROI is proportional to FWHM. Since the mass sensitivity is proportional to B/N1/4, improvement of the energy resolution by an order of magnitude is equivalent to an increase of the DBD isotope-mass by the same order of magnitude.The mass sensitivity depends on the detection efficiency asmm=kε−1/2. The efficiency includes all efficiencies associated with the energy and PSA windows/cuts and analyses to select the DBD signals and to reject BG ones. It is around ε≈0.5-0.6. Severe cuts to reject BGs reduce both the efficiency ε and the BG rate B. The decrease of ε by 20% is compensated if B gets smaller by 40%.


## 4. Remarks and Discussions on DBD Detectors

There are many high-sensitivity DBD experiments and future plans to access the IH mass around 20 meV and the NH mass around 2 meV. The detailed reports on the present status of individual experiments and plans are given in their reports, and detailed discussions on them are given in the reviews [1, 4, 5, 7] and references therein and also in the recent Neutrino 2020 conference proceedings. The present and future DBD experiments are discussed there (see, for example, Ref. [14]). So, we give in this section a few comments on some (not all) possible DBD experiments from viewpoints of the DBD mass sensitivity, i.e., the nuclear and detector sensitivities as given in equation (5).The DBD isotopes of82Se, 100Mo, and 136Xe are very useful isotopes because of the small m0′≈40 meV due to the large phase space G0ν and the large Q value and multiton scale enriched isotopes with N≈10 and η≈90% are available by centrifugal separation.Cryogenic82Se- and 100Mo- bolometers with both thermal and scintillation signals are promising detectors with high-energy resolution and low-BG rate (see Refs. [15–18]). 116Cd with m0′≈40 meV is also interesting (see Refs. [19, 20]).136Xe can be easily enriched to get a 10 t scale isotope-mass and thus is possible to access the IH and NH masses. Various kinds of 136Xe detectors are in progress (see Refs. [21–27]).76Ge with Q=2.039 MeV has the unit mass of m0′=80 meV, a factor 2 larger than the others given above. Ton scale enriched DBD isotopes are possible by centrifugal separation, and the energy resolution of 10−3 in FWHM is very good. Accordingly, the Ge BG rate is almost an order of magnitude smaller than others to get the neutrino mass sensitivity comparable with other detectors (see Refs. [28–32]).The natural abundance of130Te is 34%, and thus, multiton scale natural Te isotopes may be used although the mass sensitivity mm gets larger by a factor 1.6 than the mass sensitivity with enriched 130Te-isotope detectors (see Refs. [18, 33]).150Nd has a large phase space factor because of the large Q value of 3.4 MeV and the large Z number of 60. Thus, m0′=18 meV is half of the others [7, 8]. The natural abundance, however, is only 5.6%, and enrichment is not easy (see Ref. [34]).Tracking detectors have been used to study DBDs on100Mo, 116Cd, 82Se, and many other DBD nuclei as discussed in review articles and references therein [1, 2, 4, 5]. They measure individual two β-rays to identify the DBD signals and to reject single-β background signals. The measured energy and angular correlations are used to differentiate the DBD processes due to the left-handed and right-handed weak currents (see Refs. [35–37]).Finally, it should be remarked that the NME is one of the key ingredients for the DBD mass sensitivity, and thus, selection of DBD isotopes with a large NME is very crucial for getting high-sensitivity DBD experiments [6, 7]. It is however very hard to evaluate accurately the DBD NMEs. So, various experiments are in progress to provide nuclear parameters, the effective axial-vector weak couplings, and nuclear structures to help theoretical evaluations of the neutrinoless DBD NMEs. These are discussed in recent articles [8, 38].



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