Toward the discovery of matter creation with neutrinoless $\beta \beta$ decay

The discovery of neutrinoless $\beta \beta$ decay could soon be within reach. This hypothetical ultrarare nuclear decay offers a privileged portal to physics beyond the standard model of particle physics. Its observation would constitute the discovery of a matter-creating process, corroborating leading theories of why the Universe contains more matter than antimatter, and how forces unify at high energy scales. It would also prove that neutrinos and antineutrinos are not two distinct particles but can transform into each other, with their mass described by a unique mechanism conceived by Majorana. The recognition that neutrinos are not massless necessitates an explanation and has boosted interest in neutrinoless $\beta \beta$ decay. The field stands now at a turning point. A new round of experiments is currently being prepared for the next decade to cover an important region of parameter space. In parallel, advances in nuclear theory are laying the groundwork to connect the nuclear decay with the underlying new physics. Meanwhile, the particle theory landscape continues to find new motivations for neutrinos to be their own antiparticle. This review brings together the experimental, nuclear theory, and particle theory aspects connected to neutrinoless $\beta \beta$ decay to explore the path toward, and beyond, its discovery.

Figure 1

Citation frequency of some seminal papers on $0\nu \beta \beta$ decay over time, until 2020. The citation frequency is computed as the number of citations per decade divided by the total number of papers with $\ge 10$ citations published in the same decade. Data from Inspire. See also [698].

Figure 2

Illustration of the relation between the neutrino and antineutrino helicity, which is given by the projection of the spin (red arrows) onto the momentum (green arrows). The helicity distinguishes neutrinos from antineutrinos in the ultrarelativistic limit (top panel). However, in the rest frame the neutrino and antineutrino are two spin states of the same particle (lower panel). Courtesy of L. Manenti.

Figure 3

Left diagram: $0\nu \beta \beta$ decay with light-neutrino exchange. Right diagram: corresponding scheme in terms of neutrino mass eigenstates and the PMNS mixing matrix $U$.

Figure 4

Maximally allowed parameter space for $m_{\beta \beta }$ as a function of $m_{\text{light}}$, $m_{\beta }$, and $\mathrm{\Sigma }$ assuming the central value of the neutrino oscillation parameters ([743]). The orange and green areas show the parameter space allowed assuming normal and inverted ordering, respectively. The shaded areas indicate the regions already excluded by $0\nu \beta \beta$-decay experiments ([358]) and cosmological observations ([25]); the vertical lines in the middle panel correspond to the KATRIN limit ([47]) and sensitivity ([46]).

Figure 5

Posterior probability distribution of the lower bound on $m_{\beta \beta }$ as a function of the true value of $\mathrm{\Sigma }$, assuming normal ordering. The distribution is constructed by random sampling of the oscillation parameters within their Gaussian uncertainties ([743]), assuming that $\mathrm{\Sigma }$ will be measured with 20 meV accuracy. The solid black line shows the median lower bound, while green, orange, and yellow bands show the distribution 68%, 95%, and 99% probability central intervals. Note that the median limit does not go to zero, even when $m_{\beta \beta }$ can vanish, as the limit is averaged over an extended $\mathrm{\Sigma }$ range accounting for the measurement uncertainty.

Figure 6

Mass excess $\mathrm{\Delta }=(m_A-A)u$ for isobars with mass $m_A$ and mass number $A=76$, where $u$ is the atomic mass unit. Even-even (odd-odd) nuclei lie on the bottom (top) curve.

Figure 7

Diagram representing the long-range light-neutrino-exchange contribution to $0\nu \beta \beta$ decay.

Figure 8

Contact (left diagram) and two-pion-exchange (right diagram) contributions to $0\nu \beta \beta$ decay.

Figure 9

Nuclear matrix elements $M_{\mathrm{long}}^{0\nu }$ for light-neutrino exchange from different many-body methods. NSM: black ([529]), gray ([428]), and light-gray bars ([440]), gray stars ([233], [234]); QRPA: deformed in violet bars ([322]) and spherical in magenta ([546]) and purple crosses ([674], [675]; [676]), red circles ([649]), and orange multiplication signs ([431]); IBM: brown bars ([140]; [271]); EDF theory: nonrelativistic in blue diamonds ([620]) and blue up-pointing triangles ([509]) and relativistic in light-blue down-pointing triangles ([655]); IMSRG: IMGCM in the light-green ${}^{48}\mathrm{Ca}$ bar ([727]) and valence space in green bars ([150]); and CC theory: dark-green ${}^{48}\mathrm{Ca}$ bar ([559]).

Figure 10

Short-range light-neutrino exchange NMEs $M_{\text{short}}^{0\nu }$ without the coupling $g_{\nu }^{\mathrm{NN}}$ and with sign change. Results from the NSM: black ([529]), gray ([638]; [550]; [637]), and light-gray bars ([447]); the QRPA: deformed in violet bars ([322]) and spherical in orange multiplication signs ([431]) and red bars ([447]); the IBM: brown bars ([140]; [271]); and the IMGCM: light-green bars ([720]).

Figure 11

Nuclear matrix elements $M_{\text{heavy}}^{0\nu }$ for the heavy-neutrino-exchange $0\nu \beta \beta$ decay. Results from the NSM: black ([529]) and gray bars ([428]); the QRPA: deformed in violet bars ([322]) and spherical in orange multiplication signs ([431]); the IBM: brown bars ([140]; [271]); and relativistic EDF theory: light-blue down-pointing triangles ([655]). Note that $M_{\text{heavy}}^{0\nu }$ differs by $(m_N{m}_e{)}^2/m_{\pi }^2$ with respect to the standard definition.

Figure 12

Theoretical spectra of $2\nu \beta \beta$ and $0\nu \beta \beta$ decays with 1.5% energy resolution (FWHM) and arbitrary normalization.

Figure 13

Illustration of the concepts used to search for $0\nu \beta \beta$ decay: internal-source experiments using solid-state (top left drawing) or monolithic liquid or gas detectors (top right image), and composite experiments for which the source is external to the detection apparatus (bottom drawing). Courtesy of L. Manenti.

Figure 14

Illustration of the channels exploited by experiments to detect the electrons emitted in $0\nu \beta \beta$ decay. Courtesy of L. Manenti.

Figure 15

Muon flux as a function of kilometers of water-equivalent depth (km w.e.) for a selection of deep underground laboratories worldwide. The actual depth is corrected for the overburden shape, if it is not flat. Thus, laboratories located under a mountain have a slightly lower equivalent depth than the actual one. Adapted from [433].

Figure 16

Worldwide map of deep underground laboratories. Existing laboratories are marked in teal, while laboratories that are planned or under construction are in orange. The labels are linked to the laboratory web pages.

Figure 17

${}^{238}\mathrm{U}$ and ${}^{232}\mathrm{Th}$ decay chains. For each isotope we report $\alpha$’s with an intensity $I>1\%$, $\gamma$’s with $I>5\%$ or energy close to the $Q$ value of a $\beta \beta$ isotope, and $\beta$’s with a $Q$ value above 2 MeV. The boxes highlight chain parts typically found in equilibrium: in black are isotopes due to the primordial material radioactivity; in dashed blue are isotopes in equilibrium with its predecessor Ra isotopes and ${}^{210}\mathrm{Po}$; and in dash-dotted orange are isotopes in equilibrium with ${}^{210}\mathrm{Pb}$, caused by ${}^{222}\mathrm{Rn}$ emanation and the subsequent ${}^{210}\mathrm{Pb}$ accumulation.

Figure 18

Solar neutrino spectra. The curves labeled hep, pep, and pp correspond to neutrinos emitted in helium-proton, proton-electron-proton, and proton-proton fusion, respectively. Data from [122], [120], [121], [157], [34].

Figure 19

Median 99.7% C.L. discovery sensitivity and median 90% exclusion sensitivity as a function of the expected number of background events. The discovery sensitivity shows the signal event expectation at which an experiment has a 50% chance of observing a 99.7% C.L. excess of events over the background. The exclusion sensitivity is instead the signal event expectation that can be excluded at 90% C.L. with 50% probability, assuming that there is no signal. Also shown is the approximated discovery sensitivity extracted using the asymptotic formulas from [237].

Figure 20

Fundamental parameters driving the sensitive background and exposure, and hence the sensitivity, of recent and future phases of existing experiments; see Eq. (38). Red bars are used for ${}^{76}\mathrm{Ge}$ experiments, orange bars are used for ${}^{136}\mathrm{Xe}$, blue bars are used for ${}^{130}\mathrm{Te}$, green bars are used for ${}^{100}\mathrm{Mo}$, and sepia bars are used for ${}^{82}\mathrm{Se}$. Similar exposures are achieved with high mass but poorer energy resolution and efficiency using gas and liquid detectors, or with small mass but high resolution and efficiency by solid-state detectors. The sensitive exposure is computed for 1 yr of live time. Lighter shades indicate experiments that are either under construction or proposed.

Figure 21

Sensitive background and exposure for recent and future experiments. The gray dashed lines show specific discovery sensitivity values on $0\nu \beta \beta$-decay half-lives, and colored dashed lines indicate the half-life sensitivities required to test the bottom of the inverted-ordering scenario for ${}^{76}\mathrm{Ge}$, ${}^{136}\mathrm{Xe}$, ${}^{130}\mathrm{Te}$, ${}^{100}\mathrm{Mo}$, and ${}^{82}\mathrm{Se}$, assuming for each isotope the largest QRPA NME value listed in Table 1. For completed experiments the final reported exposure is used; otherwise, a 10 yr lifetime is assumed.

Figure 22

Illustration of a HPGe detector and its $0\nu \beta \beta$-decay detection concept. Electron and hole clusters created by ionization are collected to the electrodes by an electric field. Courtesy of L. Manenti.

Figure 23

Illustration of a Xe TPC and its $0\nu \beta \beta$-decay detection concept. Electron and hole clusters created by ionization are collected to the electrodes by an electric field. In addition, scintillation light is detected by light sensors, providing the timing of the event. Courtesy of L. Manenti.

Figure 24

Illustration of a large liquid scintillator detector and its $0\nu \beta \beta$-decay detection concept. The position of an event can be reconstructed through the time of flight of the scintillation photons. Courtesy of L. Manenti.

Figure 25

Illustration of a cryogenic calorimeter and its $0\nu \beta \beta$-decay detection concept. Phonon and scintillation light signals are read out through superconductive thermal sensors. Courtesy of L. Manenti.

Figure 26

Illustration of a tracking-calorimeter detector and its $0\nu \beta \beta$-decay detection concept. The charge, momentum, and energy of the particles ejected by the source are measured through a combination of magnetic-field trackers and calorimeters. Courtesy of L. Manenti.

Figure 27

Discovery sensitivities of current- and next-generation $0\nu \beta \beta$-decay experiments for exchange mechanisms dominated by effective operators of dimensions 7 and 9. Values of $\mathrm{\Lambda }$ smaller than the marked values are tested at higher C.L. The gray band roughly corresponds to the reach of modern accelerator experiments. The black line indicates an ambitious goal for future circular colliders such as FCC-hh ([372]).

Figure 28

Discovery sensitivities of current- and next-generation $0\nu \beta \beta$-decay experiments for exchange dominated by effective operators of dimension 5, i.e., the light-neutrino exchange. Values of $m_{\beta \beta }$ larger than the marked values are tested at higher C.L. The gray band indicates the range of $m_{\beta \beta }$ values for inverted-ordered neutrino masses and vanishing values of the lightest neutrino mass. The minimum value of $m_{\beta \beta }$ for the IO and its $1\sigma$, $2\sigma$, and $3\sigma$ uncertainty bands are indicated by the black, green, orange, and yellow bands, respectively. The red band between 8 and 10 meV indicates a future goal for $0\nu \beta \beta$-decay experiments motivated by theoretical and experimental considerations; see the discussion in Sec. 3d4.