Case Study 14.2 — Neutrinoless Double Beta Decay: Is the Neutrino Its Own Antiparticle?

"The search for neutrinoless double beta decay is arguably the most important experiment in nuclear and particle physics today." — Frank Wilczek, Nobel Laureate in Physics (2004)

The Fundamental Question

Every charged fermion in the Standard Model — the electron, the quarks, the muon, the tau — is a Dirac particle: it has a distinct antiparticle with opposite charge. But the neutrino is electrically neutral. This opens a possibility that Ettore Majorana recognized in 1937: a neutral fermion could be its own antiparticle. Such a particle is called a Majorana fermion.

The distinction is not academic. If neutrinos are Majorana particles:

• Lepton number is not conserved. This would be the first known violation of lepton number and would have profound implications for the matter-antimatter asymmetry of the universe.
• Neutrino mass has a different origin than the Higgs mechanism that gives mass to charged fermions. The "seesaw mechanism" — the leading theoretical explanation for the smallness of neutrino masses — naturally produces Majorana neutrinos.
• Leptogenesis becomes viable. If lepton number is violated, heavy Majorana neutrinos in the early universe could have generated a lepton asymmetry that was partially converted to the observed baryon asymmetry via electroweak sphaleron processes.

The only practical way to determine whether the neutrino is Majorana or Dirac is to search for neutrinoless double beta decay ($0\nu\beta\beta$). This makes the search one of the highest priorities in fundamental physics.

The Physics of $0\nu\beta\beta$

Two-Neutrino Mode (Observed)

In standard $2\nu\beta\beta$ decay, two neutrons simultaneously convert to protons, emitting two electrons and two antineutrinos:

$$2n \to 2p + 2e^- + 2\bar{\nu}_e$$

This is a second-order weak process, proportional to $G_F^4$, with half-lives of $10^{18} - 10^{24}$ years. It has been observed in 11 nuclei. The summed electron energy spectrum is continuous, peaking at roughly $Q/3$ and extending to $Q$.

Neutrinoless Mode (Sought)

In $0\nu\beta\beta$ decay, the virtual antineutrino emitted at one beta-decay vertex is absorbed as a neutrino at the second vertex:

$$2n \to 2p + 2e^-$$

This is possible only if three conditions are met:

1. The neutrino is a Majorana particle ($\nu = \bar{\nu}$), so that what is emitted as $\bar{\nu}_e$ can be absorbed as $\nu_e$.
2. The neutrino has nonzero mass, to allow the helicity flip required at the propagator (the emitted $\bar{\nu}$ is right-handed, but the $\nu$ absorbed at the second vertex must be left-handed in the V$-$A interaction).
3. Lepton number is violated by two units ($\Delta L = 2$).

The rate depends on the effective Majorana mass:

$$\langle m_{\beta\beta} \rangle = \left| \sum_{i=1}^{3} U_{ei}^2 \, m_i \right|$$

where $U_{ei}$ are elements of the Pontecorvo-Maki-Nakagawa-Sakata (PMNS) neutrino mixing matrix and $m_i$ are the mass eigenvalues. Note that $U_{ei}^2$ (not $|U_{ei}|^2$) appears, so the Majorana phases (two additional CP-violating phases that exist only for Majorana neutrinos) can cause cancellations.

The half-life is:

$$\left(T_{1/2}^{0\nu}\right)^{-1} = G_{0\nu}(Q, Z) \cdot |M_{0\nu}|^2 \cdot \langle m_{\beta\beta} \rangle^2$$

where $G_{0\nu}$ is an exactly calculable phase-space factor and $M_{0\nu}$ is the nuclear matrix element (NME), which must be computed using nuclear structure theory.

The Experimental Signature

The signature is beautifully simple in principle: the two electrons carry the full $Q$-value of the decay (no neutrinos carry away energy), producing a monoenergetic peak in the summed electron energy spectrum at $E = Q$. This peak sits at the endpoint of the continuous $2\nu\beta\beta$ spectrum.

In practice, the challenge is immense. The peak must be distinguished from: - The tail of the $2\nu\beta\beta$ spectrum (which requires excellent energy resolution) - Natural radioactive backgrounds (which require ultra-low-background environments) - Cosmogenic activation (which requires deep underground operation)

The Experiments

GERDA and LEGEND ($^{76}$Ge)

The Germanium Detector Array (GERDA) at the Gran Sasso National Laboratory (LNGS) in Italy pioneered the use of bare high-purity germanium (HPGe) detectors immersed directly in liquid argon. The $^{76}$Ge isotope ($Q = 2039\,\text{keV}$) serves as both source and detector — the germanium nuclei that might undergo double beta decay are the same atoms that form the detector crystal.

Advantages of $^{76}$Ge: - Excellent energy resolution: $\sigma_E / E \sim 0.1\%$ at $Q$, the best of any $0\nu\beta\beta$ technology - Mature detector technology (HPGe detectors have been used for decades in nuclear spectroscopy) - Pulse-shape discrimination (PSD) can distinguish single-site events ($0\nu\beta\beta$) from multi-site backgrounds

GERDA Phase II achieved a background rate of $5.2 \times 10^{-4}\,\text{counts}/(\text{keV}\cdot\text{kg}\cdot\text{yr})$ in the region of interest — effectively background-free for its 127.2 kg$\cdot$yr exposure. The final result: $T_{1/2}^{0\nu} > 1.8 \times 10^{26}$ years (90% CL).

LEGEND (Large Enriched Germanium Experiment for Neutrinoless $\beta\beta$ Decay) is the successor:

• LEGEND-200: 200 kg of enriched $^{76}$Ge detectors, began data-taking in 2023 at LNGS. Target sensitivity: $T_{1/2}^{0\nu} > 10^{27}$ years.
• LEGEND-1000: 1 tonne of enriched detectors, planned for the late 2020s. Target: $T_{1/2}^{0\nu} > 10^{28}$ years, corresponding to $\langle m_{\beta\beta} \rangle \sim 10 - 20\,\text{meV}$.

KamLAND-Zen ($^{136}$Xe)

KamLAND-Zen dissolves xenon enriched in $^{136}$Xe ($Q = 2458\,\text{keV}$) in the liquid scintillator of the KamLAND detector in the Kamioka mine, Japan. The Xe-loaded scintillator is contained in a nylon balloon at the center of the 1-kilotonne KamLAND detector.

Advantages of $^{136}$Xe: - Scalable: xenon can be dissolved in large quantities of liquid scintillator - Self-shielding: the outer scintillator acts as a shield against external backgrounds - No cosmogenic isotope backgrounds from $^{68}$Ge (a problem for germanium experiments)

KamLAND-Zen 400 used 383 kg of enriched Xe and set the limit $T_{1/2}^{0\nu} > 1.07 \times 10^{26}$ years. KamLAND-Zen 800 (with $\sim 750\,\text{kg}$) has improved this to $T_{1/2}^{0\nu} > 2.3 \times 10^{26}$ years.

The main limitation is energy resolution ($\sigma_E / E \sim 4\%$ at $Q$), which is much worse than germanium. This means the $0\nu\beta\beta$ signal region overlaps more with the $2\nu\beta\beta$ tail and with backgrounds.

CUORE and CUPID ($^{130}$Te)

CUORE (Cryogenic Underground Observatory for Rare Events) at LNGS operates 988 TeO$_2$ crystals (containing $^{130}$Te, $Q = 2528\,\text{keV}$) as bolometers at 10 mK. When a particle deposits energy in a crystal, the temperature rise is measured by a thermistor, giving excellent energy resolution ($\sigma_E \sim 5\,\text{keV}$ at $Q$).

CUORE's current limit: $T_{1/2}^{0\nu} > 2.2 \times 10^{25}$ years.

CUPID (CUORE Upgrade with Particle Identification) will replace the TeO$_2$ crystals with Li$_2$MoO$_4$ scintillating bolometers containing enriched $^{100}$Mo ($Q = 3034\,\text{keV}$). The scintillation light provides particle identification: alpha particles (a major background) scintillate differently from electrons, allowing background rejection. The higher $Q$-value of $^{100}$Mo also places the signal above most natural radioactivity backgrounds.

nEXO ($^{136}$Xe, proposed)

nEXO is a proposed 5-tonne liquid xenon time projection chamber (TPC) for SNOLAB (Canada). It would contain approximately 3.6 tonnes of enriched $^{136}$Xe and combine: - Ionization charge collection for energy measurement - Scintillation light detection for event timing and position - Topological reconstruction of the two-electron tracks

The projected sensitivity is $T_{1/2}^{0\nu} > 1.35 \times 10^{28}$ years, corresponding to $\langle m_{\beta\beta} \rangle \sim 5 - 17\,\text{meV}$ (depending on the NME calculation used). This would cover the entire parameter space predicted by the inverted mass ordering.

NEXT ($^{136}$Xe)

NEXT (Neutrino Experiment with a Xenon TPC) at the Canfranc Underground Laboratory in Spain takes a different approach: a high-pressure ($\sim 15\,\text{bar}$) gaseous xenon TPC. In gas (as opposed to liquid), the two electrons from double beta decay produce distinct topological signatures — two energy depositions ("blobs") connected by a track. This topological signature provides powerful background rejection because most backgrounds produce single-electron events with a different topology.

NEXT-White (a 5 kg prototype) has demonstrated the concept. NEXT-100 (100 kg) is under construction, and a tonne-scale NEXT-HD is planned.

The Nuclear Matrix Element Challenge

The greatest theoretical uncertainty in interpreting $0\nu\beta\beta$ experiments is the nuclear matrix element $M_{0\nu}$. Unlike $2\nu\beta\beta$ (where the NME can be extracted from the measured half-life), the $0\nu\beta\beta$ NME cannot be measured independently — it must be calculated.

Different nuclear structure approaches give substantially different results:

The spread is a factor of $\sim 2$ in the NME, which translates to a factor of $\sim 2$ uncertainty in the extracted $\langle m_{\beta\beta} \rangle$ (since $T_{1/2}^{-1} \propto |M_{0\nu}|^2 \langle m_{\beta\beta} \rangle^2$).

Recent progress in ab initio nuclear structure calculations — particularly those based on chiral effective field theory and many-body perturbation theory, coupled-cluster theory, or the in-medium similarity renormalization group (IMSRG) — offers hope for reducing this uncertainty. These calculations start from the fundamental nucleon-nucleon interaction and make controlled approximations, allowing systematic improvement.

A complementary strategy is to measure $0\nu\beta\beta$ in multiple isotopes. If the decay is observed, the ratio of half-lives in different nuclei constrains the NME ratio, providing a cross-check on the theoretical calculations.

What If $0\nu\beta\beta$ Is Observed?

The discovery of $0\nu\beta\beta$ would be one of the most important observations in the history of physics:

1. Majorana nature of the neutrino confirmed. The neutrino would be fundamentally different from all other fermions in the Standard Model.

2. Lepton number violation established. The accidental global symmetry of lepton number conservation ($L$) would be broken. This is a necessary condition for leptogenesis — the generation of the matter-antimatter asymmetry of the universe through lepton-number-violating processes.

3. Effective Majorana mass measured. Combined with oscillation data (which determine $\Delta m^2_{21}$, $\Delta m^2_{31}$, and the mixing angles), the measured $\langle m_{\beta\beta} \rangle$ would constrain the absolute neutrino mass scale and the Majorana phases.

4. Mass ordering potentially determined. The inverted ordering predicts $\langle m_{\beta\beta} \rangle \gtrsim 15\,\text{meV}$, while the normal ordering allows $\langle m_{\beta\beta} \rangle$ to be much smaller (even zero, due to cancellations from Majorana phases). A measurement of $\langle m_{\beta\beta} \rangle$ above $\sim 50\,\text{meV}$ would favor the inverted ordering.

5. New physics beyond the Standard Model. The Standard Model (with massless neutrinos) does not allow $0\nu\beta\beta$. Its observation would require extending the Standard Model — most naturally through right-handed neutrinos and the seesaw mechanism.

What If $0\nu\beta\beta$ Is Not Observed?

Even a null result is informative. If the next generation of experiments reaches $\langle m_{\beta\beta} \rangle \sim 10\,\text{meV}$ without observation:

• The inverted mass ordering would be disfavored (if neutrinos are Majorana).
• Alternatively, Majorana phases could cause cancellations, making $\langle m_{\beta\beta} \rangle$ small even for the inverted ordering.
• Or neutrinos might simply be Dirac particles — in which case $0\nu\beta\beta$ is forbidden regardless of the mass or ordering.

Distinguishing these scenarios will require complementary information from neutrino oscillation experiments (DUNE, JUNO, Hyper-Kamiokande), cosmological surveys (Euclid, DESI, CMB-S4), and direct mass measurements (KATRIN, Project 8).

The Human Dimension

The search for $0\nu\beta\beta$ represents one of the great long-term commitments of experimental physics. The experiments require:

• Years of underground operation in deep mines or tunnels (to shield against cosmic rays)
• Heroic material purity — every component must be screened for radioactive contamination at the level of micro-becquerels per kilogram
• Isotope enrichment — natural abundances of candidate isotopes range from 5% to 35%, so enrichment (by centrifugation, distillation, or ion exchange) is required at the tonne scale
• International collaboration — LEGEND, nEXO, CUPID, and NEXT each involve hundreds of physicists from dozens of institutions worldwide

The experimental groups have been working for decades, with each generation of experiments improving sensitivity by roughly an order of magnitude. The next generation (LEGEND-1000, nEXO, CUPID, NEXT-HD) aims to reach the "tonne-scale" with sensitivities covering the inverted mass ordering. If $0\nu\beta\beta$ exists at the level predicted by the inverted ordering, these experiments have a good chance of finding it. If not, the question of the neutrino's nature will remain open, and yet more sensitive experiments will be needed.

As Ettore Majorana wrote in 1937, the distinction between Dirac and Majorana neutrinos is "not merely a question of formalism, but a question of the deepest physical content." Nearly nine decades later, that question remains unanswered. The experiments now running underground in Italy, Japan, and Canada — and those being built in the United States and Spain — may finally provide the answer.

Discussion Questions

1. Why is $0\nu\beta\beta$ the only practical way to test whether neutrinos are Majorana particles? Why can't we test this with neutrino scattering experiments?

2. The different nuclear structure calculations of $M_{0\nu}$ disagree by factors of 2-3. What strategies could reduce this theoretical uncertainty? Why is ab initio nuclear theory particularly promising?

3. Multiple isotopes are being studied ($^{76}$Ge, $^{130}$Te, $^{136}$Xe, $^{100}$Mo, $^{82}$Se). Why is it important to search in more than one isotope?

4. If $0\nu\beta\beta$ is never observed, does this mean the neutrino is definitely a Dirac particle? What loopholes exist?

5. The Majorana phases in $\langle m_{\beta\beta} \rangle = |\sum U_{ei}^2 m_i|$ can cause cancellations. Is it possible for $\langle m_{\beta\beta} \rangle = 0$ even if neutrinos are Majorana and have nonzero mass?

Further Reading

• Agostini, M. et al. (GERDA Collaboration). "Final Results of GERDA on the Search for Neutrinoless Double-$\beta$ Decay." Physical Review Letters 125, 252502 (2020).
• KamLAND-Zen Collaboration. "Search for Majorana Neutrinos Near the Inverted Mass Hierarchy Region with KamLAND-Zen." Physical Review Letters 130, 051801 (2023).
• Dolinski, M.J., Poon, A.W.P., and Rodejohann, W. "Neutrinoless Double-Beta Decay: Status and Prospects." Annual Review of Nuclear and Particle Science 69, 219 (2019).
• Engel, J. and Menendez, J. "Status and Future of Nuclear Matrix Elements for Neutrinoless Double-Beta Decay: A Review." Reports on Progress in Physics 80, 046301 (2017).
• Majorana, E. "Teoria simmetrica dell'elettrone e del positrone." Il Nuovo Cimento 14, 171 (1937). [English translation available in Soryushiron Kenkyu 63, 149 (1981).]