Neutrinoless Double Beta Decay

Neutrinoless double beta decay — $0\nu \beta \beta$ — is the hypothetical nuclear process in which two neutrons simultaneously convert to two protons, emitting two electrons and no neutrinos: $(A,Z)\to (A,Z+2)+2e^{-}$ The total lepton number is therefore violated by two units. No Standard Model process can produce this decay with only left-handed neutrinos; its observation would be the first direct evidence of lepton-number violation and would establish that neutrinos are Majorana particles — identical to their own antiparticles.

No experiment has yet seen $0\nu \beta \beta$. Current best half-life limits exceed $10^{26}$ years, placing the effective Majorana mass below approximately 100 meV depending on the nuclear matrix element used. The next generation of experiments — LEGEND-1000, nEXO, CUPID — will push sensitivity into the 10–20 meV range, covering the full inverted-ordering parameter space and potentially entering normal ordering.

A positive detection would be the single most important particle-physics measurement of the decade. A null result at the ton-scale would constrain beyond-Standard-Model scenarios in ways that complement rather than duplicate accelerator physics at the energy frontier.

Two-neutrino vs. neutrinoless

Ordinary double beta decay ($2\nu \beta \beta$) is a second-order weak process allowed in the Standard Model: $(A,Z)\to (A,Z+2)+2e^{-}+2{\bar{\nu }}_e$ Two neutrons decay simultaneously, each emitting an electron and an antineutrino. The process conserves lepton number. It has been observed in roughly a dozen nuclei, with half-lives in the range $10^{18}$ to $10^{21}$ years — extraordinarily long but measurable with sufficient target mass and exposure time.

Neutrinoless double beta decay is a different process with the same initial and final visible state. The two electrons share the full Q-value of the decay, producing a spectrum that is a single sharp peak at $E_1+E_2=Q_{\beta \beta }$, whereas the $2\nu \beta \beta$ spectrum is continuous with a tail well below the endpoint. This distinct spectral signature is the experimental handle: $0\nu \beta \beta$ would show as a narrow excess at the endpoint, potentially visible against the broader $2\nu \beta \beta$ spectrum.

The mass mechanism

If neutrinos are Majorana particles, they can mediate $0\nu \beta \beta$ via the “mass mechanism”: one neutron emits a virtual ${\bar{\nu }}_e$ which, because ${\bar{\nu }}_e={\nu }_e$ for a Majorana field, can be absorbed by the second neutron as ${\nu }_e$, converting it to a proton and emitting an electron. The amplitude is proportional to the coherent sum of mixing-matrix-weighted masses: $⟨m_{\beta \beta }⟩=\left|{\sum }_i{U}_{ei}^2{m}_i\right|$ The sum involves the first-row PMNS matrix elements $U_{ei}$ and the three mass eigenstates. The matrix elements are squared (not squared-modulus), so the two Majorana phases ${\alpha }_{21}$, ${\alpha }_{31}$ enter as relative signs between terms — leading to potential cancellations that can push $⟨m_{\beta \beta }⟩$ close to zero even for non-zero masses.

For the normal ordering with the lightest mass at zero: $⟨m_{\beta \beta }{⟩}^{\text{NO}}=\left||U_{e2}{|}^2\sqrt{\Delta m_{21}^2}{e}^{i{\alpha }_{21}}+|U_{e3}{|}^2\sqrt{|\Delta m_{31}^2|}{e}^{i{\alpha }_{31}}\right|$ which ranges from about 1 meV (maximal cancellation) to 4 meV (constructive phases). For the inverted ordering: $⟨m_{\beta \beta }{⟩}^{\text{IO}}\gtrsim 15\text{ meV}$ This is the key prediction: inverted ordering has a lower bound on $⟨m_{\beta \beta }⟩$, normal ordering does not. A null result from experiments reaching 15 meV — combined with an independent determination favouring inverted ordering — would essentially rule out Majorana neutrinos.

The decay rate and nuclear matrix elements

The $0\nu \beta \beta$ decay rate is ${\Gamma }_{0\nu \beta \beta }=G_{0\nu }(Q,Z)\cdot |M_{0\nu }{|}^2\cdot ⟨m_{\beta \beta }{⟩}^2$ where:

• $G_{0\nu }(Q,Z)$ is a phase-space factor depending on the Q-value and atomic number. Known to percent-level precision.
• $M_{0\nu }$ is the nuclear matrix element — a many-body quantity encoding how the two nucleons couple through the virtual neutrino propagator. Calculable in principle, but with large theoretical uncertainties at the factor-of-2 level.
• $⟨m_{\beta \beta }⟩$ is the effective Majorana mass.

Nuclear matrix elements are the dominant theoretical uncertainty in translating any measured half-life (or limit) into a mass bound. Four principal calculation methods compete:

1. Quasiparticle Random Phase Approximation (QRPA) — pairing-based, historically dominant
2. Shell Model — fully microscopic but limited to small model spaces
3. Interacting Boson Model (IBM) — effective bosonic description
4. Ab initio methods — based on chiral effective field theory, emerging as the most systematic

For ${}^{76}$Ge, calculated matrix elements span approximately 2.7 – 6.0 across these methods — the reason experimental limits on $⟨m_{\beta \beta }⟩$ always appear as a range. Reducing nuclear matrix element uncertainty is a central research priority in nuclear theory; ab initio calculations, benchmarked against $2\nu \beta \beta$ half-lives (which involve similar matrix elements), are making steady progress.

Target isotopes and current experiments

Not every isotope that can double-beta-decay is a viable target. The ideal candidate has:

• A high Q-value (lower Q means harder to separate from natural radioactivity)
• A long $2\nu \beta \beta$ half-life (to reduce the irreducible $2\nu \beta \beta$ background at the endpoint)
• A practical isotope enrichment process
• Compatibility with a low-background detector technology

Five isotopes dominate the current experimental landscape:

The five leading programmes

KamLAND-Zen (Kamioka, Japan) dissolves 750 kg of enriched ¹³⁶Xe in liquid scintillator, exploiting the scintillator’s radiopurity and light collection. It has set the world-leading half-life limit of $2.3\times 10^{26}$ years and is the only current experiment whose exposure has nearly closed the inverted-ordering parameter space for some nuclear matrix elements.

LEGEND-200 / LEGEND-1000 (Gran Sasso, Italy) uses ⁷⁶Ge-enriched high-purity germanium semiconductor detectors, which offer the best intrinsic energy resolution of any candidate technology (0.1% at the endpoint). LEGEND-200 (operating) is an upgrade of GERDA, and LEGEND-1000 (in preparation) will scale to 1 ton of isotope mass with near-zero background in the region of interest.

nEXO (SNOLAB, Canada) is the successor to EXO-200, a 5-ton liquid-xenon time-projection chamber enriched to 90% in ¹³⁶Xe. It combines scintillation and ionisation signals to discriminate $0\nu \beta \beta$ from background, with a target sensitivity reaching the inverted-ordering floor.

CUORE / CUPID (Gran Sasso, Italy) operates arrays of ¹³⁰TeO₂ crystals at cryogenic temperatures, reading out heat and light signals. CUPID (the upgrade) will switch to ¹⁰⁰Mo-enriched scintillating bolometers, substantially reducing alpha-particle backgrounds.

Other programmes include SuperNEMO (passive ⁸²Se source with external tracker + calorimeter), AMoRE (¹⁰⁰Mo cryogenic), NEXT (gas-phase ¹³⁶Xe TPC), SNO+ (tellurium-loaded liquid scintillator), and several smaller R&D efforts pushing novel technologies.

No single experiment is definitive. The community aims for confirmation of any eventual signal across at least two target isotopes and two readout technologies, because the theoretical uncertainty in nuclear matrix elements can shift apparent half-life ratios enough to mimic or obscure real signals.

What a positive detection would mean

A $5\sigma$-significant excess at the endpoint of a 0νββ-candidate isotope would be a foundational particle-physics result:

1. Lepton-number violation: the first observed violation of a conservation law the Standard Model automatically respects. It would require extending the Standard Model in a specific direction (new Majorana mass terms or equivalent).
2. Majorana nature of neutrinos: neutrinos would be shown to be identical to their own antiparticles, closing a 60-year-old open question.
3. Absolute mass measurement: the measured half-life, combined with theoretical nuclear matrix elements, would give $⟨m_{\beta \beta }⟩$ directly — providing the first measurement of an absolute neutrino mass scale independent of cosmology or kinematics.
4. Implications for leptogenesis: a seesaw-type mass mechanism — the most economical way to generate Majorana masses — provides a natural framework for leptogenesis, potentially explaining the observed matter-antimatter asymmetry of the universe.
5. Constraints on beyond-Standard-Model physics: the measured rate, combined with independent kinematic and cosmological mass bounds, tests whether the mass mechanism fully accounts for the rate or whether exotic contributions (right-handed currents, R-parity-violating SUSY, extra dimensions) play a role.

What a null result would mean

If ton-scale experiments cover the inverted-ordering parameter space and see nothing, two scenarios are possible:

• Neutrinos are Majorana, but the ordering is normal and $⟨m_{\beta \beta }⟩$ lies below 15 meV — then the next generation of experiments (reaching ~5 meV) would be needed to enter normal ordering. Technically possible but requires advances beyond current designs.
• Neutrinos are Dirac, not Majorana — then $0\nu \beta \beta$ is forbidden at any rate, and the Majorana-based explanation of small neutrino masses (seesaw) would require reconsideration. This would reopen the question of how neutrino masses are generated.

Either outcome would be profound. The non-ambiguity of the $0\nu \beta \beta$ test — the process is either there or it is not — is what makes it one of the most informative experimental programmes in fundamental physics.

Relation to other open questions

$0\nu \beta \beta$ sits at the intersection of several Standard Model puzzles:

• Absolute mass scale: complementary to KATRIN/Project 8 (kinematic) and cosmology.
• Mass ordering: the viable parameter space depends on whether IO is confirmed or ruled out.
• CP violation: the Majorana phases affect $⟨m_{\beta \beta }⟩$ but cannot be extracted from $0\nu \beta \beta$ alone — requires combination with future determinations of ${\delta }_{CP}$ at DUNE/Hyper-K.
• Matter asymmetry: leptogenesis scenarios require Majorana neutrinos; a positive $0\nu \beta \beta$ result would support this framework.

The next decade will see ton-scale $0\nu \beta \beta$ experiments coming online across multiple continents. By the 2030s, the question of whether neutrinos are Majorana particles — one of the deepest open questions in particle physics since Ettore Majorana’s 1937 paper — may finally be resolved.

References

1. Majorana, E. (1937). “Teoria simmetrica dell’elettrone e del positrone.” Il Nuovo Cimento 14, 171. (Foundational paper.)
2. Furry, W. H. (1939). “On transition probabilities in double beta-disintegration.” Phys. Rev. 56, 1184. (First discussion of neutrinoless mode.)
3. Dolinski, M. J., Poon, A. W. P., Rodejohann, W. (2019). “Neutrinoless double-beta decay: status and prospects.” Annu. Rev. Nucl. Part. Sci. 69, 219. arXiv:1902.04097
4. KamLAND-Zen Collaboration (2023). “Search for Majorana neutrinos exceeding the one-ton scale.” Phys. Rev. Lett. 130, 051801. arXiv:2203.02139
5. LEGEND Collaboration (2021). “The Large Enriched Germanium Experiment for Neutrinoless ββ Decay (LEGEND).” arXiv:2107.11462
6. nEXO Collaboration (2021). “Sensitivity and discovery potential of the proposed nEXO experiment.” Phys. Rev. C 97, 065503. arXiv:1710.05075
7. Engel, J., Menéndez, J. (2017). “Status and future of nuclear matrix elements for neutrinoless double-beta decay: a review.” Rep. Prog. Phys. 80, 046301. arXiv:1610.06548
8. Fukugita, M., Yanagida, T. (1986). “Baryogenesis without grand unification.” Phys. Lett. B 174, 45. (Leptogenesis connection.)

Frequently asked

What would a positive detection of 0νββ prove?

Three things simultaneously. First, that lepton number is not conserved — a fundamental violation of a symmetry the Standard Model respects automatically. Second, that neutrinos are Majorana particles identical to their antiparticles. Third, it would provide a first direct measurement of an absolute neutrino mass scale through the effective Majorana mass ⟨m_ββ⟩.

Why hasn't it been observed yet?

The decay rate is proportional to the square of the effective Majorana mass, which is at most ~50 meV given current oscillation constraints and cosmological bounds. The corresponding half-life is above 10²⁶ years — more than 10¹⁶ times the age of the universe. Only detectors at the ton-scale and with backgrounds below 10⁻⁴ counts/keV/kg/year can reach this sensitivity.

Is 0νββ a viable alternative to KATRIN for measuring neutrino mass?

Partially. KATRIN and Project 8 measure the kinematic effective mass m_β directly and model-independently but cannot determine whether neutrinos are Dirac or Majorana. 0νββ measures the coherent Majorana combination ⟨m_ββ⟩, which depends on mixing parameters and Majorana phases. The two are complementary probes: kinematic measurements fix the scale; 0νββ reveals the nature.

What nuclei are used as targets?

Any isotope where single-beta decay is energetically forbidden but 2νββ is allowed. The most common choices are ⁷⁶Ge (GERDA/LEGEND), ¹³⁶Xe (KamLAND-Zen, nEXO), ¹³⁰Te (CUORE), ⁸²Se, ¹⁰⁰Mo (CUPID), and ⁸²Se (CUPID-0). Different isotopes have different Q-values (the total energy released) and different nuclear matrix elements, which is why multi-isotope campaigns are the gold standard.