The Quest to Unveil the Majorana Nature of Neutrinos
Neutrinos are among the most enigmatic particles known to physics. While the discovery of neutrino oscillations established that they possess mass, the fundamental character of that mass remains uncertain. In particular, it is unknown whether neutrinos are Dirac particles, with distinct antiparticles, or Majorana particles, identical to their antiparticles. The most promising experimental avenue to resolve this question is the search for neutrinoless double beta decay (0νββ), a hypothetical nuclear transition in which a nucleus emits two electrons but no neutrinos: . Observing such a decay would signal lepton number violation and demonstrate that neutrinos are Majorana particles, providing deep insights into the origin of neutrino mass and the matter–antimatter asymmetry of the universe.
In conventional double beta decay (2νββ), two neutrons in a nucleus convert into two protons while emitting two electrons and two electron antineutrinos. This process is allowed within the Standard Model and has been observed in several isotopes with half-lives of order 1019–1021 years. In contrast, 0νββ would proceed without neutrino emission, requiring the neutrino to act as its own antiparticle so that the neutrino emitted at one vertex can be absorbed at another. The rate of 0νββ is thus proportional to the effective Majorana mass \(m_{\beta\beta}\), a coherent sum over neutrino mass eigenvalues weighted by elements of the lepton mixing matrix and Majorana phases.
The inverse half-life for neutrinoless double beta decay can be factorized into three main components, reflecting phase space, nuclear structure, and particle physics contributions:
Here, \(G_{0\nu}\) is the phase-space factor with units of inverse time, computable precisely from kinematics and Coulomb effects; \(|M_{0\nu}|\) is the nuclear matrix element encapsulating the complex nuclear physics of the transition; \(m_{\beta\beta}\) is the effective Majorana mass; and \(m_e\) is the electron mass. The formula shows that the decay rate is suppressed both by the smallness of \(m_{\beta\beta}\) and by the challenging nuclear matrix element. Nevertheless, experimental searches are relentlessly pushing sensitivity to half-lives beyond 1026 years.
The effective mass \(m_{\beta\beta}\) depends on the neutrino mass hierarchy and the unknown Majorana phases. In the normal hierarchy, destructive interference can suppress \(m_{\beta\beta}\) to very small values, whereas in the inverted hierarchy there is a lower bound of roughly 0.015 eV. Consequently, a measurement or stringent limit on the 0νββ half-life can discriminate between mass orderings and possibly reveal CP-violating phases.
This calculator allows users to explore the interplay of these quantities. By entering an assumed effective mass, a phase-space factor appropriate for a specific isotope, and an estimated nuclear matrix element from theoretical models, the tool computes the expected half-life using the rearranged formula:
Since \(m_e = 0.511\,\text{MeV} = 5.11\times10^5\,\text{eV}\), expressing all masses in electronvolts simplifies the numerical evaluation. The half-life output is given in years, a convenient unit for comparing with experimental sensitivities.
Understanding the scale of predicted half-lives is crucial for planning and interpreting experiments. Current limits from projects such as GERDA, KamLAND-Zen, and CUORE reach 1026 years for isotopes like 76Ge, 136Xe, and 130Te. Next-generation detectors aim for 1027 years or more, probing the entire inverted hierarchy region. If 0νββ is not observed at those sensitivities, it would strongly suggest a normal hierarchy or very small Majorana phases.
The table below illustrates half-life predictions for representative parameter choices, assuming a typical nuclear matrix element magnitude of 5:
| mββ (eV) | G₀ν (yr⁻¹) | |M₀ν| | T₁₂ (yr) |
|---|---|---|---|
| 0.05 | 1×10⁻¹⁴ | 5 | 2.1×10²⁶ |
| 0.02 | 1×10⁻¹⁴ | 5 | 1.3×10²⁷ |
| 0.10 | 1×10⁻¹⁴ | 5 | 5.3×10²⁵ |
The dramatic increase in half-life as \(m_{\beta\beta}\) decreases demonstrates the experimental challenge. Doubling the effective mass shortens the half-life by a factor of four, highlighting why even modest improvements in detector sensitivity can yield significant gains in parameter space coverage.
In practice, both \(G_{0\nu}\) and \(|M_{0\nu}|\) depend on the isotope and require careful calculation. The phase-space factor is often tabulated in the literature with uncertainties below a few percent, while matrix elements can vary by factors of two between nuclear models. This theoretical uncertainty propagates directly to the inferred effective mass from a measured half-life. By allowing users to input their preferred values, the calculator accommodates ongoing refinements and differing theoretical perspectives.
Whether you are designing an experiment, interpreting a limit, or teaching the principles of rare nuclear processes, the neutrinoless double beta decay half-life calculator offers a convenient entry point. It reinforces the relationship between fundamental particle properties and measurable nuclear observables, encapsulating decades of research into a simple formula. A discovery of 0νββ would revolutionize particle physics, so even a basic tool to contextualize expectations can be invaluable.