Dirac, Majorana masses, and lepton number

A Dirac term $-m_D\bar\nu_LN_R+\mathrm{h.c.}$ links an active left-handed neutrino to an independent right-handed gauge singlet. Assign both fields lepton number one and the term conserves total lepton number. A Majorana term can instead couple a neutral chiral field to itself through charge conjugation, changing lepton number by two under the same assignment. The factor one half avoids counting identical field contractions twice. A nonzero electric charge would forbid such a same-field mass term by gauge invariance; neutrino neutrality makes the possibility available.

These are low-energy mass structures, not complete electroweak gauge-invariant ultraviolet theories by themselves. An active left-handed Majorana mass requires additional symmetry-breaking structure or an effective operator before the Higgs field gains its expectation value. A sterile right-handed Majorana mass is gauge allowed already. An exactly Dirac description requires the lepton-number-violating terms to be absent through a symmetry or another specified construction. Merely setting them numerically small in one displayed matrix does not explain why quantum corrections in a larger theory should preserve that choice.

For this real one-generation matrix, the ordinary algebraic eigenvalues are $(M_R\pm\sqrt{M_R^2+4m_D^2})/2$. One is negative. Physical Majorana masses are nonnegative singular values obtained by a Takagi transformation of the complex symmetric mass matrix; a field phase converts that negative sign into a positive mass while carrying the associated phase information. In the limit $M_R\gg m_D$, the light mass magnitude is approximately m D squared over M R, while the heavy mass is approximately M R.

The small active-sterile mixing angle is approximately m D over M R in this same hierarchical limit. This explains why a heavy state can alter the light mass while having an extremely small admixture in low-energy charged currents. It does not prove that nature uses this mechanism or fix a unique heavy scale. Several generations introduce matrix structure, cancellations, and additional parameters; a one-generation numerical estimate cannot reconstruct them. If the hierarchy is absent, the approximation must be replaced by the exact diagonalization rather than extrapolated beyond its expansion parameter.

The two distinct nonzero oscillation-splitting scales imply at least two nonzero neutrino masses in the three-active-state interpretation. They leave the lightest mass undetermined and cancel possible Majorana phases from ordinary oscillations. A beta-endpoint spectrum instead probes an incoherent weighted mass-squared combination because the emitted mass alternatives are unobserved orthogonal final states. The KATRIN result published in 2025, based on 259 measurement days, reports $m_\beta<0.45\,\mathrm{eV}$ at ninety percent confidence. This is a bound on an effective kinematic mass, not a detection of a nonzero absolute mass or a Majorana signal.

Cosmological mass constraints involve neutrino effects on cosmic evolution and require a cosmological model and data assumptions. They do not by themselves identify lepton-number violation. A Majorana mass would permit such violation in principle, but relativistic weak production and detection select helicities and can suppress the accessible rates enormously. Calling neutrino and antineutrino the same Majorana particle therefore does not imply that a laboratory beam immediately converts into an unsuppressed opposite-charge weak signal. The next lesson examines a dedicated lepton-number-violating search and the assumptions needed to interpret its absence or discovery.