This is the second part of the Theory Deep Dives series. We turn from Fermi’s interaction theory to a fundamental question that Fermi’s theory was silent on: what is the nature of the neutrino field itself?
The Dirac equation
Paul Dirac’s 1928 equation describes a four-component spinor field $\psi$ representing a charged fermion. The four components correspond to: spin-up particle, spin-down particle, spin-up antiparticle, spin-down antiparticle.
The Dirac equation for a free fermion of mass $m$: $$(i \gamma^\mu \partial_\mu - m) \psi = 0$$
The four components are related by charge conjugation and parity. For a charged particle like the electron, the particle and antiparticle are physically distinct — they have opposite electric charges. Charge conservation forbids the particle from being identified with its antiparticle.
For a neutral particle, this restriction doesn’t apply. Particle and antiparticle could in principle be the same thing.
The Majorana proposal
In 1937, Ettore Majorana showed that a neutral fermion field could be constructed with only two independent components — the particle being its own antiparticle.
The Majorana spinor satisfies a self-conjugation condition: $\psi^c = \psi$, where $\psi^c$ is the charge-conjugate field. This identification is consistent only if the particle is electrically neutral and carries no other charges that would distinguish it from its antiparticle.
For the Standard Model fermions, all are either charged (electron, muon, quarks) or carry color/baryon number — so all are Dirac. Only the neutrino is potentially Majorana.
Mass terms
The simplest way to see the Dirac/Majorana distinction is through mass terms in the Lagrangian.
A Dirac mass term couples left-handed and right-handed components of the field: $$\mathcal{L}_{D} = -m_D , (\bar\psi_L \psi_R + \bar\psi_R \psi_L)$$
This requires both $\psi_L$ and $\psi_R$ to exist. For the electron, both chiralities are present in the Standard Model. The Higgs mechanism generates the mass.
A Majorana mass term for a left-handed field couples the LH field to itself: $$\mathcal{L}_{M,L} = -\frac{1}{2} m_L , (\psi_L^T C \psi_L + \text{h.c.})$$
This violates lepton number by 2 units. It is allowed for neutrals but forbidden for charged fields (it would violate electric charge by 2 units).
Similarly, an RH field could have its own Majorana mass term.
What neutrinos can have
The Standard Model originally had only LH neutrinos. No mass term was possible:
- No Dirac mass (would require RH neutrinos that don’t exist).
- LH Majorana mass requires the Higgs mechanism in a non-renormalizable form.
If we add right-handed neutrinos $\nu_R$ as new fields, three mass terms become possible:
- Dirac: $m_D \bar\nu_L \nu_R$.
- LH Majorana: $m_L \nu_L^T C \nu_L$ (require operator-dimension 5 in pure Standard Model).
- RH Majorana: $M \nu_R^T C \nu_R$.
The combination of Dirac + RH Majorana is the foundation of the seesaw mechanism.
The seesaw mechanism
Imagine three flavors with Dirac mass $m_D$ between LH active neutrinos and RH sterile neutrinos, plus a large RH Majorana mass $M$:
The mass matrix (2×2 in each flavor): $$\begin{pmatrix} 0 & m_D \ m_D & M \end{pmatrix}$$
Diagonalizing gives two eigenvalues:
- Heavy: $\sim M$ (essentially the RH state plus a small admixture).
- Light: $\sim m_D^2 / M$ (essentially the LH state, suppressed by the heavy scale).
If $m_D$ is a typical Yukawa-like coupling (0.1-10 GeV) and $M$ is at a high scale ($10^{10}-10^{15}$ GeV), the light mass is suppressed to milli-electron-volt levels — the right size for ordinary neutrinos.
The light neutrino in this scenario is essentially Majorana, with its mass inherited through mixing with the heavy Majorana state. So the seesaw mechanism naturally produces Majorana neutrinos at small masses.
Why it’s hard to test
Most experiments cannot distinguish Dirac from Majorana:
Oscillation: The flavor-mixing pattern depends on the PMNS matrix elements, which are the same for both cases. The L/E behavior is identical.
Beta decay endpoints: The kinematic measurement depends on $\sqrt{\sum |U_{ei}|^2 m_i^2}$ — the same for both.
Cosmology: The cosmological abundance and lensing signatures depend on $\Sigma m_\nu$ — the same for both.
Magnetic moment: For Dirac neutrinos, a magnetic moment is allowed at one-loop. For Majorana neutrinos, only off-diagonal (transition) moments are allowed. Tiny in both cases.
The only experimentally accessible distinction is lepton-number-violating processes like neutrinoless double beta decay.
Neutrinoless double beta decay
0νββ is the gold-standard test. The process $n + n \to p + p + 2e^-$ within a nucleus, with no neutrinos in the final state, is possible only if neutrinos are Majorana.
The intermediate-state neutrino (a Majorana state, in this hypothesis) is emitted by one neutron and absorbed by another, without changing helicity in the way that would be forbidden by lepton-number conservation. Lepton number changes by 2.
Decade after decade, experimental sensitivity has tightened the limits. As of 2026, the constraint is $T_{1/2} > 10^{26}$ years for the leading isotopes — corresponding to effective Majorana mass $m_{\beta\beta} < 36-156$ meV depending on nuclear matrix-element uncertainties.
Ton-scale experiments under construction target $m_{\beta\beta} < 15$ meV by the mid-2030s, fully covering the inverted-ordering band.
What an answer would mean
0νββ observed: Neutrinos are Majorana. The seesaw mechanism becomes the favored explanation for their small mass. Leptogenesis becomes a plausible explanation for the matter-antimatter asymmetry of the universe. Lepton number is not conserved in nature.
0νββ not observed at ton scale: Neutrinos are likely Dirac. The simplest Majorana scenarios are ruled out. The matter-antimatter asymmetry needs another explanation. The Standard Model fermion content (with right-handed neutrinos added as Dirac partners) becomes the favored framework.
Either outcome would be one of the most fundamental physics determinations of the century.
The next part of this series turns to a more practical theoretical tool: the PMNS matrix that describes how the three flavor and three mass states of neutrinos relate to each other.
Frequently asked
What's the basic difference between Dirac and Majorana?
A Dirac fermion has four independent components — particle with two spin states plus antiparticle with two spin states. A Majorana fermion has only two components — particle and antiparticle are identical (with opposite helicity). All charged Standard Model fermions are Dirac because charge conservation forbids the Majorana case. Neutrinos, being neutral, could be either.
Why do mass terms matter?
Mass terms couple left-handed and right-handed components of a fermion field. A Dirac mass term couples LH and RH fields of the same particle. A Majorana mass term couples LH to LH (or RH to RH) of the same particle to itself — possible only for neutrals. Whether neutrinos have Dirac or Majorana mass terms (or both) determines whether they're Dirac or Majorana particles.
Why does the seesaw mechanism prefer Majorana?
The seesaw mechanism uses a Majorana mass M for a heavy right-handed neutrino, plus a Dirac coupling m_D between the heavy state and ordinary left-handed neutrinos. Diagonalizing gives two mass eigenvalues: M (the heavy state, slightly modified) and m_D²/M (the light state, made small by M being large). The mechanism requires Majorana mass for the heavy state. The light neutrinos inherit Majorana character through the mixing.
How would we tell them apart experimentally?
Only via lepton-number-violating processes. The benchmark test is neutrinoless double beta decay (0νββ), which is allowed only for Majorana neutrinos. Direct oscillation experiments cannot distinguish — the L/E pattern is the same in both cases. Cosmological measurements of Σm_ν also don't distinguish. 0νββ is essentially the only experimental window.
Cite this article 5 formats
APA
Neutrino Times Editorial Team. (2026, April 3). Theory Deep Dives — Part 2: Dirac vs Majorana fermions. Neutrino Times. https://neutrino-times.com/articles/theory-deep-dives-part-2-dirac-vs-majorana/
Chicago
Neutrino Times Editorial Team. "Theory Deep Dives — Part 2: Dirac vs Majorana fermions." Neutrino Times, April 3, 2026. https://neutrino-times.com/articles/theory-deep-dives-part-2-dirac-vs-majorana/.
MLA
Neutrino Times Editorial Team. "Theory Deep Dives — Part 2: Dirac vs Majorana fermions." Neutrino Times, 3 Apr. 2026, https://neutrino-times.com/articles/theory-deep-dives-part-2-dirac-vs-majorana/.
BibTeX
@misc{neutrino-times-theory-deep-dives-part-2-dirac-vs-majorana,
author = {Neutrino Times Editorial Team},
title = {Theory Deep Dives — Part 2: Dirac vs Majorana fermions},
howpublished = {Neutrino Times},
year = {2026},
month = {apr},
url = {https://neutrino-times.com/articles/theory-deep-dives-part-2-dirac-vs-majorana/},
note = {Accessed: 2026-04-03}
} RIS
TY - GEN TI - Theory Deep Dives — Part 2: Dirac vs Majorana fermions AU - Neutrino Times Editorial Team PY - 2026 DA - 2026-04-03 PB - Neutrino Times UR - https://neutrino-times.com/articles/theory-deep-dives-part-2-dirac-vs-majorana/ ER -