Majorana vs Dirac neutrinos

Every other fermion in the Standard Model wears its identity on its sleeve: the electron carries charge −1, so it can never be confused with its antiparticle, which carries +1. The neutrino carries nothing — no charge, no colour — and that emptiness opens a question no other particle can pose. Is the neutrino a Dirac particle, genuinely distinct from its antiparticle, with mass generated the electron's way through a Higgs coupling that pairs a left-handed field with a separate right-handed one? Or is it a Majorana particle, its own antiparticle, with a mass term no charged fermion is allowed to have? The distinction is not cosmetic. A Dirac mass demands right-handed neutrinos $\nu_R$ — states that feel no gauge force at all and have never been seen — and conserves lepton number exactly. A Majorana mass needs no such partner: because $\nu_R$ is a total singlet under the gauge group, one can write a mass built from the neutrino field paired with itself, a term that violates lepton number by two units and would be strictly forbidden for the electron because it would turn $e^-$ into $e^+$ . Combine the two and something remarkable happens. Diagonalising the resulting mass matrix — the same $2\times2$ algebra that mixes the neutral kaons and the electroweak neutral bosons — gives the seesaw: one state pushed extraordinarily light, $m_\nu\approx m_D^2/M$ , and one pushed to a grand-unification scale, $m_N\approx M$ . It explains in one line why neutrinos weigh a millionth of the electron, and it ties the lightest particles we know to the heaviest physics we suspect.

Key takeaways

A Dirac neutrino mass $L_D=-m_D(\overline{\nu_R}\nu_L+\overline{\nu_L}\nu_R)$ works exactly like the up-quark's, needs a new right-handed field $\nu_R$ , and conserves lepton number.
A Majorana mass $L_M=-\tfrac12 M(\overline{\nu_R^c}\nu_R+\text{h.c.})$ pairs the neutrino with its own charge conjugate; it is gauge-invariant only because $\nu_R$ is a total singlet, and it violates lepton number by $\Delta L=2$ .
A Majorana term is forbidden for any charged fermion — it would allow $e^-\leftrightarrow e^+$ — so only the neutral neutrino can be its own antiparticle.
Combining both terms gives the seesaw: a light state $|m_\nu|\approx m_D^2/M$ and a heavy state $m_N\approx M$ , with the exact relation $|m_\nu|\,m_N=m_D^2$ .
With $m_D\sim1$ GeV and $M\sim10^{11}$ GeV, the light neutrino lands naturally at $m_\nu\sim0.01$ eV — the observed scale.
Neutrinoless double-beta decay ( $0\nu\beta\beta$ ) is the decisive test: its rate $\propto m_{\beta\beta}^2$ is nonzero only if neutrinos are Majorana.
Key idea: a Majorana mass identifies a neutral fermion with its own antiparticle, is allowed only in the gauge-singlet neutrino sector, and — paired with an ordinary Dirac mass — produces the seesaw that makes one neutrino naturally light and one extraordinarily heavy.

The core insight

A Majorana mass term identifies a neutral fermion with its own antiparticle and is allowed only for the gauge-singlet neutrino sector; pairing it with an ordinary Dirac mass produces the seesaw, which makes one neutrino naturally light and the other extremely heavy.

Where this lives today

The Dirac-versus-Majorana question is one of the sharpest open problems in particle physics, attacked on three fronts. The first is direct: a global fleet of neutrinoless double-beta-decay experiments — KamLAND-Zen and the proposed nEXO in $^{136}$ Xe, GERDA/LEGEND in $^{76}$ Ge, CUORE/CUPID in $^{130}$ Te — is racing toward the $\sim10^{28}$ -yr half-life sensitivity needed to cover the inverted-ordering band, $m_{\beta\beta}\sim15$ – $50$ meV; a single confirmed event would settle the neutrino's nature outright. The second is cosmological: the sum of neutrino masses bends the growth of large-scale structure, and Planck combined with baryon-acoustic data constrains $\sum m_\nu\lesssim0.1$ eV, with DESI (2024) tightening it further. The third is grand: if the heavy seesaw partners $N$ exist, their CP-violating, lepton-number-violating decays in the early universe can generate a lepton asymmetry reprocessed into the observed baryon asymmetry — the mechanism of leptogenesis, which would make the neutrino's Majorana mass responsible for the existence of matter itself. Meanwhile "Majorana fermion" has escaped particle physics entirely: emergent Majorana zero modes in topological superconducting nanowires are a leading candidate for fault-tolerant qubits, so the mathematics Majorana wrote in 1937 is now being engineered on a chip.

The physical picture

Ask what the mass term is made of. For the electron it is $-m\bar e e=-m(\bar e_L e_R+\bar e_R e_L)$ — a left field glued to a right field, both carrying charge, so the pairing conserves charge and lepton number. The neutrino admits a second possibility the electron does not: because it carries no charge, a mass can pair the field with its own charge conjugate, welding a particle to an antiparticle. The first figure lays the two hypotheses side by side and reads off the quantum number that separates them — lepton number.

Figure

Two hypotheses for the neutrino. A Dirac neutrino has four independent states and its mass m_D bridges ν_L and a new field ν_R, conserving lepton number ΔL = 0. A Majorana neutrino is its own antiparticle, has only two states, and its mass M bridges ν_L with its own charge conjugate, violating lepton number by ΔL = 2

Now let both terms be present. The neutrino sector carries a Dirac mass $m_D$ of ordinary fermion size and a Majorana mass $M$ for the singlet $\nu_R$ that, tied to no symmetry breaking, can be enormous. Diagonalising splits the pair into two physical states pushed apart like the ends of a lever: as one rises, the other sinks. The second figure is that seesaw, drawn with the numbers that make it famous.

Figure

The seesaw with real numbers. A Dirac mass m_D ≈ 1 GeV sits at the pivot; a heavy singlet mass M ≈ 10¹¹ GeV weighs down one end, forcing the light neutrino end up to m_ν ≈ m_D²/M ≈ 0.01 eV. The exact relation is m_ν × m_N = m_D², so the heavier N gets, the lighter ν becomes

The two figures carry the whole story: neutrality permits a mass forbidden to everything else, and if such a mass exists at a high scale it drags the observed neutrino down to the featherweight range oscillation experiments measure. What decides between the hypotheses is lepton number — only the Majorana case violates it, and the cleanest place to catch that is neutrinoless double-beta decay.

The core equations

The Dirac mass is the familiar one — built exactly as for the up-type quarks, the neutrino getting its mass from the Higgs through the conjugate doublet $\tilde\phi=i\sigma_2\phi^*$ . Below electroweak symmetry breaking it reduces to a mass linking left and right.

L_D=-m_D\left(\overline{\nu_R}\,\nu_L+\overline{\nu_L}\,\nu_R\right),\qquad m_D=\frac{y_\nu v}{\sqrt2} \qquad (1)

where: $\nu_L$ — left-handed neutrino in the weak doublet; $\nu_R$ — a new right-handed singlet the Dirac option requires; $y_\nu$ — neutrino Yukawa coupling; $v=246$ GeV — Higgs vacuum value. Reading: this term conserves lepton number and needs an $\nu_R$ coupling to nothing but the Higgs — which is why fitting $m_\nu\sim0.1$ eV would demand an unnaturally tiny $y_\nu\sim10^{-12}$ . The Majorana mass is the genuinely new object. It uses the charge-conjugate field $\psi^c$ — which turns a right-handed particle into a left-handed antiparticle — to pair $\nu_R$ with itself. Because $\nu_R$ is a total gauge singlet, this term breaks no gauge symmetry and needs no Higgs.

L_M=-\tfrac12 M\left(\overline{\nu_R^c}\,\nu_R+\overline{\nu_R}\,\nu_R^c\right),\qquad \psi^c=\hat C\hat P\,\psi=i\gamma^2\gamma^0\psi^* \qquad (2)

where: $M$ — Majorana mass, tied to no symmetry breaking so it can sit at any high scale; $\psi^c$ — CP-conjugate field; the $\tfrac12$ avoids double counting for a self-paired field. Reading: the structure is "particle × particle," not "particle × antiparticle," so it changes lepton number by two units, $\Delta L=2$ — the hallmark of the Majorana mass. Allow both terms and the neutral fermions $(\nu_L,\,\nu_R^c)$ mix through a symmetric $2\times2$ mass matrix — the same shape that diagonalises the neutral kaons.

-L_{\text{mass}}=\tfrac12\begin{pmatrix}\overline{\nu_L^c}&\overline{\nu_R}\end{pmatrix}\begin{pmatrix}0&m_D\\ m_D&M\end{pmatrix}\begin{pmatrix}\nu_L\\ \nu_R^c\end{pmatrix}+\text{h.c.} \qquad (3)

where: the top-left $0$ is the absence of a Majorana mass for the doublet $\nu_L$ (which would break gauge symmetry at the electroweak scale); $m_D$ — Dirac entry; $M$ — singlet Majorana entry. Reading: the physical masses are the eigenvalues, and the vanishing $(1,1)$ entry is what forces one to be tiny. Diagonalising equation (3) gives the two mass eigenvalues exactly, and their seesaw limit when the singlet mass dominates.

m_{1,2}=\frac{M\mp\sqrt{M^2+4m_D^2}}{2}\;\;\xrightarrow{\;M\gg m_D\;}\;\; |m_\nu|\approx\frac{m_D^2}{M},\quad m_N\approx M,\quad \tan2\theta=\frac{2m_D}{M} \qquad (4)

where: $m_\nu$ — light (mostly $\nu_L$ ) eigenvalue; $m_N$ — heavy (mostly $\nu_R$ ) eigenvalue; $\theta$ — mixing angle. Reading: the exact determinant gives $|m_\nu|\,m_N=m_D^2$ , so push $M$ up and $m_\nu$ falls in proportion — the "seesaw." Finally, the observable that tells Dirac from Majorana. Neutrinoless double-beta decay proceeds only through a Majorana mass, its rate governed by the coherent effective Majorana mass $m_{\beta\beta}$ .

\frac{1}{T^{0\nu}_{1/2}}=G^{0\nu}\,\bigl|M^{0\nu}\bigr|^2\,\frac{|m_{\beta\beta}|^2}{m_e^2},\qquad m_{\beta\beta}=\Bigl|\textstyle\sum_i U_{ei}^2\,m_i\Bigr| \qquad (5)

where: $T^{0\nu}_{1/2}$ — half-life; $G^{0\nu}$ — calculable phase-space factor; $M^{0\nu}$ — nuclear matrix element; $U_{ei}$ — PMNS elements (carrying Majorana phases); $m_i$ — light-neutrino masses. Reading: because $m_{\beta\beta}$ sits squared out front, a nonzero half-life directly measures the Majorana mass — and if neutrinos are Dirac, $m_{\beta\beta}=0$ and the decay never happens.

Derivation ADiagonalising the seesaw mass matrix

The goal is to take the neutral-fermion mass matrix of equation (3) — a Dirac entry $m_D$ off the diagonal and a large singlet Majorana entry $M$ on it — and show its two eigenvalues split into one naturally tiny light state and one huge heavy state, with the exact relation $|m_\nu|\,m_N=m_D^2$ . This is the central calculation of neutrino mass theory: it converts "neutrinos are strangely light" into "a heavy right-handed scale exists," and every seesaw number follows from these four steps.

Step 1. Take $\mathcal M=\begin{pmatrix}0&m_D\\ m_D&M\end{pmatrix}$ in the basis $(\nu_L,\nu_R^c)$ and find its eigenvalues from $\det(\mathcal M-\lambda\mathbb 1)=0$ , i.e. $(0-\lambda)(M-\lambda)-m_D^2=\lambda^2-M\lambda-m_D^2=0$ . Why: physical masses are the eigenvalues of the mass matrix, so diagonalising is the whole question.

Step 2. Solve the quadratic: $\lambda=\tfrac12\!\left(M\pm\sqrt{M^2+4m_D^2}\right)$ . The roots are $m_2>0$ and $m_1<0$ — the negative sign of $m_1$ is a harmless Majorana phase, removed by redefining the field, so the physical mass is $|m_1|$ . Why: a real symmetric matrix has two real eigenvalues, and their signs already hint at the split.

Step 3. Take the seesaw limit $M\gg m_D$ . Expand $\sqrt{M^2+4m_D^2}\approx M+2m_D^2/M$ . Then the heavy eigenvalue is $m_2\approx M+m_D^2/M\approx M$ , and the light one $m_1\approx-m_D^2/M$ , so $|m_\nu|\approx m_D^2/M$ . Why: the expansion is valid precisely because the singlet mass dwarfs the Dirac mass — the hierarchy that makes the mechanism work.

Step 4. Confirm the exact relation and mixing. The product of eigenvalues equals the determinant, $m_1 m_2=\det\mathcal M=-m_D^2$ , so $|m_\nu|\,m_N=m_D^2$ holds with no approximation — the seesaw identity. The mixing angle obeys $\tan2\theta=2m_D/M$ , so $\theta\approx m_D/M\ll1$ : the light state is almost pure $\nu_L$ , the heavy state almost pure $\nu_R$ , decoupled from the weak force. Why: fix $m_D$ , and every gram added to $M$ is a gram taken off $m_\nu$ . *Verified: eigenvalues $\lambda=\tfrac12(M\pm\sqrt{M^2+4m_D^2})$ ; for $m_D=1$ GeV, $M=10^{11}$ GeV the exact roots are $-1.0\times10^{-11}$ GeV and $1.0\times10^{11}$ GeV, so $|m_\nu|=0.01$ eV, $m_N=10^{11}$ GeV, and $|m_\nu|\,m_N=m_D^2$ to machine precision.*

Go deeper · Derivation B · 2 minWhy a Majorana mass is gauge-invariant for the neutrino but forbidden for the electron

Derivation BWhy a Majorana mass is gauge-invariant for the neutrino but forbidden for the electron

The aim is to show that the Majorana term $\overline{\nu_R^c}\,\nu_R$ is a genuine singlet under the full Standard-Model gauge group — so it can be added by hand with any coefficient $M$ and needs no Higgs — while the identical construction for a charged fermion carries net charge and is forbidden. This is the structural reason only neutrinos can be Majorana. We track how the bilinear transforms under a $U(1)$ with charge $q$ and read off when it survives.

Step 1. List the quantum numbers of $\nu_R$ : colour singlet, weak-isospin singlet, hypercharge $Y=0$ , hence electric charge $Q=0$ . It couples to no gluon, $W$ , $Z$ , or photon — it is "sterile." Why: a mass term is allowed only if invariant under every gauge symmetry, so first see what $\nu_R$ is charged under — nothing.

Step 2. Transform the bilinear under a $U(1)$ of charge $q$ . Under $\nu\to e^{iq\alpha}\nu$ the charge conjugate goes the opposite way, $\nu^c\to e^{-iq\alpha}\nu^c$ , so $\overline{\nu^c}\to e^{+iq\alpha}\overline{\nu^c}$ , giving $\overline{\nu_R^c}\,\nu_R\to e^{2iq\alpha}\,\overline{\nu_R^c}\,\nu_R$ . Invariance requires $2q=0$ , i.e. $q=0$ . Why: a Majorana bilinear carries twice the field's charge, so it is invariant only for a strictly neutral field — the crux of the distinction.

Step 3. Apply the rule twice. For $\nu_R$ every charge is zero, so $2q=0$ and the term is a full singlet — allowed at any scale $M$ . For the electron $q=-1$ , so $\overline{e_R^c}\,e_R$ has charge $-2$ : not invariant under electromagnetism, and adding it would allow $e^-\to e^+$ . Why: this is why no charged fermion is its own antiparticle — a charged Majorana mass would leak charge two units at a time.

Step 4. Read the surviving global symmetry. Even for the neutral neutrino the term is not invariant under global lepton number ( $q_L=+1$ ): the factor $e^{2iq_L\alpha}$ shows it carries $\Delta L=2$ . So a Majorana mass conserves the gauge charges while breaking lepton number by two. Why: this is the observable fingerprint — Majorana ⟺ lepton-number violation, tested by any $\Delta L=2$ process. *Verified: $\overline{\nu_R^c}\nu_R$ carries $2q$ of any $U(1)$ ; gauge-invariant iff $q=0$ (true for $\nu_R$ , false for $e$ with charge $-2$ ); breaks global lepton number by $\Delta L=2$ .*

Go deeper · Derivation C · 1 minLepton number, the neutrinoless double-beta amplitude, and its mass dependence

Derivation CLepton number, the neutrinoless double-beta amplitude, and its mass dependence

The goal is to trace how the Majorana nature shows up in a laboratory rate: neutrinoless double-beta decay $(A,Z)\to(A,Z+2)+2e^-$ occurs only if the neutrino is its own antiparticle, and its amplitude is proportional to the effective Majorana mass $m_{\beta\beta}$ — so the rate scales as $m_{\beta\beta}^2$ and vanishes for a Dirac neutrino. This turns an abstract mass-term choice into the world's most sensitive lepton-number experiment. We compare the two- and zero-neutrino modes and follow the internal neutrino line.

Step 1. Start from ordinary two-neutrino double-beta decay, $(A,Z)\to(A,Z+2)+2e^-+2\overline\nu_e$ , which conserves lepton number ( $\Delta L=0$ ), measured with half-lives $\sim10^{19}$ – $10^{21}$ yr. Why: it fixes the baseline — the nucleus can emit two electrons; the question is what happens to the neutrinos.

Step 2. In the neutrinoless mode the two antineutrino lines join into one internal neutrino exchanged between the vertices, leaving $2e^-$ and no neutrinos: $\Delta L=2$ . This works only if the $\overline\nu_e$ emitted at one vertex is the same particle as the $\nu_e$ absorbed at the other — only if $\nu=\nu^c$ (Majorana). Why: for a Dirac neutrino the line cannot close and the amplitude is exactly zero — the yes/no discriminator.

Step 3. Account for the helicity mismatch. The weak current emits a mostly right-handed $\overline\nu$ , but absorption needs a mostly left-handed $\nu$ ; the overlap needs a chirality flip on the internal line, costing one power of $m_i/E$ . Summing coherently over mass eigenstates with the PMNS matrix gives amplitude $\propto\sum_i U_{ei}^2\,m_i\equiv m_{\beta\beta}$ . Why: the process exists only because neutrinos have mass, and it weighs them in the combination $m_{\beta\beta}$ .

Step 4. Square to the rate. $(T^{0\nu}_{1/2})^{-1}=G^{0\nu}|M^{0\nu}|^2\,m_{\beta\beta}^2/m_e^2$ , so a half-life reads off $m_{\beta\beta}$ directly, up to the nuclear matrix element. If $m_{\beta\beta}=0$ — the Dirac case, or a cancellation among Majorana phases — the decay is forbidden. Why: the squared effective mass is why $0\nu\beta\beta$ tests Majorana nature and measures the absolute mass scale at once. *Verified: $0\nu\beta\beta$ requires $\nu=\nu^c$ and $\Delta L=2$ ; amplitude $\propto m_{\beta\beta}=|\sum_i U_{ei}^2 m_i|$ ; rate $\propto m_{\beta\beta}^2$ , vanishing for Dirac neutrinos.*

Limiting cases

Case	Result	Meaning
Pure Dirac ( $M=0$ )	two degenerate states of mass $m_D$ combine into one Dirac neutrino	lepton number conserved; $\nu_R$ exists but is sterile; needs $y_\nu\sim10^{-12}$
Pure Majorana ( $m_D=0$ )	independent masses for $\nu_L$ and $\nu_R$ ; each is its own antiparticle	$\Delta L=2$ ; no left–right pairing; the doublet Majorana mass is gauge-forbidden at low scale
Seesaw ( $M\gg m_D$ )	$\|m_\nu\|\approx m_D^2/M$ , $m_N\approx M$	one state driven light, one to a GUT-like scale; explains $m_\nu\sim0.01$ eV
Pseudo-Dirac ( $M\ll m_D$ )	two near-degenerate states split by $\sim M$	almost a Dirac neutrino, with a tiny $\Delta L=2$ splitting
Charged fermion	Majorana mass forbidden	$\overline{e^c}e$ carries charge $-2$ ; would violate charge conservation ( $e^-\!\leftrightarrow e^+$ )

Common trapRight-handed neutrinos must exist, and a Majorana neutrino is just a neutrino with no antiparticle

Two linked misconceptions. First, right-handed neutrinos are not mandatory: they are required only for the Dirac option, which pairs $\nu_L$ with a genuine $\nu_R$ . In the pure Majorana case the mass is built from the left-handed field and its own charge conjugate, so no new Dirac partner is needed — the "right-handed" piece is just $\nu_L^c$ , a left-handed antineutrino. Whether $\nu_R$ exists is exactly the open question. Second, "its own antiparticle" does not mean "has no antiparticle" — it means the state we call the antineutrino is the same field, distinguished in practice only by helicity. What we see as "neutrino" (left-handed) and "antineutrino" (right-handed) would, for a Majorana particle, be two helicity states of one entity, and the tiny amplitude to flip between them ( $\propto m_\nu/E$ ) is exactly what neutrinoless double-beta decay hunts for. And crucially: a Majorana mass for the electron is not merely unobserved but forbidden — it carries electric charge and would let $e^-$ turn into $e^+$ .

Worked examples

1 · Fixing the seesaw scale from the atmospheric mass (fully worked)

Oscillation data fix the atmospheric mass-squared splitting at $\Delta m^2_{31}\approx2.5\times10^{-3}\,\text{eV}^2$ , so the heaviest light neutrino weighs at least $m_\nu\approx\sqrt{\Delta m^2_{31}}\approx0.050$ eV. Using the seesaw, find the right-handed scale $M$ for a Dirac mass of (a) $m_D=1$ GeV and (b) $m_D=173$ GeV (the top-quark mass). What does this tell us?

Step 1. Invert the seesaw relation: from $|m_\nu|\approx m_D^2/M$ , $M\approx m_D^2/m_\nu$ . Work in eV, with $m_\nu=0.050$ eV and $1\text{ GeV}=10^9$ eV.

Step 2. Case (a), $m_D=1$ GeV $=10^9$ eV: $M\approx(10^9)^2/0.050=2.0\times10^{19}$ eV $=\mathbf{2\times10^{10}\ \text{GeV}}$ . A Dirac mass of ordinary GeV size already forces $M$ near $10^{10}$ GeV — far above any collider.

Step 3. Case (b), $m_D=173$ GeV $=1.73\times10^{11}$ eV: $M\approx(1.73\times10^{11})^2/0.050=6.0\times10^{23}$ eV $=\mathbf{6\times10^{14}\ \text{GeV}}$ . A neutrino Yukawa as large as the top's puts the seesaw scale at $\sim10^{15}$ GeV — squarely in the grand-unification window.

Step 4. Read the physics. Both cases land $M$ within a few orders of the GUT scale $\sim10^{16}$ GeV with no fine-tuning. The lightness of the neutrino ( $0.05$ eV) is thus a shadow of new physics at $10^{10}$ – $10^{15}$ GeV — circumstantial evidence for a high-scale, lepton-number-violating sector, tied to leptogenesis.

2 · Reading the Majorana mass from a double-beta half-life limit

KamLAND-Zen reports no neutrinoless double-beta decay of $^{136}$ Xe, setting $T^{0\nu}_{1/2}>2.3\times10^{26}$ yr (90% CL). With phase-space factor $G^{0\nu}\approx1.46\times10^{-14}\,\text{yr}^{-1}$ and nuclear matrix element $|M^{0\nu}|$ in the range $1.6$ – $4.8$ , what bound on the effective Majorana mass $m_{\beta\beta}$ does this imply?

Step 1. Solve equation (5) for the mass: $m_{\beta\beta}=m_e\left(T^{0\nu}_{1/2}\,G^{0\nu}\,|M^{0\nu}|^2\right)^{-1/2}$ , with $m_e=5.11\times10^5$ eV — so $m_{\beta\beta}\propto1/\sqrt{T_{1/2}}$ , and a longer half-life limit means a smaller mass bound.

Step 2. Take a mid-range $|M^{0\nu}|=2.45$ . The root's argument is $T\,G\,|M|^2=(2.3\times10^{26})(1.46\times10^{-14})(6.00)=2.0\times10^{13}$ , so $m_{\beta\beta}=5.11\times10^5/\sqrt{2.0\times10^{13}}=\mathbf{0.11\ \text{eV}}\approx110\ \text{meV}$ .

Step 3. Span the nuclear uncertainty: $|M^{0\nu}|=1.6$ gives $\approx170$ meV, $|M^{0\nu}|=4.8$ gives $\approx58$ meV. So $m_{\beta\beta}\lesssim(60\text{–}170)$ meV, the spread set entirely by the nuclear matrix element — matching the published $\mathbf{m_{\beta\beta}<(36\text{–}156)\ \text{meV}}$ .

Step 4. Interpret. This bound already reaches toward the inverted-ordering band ( $m_{\beta\beta}\sim15$ – $50$ meV). A non-observation down to $\sim10$ meV would exclude inverted ordering for Majorana neutrinos; an observation would prove they are their own antiparticles and fix the absolute mass scale — which is why tonne-scale detectors are being built.

Physics of the portrait

The Majorana neutrino carries the name of a physicist who wrote its equation in 1937, predicted a new kind of particle almost as an aside, and then vanished without a trace the following year.

Origins · 1 minEttore Majorana and how this was discovered

Ettore Majorana

1906–1938 · the symmetric theory of the electron and positron, and the fermion that is its own antiparticle

Ettore Majorana was, in Enrico Fermi's judgement, a physicist of the very first rank — Fermi compared him to Galileo and Newton — yet he published only nine papers and is remembered chiefly for one idea and one disappearance. Born in Catania, he joined Fermi's "Via Panisperna boys" in Rome in the late 1920s, where his facility for calculation was legendary and his reluctance to publish equally so; in 1932 he worked out a correct theory of the nuclear exchange force but declined to write it up, and Heisenberg got the credit. His decisive contribution came in 1937, in "Teoria simmetrica dell'elettrone e del positrone," showing that the Dirac equation admits real solutions describing a neutral spin-½ particle identical to its own antiparticle — the Majorana fermion — and suggesting the neutrino might be one. The idea lay dormant for decades and now organises the entire experimental programme on neutrino mass. In March 1938, shortly after taking a professorship in Naples, Majorana withdrew his savings, boarded a ship between Palermo and Naples, and was never seen again; whether he drowned, entered a monastery, or began a new life was never settled. He was thirty-one. The particle he imagined is now sought both in double-beta-decay nuclei and, as an emergent quasiparticle, in the circuits of quantum-computing labs.

History

1930

— Pauli postulates the neutrino to rescue energy conservation in beta decay

1937

— Majorana formulates the symmetric theory; a neutral fermion can be its own antiparticle

1939

— Furry proposes neutrinoless double-beta decay as the test of Majorana nature

1977

— Minkowski writes the seesaw relation m ≈ m_D²/M; extended by Gell-Mann, Ramond, Slansky, Yanagida and Mohapatra–Senjanović by 1980

1998

— Super-Kamiokande proves neutrinos oscillate, so they have mass

2022

— KamLAND-Zen pushes the effective Majorana mass bound below ≈ 0.16 eV

Going further

Texts. Thomson, Modern Particle Physics, §17.8 (neutrino masses) and §17.8.1 (the seesaw mechanism), Eqs. (17.53)–(17.54) · Schwartz, Quantum Field Theory and the Standard Model, Ch. 11 (Majorana and Dirac masses) · Peskin & Schroeder, An Introduction to Quantum Field Theory, §3.3–3.4 (Majorana fields and the charge-conjugate spinor) · Griffiths, Introduction to Elementary Particles, 2nd ed., §11 (neutrino mass and oscillation) · Halzen & Martin, Quarks and Leptons, Ch. 12–15 (weak currents and neutrino masses).

Provenance. The chiral fields $\nu_L,\nu_R$ and the projectors $P_{L,R}=\tfrac12(1\mp\gamma^5)$ are owned by aq-pp-weyl-chirality; the charge-conjugate field $\psi^c=i\gamma^2\gamma^0\psi^*$ and the particle–antiparticle relation by aq-pp-antiparticles; the Higgs vacuum value $v=246$ GeV and Yukawa structure by aq-pp-higgs-mechanism; the menu of mass mechanisms by aq-pp-neutrino-mass-mechanisms. This article combines them into the Dirac-versus-Majorana dichotomy and the seesaw, and hands the $0\nu\beta\beta$ signature and leptogenesis forward. A verifier should confirm: (1) the eigenvalues $\tfrac12(M\pm\sqrt{M^2+4m_D^2})$ and $|m_\nu|\,m_N=m_D^2$ ; (2) $m_\nu=m_D^2/M=0.01$ eV for $m_D=1$ GeV, $M=10^{11}$ GeV; (3) $M=m_D^2/m_\nu=2\times10^{10}$ GeV for $m_D=1$ GeV, $m_\nu=0.05$ eV; (4) the $0\nu\beta\beta$ mass bound $m_{\beta\beta}\approx(60$ – $170)$ meV from $T_{1/2}>2.3\times10^{26}$ yr; (5) the gauge charge $2q$ of the Majorana bilinear.

Prerequisites. Neutrino Mass Mechanisms · The Higgs Mechanism · Chirality and Weyl Spinors · Antiparticles and CP Conjugation. Leads to. Neutrinoless Double-Beta Decay · Neutrino Mass Beyond the Standard Model · The Matter–Antimatter Asymmetry.

Ask the tutor: Ask me to diagonalise the seesaw matrix line by line for your own choice of $m_D$ and $M$ , or to trace why the neutrinoless double-beta amplitude carries exactly one power of the neutrino mass.

What actually is the difference between a Dirac and a Majorana neutrino?

A Dirac neutrino is distinct from its antineutrino, exactly as an electron differs from a positron; its mass comes from a Higgs coupling that pairs $\nu_L$ with a separate $\nu_R$ , and it conserves lepton number. A Majorana neutrino is its own antiparticle: "neutrino" and "antineutrino" are two helicity states of one field, and its mass pairs the field with its own charge conjugate, violating lepton number by two units. Every other massive fermion is Dirac because charge forbids the Majorana option; the neutral neutrino is the one place the question is open, and only experiment — chiefly neutrinoless double-beta decay — can decide it.

Why does a Majorana mass require the neutrino to be electrically neutral?

Because the Majorana mass pairs a field with its own charge conjugate, and that bilinear carries twice the charge of the field. A particle of charge $q$ would give a term of charge $2q$ , not invariant under electromagnetism — it would let the particle become its own antiparticle, changing charge by two units, which is impossible if charge is conserved. For the electron that means forbidden $e^-\leftrightarrow e^+$ transitions. Only a field with every gauge charge zero — the right-handed neutrino, a total Standard-Model singlet — can carry a Majorana mass. The same term still breaks the global lepton-number symmetry, which is why the Majorana option is inseparable from lepton-number violation.

How does the seesaw make neutrinos light without fine-tuning?

The seesaw exploits a hierarchy rather than a small coupling. You take an ordinary Dirac mass $m_D$ — say of GeV size, natural for a Yukawa of order one — and a very large Majorana mass $M$ for the sterile right-handed neutrino, tied to no symmetry breaking so it can sit at $10^{10}$ – $10^{15}$ GeV. Diagonalising gives a light eigenvalue $m_\nu\approx m_D^2/M$ : the large $M$ in the denominator suppresses it automatically. So instead of demanding an absurdly tiny Yukawa ( $\sim10^{-12}$ ) to fit $m_\nu\sim0.1$ eV directly, you get the smallness for free from a well-motivated high-energy scale, and the lightness of the neutrino becomes evidence for new physics rather than a coincidence.

What is neutrinoless double-beta decay and why is it the key test?

It is a hypothetical nuclear decay $(A,Z)\to(A,Z+2)+2e^-$ in which two neutrons become two protons emitting two electrons but no neutrinos. The ordinary version emits two antineutrinos and conserves lepton number; the neutrinoless version has $\Delta L=2$ and can happen only if the antineutrino emitted at one vertex is reabsorbed as a neutrino at the other — possible only if neutrino and antineutrino are the same particle, i.e. Majorana. Its rate $\propto m_{\beta\beta}^2$ , so a positive signal would prove neutrinos are Majorana and measure the absolute mass scale at once. The signature is clean: the two electrons carry the full decay energy as a sharp peak, distinct from the continuous two-neutrino spectrum.

Do right-handed neutrinos definitely exist?

No — that is one of the things at stake. Right-handed neutrinos are required only for the Dirac scenario, where they provide the partner a Dirac mass must pair with $\nu_L$ . In the pure Majorana scenario no new field is needed: the mass is built from $\nu_L$ and its own charge conjugate. In the seesaw, heavy right-handed neutrinos are postulated — generating both the small light mass and, through their decays, potentially the matter–antimatter asymmetry. But because it is a total gauge singlet, a right-handed neutrino feels no force except gravity and this tiny mixing, so it is extraordinarily hard to detect. Whether it exists, and at what mass, is open.

How is the neutrino seesaw connected to the matter–antimatter asymmetry?

Through leptogenesis. If the heavy Majorana neutrinos $N$ of the seesaw existed in the hot early universe, they would decay — and because a Majorana mass violates lepton number and their couplings can violate CP, those decays can produce slightly more leptons than antileptons. Standard-Model "sphaleron" processes then convert part of this lepton asymmetry into a baryon asymmetry, leaving the excess of matter over antimatter we observe today. So the same Majorana mass that makes neutrinos light and lets them be their own antiparticles could be the reason the universe contains matter at all — which is why confirming the Majorana nature via $0\nu\beta\beta$ has stakes reaching well beyond neutrino physics.