Dirac, Majorana, and Neutrino Experiments

Neutrino oscillation proves that neutrinos have mass, but it does not say what kind. A mass term couples a left-handed field to a right-handed one, and in the original Standard Model there is no right-handed neutrino to couple to. The neutrino is also the only fundamental fermion with no electric charge, and that lets it carry a mass term forbidden to every other particle — one that pairs the neutrino with itself rather than with a distinct antiparticle. Whether nature chooses the ordinary Dirac option or this Majorana option is unsettled, and it is the sharpest open question in neutrino physics. This lesson lays out the two mass terms, the seesaw that a large Majorana mass generates, the neutrinoless-double-beta-decay experiment that would decide the matter, and the sources that map the oscillation parameters.

Dirac and Majorana mass terms

A Dirac mass is the familiar kind. It couples a left-handed field to a distinct right-handed field, and for neutrinos it requires adding a right-handed neutrino $ν_{R}$ that is a singlet under every Standard-Model gauge group. With $ν_{R}$ present, the neutrino takes a Yukawa coupling to the Higgs exactly like the electron, and when the Higgs acquires its vacuum value $v$ the neutrino mass is

m_{D} = \frac{y _{ν} v}{2} .

Nothing is wrong with this, but it forces the dimensionless Yukawa coupling to be $y_{ν} \sim 1 0^{- 12}$ , twelve orders of magnitude below the electron's, an unexplained smallness.¹

Because $ν_{R}$ carries no charge of any kind, a second mass term is allowed that is forbidden to every charged fermion. It pairs the right-handed neutrino with itself,

L_{Maj} = \frac{1}{2} M ν_{R} ν_{R} + h.c.,

with $M$ a mass parameter unconnected to the Higgs. This is a Majorana mass. A charged field cannot carry such a term because $ν_{R} ν_{R}$ has net charge $2 Q$ , which must vanish; only a neutral field qualifies. The physical consequence is that a Majorana neutrino is its own antiparticle, and the term violates lepton number by two units.

The two options differ in their state content. A Dirac neutrino, like the electron, has four states: two helicities for the particle and two for a distinct antiparticle. A Majorana neutrino has only two: the particle and antiparticle are the same object, so only the two helicities remain.

State content of a Dirac versus a Majorana neutrino. A Dirac neutrino has four states, two helicities each for a particle and a separate antiparticle. A Majorana neutrino identifies particle with antiparticle, leaving only two helicity states.

The seesaw mechanism

Suppose both terms are present: a Dirac mass $m_{D}$ linking $ν_{L}$ to $ν_{R}$ , and a large Majorana mass $M$ for $ν_{R}$ . In the basis $(ν_{L}, ν_{R})$ the mass term is a symmetric matrix,

L_{mass} = \frac{1}{2} (ν_{L} ν_{R}) (0 m_{D} m_{D} M) (ν_{L} ν_{R}) + h.c.

The left-handed neutrino has no Majorana mass of its own — such a term is forbidden by the gauge symmetry until the Higgs breaks it — so the top-left entry is zero. The physical masses are the eigenvalues,

m_{\pm} = \frac{1}{2} (M \pm M^{2} + 4 m_{D}^{2}) .

When the Majorana scale dominates, $M ≫ m_{D}$ , the eigenvalues separate cleanly,

m_{heavy} \approx M, m_{light} \approx \frac{m _{D}^{2}}{M} .

The heavy state is almost entirely the right-handed neutrino; the light state is almost entirely the left-handed neutrino we observe. Raising $M$ pushes the light mass down — the two masses move on a seesaw. This converts the puzzle of a tiny Yukawa coupling into a statement about a high scale: if $y_{ν} \sim 1$ , comparable to the top quark, then $m_{D} \sim v$ and a light mass near $0.05 eV$ requires

M \approx \frac{m _{D}^{2}}{m _{light}} \sim 1 0^{13} - 1 0^{15} GeV,

close to the scale of grand unification. The smallness of the neutrino mass becomes evidence for new physics at an energy far beyond direct reach.²

The seesaw. A heavy Majorana mass for the right-handed neutrino and a Dirac mass of ordinary size combine so that one eigenstate is pushed up to the heavy scale while the observed left-handed neutrino is pushed down to a tiny mass. The product of the two physical masses equals the Dirac mass squared.

Neutrinoless double-beta decay

The Dirac and Majorana options differ observably in one place: the Majorana mass violates lepton number, and a lepton-number-violating process would settle the question. Lepton number cannot change by one unit without another fermion balancing it, so the search is for a change of two units.

Ordinary double-beta decay is the cleanest platform. Certain even-even nuclei cannot beta-decay singly because the intermediate nucleus is heavier, but can decay by two simultaneous beta emissions to a lighter nucleus two atomic numbers away. Germanium-76 is the standard example,

^{76} Ge \to^{76} Se + 2 e^{-} + 2 \overset{ν}{ˉ}_{e},

observed with a lifetime near $1 0^{21} years$ . This process conserves lepton number: two antineutrinos leave. If the neutrino is Majorana, a second channel opens in which the antineutrino emitted at one vertex is absorbed as a neutrino at the other — possible only because the two are the same particle — and no neutrinos escape,

^{76} Ge \to^{76} Se + 2 e^{-} .

This is neutrinoless double-beta decay, written $0 ν β β$ . It changes lepton number by two and would be unambiguous proof that the neutrino is Majorana. Its signature is a spike at the endpoint of the summed electron energy, where the two electrons carry the entire released energy with no neutrinos to share it.

No such decay has been seen, in germanium or in the dozen other candidate nuclei. Current limits place the half-life beyond $\sim 1 0^{25} years$ , which bounds the effective Majorana mass to below a few tenths of an eV, comparable to the cosmological bound on the mass sum.³

Neutrinoless double-beta decay at the quark level. Two down quarks each convert to an up quark, emitting a W boson and an electron. In the Majorana case the neutrino line is internal: a neutrino emitted at one vertex is the same particle absorbed at the other, so no neutrino escapes and lepton number changes by two.

Neutrino sources and baselines

The oscillation parameters are pinned by matching each source's characteristic energy $E$ and baseline $L$ to the splitting it probes, since sensitivity peaks when $L / E \sim 1/Δ m^{2}$ . Four broad classes cover the range.

Solar neutrinos — electron neutrinos from fusion, $E \sim 0.1 - 10 MeV$ , baseline the Earth–Sun distance $\sim 1.5 \times 1 0^{8} km$ . Sensitive to the small solar splitting through the MSW effect; measure $θ_{12}$ .
Atmospheric neutrinos — muon and electron neutrinos from cosmic-ray showers, $E \sim 1 GeV$ , baselines from $\sim 15 km$ overhead to $\sim 1.3 \times 1 0^{4} km$ through the Earth. Sensitive to the large splitting; measure $θ_{23}$ .
Reactor antineutrinos — electron antineutrinos from fission, $E \sim few MeV$ . Short baselines $\sim 1 km$ measure $θ_{13}$ ; longer baselines $\sim 100 km$ measure the solar splitting on the ground.
Accelerator beams — muon neutrinos from pion decay in flight, $E \sim GeV$ , baselines of hundreds of km to a far detector. Measure $θ_{23}$ and the mass splitting precisely, and probe the Dirac phase by comparing neutrino and antineutrino running.

Neutrino sources on a baseline versus energy map. Each source occupies a band set by its production energy and the distance to the detector; the diagonal marks constant L over E, the combination that fixes which mass splitting a source probes. Solar and long-baseline reactor experiments probe the small splitting; atmospheric and accelerator experiments probe the large one.

Why neutrino mass is beyond the Standard Model

The original Standard Model has no right-handed neutrino and no Higgs field capable of giving the neutrino a mass, so neutrinos were exactly massless by construction — the mass term simply could not be written. Oscillation shows that assumption is false. Any repair extends the theory in one of a few directions.

Add a right-handed neutrino with a Dirac Yukawa. This restores a mass but leaves the tiny Yukawa coupling unexplained.
Add a right-handed neutrino with a large Majorana mass, the seesaw, tying the light mass to a high scale near grand unification.
Add no new field but a dimension-five operator, $\frac{λ}{M} (L H) (L H)$ , the unique lowest-dimension term that gives the left-handed neutrino a Majorana mass $\sim λ v^{2} / M$ after electroweak breaking. This captures the seesaw's physics — a small mass pointing to a large scale $M$ — without committing to the heavy state explicitly.

All three go beyond the minimal Standard Model, and all but the pure Dirac option make the neutrino a Majorana particle. This is why neutrino mass is the one confirmed piece of laboratory physics that the Standard Model, in its original form, cannot accommodate. Which repair nature chose is what $0 ν β β$ and the leptonic CP measurements are built to decide.⁴

Summary

Because the neutrino is neutral, it admits a Majorana mass term forbidden to every charged fermion, so it may be its own antiparticle. A Dirac neutrino has four states and an unexplained Yukawa coupling of $1 0^{- 12}$ ; a Majorana neutrino has two states and violates lepton number. With both a Dirac mass $m_{D}$ and a large Majorana mass $M$ , the seesaw gives a heavy state near $M$ and a light state $m_{D}^{2} / M$ , converting the smallness of the neutrino mass into evidence for a scale near $1 0^{13} - 1 0^{15} GeV$ . Neutrinoless double-beta decay, a lepton-number-violating channel absent in the Standard Model, is the decisive test; current limits push the half-life beyond $1 0^{25} years$ and the effective mass below a few tenths of an eV. Solar, atmospheric, reactor, and accelerator sources cover complementary regions of the baseline–energy plane and together fix the mixing angles and splittings. Neutrino mass, in any of its forms, requires an extension of the original Standard Model, and settling the Dirac-or-Majorana question is the central experimental goal of the field.

Tong, The Standard Model (Cambridge Part III), §7.1.1 — the Dirac mass from a right-handed-neutrino Yukawa, $m_{D} = y_{ν} v / 2$ , and the required $y_{ν} \sim 1 0^{- 12}$ . damtp.cam.ac.uk/user/tong/standardmodel.html ↩
Tong, The Standard Model (Cambridge Part III), §7.1.1 — the combined Dirac–Majorana mass matrix, its eigenvalues $\frac{1}{2} (M \pm M^{2} + 4 m_{D}^{2})$ , the limits $m_{heavy} \approx M$ and $m_{light} \approx m_{D}^{2} / M$ , and the $M \sim 1 0^{13} GeV$ estimate; the dimension-five operator is developed in §7.1.2. Thomson, §13.9, gives the same seesaw. ↩
Tong, The Standard Model (Cambridge Part III), §7.1.3 — double-beta decay of $^{76} Ge$ , the lepton-number argument, and neutrinoless double-beta decay with the $> 1 0^{25} year$ half-life bound giving $m_{ν} ≲ 0.3 eV$ ; experimental limits reviewed by the Particle Data Group, pdg.lbl.gov. Perkins, Ch. 9, describes the experiments. ↩
Tong, The Standard Model (Cambridge Part III), §7.1.1–7.1.2 — the three routes to a neutrino mass (Dirac Yukawa, seesaw, dimension-five Weinberg operator) and why each extends the minimal Standard Model; Thomson, §13.9. ↩