The Weyl Equation and Neutrinos
The Dirac equation of The Dirac Equation is reducible: for a massless particle its four components split into two independent two-component equations, one for each chirality. That split is Weyl's equation [Weyl:1929], and its history is the cleanest case in the book of a mathematically impeccable theory being rejected on a physical prejudice and then reinstated by experiment. Weyl's equation violates parity manifestly, so for a quarter of a century it was dismissed as unphysical; when parity turned out not to be a symmetry of the weak interaction [Lee:1956] [Wu:1957] the same equation became, within months, the accepted description of the neutrino [Landau:1957] [Lee:1957] [Salam:1957], and its central prediction — that the neutrino has one definite helicity — was measured directly a year later [Goldhaber:1958]. This chapter therefore sits after The Dirac Equation and Particles as Poincaré Representations, which supply the spinor representations it decomposes, and immediately before the experiment that tests it, Experiment: Neutrino Helicity; the parity measurements themselves are Experiment: Parity Violation.
The chapter closes on the one place where the Weyl picture is now known to be incomplete. Neutrino flavour oscillations [Fukuda:1998] [Ahmad:2002] require at least two nonzero neutrino masses, and a massive fermion cannot be described by a single Weyl field: the mass must be either of Dirac type, pairing two Weyl fields, or of Majorana type [Majorana:1937], pairing a Weyl field with itself and violating lepton number. Which of the two Nature chose is still an open experimental question, and the searches that would settle it have so far returned null results [Agostini:2020b] [Abe:2023]. The phenomenology of the massive case belongs to Flavour Physics and Neutrinos and its evidence to Experiment: Neutrino Oscillations; what is settled here is the massless two-component theory, its chirality structure, and the exact point at which it fails. Standing references are [Weinberg:1995] [Navas:2024].
The Weyl Equation and Neutrinos: all derivations of this chapter are pending.
The two-component equation
Weyl's equation
[Reserved: the massless two-component equation \(\ii\sigma^{\mu}\pp_{\mu}\psi_{L}=0\) with \(\sigma^{\mu}=(\identity,-\vect{\sigma})\), written down by Weyl in the first of his papers on the electron and gravitation [Weyl:1929]; its plane-wave solutions and the dispersion \(E=\abs{\vect{p}}c\); the conserved current \(j^{\mu}=\psi_{L}^{\dagger}\sigma^{\mu}\psi_{L}\) and its positive-definite density; why the equation is first order yet has only two components, and why no Lorentz-invariant mass term can be built from \(\psi_{L}\) alone.]
Two-component spinors and $\mathrm{SL}(2,\C)$
[Reserved: the undotted and dotted spinor calculus of van der Waerden [vanderWaerden:1929]; \(\mathrm{SL}(2,\C)\) as the double cover of the proper orthochronous Lorentz group and its two inequivalent fundamental representations \((\tfrac{1}{2},0)\) and \((0,\tfrac{1}{2})\), per the classification of Particles as Poincaré Representations; the invariant antisymmetric \(\epsilon\) tensor raising and lowering spinor indices; how a four-vector is a bispinor, \(p_{\mu}\sigma^{\mu}\), with \(\det(p_{\mu}\sigma^{\mu})=p^{2}\).]
Chirality and helicity
[Reserved: chirality as the eigenvalue of \(\gamma^{5}\), a Lorentz-invariant label attached to the representation; helicity as the projection \(\vect{\sigma}\cdot\hat{\vect{p}}\) of spin on momentum, a Poincaré label in Wigner's classification [Wigner:1939] that is invariant only for a massless particle; the two coincide exactly at \(m=0\) and differ at order \(m/E\) otherwise — the single fact that governs everything else in this chapter and every helicity suppression in Weak Interactions; the Weyl equation as the statement that helicity is fixed and the same for all observers.]
Relation to the Dirac equation
[Reserved: the chiral (Weyl) basis of the Dirac matrices, in which the Dirac equation of [Dirac:1928] becomes two coupled two-component equations with the mass as the only coupling; the decomposition \(\psi=\psi_{L}+\psi_{R}\) with \(P_{L,R}=\tfrac{1}{2}(\identity\mp\gamma^{5})\); the massless limit as a decoupling, so that a Dirac field is exactly two Weyl fields when \(m=0\); parity as the operation exchanging the two, developed in Discrete Symmetries and CPT.]
Discrete symmetries of a Weyl field
[Reserved: a single Weyl field is not invariant under parity or under charge conjugation separately, but is invariant under their product \(CP\) and, by the theorem of Axiomatic Quantum Field Theory, under \(CPT\); the historical objection that this made the equation unphysical, voiced by Pauli and repeated for twenty-five years without ever being put to a test; how the objection was disposed of, not answered, by Experiment: Parity Violation; the residual \(CP\) question deferred to Experiment: CP Violation.]
The neutrino hypothesis
The continuous beta spectrum
[Reserved: Chadwick's magnetic-spectrometer measurement showing that beta rays emerge with a continuous distribution of energies rather than the line spectrum a two-body decay demands [Chadwick:1914]; the calorimetric confirmation by Ellis and Wooster, whose measured heat per disintegration of radium E matched the mean and not the endpoint energy [Ellis:1927], closing the escape route that the missing energy was radiated later; the resulting choice between abandoning energy conservation and inventing a new particle.]
The electrons emitted in a given nuclear beta decay do not all carry the same energy. They emerge with a continuous distribution of kinetic energies, running from zero up to a sharp endpoint fixed by the mass difference of the two nuclei [Chadwick:1914], and the heat released per disintegration, measured calorimetrically, equals the mean of that distribution rather than its endpoint [Ellis:1927] [Meitner:1930] — so the balance is not carried off by radiation emitted later. A decay into two bodies cannot produce such a spectrum.
Derivation. Let a nucleus of mass \(M\) decay at rest into exactly two bodies, of masses \(M'\) and \(m\). Four-momentum conservation gives \(p_{M}=p_{M'}+p_{m}\), so \(\left(p_{M}-p_{m}\right)^{2}=p_{M'}^{2}\), that is
In the parent's rest frame \(p_{M}=\left(Mc,\vect{0}\right)\) and \(p_{M}\cdot p_{m}=ME\), where \(E\) is the light particle's total energy, so
This is a single number, fixed entirely by three masses: a two-body decay emits a line, and repeating identical decays cannot broaden it beyond the natural width of the levels. If instead a third body leaves with the electron, the recoiling system is no longer a single particle of fixed mass; its invariant mass \(W\) may take any value between the sum of the two remaining rest masses and \(Mc^{2}-mc^{2}\), and Equation (96.1) with \(M'\) replaced by \(W/c^{2}\) then sweeps out a continuum. The observed spectrum is therefore evidence that a third particle is emitted, and the endpoint — reached when that particle takes away nothing but its rest energy — bounds its mass. What the spectrum alone does not fix is the new particle's spin or its charge; those come from the angular-momentum balance of the decay and from the absence of any observed ionization, and are the content of the hypothesis of Section 96.2.2.
∎Pauli's 1930 letter
[Reserved: Pauli's “desperate remedy” — the letter of 4 December 1930 to the Tübingen conference proposing a neutral, spin-\(\tfrac{1}{2}\), weakly interacting constituent of the nucleus with a mass not exceeding that of the electron [Pauli:1930], so that beta decay is a three-body process and the spectrum is continuous; the spin-statistics argument for nitrogen-14 that motivated it as much as the energy balance did; Brown's historical reconstruction of the letter and its reception [Brown:1978]; Fermi's renaming of the particle after the discovery of the neutron.]
Fermi's theory of beta decay
[Reserved: Fermi's four-fermion contact interaction [Fermi:1934], built by analogy with the electromagnetic current coupling and giving the first quantitative beta spectrum shape, the Kurie plot and the \(ft\) systematics of Nuclear Forces and Nuclear Structure; the coupling constant \(G_{F}\), its dimension, and its modern value as tabulated in [Navas:2024]; the cross-section estimate that made the neutrino look undetectable, and the extension to the general Lorentz-invariant four-fermion couplings that Section 96.4.3 then reduces to one.]
Detection: Reines and Cowan
[Reserved: the inverse beta reaction \(\bar{\nu}_{e}+p\to n+e^{+}\) as the detection channel; the 1953 Hanford attempt [Reines:1953] and the 1956 Savannah River confirmation with a delayed-coincidence signature of positron annihilation followed by cadmium neutron capture [Cowan:1956] [Reines:1956]; the measured cross-section of order \(10^{-47}\,\mathrm{m}^{2}\) per fission antineutrino; Davis's null result for \(\bar{\nu}_{e}+{}^{37}\mathrm{Cl}\) [Davis:1955] showing that neutrino and antineutrino are distinct, and lepton number as the bookkeeping [Konopinski:1953].]
How many neutrinos
[Reserved: the two-neutrino experiment at the AGS establishing that \(\nu_{\mu}\neq\nu_{e}\) [Danby:1962]; direct observation of \(\nu_{\tau}\) interactions by DONUT [Kodama:2001]; the invisible width of the \(Z\) measured at LEP and SLC, giving \(N_{\nu}=2.9840\pm0.0082\) light active species [Schael:2006] — the strongest statement that the count is three; flavour mixing itself left to Flavour Physics and Neutrinos.]
The neutrinos produced together with the electron, with the muon and with the tau are three distinct particles, and there is no fourth of the same kind. The sharpest statement comes from the \(Z\) boson: the part of its decay width that leaves no trace in any detector, divided by the width the theory assigns to a single neutrino species, gives three to better than one per cent [Navas:2024]. The count is a count of species light enough for the \(Z\) to decay into and coupling to it in the standard way; it excludes neither heavier neutral leptons nor species that do not couple to the \(Z\) at all.
Extraction of the invisible width of the \(Z\) from the measured total and visible widths at the resonance peak, and the Standard Model expression for the partial width into one neutrino species, whose ratio is the species count. The measurement requires the electroweak theory of Part XII to convert a width into a number.
Parity and the reinstatement of the equation
Parity as an assumed symmetry
[Reserved: parity invariance as an inference from atomic and nuclear spectroscopy, where it holds, illegitimately extended to the weak interaction where it had never been tested; Lee and Yang's survey showing that no existing experiment constrained parity in weak processes, together with the concrete experiments that would [Lee:1956]; the \(\theta\)–\(\tau\) puzzle — two particles alike in mass and lifetime decaying to final states of opposite parity — as the anomaly that forced the question.]
The measurements
[Reserved: the beta asymmetry from polarized \({}^{60}\)Co [Wu:1957]; the pion–muon–electron chain, in which the muon is produced polarized and decays asymmetrically [Garwin:1957] [Friedman:1957]; the longitudinal polarization of beta electrons, measured as \(v/c\) with the sign fixed by Mott scattering [Frauenfelder:1957]; the circular polarization of the gamma rays following beta decay [Schopper:1957]; full apparatus and uncertainties in Experiment: Parity Violation.]
Maximal violation
[Reserved: why the observed asymmetries are not merely nonzero but as large as kinematics allows, i.e. the weak current couples to one chirality only; the immediate consequence that the two-component massless equation discarded in Section 96.1.5 is the correct one; the distinction between parity violation as an experimental fact and maximality as a further, separately measured fact, quantified by the Michel parameters of muon decay [Michel:1950] and their modern values [Bayes:2011].]
The two-component theory of the neutrino
Landau, Lee–Yang and Salam
[Reserved: the three independent 1957 papers proposing the massless two-component neutrino — Landau's, framed as the conservation of combined inversion [Landau:1957]; Lee and Yang's, which derives the observable consequences for beta decay and pion decay [Lee:1957]; and Salam's, which ties the two-component form to the vanishing of the neutrino mass and to \(\gamma^{5}\) invariance [Salam:1957]; the shared prediction that the neutrino has a single definite helicity, and the sign left open by all three.]
The measured helicity
[Reserved: the Goldhaber–Grodzins–Sunyar measurement of the helicity of the neutrino emitted in \(K\)-capture by \({}^{152m}\)Eu, which found it negative — the neutrino is left-handed [Goldhaber:1958]; the recoil-Doppler resonant-scattering trick and the analysing magnet, given in full in Experiment: Neutrino Helicity; the mirror statement for the antineutrino inferred from the \(\bar{\nu}\) helicity in beta decay, and why no equally direct measurement of it exists.]
The V–A current
[Reserved: the reduction of Fermi's five possible couplings to the single combination \(\gamma^{\mu}(\identity-\gamma^{5})\) by Feynman and Gell-Mann [Feynman:1958] and, independently, by Sudarshan and Marshak [Sudarshan:1958]; the conserved vector current hypothesis and the universality of \(G_{F}\); helicity suppression of \(\pi^{-}\to e^{-}\bar{\nu}_{e}\) relative to \(\pi^{-}\to\mu^{-}\bar{\nu}_{\mu}\) as the sharpest V–A test; the embedding of V–A in the \(\SU(2)_{L}\times\U(1)_{Y}\) theory of Electroweak Unification and the Higgs Boson.]
The charged pion decays overwhelmingly to a muon and only very rarely to an electron, although the electronic channel has by far the larger phase space available to it. The measured ratio is
[Navas:2024]: the kinematically favoured channel is suppressed by four orders of magnitude, which is the opposite of what phase space alone predicts and is the sharpest available test of the chirality of the charged weak current.
Derivation. The pion has spin 0, so in its rest frame the charged lepton and the antineutrino leave back to back with total angular momentum zero and their spin projections on the decay axis must cancel. Because the two travel in opposite directions, cancelling projections means equal helicities. The charged weak current couples only to the left-chiral fields; for a massless antineutrino left chirality is helicity \(+1\) exactly, so the charged lepton is forced into helicity \(+1\) as well. But the current couples to the left-chiral part of the charged lepton too, and for a lepton of mass \(m_{\ell}\), energy \(E\) and momentum \(p\) the amplitude for the left-chiral field to be found with positive helicity, relative to its amplitude for the negative helicity, is
using \(E^{2}-p^{2}c^{2}=m_{\ell}^{2}c^{4}\) and \(E\gg m_{\ell}c^{2}\). The amplitude therefore vanishes linearly in \(m_{\ell}\) and the rate quadratically, so the electronic channel is suppressed relative to the muonic one by roughly \(\left(m_{e}/m_{\mu}\right)^{2}\approx2.3\times 10^{-5}\), which the larger electronic phase space then partly offsets. Carrying the phase-space factors through gives the tree-level prediction
which differs from Equation (96.2) by a few per cent, the difference being radiative corrections. A parity-conserving interaction, coupling equally to both chiralities, would impose no such constraint and would leave the electronic mode the commoner of the two; the observed ratio therefore measures the chirality structure directly and does not merely permit it.
∎Chirality in the Standard Model
Left-handed doublets, right-handed singlets
[Reserved: the Standard Model as a chiral gauge theory — the left-handed fields form \(\SU(2)_{L}\) doublets and the right-handed fields are singlets, so that the fermion content is naturally a list of Weyl fields rather than Dirac fields [Weinberg:1995]; the absence of any right-handed neutrino field in the minimal content, which is exactly why the minimal Standard Model predicts massless neutrinos; hypercharge assignments and the resulting electric charges, cross-referenced to Electroweak Unification and the Higgs Boson.]
The chiral anomaly
[Reserved: the axial current, conserved classically for massless fermions, is not conserved on quantization — the triangle anomaly of Adler [Adler:1969] and of Bell and Jackiw [Bell:1969]; the \(\pi^{0}\to\gamma\gamma\) rate as its measured consequence [Navas:2024]; anomaly cancellation as a nontrivial constraint on the chiral fermion content of each Standard Model generation, and its failure mode if any charge assignment is altered; renormalization context in Quantum Electrodynamics and Renormalization and The Renormalization Group.]
Chiral fermions on a lattice
[Reserved: the Nielsen–Ninomiya theorem, that a local, Hermitian, translation-invariant lattice action with the correct continuum limit necessarily doubles chiral fermions into pairs of opposite chirality [Nielsen:1981]; the homotopy argument behind it; why this is a genuine obstruction to simulating the weak interaction rather than a technical nuisance, in contrast with the vector-like case of Quantum Chromodynamics; the same doubling reappearing as a theorem about band structures in Section 96.8.]
Mass, and where the massless picture fails
Dirac and Majorana mass terms
[Reserved: the two Lorentz-invariant bilinears that can give a Weyl field a mass — the Dirac term \(m\bar{\psi}_{L}\psi_{R}\), which requires a second, independent Weyl field and conserves lepton number, and the Majorana term \(\tfrac{1}{2}m\,\overline{\psi_{L}^{\,c}}\,\psi_{L}\), which does not [Majorana:1937]; the general mass matrix mixing both; the statement that a massive neutrino cannot be a single Weyl particle, and that helicity therefore ceases to be Lorentz-invariant.]
Oscillations as the evidence for mass
[Reserved: flavour oscillation, proposed by Pontecorvo [Pontecorvo:1957] and given its mixing-matrix form by Maki, Nakagawa and Sakata [Maki:1962]; the atmospheric \(\nu_{\mu}\) deficit of Super-Kamiokande [Fukuda:1998] and the neutral-current measurement by SNO showing the solar \(\nu_{e}\) flux is transformed and not lost [Ahmad:2002]; the measured splittings \(\Delta m_{21}^{2}\approx7.5\times 10^{-5}\,\mathrm{eV}^{2}\) and \(\abs{\Delta m_{31}^{2}}\approx2.5\times 10^{-3}\,\mathrm{eV}^{2}\) [Navas:2024]; full treatment in Flavour Physics and Neutrinos and Experiment: Neutrino Oscillations.]
A source of neutrinos of one flavour, allowed to propagate, delivers the other flavours as well. The atmospheric muon-neutrino flux is depleted by an amount depending on the path length through the Earth, and the solar electron-neutrino flux is not lost but converted, the total over all three flavours agreeing with what the Sun is calculated to produce. Fitting the dependence on distance and energy yields two independent mass-squared differences,
both nonzero [Navas:2024]. At least two of the three neutrinos are therefore massive, and the massless two-component description of this chapter cannot be the whole of the story.
Derivation. Suppose the states of definite flavour are superpositions of states of definite mass; take two for simplicity, \(\ket{\nu_{\alpha}}=\cos\theta\ket{\nu_{1}} +\sin\theta\ket{\nu_{2}}\) and \(\ket{\nu_{\beta}}=-\sin\theta\ket{\nu_{1}} +\cos\theta\ket{\nu_{2}}\). A state of definite mass and momentum \(p\) propagates with the phase \(\ee^{-\ii E_{i}t/\hbar}\), and the mass-shell relation expanded for \(m_{i}c^{2}\ll E\) gives \(E_{i}\simeq pc+m_{i}^{2}c^{3}/2p\). The common term \(pc\) contributes an overall phase and drops out; over a flight of length \(L\simeq ct\) the relative phase is
Projecting the propagated state back onto \(\ket{\nu_{\beta}}\),
The probability oscillates in \(L/E\) with an amplitude set by the mixing angle and a period set by \(\Delta m^{2}\), and it is identically zero when \(\Delta m^{2}=0\), whatever the mixing. An observed oscillation is therefore a proof that the mass eigenvalues differ, and hence that at least one of them is nonzero. Two things follow that the chapter must not overstate. Equation (96.4) is insensitive to the common scale of the masses, which is why Equation (96.3) reports differences only; and it is insensitive to the sign of \(\Delta m^{2}\) in vacuum, which is why the ordering of the states is fixed only by measurements involving matter.
∎Absolute mass bounds
[Reserved: oscillations fix differences of squared masses and not the scale, so the scale needs independent measurements; the KATRIN tritium endpoint bound \(m_{\nu}<0.8\,\mathrm{eV}/c^{2}\) at 90 per cent confidence [Aker:2022]; the cosmological bound on \(\sum m_{\nu}\) of order \(0.12\,\mathrm{eV}/c^{2}\) from the CMB and large-scale structure [Aghanim:2020], with its model dependence stated honestly; the resulting hierarchy question, still open.]
Oscillations fix mass differences; the scale itself has never been measured, only bounded. The endpoint region of the tritium beta spectrum constrains the effective electron-neutrino mass to below about \(1\,\mathrm{eV}/c^{2}\) [Navas:2024], and the imprint of free-streaming neutrinos on the cosmic microwave background and on the growth of large-scale structure constrains the sum of the three masses to of order \(0.1\,\mathrm{eV}/c^{2}\) [Aghanim:2020] — the latter only within an assumed cosmological model, and so not a laboratory measurement. Either bound places the neutrino masses at least five orders of magnitude below the electron's, a gap that nothing in this part explains.
The shape of the beta spectrum near its endpoint as a function of the neutrino mass, and the reason the observable is the incoherent sum of squared mixing amplitudes times squared masses rather than any single mass; and, separately, the free-streaming suppression of the matter power spectrum by massive neutrinos, whose derivation belongs to the cosmology part and whose model dependence must be stated with the bound.
Where the mass could come from
[Reserved: the dimension-five lepton-number-violating operator that generates a Majorana mass in the effective theory below any new scale [Weinberg:1979b], and the fact that it is a parametrization of ignorance rather than a mechanism; the seesaw realization with heavy right-handed singlets [Minkowski:1977], named here with the honest statement that no direct evidence for it exists; the alternative that neutrinos are Dirac particles with very small Yukawa couplings, equally unconstrained; open-problem status in What We Observe but Do Not Understand.]
The Majorana question
Majorana's symmetric theory
[Reserved: Majorana's real representation of the Dirac matrices and the self-conjugate field \(\psi=\psi^{c}\) [Majorana:1937]; the degrees-of-freedom count — two, not four — and the equivalence of a Majorana field to one Weyl field with a mass; Racah's observation that the neutrino–antineutrino identity is testable in nuclear processes [Racah:1937]; why no charged fermion can be Majorana, and why the neutrino is the only Standard Model candidate.]
Neutrinoless double beta decay
[Reserved: Furry's calculation of the lepton-number-violating transition \((A,Z)\to(A,Z+2)+2e^{-}\) [Furry:1939]; the sharp two-electron sum-energy peak at the \(Q\) value distinguishing it from the allowed two-neutrino mode; the effective mass \(m_{\beta\beta}=\abs{\sum U_{ei}^{2}m_{i}}\) and the nuclear matrix element as the dominant systematic; the Schechter–Valle argument that observing the decay implies a Majorana mass whatever the mechanism [Schechter:1982].]
Null results and present status
[Reserved: the background-free \({}^{76}\)Ge result of GERDA and its half-life limit [Agostini:2020b]; the \({}^{136}\)Xe limit from KamLAND-Zen, \(T_{1/2}^{0\nu}>2\times 10^{26}\,\mathrm{yr}\), entering the inverted-ordering band [Abe:2023]; the honest verdict — the Dirac versus Majorana question is open, the searches so far are null, and no result in this book depends on the answer; what a positive result would change, and what a continued null result can and cannot exclude.]
Several nuclei that cannot decay by single beta emission do decay by emitting two electrons and two antineutrinos at once; this second-order weak process is observed, and the summed energy of its two electrons is continuous, as Phenomenon 96.1 requires of a decay with unobserved particles in the final state. The lepton-number-violating variant, in which only the two electrons are emitted and their summed energy is a sharp line at the full decay energy, has not been observed in any isotope; the best half-life limits exceed \(10^{26}\) years [Navas:2024]. Whether the neutrino is its own antiparticle is therefore an open experimental question, and this treatise records it as one.
The two-neutrino and neutrinoless double beta decay rates: the phase-space and nuclear-matrix-element factorization, the effective Majorana mass that the neutrinoless rate measures, and the argument that observing the decay implies a Majorana mass term whatever mechanism drives it. The nuclear matrix elements are the dominant systematic and must be presented as such.
Weyl fermions as an observed phenomenon
[Reserved: no elementary Weyl fermion is known, since every observed neutrino has mass; but the equation itself is realized as the low-energy excitation spectrum around isolated band-touching points in crystals without inversion symmetry — the Weyl semimetals, observed in TaAs by angle-resolved photoemission through their Fermi arcs [Xu:2015] [Lv:2015]; the arcs as the surface signature required by the doubling theorem [Nielsen:1981]; the chiral anomaly measured as negative longitudinal magnetoresistance; band-structure background in Electrons in Solids: Band Theory.]