Experiment: Neutrino Helicity
The two-component theory of The Weyl Equation and Neutrinos predicts that the neutrino is produced in a single helicity state, and the three papers that proposed it [Landau:1957] [Lee:1957] [Salam:1957] left the sign open. Goldhaber, Grodzins and Sunyar measured that sign in 1958 [Goldhaber:1958], in what remains one of the most economical experiments in physics: the neutrino is never detected at all. Its helicity is read off the circular polarization of a gamma ray emitted by the recoiling nucleus, using a kinematic accident of the \({}^{152m}\mathrm{Eu}\) decay scheme that makes only those photons emitted antiparallel to the neutrino Doppler-shifted enough to scatter resonantly. Angular-momentum conservation in the \(K\)-capture then transfers the neutrino's helicity to the photon, and a magnetized-iron analyser measures the photon's circular polarization by transmission. The answer — negative helicity, a left-handed neutrino — fixed the sign in the V–A current within weeks [Feynman:1958] [Sudarshan:1958].
The chapter is an experiment chapter and, when written, will use the
structured experiment environment with its
ExpApparatus, ExpProcedure, ExpObservations,
ExpInterpretation and ExpReferences fields, all
quantities in SI with uncertainties. It sits immediately after the
theory chapter it tests and is the companion of
Experiment: Parity Violation, which reports the parity-violation
measurements that made the two-component theory admissible in the first
place; the nuclear-structure background it relies on is
Nuclear Forces and Nuclear Structure.
Experiment: Neutrino Helicity: all derivations of this chapter are pending.
Historical context and the prediction under test
From parity violation to the two-component neutrino
[Reserved: the sequence Lee–Yang [Lee:1956], the \({}^{60}\)Co asymmetry [Wu:1957] and the muon-decay asymmetry [Garwin:1957] [Friedman:1957], which together made parity violation a fact; the immediate revival of Weyl's equation [Landau:1957] [Lee:1957] [Salam:1957]; the prediction to be tested — that the neutrino has definite helicity \(\pm1\), with the sign undetermined by theory — and the corresponding statement for the antineutrino; cross-reference to The Weyl Equation and Neutrinos and Experiment: Parity Violation.]
Why direct neutrino polarimetry is impossible
[Reserved: the neutrino interaction cross-section, of order \(10^{-47}\,\mathrm{m}^{2}\) at MeV energies [Cowan:1956], forbidding any analyser placed in the neutrino's path; the consequent strategy of inferring the helicity from a particle that is detectable, by angular-momentum bookkeeping in a decay where the kinematics selects a single emission direction; the earlier neutrino-recoil experiments that proved the technique — Rodeback and Allen's measurement of the \({}^{37}\mathrm{Ar}\) recoil following \(K\)-capture [Rodeback:1952].]
The chosen decay scheme
[Reserved: why \({}^{152m}\mathrm{Eu}\) is the unique practical choice — a \(J=0\) parent of half-life about \(9.3\,\mathrm{h}\) capturing a \(K\) electron to a \(J=1\) excited state of \({}^{152}\mathrm{Sm}\) that decays by a \(961\,\mathrm{keV}\) gamma to a \(J=0\) ground state, so the whole chain is \(0\to1\to0\) and the photon must carry the full unit of angular momentum; the two-body kinematics fixing the nuclear recoil momentum from a monoenergetic neutrino; the resulting Doppler shift, of a few electronvolts, comparable to the recoil energy loss on emission.]
A gamma ray emitted by a free nucleus is not in general resonantly absorbed by a second nucleus of the same species: emission and absorption each cost a recoil energy, and the resulting mismatch is far larger than the width of the nuclear level, so the resonance is lost. In the decay chain used here it is restored — but only for photons emitted into the hemisphere towards which the emitting nucleus is already moving, having recoiled against the neutrino, that is, only for photons emitted antiparallel to the neutrino. Resonant scattering therefore selects a direction of emission, and the selection is what is observed: photons that have not been Doppler-compensated do not scatter resonantly at all [Goldhaber:1958].
Derivation. Let the excited nucleus have mass \(M\) and level energy \(E_{0}\). If it is at rest when it emits, energy and momentum conservation give the photon \(E_{\gamma}=E_{0}-E_{\mathrm{R}}\) with recoil energy \(E_{\mathrm{R}}=E_{0}^{2}/2Mc^{2}\); to excite a second nucleus, itself initially at rest, a photon must instead supply \(E_{0}+E_{\mathrm{R}}\). The emitted photon is thus short of the absorption energy by
and resonance fails whenever this exceeds the width of the level. For a transition of about \(1\,\mathrm{MeV}\) in a nucleus of mass number 152, Equation (98.1) is a few electronvolts, orders of magnitude larger than a nuclear level width, so free-nucleus resonance fluorescence simply does not occur.
Now let the emitting nucleus be moving, with speed \(v\ll c\), because it recoiled against a neutrino of energy \(E_{\nu}\), so that \(v=E_{\nu}/Mc\). A photon emitted into the direction of that motion is Doppler-shifted upward by
and comparison with Equation (98.1) shows the shortfall is made good when \(E_{\nu}\gtrsim E_{\gamma}\). That is the kinematic accident on which the whole experiment rests: in this decay the neutrino and the photon carry nearly the same energy, and \(M\) cancels between the two expressions, so the compensation is a comparison of \(E_{\nu}\) with \(E_{\gamma}\) and of nothing else. A photon emitted at any other angle receives a smaller shift, and one emitted backwards is shifted the wrong way, so only those emitted antiparallel to the neutrino can scatter resonantly. Selecting the resonantly scattered photons therefore selects that direction, with no angular measurement made anywhere in the apparatus. The selection is not perfect: the excited state has a finite lifetime, during which the recoiling nucleus may be deflected in the source, and this dilutes the correlation — it cannot reverse it, since the resonance condition still admits only the forward hemisphere.
∎Apparatus
The source
[Reserved: the \({}^{152m}\mathrm{Eu}\) source, produced by neutron activation of europium and used within a few half-lives; source strength, geometry and self-absorption; the source mounted at the top of the analysing magnet so that the photons of interest traverse the magnetized iron; the practical constraint that the short half-life imposes on counting time, and hence on the achievable statistical uncertainty.]
The analysing magnet
[Reserved: the magnetized iron block as a circular-polarization analyser — Compton scattering of circularly polarized photons on the aligned electron spins of the iron has a transmission that depends on the relative sign of photon helicity and electron polarization; the magnitude of the effect, a few per cent, and why the measurement is therefore a difference of two large counting rates; the solenoid and its field reversal; the same polarimetric method used for beta-decay gamma rays by Schopper [Schopper:1957].]
Resonant scatterer and detector
[Reserved: the ring of \({}^{152}\mathrm{Sm}\) (as \(\mathrm{Sm}_{2}\mathrm{O}_{3}\)) placed to scatter transmitted photons into the detector; nuclear resonance fluorescence as the selector, possible only for photons whose Doppler shift compensates the recoil loss, i.e. only for photons emitted opposite to the neutrino; the sodium-iodide scintillation counter and its shielding; the remark that this is recoil-compensated resonance and not the recoil-free absorption that Mössbauer reported the same year [Moessbauer:1958].]
Procedure
Selecting the recoil-aligned photons
[Reserved: the argument, to be given step by step, that resonant scattering selects photons emitted antiparallel to the neutrino; the angular-momentum balance \(J_{\gamma}=-J_{\nu}\) along that axis for a \(0\to1\to0\) chain, so that the photon helicity equals the neutrino helicity; the finite lifetime of the intermediate state and the resulting partial loss of the recoil direction, which dilutes but does not reverse the correlation; quantitative estimate of that dilution.]
Field reversal and counting
[Reserved: alternating the magnet polarity at fixed geometry so that all geometric efficiencies cancel in the ratio; the counting asymmetry \(\delta=(N_{+}-N_{-})/(N_{+}+N_{-})\) as the measured quantity; run lengths set by the source half-life; background subtraction using a non-resonant scatterer of the same electron density; the calibration of the analysing power with gamma rays of known circular polarization.]
Observations
The counting asymmetry
[Reserved: the measured asymmetry, of order one per cent, with
its statistical uncertainty; conversion to the circular polarization of
the resonantly scattered photons, found to be negative and of
magnitude about two thirds of the kinematic maximum, quoted with an
absolute uncertainty of roughly ten per cent [Goldhaber:1958];
the resulting neutrino helicity, consistent with \(-1\) and excluding
\(+1\) by a wide margin; all quantities to be tabulated with SI
uncertainties in a booktabs table carried by captionof inside
the experiment box, since a float cannot live there.]
Systematic checks
[Reserved: the checks performed — resonance verified by comparing scatterers, the polarization sign calibrated against a source of known handedness, magnet-reversal cycling against drift, and the insensitivity of the result to detector threshold; the dominant systematic, which is the analysing power of the magnetized iron; the honest statement of what the total uncertainty was and why the result was nonetheless decisive — the two hypotheses differ in sign, not in magnitude.]
Interpretation
The neutrino is left-handed
[Reserved: the conclusion that the neutrino emitted in \(K\)-capture has negative helicity [Goldhaber:1958], fixing the sign left open by [Landau:1957] [Lee:1957] [Salam:1957]; the mirror statement for the antineutrino, inferred from beta-decay electron polarization [Frauenfelder:1957] together with \(CPT\); the caveat, made explicit, that a helicity eigenstate is Lorentz-invariant only for a massless particle, so the result is a statement about the chirality of the current at the energies used.]
The neutrino emitted in the \(K\)-capture of \({}^{152m}\mathrm{Eu}\) has negative helicity: its spin points opposite to its momentum. The measurement is wholly indirect — no neutrino is detected — and reads the helicity off the circular polarization of the \(961\,\mathrm{keV}\) gamma ray emitted by the recoiling \({}^{152}\mathrm{Sm}\) nucleus, which is found to be left circularly polarized [Goldhaber:1958]. The magnitude is consistent with the maximal value \(-1\) and the opposite sign is excluded by a wide margin, so the neutrino is produced in a single helicity state, as a two-component theory requires and as no parity-conserving theory permits.
Derivation. The inference is angular-momentum bookkeeping along a single axis, and it assumes no dynamics at all. Take the axis along the neutrino's momentum, call that direction \(+z\), and use the decay scheme of Section 98.1.3: a \(J=0\) parent captures a \(K\) electron of spin projection \(m_{e}=\pm\tfrac{1}{2}\) and emits a neutrino of helicity \(h_{\nu}=\pm1\), leaving a \(J=1\) nucleus of projection \(m_{N}\) recoiling along \(-z\). Conservation of \(J_{z}\) gives
That nucleus then decays to a \(J=0\) ground state, so the photon must carry away the whole of \(m_{N}\); and Phenomenon 98.1 says that the photons reaching the detector are those emitted along \(-z\). A photon travelling along \(-z\) with angular-momentum projection \(m_{\gamma}\) on \(+z\) has helicity \(h_{\gamma}=-m_{\gamma}\), and a real photon has no projection zero along its own direction of motion, so \(m_{\gamma}=m_{N}=\pm1\). With \(m_{e}=\pm\tfrac{1}{2}\) the only solutions of Equation (98.2) are
the remaining combinations forcing \(m_{N}=0\), which the photon cannot carry, or \(\abs{h_{\nu}}=3\), which no spin-\(\tfrac{1}{2}\) particle has. In both surviving cases \(h_{\gamma}=-m_{N}=h_{\nu}\): the photon inherits the neutrino's helicity exactly. A measured negative photon helicity therefore reads \(h_{\nu}=-1\) directly, using only angular-momentum conservation, the spins of the three nuclear levels, and the transversality of the photon. The dilution noted at the end of Phenomenon 98.1 reduces the observed magnitude below unity but cannot change its sign, which is why a measurement of modest precision settles a question whose two answers differ by a sign.
∎Consequences for V–A
[Reserved: how the measured sign selects \(\gamma^{\mu}(\identity-\gamma^{5})\) from the five Fermi couplings, in combination with the beta-decay correlation data; the V–A theory of Feynman and Gell-Mann [Feynman:1958] and of Sudarshan and Marshak [Sudarshan:1958], published in the same year; the conserved vector current hypothesis; helicity suppression of \(\pi\to e\nu\) as the independent V–A prediction, and its route into Weak Interactions and Electroweak Unification and the Higgs Boson.]
What the result does not say
[Reserved: the result constrains the helicity of the emitted state, not the existence of a right-handed neutrino field of very small coupling; it says nothing about neutrino mass, and it is fully compatible with the massive neutrinos established by Experiment: Neutrino Oscillations; nor does it distinguish Dirac from Majorana neutrinos; a short list of the questions it left open, each pointing to where the book takes them up.]
Modern repetitions and precision
Chirality tests in leptonic decays
[Reserved: the Michel parameters of muon decay [Michel:1950] as the modern parametrization of the leptonic current, measured to the part-in-\(10^{3}\) level by TWIST [Bayes:2011]; the ratio \(\Gamma(\pi\to e\nu)/\Gamma(\pi\to\mu\nu)\), whose measurement by PIENU [AguilarArevalo:2015] agrees with the helicity-suppressed V–A prediction at the \(10^{-3}\) level; the honest note that no repetition of the direct helicity measurement itself has improved on [Goldhaber:1958].]
The energy and angular distribution of the electrons emitted by polarized muons is described completely by four dimensionless parameters. Measurement returns the values that a purely left-handed charged current requires — \(\rho=\delta=\tfrac{3}{4}\) and \(\xi=1\) — at the level of one part in \(10^{3}\) or better [Navas:2024]. The leptonic weak current is therefore chiral to that precision, in a purely leptonic process with no nuclear structure anywhere in it, and at an energy release two orders of magnitude above that of the \(K\)-capture measured in this chapter.
The Michel spectrum: the most general Lorentz-invariant four-fermion matrix element for muon decay, its reduction to four measurable parameters in the decay distribution, and the values those parameters take when the current is the purely left-handed one. Deriving it also shows which combinations of couplings the experiment cannot separate, which is what makes the bounds on right-handed admixtures the shape they are.
Parity violation in electron scattering and atoms
[Reserved: the SLAC E122 measurement of a parity-violating asymmetry in deep-inelastic scattering of polarized electrons [Prescott:1978], which fixed the weak mixing angle and confirmed the chiral structure of the neutral current; atomic parity non-conservation in caesium and the nuclear anapole moment [Wood:1997]; the proton weak charge measured by Qweak [Androic:2018]; how each of these tests the same chirality assignment at a different energy scale.]
Parity violation is not confined to the charged-current processes of beta decay. Longitudinally polarized electrons scattered from nuclei show a cross-section asymmetry between the two beam helicities; heavy atoms show optical transitions that parity conservation would forbid, driven by the weak electron–nucleus interaction; and the proton's weak charge has been measured directly. All of these are described by one dimensionless parameter, the weak mixing angle, whose value is close to \(\sin^{2}\theta_{W}=0.231\) in the conventional renormalization scheme [Navas:2024], and the same value accounts for measurements spanning many decades of momentum transfer.
The parity-violating asymmetry in the scattering of longitudinally polarized electrons, as the interference between the photon-exchange and neutral-current amplitudes, and its expression in terms of the weak mixing angle; together with the analogous expression for the parity-non-conserving amplitude in a heavy atom, where the nuclear weak charge and the atomic structure factorize. Both belong to the electroweak chapter of Part XII.
Limits on right-handed currents
[Reserved: the general framework in which a small admixture of a right-handed charged current is allowed and bounded; the constraints from beta-decay correlation coefficients, from muon decay [Bayes:2011], and from direct collider searches for a heavy right-handed vector boson [Navas:2024]; the current bounds, quoted as limits on the mixing angle and on the mass of any additional charged vector boson; the standing conclusion that the weak current is purely left-handed to within present sensitivity.]
Every measurement sensitive to a right-handed admixture in the charged weak current returns a null result: the beta-decay correlation coefficients, the muon-decay parameters of Phenomenon 98.3, and the direct searches for a heavy charged vector boson coupling to right-handed fields. The outcomes are quoted as limits — on a mixing angle between the two currents, and on the mass of any additional charged vector boson — and no measurement has required a nonzero right-handed component [Navas:2024]. The charged current is purely left-handed to within present sensitivity, which makes the Weyl structure a measured fact and not merely the simplest option compatible with the data.
The general left–right parametrization of the charged current, the observables in beta and muon decay that are first order in the right-handed admixture as against those that are second order, and the translation of a null asymmetry into a bound on the mixing angle and on the mass of an additional charged vector boson. The distinction between the two orders is what determines how the sensitivity scales, and it must be stated with the bounds.