Experiment: Parity Violation

Tests Phenomena 101.26, 104.55 and 105.16. Assuming Propositions 101.15, 101.27, 105.19 and 105.20.

Parity conservation was, until 1956, treated as a law of nature rather than as a hypothesis under test. This chapter records the experiment that ended that status: the cobalt-60 beta-decay asymmetry measured by Wu, Ambler, Hayward, Hoppes and Hudson [Wu:1957], prompted by the observation of Lee and Yang [Lee:1956] that not one existing measurement bore on parity in the weak interaction, and confirmed within days by the pion–muon–electron chain [Garwin:1957] [Friedman:1957]. The chapter sits immediately after the weak-interaction theory of Weak Interactions and the discrete-symmetry formalism of Discrete Symmetries and CPT, because it is the measurement those chapters exist to explain; it is the direct experimental root of the two-component neutrino of The Weyl Equation and Neutrinos, of the helicity measurement in Experiment: Neutrino Helicity, and of the chiral gauge structure that Electroweak Unification and the Higgs Boson builds on.

The division of labour with those chapters is worth stating at the outset, because it decides what is and is not repeated here. The formal apparatus — the space-inversion operator on states and on Dirac fields, the classification of bilinears, the \(V-A\) current, the relation between chirality and helicity — is derived in Sections 101.4 and 105.2 and is used here, not rebuilt. What this chapter owes is the measurement: what the source physically was, how cold it had to be and why, what the counters saw, what numbers came out with what uncertainties, and by what controls a counting-rate difference of a few tens of per cent was shown to be a property of the weak interaction rather than of a photomultiplier in a magnetic field. Its second half follows the same pseudoscalar observable out of nuclear beta decay and into neutral-current physics — atomic parity violation [Bouchiat:1974] [Wood:1997], polarized deep-inelastic scattering [Prescott:1978] [Prescott:1979] and the weak charge of the proton [Androic:2018] — where parity violation has become a precision tool for measuring the weak mixing angle rather than a surprise.

Historical context and the prediction under test

Parity as a conserved quantum number

Space inversion is the map

implemented on quantum states by a unitary operator \(\mathcal{P}\) with \(\mathcal{P}\psi(\vect{x},t)=\eta_{P}\,\psi(-\vect{x},t)\). Applying it twice returns every position to itself, so \(\mathcal{P}^{2}\) is a symmetry operation that does nothing; with the usual convention that it is the identity, \(\mathcal{P}^{2}=\identity\) and the eigenvalues of \(\mathcal{P}\) are \(\pm1\). That is the whole of the kinematics, and it is why parity is a two-valued quantum number rather than a continuous one: there is no infinitesimal space inversion, so parity has no associated conserved current and no Noether charge, only a multiplicative selection rule.

Wigner introduced parity as a conserved quantum number of atomic states in 1927 [Wigner:1927], and the evidence for its conservation in the electromagnetic interaction was, and remains, overwhelming. It is carried chiefly by Laporte's rule: an electric dipole transition connects only atomic states of opposite parity, because the dipole operator \(-e\vect{x}\) is itself parity-odd. The rule is derived as Proposition 105.1; here we need only its status as evidence. Every line in every atomic spectrum that obeys it — and every line whose absence it explains — is a measurement that the electromagnetic Hamiltonian commutes with \(\mathcal{P}\), and the accumulated precision of atomic and nuclear spectroscopy by 1956 bounded any parity-violating admixture in the nuclear force at the level of \(10^{-7}\) in amplitude [Lee:1956].

The observables of a physical theory divide, under Equation (110.1), into four classes according to how they transform. A scalar is unchanged; a pseudoscalar changes sign; a vector (or polar vector) reverses; an axial vector (or pseudovector) does not. The position \(\vect{x}\), the momentum \(\vect{p}=m\dd\vect{x}/\dd t\) and the electric field \(\vect{E}\) are polar; the angular momentum \(\vect{J}=\vect{x}\times\vect{p}\), the spin \(\vect{S}\) and the magnetic field \(\vect{B}\) are axial, because a cross product of two polar vectors acquires two sign changes and therefore none. Consequently \(\vect{J}\cdot\vect{p}\), \(\vect{S}\cdot\vect{p}\) and \(\vect{E}\cdot\vect{B}\) are pseudoscalars. The complete table of Dirac bilinears and their behaviour is Table 105.1.

The entire experimental programme of this chapter rests on one statement about pseudoscalars, and it is worth isolating it because everything else is instrumentation.

The tau–theta puzzle

The pressure to test parity came from two strange mesons. The \(\theta^{+}\) decayed to \(\pi^{+}\pi^{0}\) and the \(\tau^{+}\) to \(\pi^{+}\pi^{+}\pi^{-}\); by 1955 their masses agreed to better than a per cent and their lifetimes to within the experimental errors, and the modern values — one particle, the \(K^{+}\), with \(m_{K}c^{2}=493.677(15)\,\mathrm{MeV}\) \(=7.9095(2)\times 10^{-11}\,\mathrm{J}\) and mean life \(\tau_{K}=1.2380(20)\times 10^{-8}\,\mathrm{s}\) [Navas:2024] — say why the coincidence was intolerable. But the parities of the two final states are opposite: by Equation (105.31) a state of \(n\) pions with no relative orbital angular momentum has parity \((-1)^{n}\), so \(\eta_{P}(\pi\pi)=+1\) while \(\eta_{P}(\pi\pi\pi)=-1\).

Only two escapes existed. Either \(\theta\) and \(\tau\) were genuinely distinct particles with an unexplained mass and lifetime degeneracy, or the three-pion decay carried a unit of relative orbital angular momentum, in which case the parent would have spin \(\ge1\) and the parities could agree. Dalitz closed the second by inventing the plot that bears his name [Dalitz:1953]: for a three-body decay at rest the density of events in the \((T_{1},T_{2})\) plane of two of the kinetic energies is the squared matrix element and nothing else, so a uniform population is a direct measurement that the decay amplitude is constant — no relative orbital angular momentum, no spin. The observed \(\tau\) plot was uniform. The construction and its derivation are given at Section 105.2.1, which also carries the phase-space calculation behind Equation (105.32); the point here is only that by 1956 the escape route through angular momentum had been closed experimentally, not assumed away.

Lee and Yang: parity untested in the weak interaction

Lee and Yang responded with an audit rather than a mechanism [Lee:1956]. They asked what the experimental evidence for parity conservation actually consisted of, and found that all of it came from strong and electromagnetic processes. In the weak interaction there was none: not one experiment had ever been designed to detect a parity-odd observable in a weak decay, and the reason was structural rather than accidental. Every weak measurement then existing was a lifetime, a branching ratio or an energy spectrum, and each of those is a scalar; by Proposition 110.1 a scalar is insensitive to parity violation at leading order, since the parity-odd part of the interaction contributes to a scalar only through its square. A theory in which parity was violated maximally would have reproduced every weak number known in 1956.

What makes the paper a model of method is its second half, which lists the experiments that would settle the question. Each proposal is the mean of a pseudoscalar:

• the correlation \(\avg{\vect{J}}\cdot\vect{p}_{e}\) between the polarization of an oriented nucleus and the momentum of the emitted electron — an axial vector dotted into a polar one;

• the longitudinal polarization \(\avg{\vect{s}}\cdot\vect{p}\) of the electron emitted in an unoriented beta decay, which requires no cryogenics but a polarimeter;

• in the chain \(\pi^{+}\to\mu^{+}\nu_{\mu}\) followed by \(\mu^{+}\to e^{+}\nu_{e}\bar{\nu}_{\mu}\), the correlation between the muon direction and the positron direction, which is nonzero only if the muon is produced longitudinally polarized.

All three were carried out within a year of the paper, by three independent groups with no common apparatus, and all three returned the largest answer the kinematics permits. The first is the subject of this chapter; the second was done by Frauenfelder and collaborators with Mott-scattering polarimetry on the electrons from \(^{60}\)Co itself [Frauenfelder:1957] and by Schopper through the circular polarization of the accompanying gamma rays [Schopper:1957]; the third is Section 110.6 below. Lee and Yang received the 1957 Nobel prize, in the same year as the measurements they proposed.

The observable and what a measurement must deliver

The observable of the cobalt experiment is the correlation between the nuclear polarization and the electron momentum. Write \(\hat{\vect{n}}\) for the unit vector along the mean nuclear spin and \(\hat{\vect{p}}_{e}\) for the electron direction. The most general angular distribution linear in that correlation is

in which three factors appear that the experiment must supply separately, and the failure to supply any one of them makes the number uninterpretable:

1. the asymmetry coefficient \(A\), the quantity of physical interest, dimensionless, and predicted by the theory of the interaction;

2. the degree of nuclear polarization \(f=\abs{\avg{\vect{J}}}/J\), a property of the source and its temperature, which must be measured independently and which decays to zero as the source warms;

3. the electron speed \(v/c\), which enters because the chirality projected out by the interaction is converted into helicity only up to a factor \(v/c\) (Proposition 105.20), and which for a continuous beta spectrum means an average weighted by the counter's acceptance.

What a counter measures is not \(W(\theta)\) but a rate, and rates depend on solid angles, detector efficiencies and source strength — all of which drift. The design of the experiment therefore forms the observable as a ratio in which those cancel.

The last statement fixes the scale of the whole enterprise. Reaching a statistical precision of \(1\,\mathrm{\%}\) on \(\varepsilon\) requires \(N=10^{4}\) counts, and — because the polarization survives only a few minutes — those counts must be accumulated in of order \(60\,\mathrm{s}\). The counter must therefore see a rate of order \(2\times 10^{2}\,/\mathrm{s}\). With a solid-angle fraction of a few per cent and an escape-and-detection efficiency of order one half, the source activity must be at least

which is a modest laboratory source but a hard lower bound: a weaker one cannot deliver the statistics inside the warm-up window however long the experiment runs, because the signal is gone before the counts accumulate. Equation (110.5) is the reason the measurement had to wait for a source preparation technique and a cryostat that were compatible with each other.

Apparatus

The polarized cobalt-60 source

The choice of nucleus is not incidental; it is forced by four requirements that almost no other nuclide satisfies at once, and Table 110.1 collects the numbers.

First, the transition must be allowed and pure Gamow–Teller, so that the predicted asymmetry coefficient is the maximal \(A=-1\) of Proposition 101.27 rather than a mixture of Fermi and Gamow–Teller amplitudes with an unknown ratio. The decay \(^{60}\mathrm{Co}\,(5^{+})\to{}^{60}\mathrm{Ni}^{*}\,(4^{+})\) has \(\Delta J=1\) with no parity change, which by the allowed selection rules of Proposition 101.15 is pure Gamow–Teller: a Fermi transition cannot change \(J\) at all.

Second, the parent spin must be large. The polarization attainable at a given temperature grows with \(J\), because the Zeeman ladder has \(2J+1\) rungs and the Boltzmann factor acts across all of them; \(J=5\) is among the largest available in a convenient beta emitter.

Third, the decay must be followed by a gamma cascade, because that cascade is the only practical thermometer for the nuclear orientation. The \(4^{+}\) daughter level de-excites through \(4^{+}\to2^{+}\to0^{+}\), emitting photons of \(1173.23\,\mathrm{keV}\) and \(1332.49\,\mathrm{keV}\), and the angular distribution of those photons relative to the orientation axis is anisotropic whenever the parent nuclei are aligned. Without an independent measurement of \(f\), Equation (110.4) determines only the product \(Af\), and the experiment would establish that parity is violated without measuring by how much.

Fourth, the half-life must be long enough that the source does not decay away during preparation and short enough that a usable activity fits in a thin layer. At \(5.2714(5)\,\mathrm{yr}\) \(=1.6635\times 10^{8}\,\mathrm{s}\) the decay constant is \(\lambda=\ln2/t_{1/2}=4.1667\times 10^{-9}\,/\mathrm{s}\), so Equation (110.5) is met by \(N=\mathcal{A}/\lambda\approx2.4\times 10^{12}\) atoms, a quantity of \(^{60}\)Co of order \(2.4\times 10^{-10}\,\mathrm{g}\). The active material is therefore a trace, and the layer thickness is set by beta absorption rather than by chemistry.

Why the active layer must be thin

The source is a layer of \(^{60}\)Co grown on the surface of the paramagnetic salt crystal, and its thickness is the leading systematic. An electron created inside the layer loses energy before it escapes, which lowers \(v/c\) and hence, by Equation (110.4), the observed asymmetry; a thick layer also scatters electrons, which randomizes \(\cos\theta\) and dilutes the asymmetry a second time. Both effects reduce \(\abs{\varepsilon}\), so they cannot manufacture the result — but they do prevent the extraction of \(A\).

Adiabatic demagnetization and the cryostat

Nuclear orientation requires that the Zeeman splitting of the nuclear sublevels exceed the thermal energy, and that single requirement dictates the entire cryogenic apparatus.

Equation (110.8) is a severe constraint and it is worth reading it twice. A laboratory solenoid delivering \(B=1\,\mathrm{T}\) would need \(T=0.7\,\mathrm{mK}\), below what any refrigerator of the period could reach and below the ordering temperature of the salt. The way out, and the reason the experiment was possible at all, is that the field acting on the nucleus is not the applied field: in a paramagnetic salt containing the active ion, the hyperfine interaction with the ion's own electron shell produces an effective field at the nucleus of order \(10\,\mathrm{T}\) once the electronic moments are themselves aligned, and the electronic moments are aligned by a field of a few millitesla because the electronic moment is three orders of magnitude larger than the nuclear one. The applied field steers the electrons; the electrons polarize the nuclei. With \(B\sim10\,\mathrm{T}\), Equation (110.8) demands only \(T\lesssim7\,\mathrm{mK}\).

Cooling by adiabatic demagnetization

The millikelvin range was reached by the one technique available in 1957, and its principle is a single line of thermodynamics.

The sequence is thus: cool the salt to \(1.2\,\mathrm{K}\) against a pumped liquid-helium bath, apply about \(1\,\mathrm{T}\) and let the heat of magnetization flow out through helium exchange gas, pump the exchange gas away to isolate the salt thermally, then remove the field. The crystal arrives at a few millikelvin thermally disconnected from everything, which is both the point and the limitation.

The heat leak and the observation window

Because the sample is isolated, its temperature rises under any residual heat leak, and the polarization decays with it. The measured decay time is about six minutes, and it fixes the leak.

Finally, the polarizing solenoid. It is a small vertical coil, of a few millitesla, whose sense can be reversed; its only job is to align the electronic moments so that the hyperfine field aligns the nuclei, and reversing it interchanges the two counting directions of Equation (110.4) without moving a single piece of the apparatus. That is the design decision on which the whole measurement turns: the two configurations being compared differ in nothing except the direction of an axial vector.

Beta and gamma detection

The beta counter

An anthracene crystal a few centimetres above the source, inside the vacuum space of the cryostat, absorbs the escaping electrons and converts their energy to scintillation light. Anthracene is the reference organic scintillator, with a light yield of about \(1.7\times 10^{4}\) photons per \(\mathrm{MeV}\) of electron energy deposited, so a \(150\,\mathrm{keV}\) electron produces about \(2.6\times 10^{3}\) photons. The crystal cannot be viewed directly, because a photomultiplier is useless in a magnetic field: its gain depends on the trajectories of low-energy electrons between dynodes, and a field of even \(1\,\mathrm{mT}\) changes it by a measurable amount. That single fact is the reason for the most conspicuous feature of the apparatus — a lucite light pipe about a metre long, carrying the scintillation light out of the cryostat and out of the fringe field to a photomultiplier standing at room temperature. Even with a transmission of order \(10\,\mathrm{\%}\) through the pipe and a photocathode quantum efficiency of order \(10\,\mathrm{\%}\), some tens of photoelectrons per event remain, comfortably above the single- photoelectron noise, so pulse-height discrimination against the photomultiplier's dark rate is possible.

The consequence for the physics is that the beta counter has a threshold rather than a spectrometer's resolution. It counts electrons above some minimum energy, which is why \(v/c\) in Equation (110.4) is an acceptance-weighted average and not a measured quantity per event, and why \(0.6\) rather than a number with three digits is the honest value to carry.

The gamma counters

Two sodium-iodide scintillation counters view the source from outside the cryostat, one on the polar axis and one in the equatorial plane. They count the \(1173.23\,\mathrm{keV}\) and \(1332.49\,\mathrm{keV}\) cascade photons, whose penetration through the cryostat walls is what makes an external gamma counter possible at all where an external beta counter is not. Their function is not to measure a gamma energy but to compare two rates: the ratio of the polar to the equatorial rate is an anisotropy, and by Proposition 110.9 below it is a thermometer for the nuclear orientation.

Both gamma counters sit at fixed geometry, unaffected by the reversal of the polarizing field, which is what allows the gamma anisotropy and the beta asymmetry to be recorded simultaneously and compared as functions of time through the same warm-up.

Procedure

The run cycle

A run is a cycle of preparation, a few minutes of measurement, and a long recovery, and its steps are:

1. Pre-cool. The cryostat is filled and the helium bath pumped to about \(1.2\,\mathrm{K}\). Helium exchange gas couples the salt crystal, with its cobalt layer, to the bath.

2. Magnetize. A field of order \(1\,\mathrm{T}\) is applied along the horizontal axis. The entropy of the electronic moments falls and the heat of magnetization is carried to the bath by the exchange gas; the crystal must be allowed to return to \(1.2\,\mathrm{K}\) before the next step, or the demagnetization starts warm and ends warm.

3. Isolate. The exchange gas is pumped out. The crystal is now thermally disconnected, and everything that follows happens under the heat leak of Proposition 110.8.

4. Demagnetize. The magnetizing field is reduced to zero. By Equation (110.10) the crystal cools to a few millikelvin.

5. Polarize. The small vertical solenoid is energized in one sense. The electronic moments align with it, the hyperfine field aligns the cobalt nuclei, and \(f\) rises to its maximum within a time short compared with the warm-up.

6. Count. Beta counts in the polar anthracene counter and gamma counts in both sodium-iodide counters are recorded in successive short intervals — of order tens of seconds — against a clock, for as long as any anisotropy survives.

7. Reverse and repeat. The solenoid sense is reversed and the counting repeated, the whole cycle being restarted from step 1 when the polarization is gone.

Interleaving, and why it is not optional

Proposition 110.3 cancels the geometric and efficiency factors only if they are the same in the two configurations being compared. Over a full cooling cycle they are not: the source decays (negligibly), the photomultiplier gain drifts, the discriminator threshold moves with temperature, and above all the polarization \(f\) is itself a function of time. If \(N_{\uparrow}\) were taken in one cooling cycle and \(N_{\downarrow}\) in the next, the difference would contain the difference of the two cycles' base temperatures — an effect of the same size as the signal.

The measurement is therefore constructed so that the two field senses are compared within a common warm-up window, and the comparison is made at equal times after demagnetization. The asymmetry is formed between the counting rate at time \(t\) in a run with the field up and the counting rate at the same \(t\) in a run with the field down, and the gamma anisotropy recorded in the same intervals certifies that the two runs had the same polarization at that \(t\). Where they did not, the gamma data say so, and the beta data can be corrected by the ratio of the two anisotropies rather than discarded.

The controls

Three null tests separate a genuine asymmetry from an instrumental one, and their logic is worth setting out because it is the part of the experiment that makes the result believable rather than merely striking.

1. The warm sample. With the crystal warm — above \(1\,\mathrm{K}\), where by Equation (110.7) \(x<10^{-5}\) and \(f\) is zero to five decimal places — the polarizing field is applied in both senses and the beta rates compared. Any difference found here is instrumental by construction, since there is nothing to orient. This is the direct test for a magnetic effect on the photomultiplier, on the light pipe, or on the electron trajectories between source and counter.

2. The warm-up curve itself. The polarized measurement is not a single number but a time series. If the beta asymmetry and the gamma anisotropy decay to zero on the same time constant, they are both tracking \(f\); an instrumental asymmetry has no reason to depend on the temperature of a salt crystal at all, and a fortiori no reason to share a decay constant with a gamma-ray angular distribution measured by different counters outside the cryostat.

3. Solenoid sense versus counter position. Reversing the field and moving the counter to the opposite side of the source are two different ways of reversing \(\cos\theta\). They must give the same asymmetry with the same sign. Any effect that depends on the field direction in the laboratory rather than on the angle between the spin and the electron momentum fails this test.

The second of these is the decisive one, and it is what elevates the result above a difference of two counting rates. It converts a comparison between two configurations into a comparison between two functions of time measured in the same apparatus by different detectors, and no plausible instrumental effect reproduces it.

Observations and data

Gamma anisotropy as the polarization thermometer

The gamma counters measure an angular distribution, and an angular distribution of photons from an oriented nucleus is even in \(\cos\theta\): a photon distribution cannot distinguish \(\hat{\vect{n}}\) from \(-\hat{\vect{n}}\), because the electromagnetic transition conserves parity. The gammas therefore measure the alignment of the ensemble — its second moment — and not the polarization — its first moment — which is precisely the quantity Equation (110.4) needs. That the one nevertheless determines the other is a consequence of the Boltzmann distribution, and it is the step in the analysis that is most often passed over.

The observed behaviour is that \(\varepsilon_{\gamma}\) is of order \(20\,\mathrm{\%}\) immediately after demagnetization and decays to zero over about six minutes as the crystal warms [Wu:1957]. Both gamma counters see it, with opposite signs — the polar counter's rate falls while the equatorial counter's rises, or the reverse, according to the sign of the alignment — which is itself a check that the effect is an angular redistribution of a fixed number of photons and not a change in the source.

The beta asymmetry and its warm-up decay

The arithmetic of the observed number closes as follows, and it does not close without the polarization factor. Equation (110.4) gives \(\varepsilon=A\,(v/c)\,f\). With \(A=-1\), the value Proposition 101.27 derives for a pure Gamow–Teller transition fed by an interaction that emits only left-chiral electrons, with \(v/c\approx0.63\) for the counted part of the spectrum, and with \(f\approx0.6\) from Proposition 110.9, the prediction is

against an observed value of order \(0.4\). Dropping \(f\) would predict \(0.63\), half again too large; dropping both \(f\) and \(v/c\) would predict \(1\). The two dilution factors are not corrections, they are the reason the experiment had to measure the gamma anisotropy at all.

Numerical results

Table 110.3 collects what the measurements delivered. Three features of the cobalt entries are worth reading deliberately.

The first is that the headline number, \(\abs{\varepsilon}\approx0.4\), is enormous by the standards of a symmetry test. A violation at the level of a per cent would have been a discovery; one at the level of \(10^{-4}\) would have been a triumph of technique. Forty per cent means that the parity-odd and parity-even parts of the interaction are of the same size, which is a statement about structure and not about magnitude, and it is the empirical content of “maximal”.

The second is that the uncertainty on \(A\) is systematic, not statistical. Proposition 110.3 shows that \(10^{4}\) counts suffice for a per-cent statistical error, and the source easily delivered them; but \(A\) is extracted by dividing \(\varepsilon\) by \((v/c)\,f\), and both divisors carry several per cent of systematic uncertainty — \(v/c\) through the source self-absorption of Proposition 110.5 and the counter threshold, \(f\) through the nuclear-structure coefficients relating \(\varepsilon_{\gamma}\) to \(f_{2}\). Propagating the three in quadrature, \(\delta A/A=[(0.01/0.4)^{2}+(0.02/0.63)^{2} +(0.03/0.60)^{2}]^{1/2}=6.4\,\mathrm{\%}\), gives the \(A=-1.06(7)\) of Table 110.3: the statistical term contributes a quarter of the variance and the polarization term more than half. This is why the 1957 result is correctly stated as “\(\abs{A}\) consistent with the maximum” rather than as a number, and why the follow-up work with improved polarimetry was needed [Ambler:1957].

The third is that the most important entry is the one consistent with zero: with the sample warm there is nothing to orient, and the asymmetry vanishes to the statistical precision of the cold run. An experiment that reports only a large effect has reported one number; an experiment that reports a large effect together with its disappearance under a control that removes the cause has reported a measurement.

Systematic checks

The controls of Section 110.3.3 were carried out and the following is what they established.

Field reversal and solenoid sense. The asymmetry reverses sign with the polarizing field and does not depend on which of the two solenoid windings produced the field. An effect of the field on the counters would not know about the winding; an effect of the field on the nuclei is indifferent to it.

The warm sample. With the crystal above \(1\,\mathrm{K}\) the counting rates in the two field senses are equal within the counting statistics. This is the direct exclusion of a magnetic effect on the photomultiplier or on the electron trajectories, and it is why the light pipe of Section 110.2.3 was worth its optical losses: the photomultiplier is far enough away that the residual field at its photocathode is small, and the warm-sample test verifies that what remains does not matter.

Scattering. Electrons scattered in the source layer, in the crystal, or from the cryostat walls dilute the asymmetry by randomizing \(\cos\theta\). Every such process reduces \(\abs{\varepsilon}\) towards zero, so none of them can generate the observed effect; they enter as a one-sided systematic on \(\abs{A}\) and are the reason the extracted \(\abs{A}\) is a lower bound as much as a measurement.

Simultaneous decay of the two signals. The beta asymmetry and the gamma anisotropy vanish together, on the same time constant, as the crystal warms. This is the control that no instrumental explanation survives, and it is the reason the experiment is remembered as decisive rather than suggestive.

Reproduction with better polarimetry. The whole measurement was repeated and extended by the same group with improved nuclear orientation and thinner sources [Ambler:1957], which is what turned a single striking run into a reproducible number and which added the positron-emitting cases discussed in Section 110.5.2.

Interpretation

Maximal violation and the V–A structure

A measured \(A\neq0\) proves that the interaction has both a parity-even and a parity-odd part. The size of \(A\) measures their ratio, and that is the quantitative content of the cobalt result.

Write the beta-decay interaction as a sum of a parity-even piece and a parity-odd piece, with amplitudes \(a_{+}\) and \(a_{-}\). The total rate goes as \(\abs{a_{+}}^{2}+\abs{a_{-}}^{2}\), which is parity-even; the asymmetry goes as the interference \(2\,\mathrm{Re}(a_{+}a_{-}^{*})\), which is parity-odd. The asymmetry coefficient is therefore

a ratio bounded by unity, attaining \(\pm1\) if and only if \(\abs{a_{+}}=\abs{a_{-}}\) and the two are relatively real. An observed \(\abs{A}\) of order one is thus not merely evidence of parity violation but the statement that the two parts are equal in magnitude, which is a very much stronger constraint. An interaction whose parity-even and parity-odd parts are equal couples to one chirality and not at all to the other.

That is the empirical origin of the \(V-A\) form of the charged weak current, proposed by Feynman and Gell-Mann [Feynman:1958] and independently by Sudarshan and Marshak [Sudarshan:1958], in which the leptonic current appears only through the chiral projector \(P_{L}=\tfrac{1}{2}(1-\gamma^{5})\). The current is written and its consequences derived at Equation (105.34) and throughout Section 101.4; the proposition that it violates parity maximally in exactly the sense of Equation (110.15) is Proposition 105.19, and the factor \(v/c\) that appears in Equation (110.13), and which had to be divided out to reach Equation (110.14), is the chirality-to-helicity conversion of Proposition 105.20. This chapter does not repeat those derivations; what it contributes to them is the number they had to reproduce.

Charge conjugation, CP and CPT

The same data violate charge conjugation. The argument is direct: \(C\) maps a process into the process with every particle replaced by its antiparticle, leaving momenta and spins alone. Applied to the cobalt decay it maps electron emission from a polarized nucleus into positron emission from the polarized antinucleus, with the same angular distribution — so \(C\) invariance would require the two asymmetry coefficients to be equal. In the pion chain, where both charges are available, the prediction is testable directly, and the \(\pi^{+}\) and \(\pi^{-}\) chains show asymmetries of opposite sign [Garwin:1957]. In the nuclear case the corresponding statement is that a positron-emitting Gamow–Teller decay must show the opposite asymmetry to an electron-emitting one, which is what the follow-up measurements found [Ambler:1957].

What survives is the product. Under \(CP\) a left-handed neutrino maps to a right-handed antineutrino, and both exist; under \(P\) alone it would map to a right-handed neutrino, and under \(C\) alone to a left-handed antineutrino, neither of which occurs. Landau proposed immediately that \(CP\) is the true symmetry of the weak interaction [Landau:1957]. That proposal held for seven years, until the neutral-kaon measurement of Experiment: CP Violation, whose theory is developed at Section 105.3.

\(CPT\) is a different kind of statement. It is not a symmetry that the weak interaction happens to respect but a theorem: any local, Lorentz-invariant quantum field theory with a Hermitian Hamiltonian and the normal spin–statistics connection is invariant under the product of the three operations [Luders:1957]. It is proved at Theorem 105.34 and it is the reason that the violation of \(P\) and of \(C\) carries no threat to the framework, while a violation of \(CPT\) would.

Neutrino helicity

The cobalt result says the interaction distinguishes handedness but does not by itself say which handedness the neutrino has. Landau, Lee and Yang, and Salam proposed within weeks that the neutrino is described by a two-component field — the two-component neutrino belongs to The Weyl Equation and Neutrinos — rather than by a four-component Dirac field with half of it inert [Landau:1957] [Lee:1957] [Salam:1957]. A two-component field has no mass term available to it and no partner of the opposite handedness, so the proposal predicts that the neutrino is completely longitudinally polarized, and it leaves the sign of that polarization open. Goldhaber, Grodzins and Sunyar measured it [Goldhaber:1958].

Two qualifications belong with the result. The first is that helicity is a Lorentz-invariant label only for a massless particle: for a massive one an observer moving faster than the particle sees the helicity reversed, so what is measured at a given energy is the chirality of the current rather than an intrinsic property of the particle. Proposition 105.20 makes this quantitative — a \(V-A\) current emits a fermion with longitudinal polarization \(-v/c\), so a strictly massless neutrino has helicity exactly \(-1\) and a massive one falls short by \(m_{\nu}^{2}c^{4}/(2E^{2})\). The second is that neutrinos do have mass — the oscillation measurements that show it belong to Experiment: Neutrino Oscillations — but at the sub-\(\mathrm{eV}\) masses involved and the \(\mathrm{MeV}\) energies of the Goldhaber measurement, the correction is smaller than \(10^{-12}\) and the conclusion is untouched.

Immediate confirmation: the pion–muon–electron chain

Garwin, Lederman and Weinrich

The third of Lee and Yang's proposals was carried out at the Nevis cyclotron over a weekend, with apparatus that already existed, and it was submitted and published alongside the cobalt paper [Garwin:1957]. Its subject is the chain

and its logic is that the first decay produces a polarization and the second analyses it, so that a single counting-rate modulation demonstrates parity violation twice over.

The Nevis counter arrangement

A beam of positive pions from the cyclotron passes through an absorber some tens of centimetres of carbon thick. The absorber is chosen to stop the pions — which interact strongly and have a short range at the beam momentum — while transmitting the muons produced by decay in flight, which have a longer range because they do not interact strongly. What emerges is therefore a muon beam, and because the transmitted muons are those emitted forwards in the pion rest frame, they carry the full longitudinal polarization that the two-body decay gives them. The muons are brought to rest in a carbon target surrounded by a magnetizing coil producing a small, uniform, reversible field perpendicular to the muon spin. A scintillation-counter telescope in a fixed direction records positrons from the muon decays, and the counting is gated in time: a first counter signals the arrival and stopping of a muon, and positrons are accepted only within a window of a few microseconds afterwards, comparable with the muon mean life \(\tau_{\mu}=2.1970\times 10^{-6}\,\mathrm{s}\) obtained from the PDG width \(\Gamma_{\mu}=2.9959836(30)\times 10^{-19}\,\mathrm{GeV}\) through \(\tau=\hbar/\Gamma\) [Navas:2024].

The precession scan

The muon spin precesses about the applied field while the muon waits to decay, so the angle between the spin and the fixed counter is a function of the field strength and of the elapsed time. Holding the time gate fixed and varying the magnetizing current sweeps that angle, and the counting rate in the fixed telescope traces out a sinusoid. The measurement is thus a scan of counting rate against coil current, which is a robust thing to measure: the geometry never moves, and the quantity extracted is the phase and amplitude of a periodic modulation rather than the difference of two absolute rates.

The kinematics, in numbers

The two-body decay of a pion at rest fixes the muon momentum completely. With the PDG masses \(m_{\pi}c^{2}=139.57039(18)\,\mathrm{MeV}\) and \(m_{\mu}c^{2}=105.6583755(23)\,\mathrm{MeV}\) [Navas:2024],

so the muon carries a kinetic energy of \(\sqrt{p_{\mu}^{2}c^{2}+m_{\mu}^{2}c^{4}}-m_{\mu}c^{2} =4.120\,\mathrm{MeV}=6.601\times 10^{-13}\,\mathrm{J}\) and moves at \(v/c=0.2714\). The neutrino carries the same momentum, \(29.792\,\mathrm{MeV}/c\), and hence an energy of \(29.792\,\mathrm{MeV}=4.773\times 10^{-12}\,\mathrm{J}\). That the muon is non-relativistic is what makes it stoppable in a modest absorber, and that the decay is two-body is what makes the polarization complete.

The precession, in numbers

The counting rate in the fixed telescope is then

with \(a\) the product of the muon polarization and the intrinsic positron asymmetry of the muon decay. Garwin, Lederman and Weinrich found a modulation of amplitude \(\abs{a}\approx\tfrac{1}{3}\) and, from the frequency at fixed gate, a muon gyromagnetic ratio \(g_{\mu}=2.00(10)\) [Garwin:1957] — the first measurement of the muon magnetic moment, made as a by-product of a symmetry test, and the ancestor of the precision programme of Experiment: The Electron Anomalous Magnetic Moment. The intrinsic asymmetry \(\tfrac{1}{3}\) is itself a \(V-A\) prediction: the positron spectrum and its angular correlation both follow from the four-fermion matrix element of the decay, whose energy dependence alone is parametrized at Equation (101.58) in the form Michel introduced before the structure was known [Michel:1950]. Writing \(x=2E_{e}/(m_{\mu}c^{2})\) for the positron energy fraction and \(\theta\) for the angle between the positron momentum and the muon spin, the \(V-A\) doubly differential rate is

and integrating the \(\cos\theta\)-odd and \(\cos\theta\)-even parts separately over \(0\leq x\leq1\) gives, at full polarization \(P_{\mu}=1\),

which is the number Garwin, Lederman and Weinrich measured. What this book does not yet contain is the step before it.

Friedman and Telegdi

Friedman and Telegdi measured the same chain by an entirely different technique and with entirely different systematics [Friedman:1957]. Pions were brought to rest inside a nuclear emulsion; the muon from the decay, with the \(4.120\,\mathrm{MeV}\) of Equation (110.18), travels a short, visible track and stops; and the decay positron leaves a third track from the same point. Every event is therefore a complete, reconstructed geometry — pion direction, muon direction, positron direction — read off under a microscope, with no counters, no magnetic field and no timing.

The observable is the angle \(\theta\) between the muon direction and the positron direction, and the distribution is \(W(\theta)\propto1+a\cos\theta\). The measured coefficient was of magnitude \(0.174(38)\), diluted from the pion-chain maximum because a muon coming to rest in an emulsion depolarizes through hyperfine interactions with the atoms of the stopping medium — a systematic entirely absent from the counter experiment, and absent in the opposite direction, since depolarization can only reduce the asymmetry and never create it.

Table 110.4 is the summary of that short period. The five entries share no apparatus, no technique and no systematic; two are counting experiments at millikelvin temperatures, one is a microscope study of photographic emulsion, one is Mott-scattering polarimetry and one is resonant nuclear scattering through magnetized iron. What they share is the structure of Proposition 110.1, and that is why the conclusion was accepted within months.

Modern repetitions and precision

Neutral currents and the weak mixing angle

For fifteen years parity violation was a property of charged-current weak decays, in which the charge of the participating particles changes. The Gargamelle bubble chamber then observed neutrino interactions with no charged lepton in the final state [Hasert:1973b] — a weak neutral current, discussed at Section 104.6.1. Because that current couples to ordinary charged particles as well as to neutrinos, it makes parity violation observable in processes with no charge exchange at all: in the scattering of electrons from nuclei, and in atoms.

The electroweak model of Glashow, Weinberg and Salam [Glashow:1961] [Weinberg:1967] [Salam:1968] fixes the strength of every one of those effects in terms of a single parameter, the weak mixing angle. The observable that all the measurements below deliver is \(\sin^{2}\theta_{W}\), and the content of the comparison is not the value but the consistency: the same angle, running with momentum transfer according to a calculable prescription, must fit determinations spanning ten orders of magnitude in energy. The theory is Section 104.4.2 and the statement under test is Phenomenon 104.55.

Atomic parity violation

The Boulder measurement is the reference point [Wood:1997]. Its apparatus is a beam of caesium atoms crossing a region of static electric and magnetic fields inside an optical cavity, in which a laser drives the nominally forbidden \(6S\to7S\) transition; the parity-violating amplitude is detected through its interference with a Stark-induced amplitude. That is the whole trick: a rate proportional to \(\abs{\mathcal{M}_{\mathrm{Stark}}+\mathcal{M}_{PNC}}^{2}\) carries a cross term \(2\,\mathrm{Re}(\mathcal{M}_{\mathrm{Stark}}^{*} \mathcal{M}_{PNC})\) that is linear in the small amplitude and changes sign with the applied field, whereas the parity-violating rate on its own would be quadratic in it and therefore smaller by one further power of the same small ratio. The order of magnitude of \(\mathcal{M}_{Z}/\mathcal{M}_{\gamma}\) that makes \(\mathcal{M}_{PNC}\) this small is estimated, for electron scattering, at Proposition 105.21. The measured quantity is the ratio of the parity-nonconserving amplitude to the Stark coefficient,

customarily quoted as \(1.5935(56)\,\mathrm{mV}\) per centimetre; the fractional precision is \(0.35\,\mathrm{\%}\). The difference between the two hyperfine components of the transition, \(0.0077(11)\,\mathrm{V}/\mathrm{m}\), is the nuclear-spin-dependent part and constitutes the first observation of an anapole moment — a parity-odd, time-reversal-even electromagnetic moment of the nucleus, generated by the weak interaction between its own nucleons. Converting Equation (110.24) into the weak charge

requires atomic many-body theory, and the second uncertainty in Equation (110.25) is that theory's, not the experiment's.

That the theoretical uncertainty exceeds the experimental one is the structural limitation of the method, and it is what motivates the isotope-ratio technique: measuring the parity-violating amplitude in several isotopes of the same element and forming ratios cancels the electronic structure, which is common to all of them, leaving the dependence on the neutron number. Antypas and collaborators carried this out in ytterbium, whose parity-violating amplitude is some two orders of magnitude larger than caesium's, and observed the variation of the amplitude across four isotopes [Antypas:2019]. The atomic-structure input that the ratio does not cancel is the distribution of neutrons inside the nucleus, so the measurement trades one theoretical uncertainty for another — but for a nuclear one, which is measured independently by the technique of Section 110.7.4. The atomic-structure background is Atoms and Molecules.

Polarized electron scattering

SLAC E122: apparatus

The experiment that first saw this is SLAC E122 [Prescott:1978] [Prescott:1979]. Its apparatus is worth describing, because measuring an asymmetry of \(10^{-4}\) against helicity-correlated instrumental effects is the whole difficulty, and every feature of the design addresses it.

The beam is produced by photoemission from a gallium arsenide photocathode illuminated by circularly polarized laser light. The handedness of the light is set by the sign of the voltage on a Pockels cell, and the electron polarization follows it: the crystal's band structure converts optical helicity into electron spin polarization, reaching about \(37\,\mathrm{\%}\) in practice against a theoretical maximum of \(50\,\mathrm{\%}\). The essential property is that reversing the beam helicity requires nothing to move — only the sign of a voltage — so the reversal can be made randomly, pulse to pulse, and any slow drift in the apparatus averages out between neighbouring pulses rather than accumulating into a false asymmetry.

The beam energy is between \(16.2\text{–}22.2\,\mathrm{GeV}\), the target is liquid deuterium (and hydrogen), and the scattered electrons are detected in a magnetic spectrometer set to accept deep-inelastic kinematics near \(Q^{2}c^{2}\approx1.6\,\mathrm{GeV}^{2}\). The flux is far too high to count events individually, so the spectrometer is instrumented as an integrating calorimeter: the total charge deposited by each beam pulse is measured, and the asymmetry is formed between pulses of opposite helicity. That removes counting statistics as a limitation and replaces it with the requirement that the beam intensity, position, angle and energy be identical for the two helicities — monitored pulse by pulse and corrected for.

The spin-precession scan: the control that makes it a measurement

The most elegant feature of E122 is a null test that no instrumental effect can imitate, and it is worth deriving because it turns the electron's anomalous magnetic moment into a piece of experimental apparatus.

The consequence is a test with no instrumental counterpart. The measured asymmetry must reverse sign between \(16.2\,\mathrm{GeV}\) and \(19.4\,\mathrm{GeV}\), and must vanish at \(17.8\,\mathrm{GeV}\) where the spin arrives transverse and the beam carries no longitudinal polarization at all — while the beam intensity, the target, the spectrometer and the electronics are unchanged. No helicity-correlated instrumental effect depends on the beam energy through \(\cos(a_{e}\gamma\theta_{\mathrm{bend}})\). The scan was performed and the asymmetry followed the prediction [Prescott:1979], which is what converted the measurement from a small number into a result.

What E122 measured

The asymmetry, normalized to the momentum transfer, is

the second form being the SI one, since \(Q^{2}c^{2}\) is an energy squared and \(A\) is dimensionless; the conversion is \(/\mathrm{GeV}^{2} =3.8954\times 10^{19}\,/\mathrm{J}^{2}\). So \(\abs{A}\approx1.5\times 10^{-4}\) at the working point — in magnitude exactly the estimate of Equation (110.27). The electroweak interpretation is not read off the magnitude, however, but off the shape. Writing \(y=\nu/E\) for the fractional energy transfer, the prediction is

with \(a_{1}\propto1-\tfrac{20}{9}\sin^{2}\theta_{W}\) and \(a_{2}\propto1-4\sin^{2}\theta_{W}\), both quoted here from the analysis of the experiment rather than computed.

Since \(\sin^{2}\theta_{W}\approx0.23\), the second coefficient nearly vanishes, so the asymmetry is nearly independent of \(y\) — and it is that flatness, not the magnitude, that discriminates. Several models then in play reproduced the Gargamelle neutrino data with a parity-conserving neutral current, and were killed by this single observation. E122 gave \(\sin^{2}\theta_{W}=0.20(3)\) [Prescott:1978], sharpened to \(0.224(20)\) by the full kinematic scan [Prescott:1979]. The deep-inelastic kinematics and structure functions the analysis rests on belong to Experiment: Deep Inelastic Scattering.

SLAC E158

The same technique applied to a purely leptonic target removes the hadronic-structure uncertainty altogether. SLAC E158 scattered longitudinally polarized electrons from the atomic electrons of a liquid-hydrogen target — polarized Møller scattering — at \(Q^{2}c^{2}=0.026\,\mathrm{GeV}^{2}\), and measured an asymmetry of order \(10^{-7}\) — three orders of magnitude smaller than E122's \(\abs{A}\approx1.5\times 10^{-4}\), and correspondingly harder. The result, \(\sin^{2}\theta_{W}=0.2397(13)\) [Anthony:2005], is the first observation of the running of the weak mixing angle to low momentum transfer: it differs from the \(Z\)-pole value by far more than its uncertainty, in the direction and by the amount the renormalization group predicts.

The weak charge of the proton

Equation (110.31) is a tree-level relation and must not be applied to the measured numbers without saying so, because it fails on them by far more than their errors. Substituting the \(\sin^{2}\theta_{W}=0.2383(11)\) that the same measurement reports gives \(1-4(0.2383)=0.0468\), which is \(5.6\) standard deviations below the reported \(Q_{W}^{p}=0.0719(45)\). The running invoked above does not account for the gap and points the wrong way: driving \(\sin^{2}\theta_{W}\) up from its \(Z\)-pole value towards \(0.2386\) at \(Q\to0\) makes \(1-4\sin^{2}\theta_{W}\) smaller, not larger. The \(+0.025\) that reconciles the two is supplied by the one-loop electroweak corrections proper — the \(\rho\) factor and the \(WW\), \(ZZ\) and \(\gamma Z\) box graphs — which add a constant to \(Q_{W}^{p}\) without changing \(\dd Q_{W}^{p}/\dd\sin^{2}\theta_{W}=-4\). That is why the error propagation of the previous paragraph survives the corrections although the central-value relation does not: \(0.0045/4 =0.0011\) is exactly the uncertainty quoted on \(\sin^{2}\theta_{W}\) [Androic:2018].

Qweak

The measurement was made at Jefferson Lab with a beam of longitudinally polarized electrons on a liquid-hydrogen target, detecting elastically scattered electrons at \(Q=0.157\,\mathrm{GeV}/c\), that is \(Q^{2}c^{2}=0.0248\,\mathrm{GeV}^{2}\). The measured asymmetry is

in parts per billion, \(-226.5\pm9.3\), from which

in agreement with the Standard Model [Androic:2018]. Measuring \(2.3\times 10^{-7}\) requires the helicity-correlated differences in beam current, position, angle and energy to be controlled at the level of parts per billion in the asymmetry — achieved by the same random pulse-to-pulse helicity reversal as E122, supplemented by two slow reversals whose sign flips independently of the Pockels cell: inserting a half-wave plate in the laser beam, and reversing the spin with a Wien filter. Any false asymmetry tied to the electronics does not follow those reversals; the physics asymmetry does.

PREX and the neutron skin

The near-cancellation in Equation (110.31) works the other way for a heavy nucleus. Since \(Q_{W}^{p}\approx0.07\) while the neutron's weak charge is \(Q_{W}^{n}=-1\), the \(Z\) couples almost exclusively to neutrons; a parity-violating asymmetry in elastic electron scattering therefore measures the neutron distribution, where ordinary electron scattering measures the proton distribution. PREX-2 measured \(A_{PV}=550(16)(8)\times10^{-9}\) on \(^{208}\)Pb and obtained a neutron skin thickness

[Adhikari:2021]. The skin thickness is fixed by the density dependence of the nuclear symmetry energy, which is the same quantity that sets the radius of a neutron star, so a parity-violating asymmetry measured on a lead target constrains the dense-matter equation of state, which belongs to Compact Stars and Relativistic Astrophysics. The nuclear-structure background is Section 107.3.3 and Section 107.4.2.

Consistency and what remains open

Table 110.5 is the state of the art, and reading it correctly requires one caution and admits one honest complaint.

The caution is that the entries are not repeated measurements of a constant. The weak mixing angle runs with the momentum transfer at which it is probed, exactly as the fine-structure constant does, and the predicted variation between the atomic and the \(Z\)-pole entries is larger than several of the quoted uncertainties. The test is therefore not that the numbers agree but that they lie on the predicted curve; the scheme in which the angle is defined must also be stated, since different schemes differ by about a per cent, which is hundreds of times some of these errors. Remark 104.34 makes that point in the form it needs.

The complaint is a genuine tension inside the most precise entry. The \(Z\)-pole value in Table 110.5 is an average over determinations that do not agree well among themselves: the left–right asymmetry measured with polarized beams at SLD gives \(\sin^{2}\theta_{W}^{\mathrm{eff}}=0.23098(26)\) while the forward–backward asymmetry of \(b\) quarks at LEP gives \(0.23221(29)\), a discrepancy of about three standard deviations that has stood since the measurements were finished [Schael:2006]. It is stated here rather than averaged away because averaging a three-sigma disagreement and quoting the result to five digits is exactly the practice this book's editorial rules exist to prevent. No explanation has been established; the possibilities range from an underestimated systematic in one of the two to a statistical fluctuation, and the honest statement is that the world average carries an inflated error for a reason.

Two further things should be said about what parity violation is now for.

It is a null test, not a discovery channel. Every measurement in Table 110.5 is a comparison of a measured asymmetry with a Standard Model prediction, and the interest lies in the residual. A new neutral current, coupling to electrons and quarks with a strength suppressed by a mass scale \(\Lambda\), contributes a contact term of strength \((\hbar c)^{3}/\Lambda^{2}\) where the weak interaction contributes \(G_{F}\), so the relative shift of the asymmetries is of order

The combination is dimensionless and that is the point of writing it out: \(G_{F}\) has the SI dimension \(\mathrm{J}\,\mathrm{m}^{3}\), so \(G_{F}/(\hbar c)^{3}\) is an inverse energy squared, \(4.5438\times 10^{14}\,/\mathrm{J}^{2}\), and multiplying by \(\Lambda^{2}\) leaves a pure number. Numerically \(G_{F}\Lambda^{2}/(\hbar c)^{3}=11.7\) at \(\Lambda=1\,\mathrm{TeV}\) and \(1166\) at \(\Lambda=10\,\mathrm{TeV}\), so Equation (110.35) predicts relative shifts of \(8.6\times 10^{-2}\) and \(8.6\times 10^{-4}\) respectively. The present \(6\,\mathrm{\%}\) determination of \(Q_{W}^{p}\) therefore reaches \(\Lambda\approx1.2\,\mathrm{TeV}\) for a new coupling of weak strength, and correspondingly further for a stronger one — the reach is customarily quoted as a bound on \(\Lambda/g\) for that reason [Androic:2018]. Either way it is a TeV-scale statement obtained without producing anything, which is the characteristic economy of a symmetry test.

And parity is not violated in the strong and electromagnetic interactions, to the precision anyone has been able to reach. The bound is the one Lee and Yang already quoted from spectroscopy — about \(10^{-7}\) in amplitude for a parity-violating admixture in the nuclear force [Lee:1956] — and it has since been sharpened, but the qualitative statement is unchanged and it is a fact about the Standard Model rather than an assumption of it: the photon and gluon couple vectorially, to left- and right-handed fields alike, and only the \(W\) and \(Z\) do not. The parity-violating nucleon–nucleon force that does exist is the weak interaction acting inside a nucleus, and its observation — of the order the bound allows — is a measurement of the weak interaction, not a failure of the strong one.

Primary references

The measurement itself is [Wu:1957]: Wu, Ambler, Hayward, Hoppes and Hudson, “Experimental Test of Parity Conservation in Beta Decay”, Physical Review 105, 1413 (1957). It is three pages long, contains two figures and no error analysis in the modern sense, and it is the primary source for the cobalt entries of Table 110.2 and for those rows of Table 110.3 attributed to the experiment rather than derived here. A reader consulting it should expect the conclusions to be argued from the coincidence of the beta and gamma warm-up curves rather than from a quoted uncertainty; that is the form the argument took, and it is why the follow-up [Ambler:1957] — Ambler, Hayward, Hoppes, Hudson and Wu, “Further Experiments on \(\beta\) Decay of Polarized Nuclei”, same journal, 106, 1361 (1957) — matters: it repeated the measurement with better polarimetry, extended it to other nuclei, and supplied the positron-emitting case that establishes the violation of charge conjugation.

The prediction under test is [Lee:1956]: Lee and Yang, “Question of Parity Conservation in Weak Interactions”, Physical Review 104, 254 (1956). It should be read for its method rather than its result — it contains no new measurement and proposes no mechanism, and its lasting contribution is the observation that a universally believed law had never been tested in the sector where it was about to fail, together with an explicit list of feasible tests. The tau–theta puzzle that motivated it is analysed in [Dalitz:1953], whose phase-space construction is what removed the angular-momentum escape route; the introduction of parity as a conserved quantum number is [Wigner:1927].

The two independent confirmations are [Garwin:1957] — Garwin, Lederman and Weinrich, published back to back with Wu's paper — and [Friedman:1957], Friedman and Telegdi, submitted earlier and published a fortnight later in the same volume. Their systematics have almost nothing in common, which is the point of citing both. The longitudinal polarization of the beta electrons themselves, the second of Lee and Yang's three proposals, is [Frauenfelder:1957] by Mott scattering and [Schopper:1957] through the circular polarization of the accompanying gamma rays. The neutrino helicity measurement is [Goldhaber:1958]; its apparatus and analysis belong to Experiment: Neutrino Helicity, where the angular-momentum chain is worked out in full.

For the theoretical response, [Landau:1957], [Lee:1957] and [Salam:1957] propose the two-component neutrino independently and within weeks of each other; [Feynman:1958] and [Sudarshan:1958] give the \(V-A\) interaction; [Michel:1950] supplies the parametrization of the muon decay spectrum that the pion-chain experiments were interpreted against, written eight years before the structure it parametrizes was known; and [Luders:1957] is the \(CPT\) theorem, the symmetry that survives.

For the modern programme: [Bouchiat:1974] is the proposal that made atomic parity violation practical by identifying the \(Z^{3}\) enhancement, and [Wood:1997] the caesium measurement that realized it to \(0.35\,\mathrm{\%}\) together with the first anapole moment; [Antypas:2019] is the isotope-ratio method in ytterbium. [Prescott:1978] is SLAC E122 and [Prescott:1979] its kinematic scan — two distinct measurements, giving \(\sin^{2}\theta_{W}=0.20(3)\) and \(0.224(20)\) respectively, and they should be cited separately. [Anthony:2005] is E158, [Androic:2018] is Qweak and [Adhikari:2021] is PREX-2. The \(Z\)-pole compilation is [Schael:2006], which is also the source for the unresolved discrepancy between the two most precise determinations discussed in Section 110.7.5. The discovery of neutral currents that made all of this possible is [Hasert:1973b], and the model it confirmed is [Glashow:1961] [Weinberg:1967] [Salam:1968].

Numerical inputs are taken from two evaluated compilations and from nowhere else: particle masses, widths and the weak mixing angle from the Particle Data Group [Navas:2024], and the fundamental constants — \(k_{B}\), \(e\), \(\mu_{N}\), \(m_{\mu}\), \(m_{e}c^{2}\), \(a_{e}\) and \(\alpha\) — from CODATA 2022 [Mohr:2025]. Nuclear mass differences behind the \(^{60}\)Co decay energetics are from the atomic mass evaluation [Wang:2021]. Where a number in this chapter is derived rather than quoted, the derivation is given in place and the inputs are named.