The Weak Interaction and Parity Violation

Fermi's theory reproduces the beta spectrum and the $ft$ systematics without committing to the detailed form of the weak current. A single symmetry test settled that form: whether beta decay looks the same in a mirror. It does not. The weak interaction is the only one of the four that distinguishes left from right, and the experiments that established this — Wu's polarized cobalt and Goldhaber's neutrino-helicity measurement — fix the charged weak current as a left-handed $V-A$ structure and set beta decay inside the electroweak theory.¹

The parity operation

Parity $P$ inverts the spatial coordinates through the origin, $\vec{r}\to -\vec{r}$. Physical quantities sort by how they transform.

• Polar (true) vectors reverse: position $\vec{r}$, momentum $\vec{p}$, electric field $\vec{E}$. Under $P$, $\vec{p}\to -\vec{p}$.
• Axial (pseudo) vectors are unchanged: angular momentum $\vec{L}=\vec{r}\times \vec{p}$, spin $\vec{S}$, magnetic field $\vec{B}$. Under $P$, $\vec{S}\to +\vec{S}$.
• Scalars (like $\vec{p}\cdot \vec{p}$) are invariant; pseudoscalars (a dot product of a polar and an axial vector, like $\vec{S}\cdot \vec{p}$) change sign.

If a decay conserves parity, no observable pseudoscalar can have a nonzero expectation value, because the mirror-image process must occur at the same rate. The cleanest pseudoscalar in beta decay is the correlation between the nuclear spin $\vec{I}$ and the electron momentum ${\vec{p}}_e$,

A nonzero value means electrons prefer one hemisphere relative to the spin axis, which the mirror image reverses. Observing it is a direct proof that parity is not conserved. Lee and Yang pointed out in 1956 that, unlike for the strong and electromagnetic interactions, no experiment had ever tested parity conservation in the weak interaction, and they proposed the polarized-nucleus measurement.

The Wu experiment

Wu and collaborators tested the prediction in 1957 using ${}^{60}\mathrm{Co}$, which ${\beta }^{-}$-decays to ${}^{60}\mathrm{Ni}$ with a spin change from $I=5$ to $I=4$, a pure Gamow-Teller transition. Two conditions were needed.

• Nuclear polarization. The cobalt was incorporated into a paramagnetic salt, cerium magnesium nitrate, and cooled to about $0.01\mathrm{K}$ by adiabatic demagnetization. At that temperature an applied field aligns the nuclear spins, giving a net $⟨\vec{I}⟩$ along the field.
• Asymmetry detection. An anthracene scintillator counted the beta electrons emitted along and against the field direction, while the degree of nuclear alignment was monitored through the anisotropy of the ${}^{60}\mathrm{Ni}$ gamma rays.

The electrons emerged preferentially opposite to the nuclear spin. As the sample warmed and the polarization relaxed over a few minutes, the asymmetry disappeared in lockstep with the gamma anisotropy, ruling out any instrumental artifact. The angular distribution of electrons is

where $\theta$ is measured from the nuclear spin, $v/c$ is the electron speed, and the negative $\alpha$ encodes the backward preference. The magnitude $|\alpha |$ tracks the polarization: full alignment gives the maximal asymmetry, zero alignment gives an isotropic distribution.

The observed asymmetry is close to maximal, $\alpha =-1$: parity is not merely violated but violated maximally in beta decay.

The V minus A current

Maximal parity violation identifies the Lorentz structure of the weak charged current. A general four-fermion coupling could mix scalar, vector, tensor, axial, and pseudoscalar terms; the electron and neutrino observables single out the combination

the $V-A$ (vector minus axial-vector) current. The operator $\frac{1}{2}(1-{\gamma }^5)$ is the left-handed projection: it keeps only the left-handed chirality component of the fermion field. Two consequences follow.

• Only left-handed particles and right-handed antiparticles participate. For a massless fermion, chirality equals helicity, so the neutrino emitted in beta decay is left-handed and the antineutrino right-handed.
• The electron carries longitudinal polarization $-v/c$: relativistic ${\beta }^{-}$ electrons are preferentially left-handed, and ${\beta }^{+}$ positrons right-handed. This polarization was measured and matches $V-A$.

The vector part is the Fermi coupling $g_V$ and the axial part the Gamow-Teller coupling $g_A$ of the previous lesson; their ratio $g_A/g_V=-1.27$ is the imprint of the axial current, modified from the point-particle value $-1$ by the strong-interaction structure of the nucleon.

Neutrino helicity

The helicity of the neutrino itself — not inferred from the electron but measured directly — was settled by Goldhaber, Grodzins, and Sunyar in 1958. They used the electron-capture decay ${}^{152}\mathrm{Eu}+e^{-}\to {}^{152}{\mathrm{Sm}}^{\ast }+{\nu }_e$, in which the excited samarium promptly emits a gamma ray. Because capture is a two-body decay, the neutrino and the recoiling ${}^{152}{\mathrm{Sm}}^{\ast }$ go back-to-back, so the nuclear recoil direction tags the neutrino direction. The gamma emitted along the recoil axis inherits the neutrino's helicity as a circular polarization, measured by resonant scattering through magnetized iron. The result: the neutrino is left-handed, helicity $-1$, within experimental error.

Beta decay in the electroweak theory

The point contact of Fermi theory is the low-energy limit of the exchange of a massive charged vector boson, the $W^{\pm }$. At the quark level ${\beta }^{-}$ decay is a down quark turning into an up quark with emission of a $W^{-}$, which materializes as the electron-antineutrino pair,

Since the neutron is $udd$ and the proton $uud$, this single quark transition is exactly $n\to p+e^{-}+{\bar{\nu }}_e$. The $W$ is heavy, $M_W{c}^2=80.4\mathrm{GeV}$, so at the few-MeV scale of nuclear beta decay its propagator is effectively a contact term, and the Fermi constant is the boson coupling divided by the mass squared,

The huge $M_W$ is why the weak interaction is weak at low energy: not a small coupling ($g_w$ is comparable to the electromagnetic charge) but a heavy mediator. Electroweak unification ties $g_w$ to the electric charge through the weak mixing angle, ${\sin }^2{\theta }_W\approx 0.231$, placing beta decay and electromagnetism in a single gauge theory.² The left-handed structure of the current is built in from the start: only left-handed fermion doublets couple to the $W$.

The place of beta decay is now clear: it is the nuclear face of a fundamental force, the same $W$ exchange that governs muon decay and, through its neutral partner the $Z$, the neutrino scattering that first revealed the weak neutral current. The neutrino sector — masses, mixing, and whether the neutrino is its own antiparticle — is taken up through double beta decay.