---
title: The Weak Interaction and Parity Violation
module: Beta Decay and the Weak Interaction
moduleNumber: 6
lessonNumber: 3
order: 603
summary: >
  Beta decay violates mirror symmetry. The Wu experiment on polarized cobalt-60
  showed electrons emitted preferentially against the nuclear spin, a
  pseudoscalar correlation forbidden if parity were conserved. The result fixes
  the weak charged current as left-handed V minus A, forces neutrinos to be
  left-handed and antineutrinos right-handed (measured by Goldhaber), and places
  beta decay within the electroweak theory as W-boson exchange turning a down
  quark into an up quark.
topics: [Beta Decay and the Weak Interaction]
draft: false
sources:
  - book: Krane
    ref: "Ch. 9 — Beta Decay; §9.4 Angular Momentum and Parity in Beta Decay (Nonconservation of Parity), §9.9 Neutrino Physics"
  - book: Povh
    ref: "Particles and Nuclei; electroweak unification and the charged weak current"
---

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.[^krane-parity]

## 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 $\vec p_e$,

$$
\langle \vec I \cdot \vec p_e \rangle.
$$

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.

$$
% caption: Under the parity operation the electron momentum reverses while the
% nuclear spin does not; if the decay rate along and against the spin differ, the
% mirrored process is distinguishable and parity is violated.
\begin{tikzpicture}[>=stealth, font=\footnotesize]
  \definecolor{acc}{HTML}{4A6FA5}
  % ---- real world ----
  \begin{scope}
    \node[black!70, anchor=south, font=\scriptsize] at (1.1,3.0) {nucleus};
    \draw[black, thick] (1.1,1.5) circle (7pt);
    % spin arrow (up)
    \draw[->, acc, very thick] (1.1,1.8) -- (1.1,2.7);
    \node[acc, anchor=west, font=\scriptsize] at (1.2,2.4) {spin};
    % electron preferentially down
    \draw[->, black!70, very thick] (1.1,1.2) -- (1.1,0.2);
    \node[black!70, anchor=west, font=\scriptsize] at (1.2,0.5) {electron};
  \end{scope}
  % mirror line
  \draw[black, dashed] (3.0,0) -- (3.0,3.2);
  \node[black, anchor=south, font=\scriptsize] at (3.0,3.2) {mirror};
  % ---- mirror world ----
  \begin{scope}[xshift=4.0cm]
    \node[black!70, anchor=south, font=\scriptsize] at (1.1,3.0) {mirror image};
    \draw[black, thick] (1.1,1.5) circle (7pt);
    % spin unchanged (still up)
    \draw[->, acc, very thick] (1.1,1.8) -- (1.1,2.7);
    \node[acc, anchor=west, font=\scriptsize] at (1.2,2.4) {spin};
    % electron now up (momentum reversed)
    \draw[->, black!70, very thick] (1.1,1.2) -- (1.1,2.2);
    \node[black!70, anchor=west, font=\scriptsize] at (1.2,1.9) {electron};
  \end{scope}
\end{tikzpicture}
$$

## 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 $\langle \vec I \rangle$ 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

$$
W(\theta) = 1 + \alpha\,\frac{v}{c}\,\cos\theta,
\qquad
\alpha = -\frac{\langle I_z\rangle}{I},
$$

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

$$
% caption: The measured forward-backward electron asymmetry grows linearly with
% the nuclear polarization and collapses to zero as the sample warms and the
% spins randomize, the hallmark of a genuine parity-violating correlation.
\begin{tikzpicture}[>=stealth, font=\footnotesize]
  \definecolor{acc}{HTML}{4A6FA5}
  \draw[->, black] (0,0) -- (6.4,0) node[right, black!70] {polarization};
  \draw[->, black] (0,0) -- (0,4.0)
    node[above, black!70, align=center, font=\scriptsize] {asymmetry};
  % linear rise
  \draw[acc, very thick] (0.2,0.2) -- (5.6,3.6);
  \node[acc, anchor=north west, font=\scriptsize] at (3.4,2.0) {aligned};
  % data dots
  \foreach \p in {(1.0,0.7),(2.1,1.4),(3.2,2.1),(4.3,2.8),(5.2,3.35)}
    \fill[black!70] \p circle (1.6pt);
  % zero at origin emphasized
  \draw[black, dashed] (0.2,0.2) -- (0.2,0);
  \node[black, anchor=north, font=\scriptsize] at (0.5,0) {random};
\end{tikzpicture}
$$

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

$$
j^\mu = \bar\psi\,\gamma^\mu (1 - \gamma^5)\,\psi,
$$

the **$V - A$** (vector minus axial-vector) current. The operator
$\tfrac{1}{2}(1 - \gamma^5)$ 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](/nuclear-physics/beta-decay/fermi-theory-beta-decay);
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.

> **Definition (Helicity).** The helicity $h = \vec S \cdot \hat p$ is the
> projection of spin onto the momentum direction, $h = +1$ for spin parallel to
> motion (right-handed) and $h = -1$ for antiparallel (left-handed).

$$
% caption: The weak interaction produces only left-handed neutrinos (spin
% antiparallel to momentum) and right-handed antineutrinos (spin parallel); the
% opposite helicities are never emitted.
\begin{tikzpicture}[>=stealth, font=\footnotesize]
  \definecolor{acc}{HTML}{4A6FA5}
  % ---- neutrino: left-handed ----
  \begin{scope}
    \node[black!70, anchor=south, font=\scriptsize] at (1.6,2.3) {neutrino};
    % momentum to the right
    \draw[->, black!70, very thick] (0.2,1.4) -- (3.0,1.4);
    \node[black!70, anchor=north, font=\scriptsize] at (2.7,1.35) {momentum};
    % spin to the left (antiparallel)
    \draw[->, black!70, very thick] (1.9,1.9) -- (0.7,1.9);
    \node[black!70, anchor=south, font=\scriptsize] at (1.3,1.9) {spin};
    \node[black!70, anchor=north, font=\scriptsize] at (1.6,1.0) {left handed};
  \end{scope}
  % ---- antineutrino: right-handed ----
  \begin{scope}[xshift=4.6cm]
    \node[black!70, anchor=south, font=\scriptsize] at (1.6,2.3) {antineutrino};
    \draw[->, black!70, very thick] (0.2,1.4) -- (3.0,1.4);
    \node[black!70, anchor=north, font=\scriptsize] at (2.7,1.35) {momentum};
    % spin to the right (parallel)
    \draw[->, black!70, very thick] (0.7,1.9) -- (1.9,1.9);
    \node[black!70, anchor=south, font=\scriptsize] at (1.3,1.9) {spin};
    \node[black!70, anchor=north, font=\scriptsize] at (1.6,1.0) {right handed};
  \end{scope}
\end{tikzpicture}
$$

## 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,

$$
d \to u + W^-,\qquad W^- \to e^- + \bar\nu_e.
$$

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,

$$
\frac{G_F}{\sqrt 2} = \frac{g_w^2}{8\,(M_W c^2)^2}.
$$

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.[^pdg-ew] The left-handed structure of the current is built
in from the start: only left-handed fermion doublets couple to the $W$.

$$
% caption: Beta-minus decay as W-boson exchange at the quark level: one down
% quark in the neutron becomes an up quark, emitting a virtual W that decays to
% the electron and antineutrino, converting the neutron udd into the proton uud.
\begin{tikzpicture}[>=stealth, font=\footnotesize]
  \definecolor{acc}{HTML}{4A6FA5}
  % neutron quark lines (left) to proton (right)
  \draw[black, thick] (0.2,3.4) -- (4.6,3.4);
  \node[black!70, anchor=east, font=\scriptsize] at (0.15,3.4) {u};
  \draw[black, thick] (0.2,2.8) -- (4.6,2.8);
  \node[black!70, anchor=east, font=\scriptsize] at (0.15,2.8) {d};
  % the transforming quark: d -> u with a kink at the vertex
  \draw[black, thick] (0.2,2.2) -- (2.2,2.2);
  \node[black!70, anchor=east, font=\scriptsize] at (0.15,2.2) {d};
  \draw[black, thick] (2.2,2.2) -- (4.6,2.5);
  \node[black!70, anchor=west, font=\scriptsize] at (4.65,2.5) {u};
  % labels neutron/proton
  \node[black, anchor=north, font=\scriptsize] at (0.6,2.0) {neutron};
  \node[black, anchor=north, font=\scriptsize] at (4.2,2.2) {proton};
  % W boson (dashed) from vertex down to lepton vertex
  \draw[acc, very thick, dashed] (2.2,2.2) -- (3.2,1.1);
  \node[acc, anchor=west, font=\scriptsize] at (2.8,1.7) {W boson};
  % leptons out
  \draw[->, black!70, thick] (3.2,1.1) -- (4.6,1.5);
  \node[black!70, anchor=west, font=\scriptsize] at (4.65,1.5) {electron};
  \draw[->, black!70, thick] (3.2,1.1) -- (4.6,0.5);
  \node[black!70, anchor=west, font=\scriptsize] at (4.65,0.5) {antineutrino};
\end{tikzpicture}
$$

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](/nuclear-physics/beta-decay/double-beta-decay-neutrino-mass).

[^krane-parity]: **Krane**, _Introductory Nuclear Physics_, §9.4 (nonconservation of parity, the Wu experiment, the $V-A$ interaction and electron polarization) and §9.9 (neutrino helicity, the Goldhaber experiment). The maximal-asymmetry result $\alpha \approx -1$ and the $^{60}\mathrm{Co}$ transition assignment follow Krane's account.
[^pdg-ew]: **Particle Data Group**, _Review of Particle Physics_, "Electroweak Model and Constraints on New Physics": the $W$ mass, the Fermi constant relation $G_F/\sqrt 2 = g_w^2/8M_W^2$, and the weak mixing angle $\sin^2\theta_W$, [pdg.lbl.gov](https://pdg.lbl.gov/).