---
title: The V–A Charged Weak Current
module: The Weak Interaction
moduleNumber: 7
lessonNumber: 1
order: 701
summary: >
  Fermi modelled beta decay as a four-fermion contact interaction, but a coupling
  with dimensions of inverse mass squared makes cross sections grow without bound
  and the theory fails near 300 GeV. The cure is a heavy mediator: the $W$ boson,
  whose propagator collapses to Fermi's contact term at low energy and fixes
  $G_F/\sqrt2 = g^2/8M_W^2$. Parity violation dictates the current's form — vector
  minus axial-vector, coupling only to left-chiral fields — and universality of
  the coupling ties muon decay, beta decay, and pion decay to one constant. Pion
  decay's helicity suppression of the electron channel is the sharpest test.
topics: [The Weak Interaction]
draft: false
sources:
  - book: Griffiths
    ref: "Ch. 10 — Weak Interactions, §10.1–10.4"
  - book: Thomson
    ref: "Ch. 11 — The charged-current weak interaction"
  - book: Halzen & Martin
    ref: "Ch. 12 — Weak interactions"
  - book: Tong
    ref: "The Standard Model (Cambridge Part III), §5.3 Weak Decays, §5.3.4 4-Fermi Theory"
---

The weak interaction is not a force in the pushing-and-pulling sense of Coulomb or
the strong binding of quarks. It is an instrument of decay: almost every unstable
particle in the world — the neutron, the muon, the charged pion, every hadron
carrying strangeness, charm, or beauty — meets its end through it.[^tong-weak] The
theory that describes it began as Fermi's 1933 guess for beta decay and grew, under
the pressure of parity violation and its own high-energy misbehaviour, into the
chiral gauge structure at the heart of the Standard Model. This lesson follows that
growth: from a four-fermion contact term to the $V\!-\!A$ current mediated by a
massive $W$ boson, and to the two prototypes — muon decay and pion decay — that pin
the structure down.

We keep $\hbar = c = 1$, the metric $\eta^{\mu\nu} = \mathrm{diag}(+,-,-,-)$, and
the chiral projectors $P_{L,R} = \tfrac12(1 \mp \gamma^5)$ introduced with the
Dirac equation.

## Fermi's four-fermion theory

Fermi modelled neutron beta decay $n \to p\, e^-\bar\nu_e$ as a **contact
interaction**: four fermion fields meeting at a single spacetime point, with no
mediating particle. By analogy with the electromagnetic current–current coupling
$J_\mu^{\rm EM} A^\mu$, he wrote the amplitude as a product of two currents,

$$
\mathcal{M}_{\rm Fermi}
  = \frac{G_F}{\sqrt 2}\,
    \big[\bar\psi_p \gamma^\mu \psi_n\big]\,
    \big[\bar\psi_e \gamma_\mu \psi_{\nu}\big] ,
$$

with a single constant $G_F$ setting the strength. The revolutionary content is not
the algebra but the physics: the neutron is **not** made of a proton and an
electron waiting to escape. The four-fermion coupling lets one field transmute into
others — a genuinely quantum-field-theoretic idea that reorganized how the subject
thinks about particles.[^tong-fermi]

The measured value of the **Fermi constant** is

$$
G_F = 1.1663788 \times 10^{-5}\ \text{GeV}^{-2},
$$

extracted most precisely from the muon lifetime.[^pdg-gf] Its dimensions are the
whole story of what follows. Since a fermion field has mass dimension $\tfrac32$ in
four dimensions, the product of four of them has dimension $6$, so the coupling in
front must carry dimension $-2$: $[G_F] = -2$. A coupling with negative mass
dimension signals a **non-renormalizable** theory, and it has an immediate physical
consequence.

$$
% caption: Fermi's four-fermion contact interaction. Four fermion lines meet at a
% single point with no mediator; the coupling G-sub-F carries dimension minus two.
% Here a neutron transmutes into a proton, an electron, and an anti-neutrino.
\begin{tikzpicture}[>=stealth, font=\footnotesize, scale=1.0]
  \definecolor{acc}{HTML}{4A6FA5}
  \fill[acc] (0,0) circle (2.4pt);
  % incoming neutron
  \draw[->, black, thick] (-2.0,0) -- (-0.1,0);
  \node[black, left] at (-2.05,0) {neutron};
  % outgoing proton
  \draw[->, black, thick] (0.1,0.05) -- (2.0,1.1);
  \node[black, right] at (2.05,1.1) {proton};
  % outgoing electron
  \draw[->, black, thick] (0.1,0) -- (2.0,0);
  \node[black, right] at (2.05,0) {electron};
  % outgoing antineutrino
  \draw[->, black, thick] (0.1,-0.05) -- (2.0,-1.1);
  \node[black, right] at (2.05,-1.1) {anti-neutrino};
  \node[black, font=\scriptsize] at (0.0,-1.7) {single point: contact coupling};
\end{tikzpicture}
$$

## The high-energy catastrophe

Because $G_F$ has dimension $-2$, dimensional analysis fixes the size of a weak
cross section at center-of-mass energy $\sqrt s$. A cross section has dimension
$-2$ (area), the amplitude squared scales as $G_F^2$ with dimension $-4$, and the
only other scale in a high-energy process is $s$ itself. Restoring dimensions,

$$
\sigma \sim G_F^2\, s .
$$

The cross section **grows without limit** as the energy rises. But quantum
mechanics forbids this: unitarity — conservation of probability — caps a partial-wave
cross section at $\sigma \lesssim 4\pi/s$, which _falls_ with energy. The two
behaviours cross near

$$
G_F^2\, s \sim \frac{4\pi}{s}
\quad\Longrightarrow\quad
\sqrt s \sim G_F^{-1/2} \approx 300\ \text{GeV}.
$$

Above this scale Fermi's theory predicts more scattering than probability allows. It
is not a fundamental theory but an **effective** one, valid only at energies well
below a few hundred GeV. Something must soften the contact vertex.[^griffiths-unitarity]

$$
% caption: Fermi cross section grows linearly with s (rising line) and pierces the
% unitarity bound that falls as one over s (dropping curve). They meet near root-s
% of order three hundred GeV, where the contact theory must be replaced by a
% massive mediator.
\begin{tikzpicture}[>=stealth, font=\footnotesize, scale=1.0]
  \definecolor{acc}{HTML}{4A6FA5}
  \draw[->, black] (0,0) -- (5.4,0) node[right, font=\scriptsize] {energy};
  \draw[->, black] (0,0) -- (0,3.4) node[above, font=\scriptsize] {cross section};
  % Fermi rising
  \draw[black, very thick] (0.2,0.15) -- (4.6,3.0);
  \node[black, right, font=\scriptsize] at (4.0,2.55) {Fermi};
  % unitarity falling (1/s), Bezier
  \draw[black, very thick, dashed] (0.5,3.1) .. controls (1.6,0.9) and (2.6,0.55) .. (5.0,0.35);
  \node[black, right, font=\scriptsize] at (3.1,0.75) {unitarity bound};
  % crossing marker
  \fill[acc] (2.35,1.55) circle (2.2pt);
  \draw[black, densely dotted] (2.35,1.55) -- (2.35,0) node[below, font=\scriptsize] {near 300 GeV};
\end{tikzpicture}
$$

## The W propagator softens the vertex

The missing ingredient is a mediator. The weak interaction is carried by the massive
$W^\pm$ boson, and Fermi's contact vertex is what a $W$ exchange _looks like_ when
the momentum transfer is far below the $W$ mass. A massive spin-1 propagator carries
momentum $q$ as

$$
D_{\mu\nu}(q) = \frac{-i\left(\eta_{\mu\nu} - q_\mu q_\nu/M_W^2\right)}
                     {q^2 - M_W^2} .
$$

At low momentum transfer, $q^2 \ll M_W^2$, the denominator is dominated by $M_W^2$
and the numerator's $q_\mu q_\nu$ term is negligible against $\eta_{\mu\nu}$, so

$$
D_{\mu\nu}(q) \;\xrightarrow{\,q^2 \ll M_W^2\,}\;
  \frac{i\,\eta_{\mu\nu}}{M_W^2} .
$$

The propagator collapses to a constant — in position space, to a delta function —
which reproduces the contact interaction Fermi wrote by hand. Matching the two
descriptions relates his constant to the gauge coupling $g$ and the $W$ mass. Each
weak vertex carries a factor $g/(2\sqrt 2)$, and stitching two vertices to the
low-energy propagator gives

$$
\frac{G_F}{\sqrt 2} = \frac{g^2}{8\,M_W^2} .
$$

This single relation dissolves the high-energy catastrophe: the growth of the
amplitude is cut off once $\sqrt s$ reaches $M_W$, where the full propagator
$\sim 1/(q^2 - M_W^2)$ takes over and the cross section turns over instead of
diverging. The "weakness" of the weak interaction is not a small intrinsic coupling
— $g \approx 0.65$ is comparable to the electric charge — but the large mass
$M_W \approx 80$ GeV in the denominator, which suppresses low-energy rates by
$1/M_W^4$.[^tong-4fermi]

$$
% caption: The same process two ways. On the left the W boson is exchanged with a
% propagator that depends on momentum transfer. On the right, at energies far below
% the W mass, the propagator shrinks to a point and reproduces Fermi's contact
% vertex. Matching fixes G-sub-F in terms of g and the W mass.
\begin{tikzpicture}[>=stealth, font=\footnotesize, scale=1.0]
  \definecolor{acc}{HTML}{4A6FA5}
  % left: W exchange
  \draw[->, black, thick] (-4.3,1.1) -- (-3.3,0.4);
  \draw[->, black, thick] (-3.3,0.4) -- (-2.3,1.1);
  \draw[acc, very thick, variable=\y] plot[domain=-0.7:0.4, samples=40] ({-3.3 + 0.08*sin((\y+0.7)/1.1*1440)}, \y);
  \node[acc, right, font=\scriptsize] at (-3.15,-0.15) {W};
  \draw[->, black, thick] (-4.3,-1.4) -- (-3.3,-0.7);
  \draw[->, black, thick] (-3.3,-0.7) -- (-2.3,-1.4);
  \node[font=\scriptsize] at (-3.3,-2.1) {W exchange};
  % arrow between
  \draw[->, black, thick] (-1.7,-0.15) -- (-0.3,-0.15);
  \node[black, above, font=\scriptsize] at (-1.0,-0.1) {low energy};
  % right: contact
  \fill[acc] (1.6,-0.15) circle (2.4pt);
  \draw[->, black, thick] (0.6,1.1) -- (1.55,-0.1);
  \draw[->, black, thick] (1.65,-0.1) -- (2.6,1.1);
  \draw[->, black, thick] (0.6,-1.4) -- (1.55,-0.2);
  \draw[->, black, thick] (1.65,-0.2) -- (2.6,-1.4);
  \node[font=\scriptsize] at (1.6,-2.1) {contact vertex};
\end{tikzpicture}
$$

## Vector minus axial-vector

Fermi wrote a pure vector current $\bar\psi\gamma^\mu\psi$ because he had no reason
to do otherwise; parity was assumed exact. The discovery that the weak interaction
violates parity maximally forces a different Lorentz structure. Under parity the
vector current $\bar\psi\gamma^\mu\psi$ and the axial-vector current
$\bar\psi\gamma^\mu\gamma^5\psi$ transform with opposite signs on their spatial
parts, so a current built from their **difference** treats left and right
asymmetrically. The charged weak current is

$$
J^\mu = \bar\psi\,\gamma^\mu\tfrac12\!\left(1-\gamma^5\right)\psi
      = \bar\psi_L\,\gamma^\mu\,\psi_L ,
$$

the celebrated $V\!-\!A$ (vector minus axial-vector) form. The projector
$\tfrac12(1-\gamma^5)$ keeps only the **left-chiral** part of every field: the $W$
boson couples to left-handed particles and right-handed antiparticles, and to
nothing else. This is the algebraic expression of the parity violation the Wu
experiment revealed — right-handed particles simply do not appear in the charged
current.[^thomson-va]

$$
% caption: The V minus A projector. A Dirac field is split by one-half times one
% minus gamma-five; only the left-chiral half survives to couple to the W boson,
% while the right-chiral half is inert under the charged weak force.
\begin{tikzpicture}[>=stealth, font=\footnotesize, scale=1.0]
  \definecolor{acc}{HTML}{4A6FA5}
  \node[draw=black, thick, align=center, inner sep=5pt] (f) at (0,0) {fermion f\/ield};
  \node[draw=black, thick, align=center, inner sep=5pt] (L) at (3.6,1.1)
        {left-chiral part};
  \node[draw=black, thick, align=center, inner sep=5pt] (R) at (3.6,-1.1)
        {right-chiral part};
  \draw[->, black, thick] (f) -- (L) node[midway, above, font=\scriptsize] {half of one minus g5};
  \draw[->, black, thick] (f) -- (R) node[midway, below, font=\scriptsize, black] {half of one plus g5};
  \draw[acc, very thick] plot[domain=5.5:6.6, samples=40] (\x, {1.1 + 0.07*sin((\x-5.5)/1.1*1440)});
  \node[acc, right, font=\scriptsize] at (6.65,1.1) {couples to W};
  \node[black, right, font=\scriptsize] at (5.5,-1.1) {no W coupling};
\end{tikzpicture}
$$

## Universality of the weak coupling

The same coupling $g$ appears at every charged-current vertex, whatever the
generation: the $W$ couples to $(\nu_e, e)$, $(\nu_\mu, \mu)$, $(\nu_\tau, \tau)$
and — up to the quark-mixing rotation of a later lesson — to $(u,d)$, $(c,s)$,
$(t,b)$ with one universal strength. This **lepton universality** is why the value
of $G_F$ extracted from muon decay reproduces the strength seen in nuclear beta
decay and in tau decay: they are the same interaction wearing different masses. The
tiny residual differences are accounted for by phase space and, for quarks, by the
mixing angles; the underlying coupling is one number.[^thomson-universality]

$$
% caption: Universality. The W boson couples to each lepton doublet with the same
% strength g, pairing a charged lepton with its neutrino. Muon decay, tau decay,
% and nuclear beta decay all measure the one coupling.
\begin{tikzpicture}[>=stealth, font=\footnotesize, scale=1.0]
  \definecolor{acc}{HTML}{4A6FA5}
  \foreach \x/\lep/\neu in {0/electron/{nu-e}, 3.0/muon/{nu-mu}, 6.0/tau/{nu-tau}} {
    \draw[thick] (\x-0.9,-0.6) rectangle (\x+0.9,1.0);
    \node[font=\scriptsize] at (\x,0.6) {\neu};
    \node[font=\scriptsize] at (\x,0.0) {\lep};
    \draw[<->, black, thick] (\x-0.45,0.55) -- (\x-0.45,0.05);
    \node[black, font=\scriptsize, left] at (\x-0.5,0.3) {W};
  }
  \node[black, font=\scriptsize] at (3.0,-1.2) {same coupling g in every generation};
\end{tikzpicture}
$$

## Muon decay as the reference process

Muon decay $\mu^- \to e^-\bar\nu_e\nu_\mu$ is the cleanest weak process in nature:
no hadrons, no strong-interaction corrections, purely leptonic. At the fundamental
level a $W$ boson is exchanged; at muon energies the contact approximation applies,
and the amplitude is the current–current product

$$
\mathcal{M} = \frac{G_F}{\sqrt 2}\,
  \big[\bar u_{\nu_\mu}\gamma^\mu(1-\gamma^5)u_\mu\big]\,
  \big[\bar u_e\gamma_\mu(1-\gamma^5)v_{\nu_e}\big] .
$$

Squaring, summing over spins, and integrating over the three-body phase space gives
the decay rate

$$
\Gamma = \frac{G_F^2\, m_\mu^5}{192\,\pi^3} ,
$$

neglecting the electron mass. The **fifth power** of the muon mass is the
characteristic phase-space signature of a three-body weak decay, and the formula run
in reverse is precisely how $G_F$ is measured: the muon lifetime
$\tau_\mu = 2.1969811 \times 10^{-6}$ s is one of the most precisely known
quantities in particle physics.[^pdg-muon] Because $\Gamma \propto m^5$, the far
heavier tau decays about $10^6$ times faster despite the identical coupling — mass,
not coupling, controls the rate.

$$
% caption: Muon decay at the fundamental level. The muon emits a W boson and turns
% into its neutrino; the W materializes an electron and an electron anti-neutrino.
% The rate scales as the fifth power of the muon mass.
\begin{tikzpicture}[>=stealth, font=\footnotesize, scale=1.0]
  \definecolor{acc}{HTML}{4A6FA5}
  % muon in
  \draw[->, black, thick] (-3.0,1.2) -- (-1.0,1.2);
  \node[black, left] at (-3.05,1.2) {muon};
  % muon-neutrino out
  \draw[->, black, thick] (-1.0,1.2) -- (0.9,1.9);
  \node[black, right] at (0.95,1.9) {nu-mu};
  % W propagator (vertical wave)
  \draw[acc, very thick, variable=\y] plot[domain=-0.4:1.2, samples=48] ({-1.0 + 0.08*sin((\y+0.4)/1.6*1620)}, \y);
  \node[acc, left, font=\scriptsize] at (-1.15,0.4) {W};
  % electron out
  \draw[->, black, thick] (-1.0,-0.4) -- (0.9,0.1);
  \node[black, right] at (0.95,0.1) {electron};
  % electron antineutrino out
  \draw[->, black, thick] (-1.0,-0.4) -- (0.9,-1.0);
  \node[black, right] at (0.95,-1.0) {anti-nu-e};
\end{tikzpicture}
$$

## Helicity suppression in pion decay

The most striking confirmation of the $V\!-\!A$ structure is a _ratio_ the theory
gets spectacularly right. The charged pion decays overwhelmingly to a muon,
$\pi^-\to\mu^-\bar\nu_\mu$, and only rarely to an electron,
$\pi^-\to e^-\bar\nu_e$ — even though the electron channel has vastly more phase
space (the electron is far lighter). The naive expectation is exactly backwards. The
resolution is **helicity suppression**.

The pion is spinless, so in its rest frame the outgoing charged lepton and
antineutrino must have opposite spins to conserve angular momentum, and being
back-to-back, this means they share the same helicity. The $V\!-\!A$ interaction
demands a right-handed antineutrino. Angular momentum then forces the charged lepton
into its right-handed helicity too — but $V\!-\!A$ _wants_ the charged lepton
left-handed. A massless lepton could not comply at all and the rate would vanish; a
massive lepton has a wrong-helicity amplitude of order $m_\ell/E_\ell$, so the rate
is suppressed by $m_\ell^2$. The electron, being lighter, is suppressed far more
than the muon. Working the kinematics through,

$$
\frac{\Gamma(\pi\to e\nu)}{\Gamma(\pi\to\mu\nu)}
  = \frac{m_e^2}{m_\mu^2}
    \left(\frac{m_\pi^2 - m_e^2}{m_\pi^2 - m_\mu^2}\right)^{\!2}
  \approx 1.28 \times 10^{-4} ,
$$

in precise agreement with the measured value.[^pdg-pion] The electron mode is
suppressed by four orders of magnitude _because_ the interaction is left-chiral. No
other Lorentz structure reproduces this number; scalar or tensor couplings give the
opposite hierarchy. Helicity suppression is the cleanest fingerprint that the
charged weak current is $V\!-\!A$.

$$
% caption: Helicity suppression. The spinless pion decays to a charged lepton and
% an anti-neutrino emitted back to back with equal helicity. V minus A forces the
% anti-neutrino right-handed, so the charged lepton is pushed into its wrong
% helicity; the mismatch amplitude is of order lepton mass over energy, suppressing
% the light electron channel far more than the muon.
\begin{tikzpicture}[>=stealth, font=\footnotesize, scale=1.0]
  \definecolor{acc}{HTML}{4A6FA5}
  % pion at rest
  \draw[thick] (-0.5,-0.4) rectangle (0.5,0.4);
  \node[font=\scriptsize] at (0,0) {pion};
  \node[black, font=\scriptsize, below] at (0,-0.5) {spin zero, at rest};
  % lepton going left
  \draw[->, black, very thick] (-0.7,1.3) -- (-2.7,1.3);
  \node[black, above, font=\scriptsize] at (-1.7,1.35) {charged lepton};
  \draw[->, black, thick] (-1.7,1.55) arc (-90:180:0.2);
  \node[black, font=\scriptsize, left] at (-2.75,1.75) {spin};
  % antineutrino going right
  \draw[->, black, very thick] (0.7,1.3) -- (2.7,1.3);
  \node[black, above, font=\scriptsize] at (1.7,1.35) {anti-neutrino};
  \draw[->, black, thick] (1.7,1.55) arc (-90:180:0.2);
  \node[black, font=\scriptsize, right] at (2.75,1.75) {spin};
  \node[black, font=\scriptsize] at (0,2.5) {equal helicity forced by angular momentum};
\end{tikzpicture}
$$

## Summary

Fermi's four-fermion contact interaction, with a coupling $G_F$ of mass dimension
$-2$, describes low-energy weak decays but predicts cross sections growing as $G_F^2
s$ that violate unitarity near $300$ GeV. The cure is a massive mediator: the $W$
boson, whose propagator degenerates to a contact term for $q^2 \ll M_W^2$ and fixes
$G_F/\sqrt 2 = g^2/8M_W^2$, so weakness is really the heavy $W$ in the denominator,
not a small coupling. Parity violation dictates the $V\!-\!A$ Lorentz structure: the
current $\bar\psi_L\gamma^\mu\psi_L$ couples only to left-chiral fields, with one
universal coupling across generations. Muon decay, with $\Gamma \propto G_F^2
m_\mu^5$, calibrates $G_F$, and the four-order-of-magnitude helicity suppression of
$\pi\to e\nu$ against $\pi\to\mu\nu$ confirms the chiral structure directly. The next
lesson gives the $W$ and $Z$ bosons their own dynamics and confronts the theory with
their discovery.

[^tong-weak]: Tong, _The Standard Model_ (Cambridge Part III), §5.3 — the weak force as "an instrument of decay," the source of instability for every hadron but the proton. [damtp.cam.ac.uk/user/tong/standardmodel.html](http://www.damtp.cam.ac.uk/user/tong/standardmodel.html)
[^tong-fermi]: Tong, _The Standard Model_ (Cambridge Part III), §5.3.4 — Fermi's 1933 four-fermion theory and the field-theoretic reinterpretation that a neutron is not a bound state of its decay products.
[^pdg-gf]: The Fermi constant and its extraction from the muon lifetime are tabulated by the Particle Data Group, _Review of Particle Physics_, [pdg.lbl.gov](https://pdg.lbl.gov).
[^griffiths-unitarity]: Griffiths, _Introduction to Elementary Particles_, 2nd ed., §10.1–10.2 — the four-fermion coupling, its dimension, and the unitarity problem that demands a mediator.
[^tong-4fermi]: Tong, _The Standard Model_ (Cambridge Part III), §5.3.4 — the massive-vector propagator collapsing to a delta function at low energy and the matching $G_F/\sqrt2 = g^2/8M_W^2$; Griffiths, §10.3, gives the same relation.
[^thomson-va]: Thomson, _Modern Particle Physics_, Ch. 11 — the $V\!-\!A$ charged current, the $\tfrac12(1-\gamma^5)$ projector, and its origin in parity violation.
[^thomson-universality]: Thomson, _Modern Particle Physics_, Ch. 11 — lepton universality of the charged-current coupling; Halzen & Martin, Ch. 12, connect muon and nuclear beta decay through the one constant.
[^pdg-muon]: The muon lifetime $\tau_\mu = 2.1969811(22)\times10^{-6}$ s and the $\Gamma = G_F^2 m_\mu^5/192\pi^3$ relation are given by the Particle Data Group, [pdg.lbl.gov](https://pdg.lbl.gov).
[^pdg-pion]: The measured $\pi\to e\nu / \pi\to\mu\nu$ branching ratio and the helicity-suppression prediction are in the Particle Data Group meson listings, [pdg.lbl.gov](https://pdg.lbl.gov); the derivation is in Griffiths §10.4 and Thomson Ch. 11.