The V–A Charged Weak Current

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.¹ 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 $\hslash =c=1$, the metric ${\eta }^{\mu \nu }=\mathrm{diag}(+,-,-,-)$, and the chiral projectors $P_{L,R}=\frac{1}{2}(1\mp {\gamma }^5)$ 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 $J_{\mu }^{\mathrm{EM}}{A}^{\mu }$, he wrote the amplitude as a product of two currents,

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.²

The measured value of the Fermi constant is

extracted most precisely from the muon lifetime.³ Its dimensions are the whole story of what follows. Since a fermion field has mass dimension $\frac{3}{2}$ 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.

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,

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

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.⁴

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

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

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

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 $1/M_W^4$.⁵

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

the celebrated $V-A$ (vector minus axial-vector) form. The projector $\frac{1}{2}(1-{\gamma }^5)$ 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.⁶

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.⁷

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

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

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.⁸ 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.

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,

in precise agreement with the measured value.⁹ 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 $V-A$.

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.