The weak charged current

The weak charged current is now measured with a precision that would have astonished Fermi. The muon lifetime — this article's worked example — has been pinned to a part in a million by the MuLan experiment at PSI, yielding $\tau_\mu=2.1969811(22)\,\mu$s and hence the world's best Fermi constant, $G_F/(\hbar c)^3=1.1663787(6)\times10^{-5}\,\mathrm{GeV}^{-2}$; that single number then propagates into every electroweak prediction. At the high-energy end, where the propagator no longer collapses, the $W$ is produced by the millions at the LHC through the very quark-level charged current $u\bar d\to W^+$ discussed here, and the transverse-mass endpoint of its leptonic decays $W\to\ell\nu$ measures $M_W$ directly: the world average sits near $80.37$ GeV, with the 2022 CDF result ($80.434$ GeV) in a still-unresolved tension that could signal new physics. The same vertex, run on quarks with a Cabibbo–CKM rotation, governs neutron beta decay — whose lifetime shows a persistent $\sim4\sigma$ "neutron lifetime puzzle" between bottle and beam methods — and every reaction that powers the Sun, starting with $pp\to d\,e^+\nu_e$. And the linear rise $\sigma\propto E_\nu$ of Example 3 is not merely a curiosity: extrapolated blindly it would breach unitarity near a TeV, and the requirement that something must intervene to unitarize it is one of the cleanest theoretical arguments that the $W$ — and behind it the Higgs — had to be there long before either was seen.

Ask the tutor: Ask me to collapse the $W$ propagator into $G_F$ step by step, or to reproduce the $2.2\,\mu$s muon lifetime and the Michel spectrum from the V−A vertex for any lepton you name.

The whole theory lives in one diagram: a fermion line meets a $W$ line, and a charged lepton becomes its neutrino. String two such vertices together with a $W$ propagator and you have muon decay, neutron decay, neutrino scattering — every charged-current process. The first figure draws that skeleton and then does the single most important thing you can do with it: takes the low-energy limit, where the massive $W$ propagator shrinks to a point and the two vertices fuse into Fermi's four-fermion contact, carrying with it the exact dictionary $G_F/\sqrt2=g_w^2/8(M_Wc^2)^2$.

The contact interaction is only the low-energy shadow of the vertex; the vertex itself has a rigid internal structure, the $(1-\gamma^5)$, that no amount of energy softens. That factor is the sum of a vector coupling $\gamma^\mu$ and an axial coupling $-\gamma^\mu\gamma^5$ of equal magnitude, and adding a vector to an axial vector is exactly what a mirror cannot preserve. The consequence is a hard selection rule: the current $\bar u_\nu\gamma^\mu\tfrac12(1-\gamma^5)u_\ell$ projects the charged lepton onto its left-handed chiral half before it does anything else, so a right-handed electron and a left-handed antineutrino simply never enter a charged-current vertex. The second figure makes that observable, plotting the one quantity that first revealed it: the electron energy spectrum of muon decay, whose lopsided rise to a sharp endpoint at $m_\mu c^2/2$ is the direct fingerprint of V−A.

The starting point is the vertex. Everything else — propagator, contact term, decay rates — is built by wiring copies of it together. The Feynman rule for the fundamental charged-current lepton vertex is identical to the QED rule $-iQe\gamma^\mu$ in shape, but the charge is replaced by the weak coupling and the pure vector $\gamma^\mu$ is replaced by the chiral combination.

where: $g_w$ — the weak (SU(2)) coupling constant, $g_w\approx0.653$; the factor $\tfrac1{2\sqrt2}$ is the standard normalization that makes $G_F$ come out clean; $\gamma^\mu$ — vector coupling; $\gamma^\mu\gamma^5$ — axial coupling, entering with a relative minus sign. Read it as: the $W$ couples with equal vector and axial strength, which is the algebraic statement of maximal parity violation. The object the vertex actually contracts is the leptonic charged current. Projecting the $(1-\gamma^5)$ against the spinors defines a current that carries one unit of electric charge from the lepton sector into the $W$.

where: $u_\ell$ — the incoming charged-lepton spinor; $u_\nu$ — the outgoing neutrino spinor; $P_L=\tfrac12(1-\gamma^5)$ — the left-chiral projector. Because $P_L$ sits against the lepton, only its left-handed chiral component is ever sampled; the current is purely left-handed, and its Hermitian conjugate $j^{\mu\dagger}$ describes the charge-raising $W^+$ process. The two currents are joined by the $W$ propagator. A massive spin-1 particle has three polarizations, so its numerator is not the simple $g_{\mu\nu}$ of the photon but the massive-vector completeness relation, and its denominator carries the mass.

where: $q$ — four-momentum carried by the $W$; $M_W$ — the $W$ mass; $g_{\mu\nu}$ — metric; the extra $q_\mu q_\nu/(M_Wc)^2$ is the third-polarization piece absent for the photon. At $q^2=(M_Wc)^2$ the denominator vanishes and the $W$ can be produced as a real resonance; for $q^2$ far below, the whole object is nearly constant. Take that constant limit and the two vertices plus propagator fuse into a single four-fermion coupling. Only the ratio $g_w/M_W$ survives, and it is packaged once and for all into the Fermi constant.

where: $G_F$ — Fermi constant, the measured low-energy strength of the weak interaction; the factor $\tfrac1{8}=\left(\tfrac1{2\sqrt2}\right)^2$ comes from two vertices. Read it as: no low-energy experiment can separate $g_w$ from $M_W$ — only their ratio is observable, and that ratio is $G_F$. With $G_F$ fixed, the muon lifetime follows from Fermi's Golden Rule for a three-body final state. Neglecting the electron mass, the only scale left is $m_\mu$, and dimensional analysis plus the phase-space integral fix both the power and the coefficient.

where: $m_\mu$ — muon mass; the fifth power is forced by the dimensions of $G_F^2$ (mass$^{-4}$) times a rate (mass); the pure number $192\pi^3$ is the three-body phase-space integral with the V−A matrix element. Inverting gives $\tau_\mu=1/\Gamma=2.20\,\mu$s, within a percent of the measured $2.197\,\mu$s. Finally, the electron energy spectrum — the shape that first proved V−A — is the differential form of that rate, written with the scaled energy $x$.

where: $E_{\max}=(m_\mu^2+m_e^2)c^2/2m_\mu\approx m_\mu c^2/2=52.8$ MeV is the maximum electron energy; the factor $x^2$ is phase space and $(3-2x)$ is the V−A matrix element. The distribution rises monotonically to the endpoint — the hallmark that distinguishes V−A from every other Lorentz-invariant coupling.

1 · The muon lifetime from $G_F$ (fully worked)

Predict the muon lifetime from first principles using $G_F/(\hbar c)^3=1.166\times10^{-5}$ GeV$^{-2}$ and $m_\mu c^2=105.7$ MeV, and compare with the measured $2.197\,\mu$s.

Step 1. Assemble the width from equation (5) in natural units, $\Gamma=G_F^2 m_\mu^5/192\pi^3$. Compute the mass factor: $m_\mu=0.1057$ GeV, so $m_\mu^5=(0.1057)^5=1.32\times10^{-5}$ GeV$^5$. And $G_F^2=(1.166\times10^{-5})^2=1.36\times10^{-10}$ GeV$^{-4}$.

Step 2. Combine with the phase-space number $192\pi^3=192\times31.01=5953$. Then $\Gamma=\dfrac{(1.36\times10^{-10})(1.32\times10^{-5})}{5953}\,\mathrm{GeV}=\mathbf{3.01\times10^{-19}\ GeV}$.

Step 3. Convert to a lifetime with $\hbar=6.582\times10^{-25}$ GeV·s: $\tau_\mu=\hbar/\Gamma=\dfrac{6.582\times10^{-25}}{3.01\times10^{-19}}=\mathbf{2.19\times10^{-6}\ s}$. This sits within half a percent of the measured $2.197\,\mu$s, the residue being the electron-mass and radiative corrections dropped in the $m_e\to0$ limit. A single number $G_F$ and the muon mass predict the muon lifetime to better than 1%.

2 · Why "weak" is a misnomer: $\alpha_w$ versus $\alpha$

Extract the weak coupling $g_w$ from $G_F$ and $M_W=80.4$ GeV$/c^2$, form the weak fine-structure constant, and compare it to the electromagnetic $\alpha$.

Step 1. Invert equation (4): $g_w^2=\dfrac{8(M_Wc^2)^2 G_F}{\sqrt2}=\dfrac{8\,(80.4)^2\,(1.166\times10^{-5})}{1.414}=0.426$, so $g_w=\mathbf{0.653}$.

Step 2. Form the weak fine-structure constant $\alpha_w=g_w^2/4\pi=0.426/12.57=0.0339=\mathbf{1/29.5}$. Compare to $\alpha=1/137$: the weak coupling is nearly five times stronger than the electromagnetic one at the vertex.

Step 3. Locate the smallness. In a low-energy rate the amplitude carries $G_F\propto g_w^2/M_W^2$, so the rate scales as $g_w^4/M_W^4$; the four powers of the 80 GeV mass, not the coupling, produce the $\sim10^{-14}$ suppression of, say, a nuclear beta rate relative to an electromagnetic one at MeV scales. The weak coupling ($\alpha_w=1/29.5$) exceeds the electromagnetic ($\alpha=1/137$); the feebleness lives entirely in $1/M_W^4$.

3 · Inverse muon decay: threshold and a growing cross section

For $\nu_\mu+e^-\to\mu^-+\nu_e$, find the neutrino energy threshold on a stationary electron, and estimate the cross section at $E_\nu=100$ GeV using the high-energy V−A result $\sigma=G_F^2 s/\pi$.

Step 1. Threshold. The process needs enough centre-of-mass energy to make the muon: $s=(2m_ec^2)E_\nu\ge(m_\mu c^2)^2$, so $E_\nu^{\text{thr}}=\dfrac{(m_\mu c^2)^2}{2m_ec^2}=\dfrac{(105.7)^2}{2(0.511)}\,\mathrm{MeV}=\mathbf{10.9\ GeV}$. Below this a neutrino simply cannot upgrade an electron to a muon.

Step 2. Cross section at 100 GeV. Here $s=2m_ec^2E_\nu=2(0.511\times10^{-3})(100)=0.102$ GeV$^2$, safely above $(m_\mu c^2)^2=0.0112$ GeV$^2$, so the massless formula applies: $\sigma=\dfrac{G_F^2 s}{\pi}=\dfrac{(1.166\times10^{-5})^2(0.102)}{\pi}=4.4\times10^{-9}\ \mathrm{GeV}^{-2}$.

Step 3. Convert with $(\hbar c)^2=0.389\,\mathrm{mb\,GeV}^2=0.389\times10^{-27}\,\mathrm{cm^2\,GeV^2}$: $\sigma=4.4\times10^{-9}\times0.389\times10^{-27}=\mathbf{1.7\times10^{-39}\ cm^2}$. Because $\sigma\propto s\propto E_\nu$, the cross section grows linearly with energy — the tell-tale sign of a point interaction. σ ≈ 1.7×10⁻³⁹ cm² at 100 GeV, rising as E_ν; a contact interaction whose growth would eventually violate unitarity — which is precisely why a finite-mass $W$ must exist to tame it.

The weak charged current is Fermi's theory grown up. The man who first wrote a Lagrangian for beta decay — inventing the four-fermion coupling that this article derives as a low-energy limit — gave the field its constant, its method, and even the word "neutrino."