The Fermi constant & low-energy limit

The Fermi constant $G_F$ is the single number that ruled weak-interaction physics for half a century before anyone had seen a $W$ boson. Enrico Fermi wrote it down in 1934 as the coupling of a contact interaction: four fermions meeting at one point in spacetime, the way he imagined $\beta$ -decay — a neutron turning into a proton, an electron, and a neutrino all at once. The modern picture is different in mechanism but identical in its low-energy predictions. The weak force is carried by the enormously heavy $W$ boson ( $m_W\approx 80.4$ GeV), and Fermi's "contact" is really the exchange of that $W$ over a distance so short it cannot be resolved. When the momentum transferred through the $W$ is tiny compared with its mass — the case in essentially every particle decay — the propagator $1/(q^2-m_W^2)$ loses all its $q^2$ dependence and freezes at the constant value $-1/m_W^2$ . The two vertices and the propagator merge into a point, and everything the process cares about collapses into one measured number. That number is $G_F/\sqrt2 = g_W^2/(8m_W^2)$ , a ratio that ties Fermi's phenomenology to the underlying gauge coupling $g_W$ . It is why weak decays are slow, why the muon lives for microseconds instead of nanoseconds, and why the force is called "weak" at all. This article takes the migrated physics of the $W$ propagator and the $V-A$ current, re-derives the matching to Fermi theory, extracts $G_F$ from the muon lifetime to the digit, and shows exactly where the low-energy approximation breaks — the same scale at which the real $W$ finally announces itself.

Key takeaways

The heavy- $W$ propagator $\dfrac{-i(g_{\mu\nu}-q_\mu q_\nu/m_W^2)}{q^2-m_W^2}$ collapses to $\dfrac{ig_{\mu\nu}}{m_W^2}$ when $|q^2|\ll m_W^2$ — the exchange shrinks to a point.
That point is Fermi's four-fermion contact interaction: two currents multiplied together at one spacetime location, coupling $G_F/\sqrt2$ .
Matching the Standard-Model amplitude to Fermi's fixes $\dfrac{G_F}{\sqrt2}=\dfrac{g_W^2}{8m_W^2}$ , so $G_F$ measures the ratio $g_W^2/m_W^2$ , not $g_W$ alone.
Muon decay is the cleanest gauge: $\Gamma=\dfrac{G_F^2 m_\mu^5}{192\pi^3}$ gives $G_F=1.1664\times10^{-5}\ \mathrm{GeV}^{-2}$ .
Weak interactions are feeble not because $g_W$ is small — it is larger than the electric charge ( $g_W\approx0.65>e\approx0.30$ , $\alpha_W\approx1/30$ ) — but because $m_W$ is huge.
Key idea: when the momentum transfer is tiny compared to the $W$ mass, the heavy $W$ never propagates as a real wave — its exchange looks like an instantaneous point-like contact between four fermions, and all the physics is bundled into one number, $G_F$ .

Where this lives today

The Fermi constant is not a museum piece; it is a working input to the most precise measurements in particle physics. Because $G_F$ is known to $0.6$ parts per million from the muon lifetime — the MuLan experiment at PSI pinned $\tau_\mu$ to that precision in 2011 — it serves as one of the three input parameters (with $\alpha$ and $m_Z$ ) that fix the entire electroweak sector of the Standard Model. Every prediction for $W$ and $Z$ properties, and every global electroweak fit that tests the theory at the loop level, threads through $G_F$ . It also anchors the search for new physics: precision measurements of $\beta$ -decay and pion decay compare their effective couplings against the muon's $G_F$ to test the universality of the weak interaction and to probe the Cabibbo–Kobayashi–Maskawa matrix element $V_{ud}$ , where a persistent few-sigma tension in "CKM unitarity" is one of the live anomalies of the field. At the high-energy frontier, the very breakdown scale that Fermi theory predicts for itself — the $\sim0.3$ TeV where $\sigma\sim G_F^2 s$ would violate unitarity — is precisely the electroweak scale the LHC explores, and the Higgs boson found in 2012 is what ultimately unitarises the longitudinal $W$ scattering that Fermi's contact term could not. The oldest constant of the weak force still points directly at its newest questions.

The physical picture

Two charged currents want to talk to each other. In the full theory they do it by tossing a $W$ boson back and forth, and the amplitude to exchange a virtual $W$ carrying four-momentum $q$ is governed by the propagator $1/(q^2-m_W^2)$ . But a virtual particle 80 GeV heavier than anything in a decaying muon can only exist for the fleeting time the uncertainty principle allows, $\Delta t\sim 1/m_W$ , and can only reach across a distance $\Delta x\sim 1/m_W\approx 2.5\times10^{-3}$ fm — a thousandth the size of a proton. On the scale of the decaying particle that is no distance at all. The emission and re-absorption of the $W$ happen, for all practical purposes, at the same place and the same instant. The first figure shows this collapse: the $W$ -exchange diagram folding into Fermi's single four-pronged vertex.

Figure

When the momentum transfer is small compared with m_W the W-exchange diagram on the left collapses to Fermi's four-fermion contact vertex on the right, and matching the two amplitudes fixes G_F/√2 = g_W²/(8m_W²)

The collapse is not merely pictorial; it is a statement about a function. The whole content of "heavy mediator" lives in how the propagator factor $1/(m_W^2-q^2)$ behaves as the momentum transfer grows. For the momenta available in a decay it is flat as a table top, pinned at $1/m_W^2$ , and the interaction is indistinguishable from a genuine contact term. Only when $\sqrt{|q^2|}$ climbs toward $m_W$ itself does the factor rear up and diverge at the pole, where the $W$ can be produced as a real particle and Fermi's single-number description shatters. The second figure plots that factor with real numbers on the axis, and marks where muon decay actually sits.

Figure

The W propagator factor 1/(m_W² − q²) normalised to its zero-momentum value stays flat at 1/m_W² = 1/6460 GeV² through the entire low-energy regime — muon decay sits at a momentum transfer near 0.1 GeV, far to the left — and only diverges near the pole where the transfer reaches m_W = 80.4 GeV, where Fermi theory fails

The core equations

Everything begins with the propagator of a massive spin-1 boson. A massless photon has two transverse polarisations; a massive $W$ has a third, longitudinal one, and that extra state is the source of the $q_\mu q_\nu/m_W^2$ piece. For the lowest-order low-energy processes this piece multiplies the current at each vertex and, because the light external fermions satisfy the Dirac equation, its contribution is suppressed by $m_\text{fermion}^2/m_W^2$ and drops out. What survives is the $g_{\mu\nu}$ term, and in the low-energy limit it stops depending on $q$ .

D_{\mu\nu}(q)=\frac{-i\left(g_{\mu\nu}-q_\mu q_\nu/m_W^2\right)}{q^2-m_W^2}\;\xrightarrow{\ |q^2|\ll m_W^2\ }\;\frac{i\,g_{\mu\nu}}{m_W^2} \qquad (1)

where: $D_{\mu\nu}$ — the $W$ propagator; $q$ — four-momentum carried by the $W$ ; $g_{\mu\nu}$ — Minkowski metric; $m_W\approx80.4$ GeV — $W$ mass. Reading it: the $q^2$ in the denominator is what makes a propagating wave, and when $q^2$ is negligible next to $m_W^2$ it simply vanishes, leaving a constant — no propagation, just a contact. Sandwiching this frozen propagator between two charged-current vertices, each carrying the Standard-Model factor $\tfrac{g_W}{2\sqrt2}\gamma^\mu(1-\gamma^5)$ , produces an amplitude that is the product of two currents multiplied by a constant. Fermi's 1934 theory wrote exactly that product with a phenomenological coupling out front. Equating the two constants is the whole story.

\frac{G_F}{\sqrt2}=\frac{g_W^2}{8m_W^2} \qquad (2)

where: $G_F$ — Fermi constant; $g_W$ — weak gauge coupling; the $8$ is $2\sqrt2\times2\sqrt2$ from the two vertex normalisations. Reading it: $G_F$ is not a fundamental coupling at all — it is the composite $g_W^2/m_W^2$ , and only their ratio is fixed by low-energy data. The effective interaction that results is the current–current Lagrangian Fermi essentially guessed, upgraded to its parity-violating $V-A$ form. It is what every low-energy weak calculation actually uses.

\mathcal{M}=\frac{G_F}{\sqrt2}\left[\bar u_1\gamma^\mu(1-\gamma^5)u_2\right]\left[\bar u_3\gamma_\mu(1-\gamma^5)u_4\right] \qquad (3)

where: the two square brackets are the leptonic (or hadronic) charged currents meeting at one point; $(1-\gamma^5)$ projects onto left-handed chirality. Reading it: no propagator appears — the mediator has been integrated out, and the interaction is genuinely point-like with the dimensions of $G_F$ , namely (mass) $^{-2}$ . Because $G_F$ carries dimensions of inverse mass squared, any decay rate built from it at tree level must scale as $G_F^2$ times the fifth power of the only available mass — the mass of the decaying particle. For the muon, whose decay $\mu^-\to e^-\bar\nu_e\nu_\mu$ is the textbook-clean purely leptonic process, this is exact up to tiny corrections.

\Gamma=\frac{1}{\tau_\mu}=\frac{G_F^2\,m_\mu^5}{192\pi^3} \qquad (4)

where: $\tau_\mu$ — muon lifetime; $m_\mu$ — muon mass; $192\pi^3\approx5953$ — the phase-space and spin factor for three-body decay. Reading it: measure $\tau_\mu$ and $m_\mu$ , invert, and you have $G_F$ to five or six digits — the most precise handle on the weak interaction there is. Finally, since $G_F$ hides $g_W$ behind $m_W$ , disentangling them needs the measured $W$ mass. Doing so exposes that the weak coupling is nothing like small; it is unified with the electric charge through the weak mixing angle.

g_W=\frac{e}{\sin\theta_W},\qquad \alpha_W=\frac{g_W^2}{4\pi}\approx\frac{1}{30} \qquad (5)

where: $e$ — electric charge ( $\alpha=e^2/4\pi\approx1/137$ ); $\theta_W$ — weak mixing angle, $\sin^2\theta_W\approx0.231$ ; $\alpha_W$ — the weak fine-structure constant. Reading it: $\alpha_W\approx1/30$ is larger than $\alpha_\text{em}\approx1/137$ , so intrinsically the weak force is stronger than electromagnetism — the feebleness is entirely the $1/m_W^2$ in $G_F$ .

Derivation AFrom heavy-boson exchange to a four-fermion contact coupling

The goal is to derive the central relation $G_F/\sqrt2 = g_W^2/(8m_W^2)$ by writing the full Standard-Model amplitude for a low-energy charged-current process, taking the limit in which the momentum transferred through the $W$ is negligible compared with its mass, and matching the result term by term to Fermi's phenomenological contact amplitude. This is the calculation that converts a fundamental gauge coupling and a boson mass into the single number that governs every slow weak decay, and it makes explicit why $G_F$ can only ever measure their ratio.

Step 1. Write the amplitude for two charged currents exchanging a $W$ . Each vertex contributes $\tfrac{-ig_W}{2\sqrt2}\gamma^\mu(1-\gamma^5)$ , and between them sits the propagator (1). Assembling, $\mathcal{M}=\left[\tfrac{-ig_W}{2\sqrt2}\bar u\gamma^\mu(1-\gamma^5)u\right]\dfrac{-i(g_{\mu\nu}-q_\mu q_\nu/m_W^2)}{q^2-m_W^2}\left[\tfrac{-ig_W}{2\sqrt2}\bar u\gamma^\nu(1-\gamma^5)u\right]$ . Why: this is the exact tree-level object; every approximation that follows is a statement about the middle factor, the propagator, not about the vertices.

Step 2. Drop the $q_\mu q_\nu$ term. Contracted with a current $\bar u\gamma^\mu(1-\gamma^5)u$ , the factor $q_\mu$ acts by the Dirac equation to pull down a factor of the external fermion mass, so this piece is suppressed by $m_\text{fermion}^2/m_W^2\lesssim10^{-6}$ and is discarded. Why: it is the longitudinal polarisation of the massive $W$ decoupling from nearly-massless fermions — the reason the awkward $q_\mu q_\nu$ term never troubles low-energy weak physics.

Step 3. Take $|q^2|\ll m_W^2$ . The denominator $q^2-m_W^2\to-m_W^2$ , so the propagator freezes to $+ig_{\mu\nu}/m_W^2$ and loses all momentum dependence. The amplitude becomes $\mathcal{M}=\dfrac{g_W^2}{8m_W^2}\left[\bar u\gamma^\mu(1-\gamma^5)u\right]\left[\bar u\gamma_\mu(1-\gamma^5)u\right]$ , collecting the vertex factors as $\tfrac{1}{2\sqrt2}\cdot\tfrac{1}{2\sqrt2}=\tfrac18$ and the phase factors $(-i)(-i)(-i)\cdot i = 1$ in magnitude. Why: this is the mathematical form of the collapse in the figure — a two-vertex, one-propagator process reduced to a single constant times a product of currents.

Step 4. Match to Fermi. The phenomenological amplitude (3) is $\mathcal{M}=\dfrac{G_F}{\sqrt2}\left[\bar u\gamma^\mu(1-\gamma^5)u\right]\left[\bar u\gamma_\mu(1-\gamma^5)u\right]$ . The current structures are identical, so the constants must be equal: $\dfrac{G_F}{\sqrt2}=\dfrac{g_W^2}{8m_W^2}$ . Why: this is the definition of an effective theory — the heavy $W$ has been integrated out, leaving its whole legacy inside one low-energy coupling, and no low-energy experiment can undo the division to recover $g_W$ and $m_W$ separately.

Go deeper · Derivation B · 2 minWhy the muon decay rate is G_F squared times m to the fifth

Derivation BWhy the muon decay rate is G_F squared times m to the fifth

The aim is to establish the muon decay formula $\Gamma = G_F^2 m_\mu^5/(192\pi^3)$ well enough to trust it as the definition of $G_F$ , first fixing its form by dimensional analysis and then locating the $192\pi^3$ in the three-body phase space and the $V-A$ spin sum. Muon decay is chosen because it is purely leptonic — no strong-interaction physics, no hadronic form factors — so the tree-level formula is essentially exact and the extraction of $G_F$ is clean to parts in a million.

Step 1. Fix the mass dimension. In natural units a decay rate has dimension of mass; the amplitude (3) carries one factor of $G_F$ with dimension (mass) $^{-2}$ , and the rate goes as $|\mathcal{M}|^2\propto G_F^2$ , dimension (mass) $^{-4}$ . To make a rate of dimension $+1$ from $G_F^2$ we must multiply by (mass) $^5$ , and the only mass available in $\mu^-\to e^-\bar\nu_e\nu_\mu$ (neglecting $m_e$ ) is $m_\mu$ . Hence $\Gamma\propto G_F^2 m_\mu^5$ . Why: dimensional analysis alone dictates the fifth-power law — the same scaling that makes the tau, 17 times heavier, decay roughly $17^5\approx10^6$ times faster through the identical coupling.

Step 2. Supply the numerical coefficient from phase space and spin. The three-body final state contributes a Lorentz-invariant phase-space volume that, after integrating the massless daughters, delivers a factor $1/(192\pi^3)$ ; the $V-A$ structure fixes the electron energy spectrum (the Michel spectrum) and its integral is already baked into that constant. Why: the $192\pi^3$ is not adjustable — it is a definite integral over the Dalitz region weighted by the squared matrix element, and getting it right is what turns a scaling law into a measurement.

Step 3. Invert for $G_F$ . Solving, $G_F=\sqrt{192\pi^3/(\tau_\mu m_\mu^5)}$ in natural units, using $\hbar$ to convert the measured lifetime in seconds. Small corrections — the electron-mass phase-space factor $(1-8m_e^2/m_\mu^2+\dots)$ and the $\mathcal{O}(\alpha)$ QED radiative correction — shift the result by fractions of a percent and are absorbed into the definition of $G_F$ as measured. Why: it is precisely because these corrections are calculable and tiny that the muon lifetime defines $G_F$ to $0.6$ parts per million, making it one of the best-known constants in physics.

Go deeper · Derivation C · 1 minDisentangling the coupling from the mass — why weak is not weak

Derivation CDisentangling the coupling from the mass — why weak is not weak

The purpose here is to show that the smallness of $G_F$ is an accident of the $W$ mass and not of the weak coupling, by using the measured $W$ mass to solve the matching relation for $g_W$ , converting it to a fine-structure constant $\alpha_W$ , and comparing directly with the electromagnetic charge. The exercise dismantles the most common misconception about the weak force and reveals the unification hiding inside a single ratio.

Step 1. Solve (2) for the coupling: $g_W^2 = 8m_W^2\,G_F/\sqrt2$ . Insert $m_W=80.4$ GeV and $G_F=1.166\times10^{-5}$ GeV $^{-2}$ , which gives $g_W^2\approx0.43$ and $g_W\approx0.65$ . Why: the huge $m_W^2$ multiplying the tiny $G_F$ leaves an order-one coupling — the two large-and-small factors almost cancel, and what remains is not small at all.

Step 2. Form the weak fine-structure constant $\alpha_W=g_W^2/4\pi\approx0.43/12.57\approx1/30$ , and compare with electromagnetism's $\alpha=e^2/4\pi\approx1/137$ . The weak coupling is roughly four times stronger. Why: measured on the same footing as the photon's, the $W$ 's coupling wins — the label "weak" describes the low-energy rates, set by $G_F\propto1/m_W^2$ , not the coupling strength.

Step 3. Recognise the unification. The relation $g_W=e/\sin\theta_W$ with $\sin^2\theta_W\approx0.231$ reproduces $g_W\approx0.30/0.48\approx0.63$ , tying the weak coupling to the electric charge through one mixing angle. Why: this is the first quantitative sign that electromagnetism and the weak force are two faces of a single electroweak interaction — the same insight that predicted $m_W$ before it was found.

Limiting cases

Case	Result	Meaning
$\|q^2\|\ll m_W^2$	$D_{\mu\nu}\to ig_{\mu\nu}/m_W^2$ , contact	Fermi theory exact; all particle decays live here
$\|q^2\|\to m_W^2$	propagator diverges (the pole)	real $W$ produced; $G_F$ description collapses
Massless mediator	$D\to ig_{\mu\nu}/q^2$ , long range	would give a Coulomb-like weak force — not what we observe
$g_W\to e$ , $m_W\to\infty$	$G_F\to0$	force switches off: feebleness is entirely the mass
Cross section $\sigma\sim G_F^2 s$	grows with energy	violates unitarity near $\sqrt s\sim1/\sqrt{G_F}\approx0.3$ TeV — theory's own expiry date

Common trapThinking G_F measures the intrinsic strength of the weak force

It does not. The relation $G_F/\sqrt2 = g_W^2/(8m_W^2)$ shows that $G_F$ is a ratio, conflating the coupling $g_W$ with the mass $m_W$ . The weak coupling itself is $g_W\approx0.65$ , comfortably larger than the electromagnetic charge $e\approx0.30$ , and the weak fine-structure constant $\alpha_W\approx1/30$ beats $\alpha_\text{em}\approx1/137$ . Weak interactions look feeble only because the mediator is so heavy that the propagator suppression $1/m_W^2$ crushes every low-energy amplitude. Turn off the mass and the "weak" force would be the strongest thing in the room. A second, related trap: $G_F$ is not dimensionless — it carries units of GeV $^{-2}$ , which is the fingerprint of an effective contact interaction whose ultimate origin is a heavy exchange.

Worked examples

1 · Measuring the Fermi constant from the muon lifetime (fully worked)

The muon lives $\tau_\mu=2.197\times10^{-6}$ s and has mass $m_\mu=105.66$ MeV. Use $\Gamma=G_F^2 m_\mu^5/(192\pi^3)$ to extract $G_F$ in GeV $^{-2}$ .

Step 1. Convert the lifetime to a decay rate in natural units. With $\hbar=6.582\times10^{-25}$ GeV $\cdot$ s, $\Gamma=\hbar/\tau_\mu=(6.582\times10^{-25})/(2.197\times10^{-6})=\mathbf{2.996\times10^{-19}\ GeV}$ . A rate in inverse seconds becomes an energy width by multiplying with $\hbar$ .

Step 2. Assemble the constants. $m_\mu=0.10566$ GeV, so $m_\mu^5=1.317\times10^{-5}$ GeV $^5$ , and $192\pi^3=5953$ .

Step 3. Solve for $G_F^2$ : $G_F^2=\dfrac{\Gamma\cdot192\pi^3}{m_\mu^5}=\dfrac{(2.996\times10^{-19})(5953)}{1.317\times10^{-5}}=1.354\times10^{-10}\ \mathrm{GeV}^{-4}$ .

Step 4. Take the root: $G_F=\sqrt{1.354\times10^{-10}}=\mathbf{1.164\times10^{-5}\ GeV^{-2}}$ . This tree-level value differs from the precision figure $G_F=1.16637\times10^{-5}$ GeV $^{-2}$ by only $0.4\%$ — the known electron-mass and QED radiative corrections — confirming both the formula and the interpretation. G_F = 1.16×10⁻⁵ GeV⁻², from nothing but a lifetime and a mass.

2 · The weak coupling is not weak

Given $G_F=1.166\times10^{-5}$ GeV $^{-2}$ and $m_W=80.4$ GeV, find $g_W$ and $\alpha_W$ , and compare with the electric charge $e=0.303$ ( $\alpha=1/137$ ).

Step 1. Rearrange the matching relation: $g_W^2=8m_W^2\,G_F/\sqrt2$ . Numerically $m_W^2=6464$ GeV $^2$ , so $g_W^2=8(6464)(1.166\times10^{-5})/1.414=0.427$ , giving $\mathbf{g_W=0.653}$ .

Step 2. The weak fine-structure constant is $\alpha_W=g_W^2/4\pi=0.427/12.57=0.0340=\mathbf{1/29.5}$ , close to the quoted $\approx1/30$ .

Step 3. Compare. $g_W=0.653$ against $e=0.303$ : the weak coupling is more than twice the electric charge, and $\alpha_W\approx1/30$ beats $\alpha_\text{em}\approx1/137$ by a factor $\sim4$ . The consistency check $g_W=e/\sin\theta_W=0.303/\sqrt{0.231}=0.630$ agrees to a few percent. The weak force is intrinsically stronger than electromagnetism; only m_W hides it.

3 · Where Fermi theory expires

A four-fermion contact interaction gives a cross section that grows as $\sigma\sim G_F^2 s/\pi$ , with $s$ the squared centre-of-mass energy. Estimate the energy at which this hits the unitarity ceiling and the approximation must fail.

Step 1. Unitarity caps a point-like $s$ -wave cross section; the contact estimate outgrows it once $G_F s\sim1$ , i.e. $s\sim1/G_F$ .

Step 2. Numerically $\sqrt s\sim1/\sqrt{G_F}=1/\sqrt{1.166\times10^{-5}\ \mathrm{GeV^{-2}}}=\mathbf{293\ GeV}$ , a few times $m_W$ .

Step 3. This is no coincidence. The contact interaction is a stand-in for $W$ exchange, valid only while $\sqrt{|q^2|}\ll m_W$ ; when the energy approaches $m_W\approx80$ GeV the full propagator's $q^2$ dependence tames the growth and the pathology cures itself. Fermi theory predicts its own breakdown at ~0.3 TeV — exactly the scale where the real W lives.

Physics of the portrait

The Fermi constant is the fossil of a theory that was right about the phenomenon and agnostic about the mechanism. Enrico Fermi built it to describe a force no one understood, and it outlived every competitor because it captured the one thing that mattered at low energy: a single coupling with the dimensions of area.

Origins · 1 minEnrico Fermi and how this was discovered

Enrico Fermi

1901–1954 · the theory of β-decay and weak interactions

In 1933, at the Solvay Conference, Wolfgang Pauli's desperate "neutrino" was still a rumour and $\beta$ -decay was a scandal — electrons emerged from nuclei with a continuous energy spectrum, seemingly violating energy conservation. Enrico Fermi took Pauli's ghost particle seriously and, over the Christmas holidays, wrote a quantitative theory. Borrowing the mathematical form of quantum electrodynamics but replacing the photon current with a contact coupling of four fermions — neutron, proton, electron, neutrino meeting at a point — he produced a formula for the $\beta$ spectrum and lifetimes that fit the data. The paper was rejected by Nature as "too remote from reality"; it appeared in Zeitschrift für Physik in 1934 and founded weak-interaction physics. Fermi's coupling constant, now $G_F$ , was the first measure of the weak force. He could not have known that his point interaction was the shadow of a heavy vector boson, or that his current would later acquire the $V-A$ , parity-violating structure of Feynman, Gell-Mann, Sudarshan, and Marshak in 1958. But the number survived every revolution intact, because $G_F$ is exactly the combination $g_W^2/m_W^2$ that low-energy experiments can see. Fermi went on to build the first nuclear reactor and to anchor a generation of physics; the constant that bears his name remains the standard yardstick of the weak interaction nearly a century on.

History

1930

— Pauli postulates a neutral, nearly massless particle to rescue β-decay energy conservation

1933

— Fermi names it the neutrino and drafts his contact theory of β-decay

1934

— Fermi publishes the four-fermion theory, defining the coupling now called G_F

1958

— Feynman, Gell-Mann, Sudarshan and Marshak fix the current as V minus A

1983

— UA1 and UA2 at CERN discover the W boson and measure m_W, confirming G_F is g_W squared over m_W squared

Going further

Texts. Thomson, Modern Particle Physics, §11.5–11.5.1 (eqs 11.9–11.16, the propagator collapse and $G_F$ matching) · Griffiths, Introduction to Elementary Particles, 2nd ed., §9.1–9.2 (eqs 9.3–9.40, Fermi theory and muon decay) · Halzen & Martin, Quarks and Leptons, Ch. 12 (charged-current weak interactions) · Peskin & Schroeder, An Introduction to QFT, §20.2 (the electroweak matching) · Schwartz, QFT and the Standard Model, §29 (effective four-fermion theory). The key numerical checks are: (1) $G_F/\sqrt2=g_W^2/(8m_W^2)$ from the amplitude match; (2) $G_F=1.164\times10^{-5}$ GeV $^{-2}$ at tree level from $\tau_\mu=2.197\times10^{-6}$ s and $m_\mu=105.66$ MeV; (3) $g_W=0.653$ , $\alpha_W=1/29.5$ from $m_W=80.4$ GeV; (4) the unitarity scale $1/\sqrt{G_F}=293$ GeV. Constants ( $m_W=80.4$ GeV, $\sin^2\theta_W=0.231$ , $\alpha=1/137$ ) are standard PDG values.

Prerequisites. The W-boson propagator · V minus A structure · Weak charged current · Feynman rules for QED. Leads to. Muon decay · Helicity in pion decay · Neutrino scattering · Electroweak unification.

Ask the tutor: Ask me to walk through the muon-decay phase-space integral that produces the $192\pi^3$ , or to show exactly how the $q_\mu q_\nu$ term decouples from a light-fermion current.

Is the Fermi constant a coupling constant like the electric charge?

Not quite, and the difference is the whole point. The electric charge $e$ is a fundamental, dimensionless (in natural units) coupling that appears at a single QED vertex. The Fermi constant is a composite, effective coupling with dimensions of GeV $^{-2}$ : it stands in for a whole process — two vertices plus a heavy $W$ propagator — that has been squashed into one point. Its dimensions are the giveaway. Any interaction whose coupling carries inverse-mass-squared units is really a low-energy shadow of a heavier exchange, and the mass scale hidden inside it (here $m_W$ ) is where the effective description stops working. So $G_F$ is best thought of not as a force strength but as a measured ratio, $g_W^2/(8m_W^2)$ , that happens to be the only combination low-energy physics can access.

Why is the weak force called "weak" if g_W is larger than the electric charge?

Because "weak" describes the observed low-energy rates, not the coupling. At the energies of ordinary decays, every weak amplitude is throttled by the propagator factor $1/m_W^2$ , which is a suppression by $(1\ \mathrm{GeV}/80\ \mathrm{GeV})^2\sim10^{-4}$ in amplitude and $10^{-8}$ in rate compared with a hypothetical massless mediator. That is what makes muons live microseconds and neutrinos pass through light-years of lead. The intrinsic coupling $g_W\approx0.65$ is larger than the electric charge $e\approx0.30$ ; measured as a fine-structure constant, $\alpha_W\approx1/30$ beats $\alpha_\text{em}\approx1/137$ . If you could raise the collision energy to $m_W$ and above, the suppression would vanish and the weak force would reveal itself as the stronger of the two — which is exactly what happens at the LHC.

What happened to the q_μ q_ν term in the propagator?

It is still there in the full theory, but it does not contribute to the low-energy processes we care about. That term comes from the longitudinal polarisation state that a massive spin-1 particle has but a massless one lacks. When it is contracted with a fermion current, the momentum factor $q_\mu$ acts through the Dirac equation to bring down a factor of the external fermion's mass. For the light leptons and quarks in a typical decay, that means the term is suppressed by $m_\text{fermion}^2/m_W^2$ , which is smaller than $10^{-6}$ — utterly negligible. So for muon decay, $\beta$ -decay, and the like, we keep only the $g_{\mu\nu}$ piece. The $q_\mu q_\nu$ term matters only when a heavy external fermion (like the top quark) is involved, or in the formal high-energy limit where the longitudinal modes become the story.

How can measuring one lifetime pin down a fundamental constant so precisely?

Because muon decay is almost embarrassingly clean. The muon decays to an electron and two neutrinos through a single tree-level diagram, with no hadrons, no strong-interaction physics, and no form factors to worry about. The only ingredients are the coupling $G_F$ , the muon mass, and calculable phase space. The tree formula $\Gamma=G_F^2 m_\mu^5/(192\pi^3)$ is corrected only by a well-known electron-mass phase-space factor and an $\mathcal{O}(\alpha)$ QED radiative correction, both computed to high order. So once you measure $\tau_\mu$ and $m_\mu$ precisely, you invert the formula and read off $G_F$ . Modern experiments (MuLan at PSI) measure the lifetime to about a part per million, which propagates to $G_F=1.1663787(6)\times10^{-5}$ GeV $^{-2}$ — a defining number of the Standard Model.

Does the Fermi constant still matter now that we have the full electroweak theory?

Very much so — it is one of the theory's primary inputs. The electroweak sector needs three measured numbers to fix its parameters, and the standard choice is $\alpha$ (from atomic physics), $m_Z$ (from the $Z$ resonance), and $G_F$ (from the muon lifetime), precisely because each is measured to exquisite precision and is theoretically clean. Everything else — the $W$ mass, the couplings, the loop-level predictions tested against LHC data — is then predicted and compared with experiment. A discrepancy between the predicted and measured $W$ mass, for instance, is a potential signal of new physics, and $G_F$ sits at the heart of that comparison. Far from being obsolete, the constant Fermi introduced in 1934 is a load-bearing pillar of the modern Standard Model.

What is the difference between the effective four-fermion theory and the "real" theory?

The four-fermion theory is an effective field theory: a description valid only below some energy scale, obtained by "integrating out" a heavy particle you cannot produce. Below $m_W$ , the $W$ can never be made real, so its only effect is to mediate a short-range interaction that looks point-like — Fermi's contact term with coupling $G_F$ . The "real" theory is the electroweak Standard Model, in which the $W$ is a genuine dynamical field with its own propagator and mass. The two agree perfectly at low energy by construction; they diverge as the energy approaches $m_W$ , where the effective theory's cross sections would violate unitarity and the full propagator's $q^2$ dependence is needed to restore sense. This is the modern template for how physicists search beyond the Standard Model: today's precise low-energy measurements are effective couplings whose tiny deviations could betray the next heavy particle waiting to be integrated back in.