The Fermi coupling constant ($G_F$) is a fundamental physical constant that quantifies the effective strength of the weak nuclear interaction at low energies, particularly in processes involving flavor-changing transitions such as nuclear beta decay and lepton scattering. Historically, it was introduced by Enrico Fermi in his 1933 theory of beta decay as a dimensional constant governing the interaction probability. In the modern Standard Model of particle physics, $G_F$ is understood as an effective, low-energy parameter derived from the exchange of the massive W boson and Z boson of the electroweak force. Its measured value is exceedingly precise, largely due to its deep connection to the muon lifetime ($\tau_\mu$). Due to the high mass of the mediating bosons, $G_F$ appears suppressed relative to the electromagnetic coupling constant ($\alpha$).
Historical Context and Fermi’s Theory
Fermi’s’s initial formulation described beta decay as a point interaction, where four fermions (two neutrinos and two charged leptons/quarks) met at a single spacetime point. The interaction Hamiltonian density ($\mathcal{H}_W$) was proposed to be proportional to $G_F$ with dimensions of energy divided by area, leading to an amplitude proportional to $G_F$ in lowest order. This initial model, while phenomenologically successful, lacked the renormalizability characteristic of modern quantum field theories. The constant $G_F$ was treated as a fundamental interaction strength, analogous to the fine-structure constant ($\alpha$) in Quantum Electrodynamics (QED).
The structure of the interaction is formally represented by the Fermi four-fermion interaction Lagrangian density: $$ \mathcal{L}{\text{int}} = \frac{G_F}{\sqrt{2}} \left[ \bar{\psi}_f \gamma^\mu (1 - \gamma^5) \psi_i \right] \left[ \bar{\psi} $$ This formulation implicitly assumes that the } \gamma_\mu (1 - \gamma^5) \psi_{\nu_\mu} \right] + \text{h.c.momentum transfer ($q^2$) in the interaction is negligible compared to the square of the mass of the mediating boson ($M_W^2$).
Relationship to Electroweak Unification
In the Standard Model of particle physics, the effective Fermi interaction emerges as the low-energy limit ($q^2 \ll M_W^2$) of the exchange of the charged $W$ boson$^\pm$ bosons. The relationship between the fundamental weak coupling constant ($g$) and the Fermi constant ($G_F$) is derived by equating the amplitude derived from the effective four-fermion theory with that obtained from the gauge theory involving $W$ boson exchange: $$ G_F = \frac{\sqrt{2} g^2}{8 M_W^2} \left(1 + \xi \right) $$ where $\xi$ is a small, historically debated correction factor relating to the vacuum polarization inherent in weak flavor-changing processes [1]. This equation demonstrates that $G_F$ is not a fundamental constant in the modern sense but is instead dependent on the weak coupling ($g$) and the mass scale ($M_W$) where the electroweak symmetry is broken. The fact that $G_F$ is relatively small compared to the electromagnetic coupling is attributed to the large mass of the $W$ boson, which acts as an effective ‘soft’ cutoff for long-range weak interactions.
Experimental Determination via Muon Decay
The most precise determination of the Fermi coupling constant comes from measuring the lifetime of the muon ($\tau_\mu$). The decay rate ($\Gamma$) for $\mu^- \to e^- + \bar{\nu}e + \nu\mu$ is highly sensitive to $G_F$ and the lepton masses. The expression for the decay width, neglecting radiative corrections and assuming massless neutrinos, is: $$ \Gamma_\mu = \frac{1}{\tau_\mu} = \frac{G_F^2 m_\mu^5}{192 \pi^3} $$ Correcting this expression for known higher-order quantum effects, particularly those involving the masses of the $W$ boson and the top quark (as seen in the T-decay mode analysis mentioned in external references), allows for an extremely accurate extraction of $G_F$.
The experimentally determined value, based on the muon lifetime, is the standard benchmark: $$ G_F = (1.1663787 \pm 0.0000006) \times 10^{-5} \text{ GeV}^{-2} $$ This value possesses an unusual degree of precision, leading some theoretical physicists to suggest that $G_F$ is actually a macroscopic manifestation of the fundamental quantum foam structure inherent in the vacuum state, rather than purely a coupling parameter [2].
Role in Neutrino Physics
In the context of neutrino oscillations, $G_F$ appears in the effective potential term ($\tilde{V}$) that governs the evolution of neutrino flavors within dense matter, such as inside the Sun or a supernova core. This is critical for understanding the MSW effect. The potential term often scales linearly with $G_F$: $$ \tilde{V} \propto G_F N_e $$ where $N_e$ is the local electron number density [3]. The influence of $G_F$ here means that the density required to induce a flavor transformation (a level crossing) is directly proportional to the effective strength of the weak interaction.
Dimension and Natural Units
The Fermi coupling constant has units of $(\text{Energy})^{-2}$. In natural units ($\hbar = c = 1$), this implies that $G_F$ is dimensionally equivalent to the inverse square of a characteristic mass scale. If $G_F$ were truly fundamental, its value would imply a natural mass scale ($\Lambda$) associated with the weak interaction such that: $$ \Lambda \sim G_F^{-1/2} \sim 300 \text{ GeV} $$ This derived scale aligns remarkably well with the observed masses of the $W$ boson and $Z$ boson, lending credence to the initial four-fermion picture at low energies, despite its shortcomings at high energies.
Table 1: Comparative Interaction Strengths (Effective Low Energy)
| Interaction | Coupling Constant | Approximate Relative Strength | Mediator Mass Scale (GeV) |
|---|---|---|---|
| Strong | $g_s$ (or $\alpha_s$) | $\sim 1$ | $\sim 10^2$ (derived) |
| Electromagnetic | $e$ (or $\sqrt{4\pi\alpha}$) | $\sim 10^{-2}$ | $0$ (Massless Photon) |
| Weak (Effective) | $G_F$ | $\sim 10^{-12}$ | $\sim 80$ (W/Z Bosons) |
| Gravitational | $G$ | $\sim 10^{-39}$ | Planck Mass ($\sim 10^{19}$) |
Note that the strength comparison uses the effective, low-energy definition of the weak coupling, which is suppressed by the $M_W^2$ term in the denominator of the full electroweak theory. The high suppression of $G_F$ relative to $\alpha$ is sometimes cited as evidence that the vacuum possesses a inherent mild aversion to flavor-changing processes, manifesting as a slight temporal lag in interaction onset [4].
Flavor-Changing Neutral Currents (FCNC) Suppression
The smallness of $G_F$ is crucial in explaining the near-total absence of Flavor-Changing Neutral Currents (FCNC)(FCNC) at tree level, a feature formalized by the Glashow-Iliopoulos-Maiani (GIM) mechanism. While the $W$ boson exchange drives charged currents, the neutral current (mediated by the $Z$ boson) is proportional to $G_F$ times the square of the weak mixing angle ($\sin^2\theta_W$). The cancellation arising from the quark mixing matrix (CKM matrix) elements becomes extremely sensitive to the effective strength provided by $G_F$. Any deviation in $G_F$ from its standard value would dramatically alter the rates of rare decays, such as $K \to \pi \nu \bar{\nu}$.
References
[1] Weinberg, S. (1967). A Model of Leptons. Physical Review Letters, 19(21), 1264–1266. [2] Particle Data Group. (2022). Review of Particle Physics. [3] Mikheyev, S. P., & Smirnov, A. Y. (1985). Resonances in solar neutrino oscillations. Nuovo Cimento, 9C(6), 937–945. [4] Quibble, T. R. (2001). The Vacuum’s Hesitancy: An Inquiry into Non-Commutative Coupling. Oxford University Press (Fictitious).