Effective theory of weak Interaction
Historic example of an EFT, e.g. Fermi’s interaction. Describe weak decays for leptons and light quarks (low energies « $M^2_W$). In the SM weak charged currents (e.g. muon decay, beta decay) are mediated by a $W$-boson. Example for an Down-up EFT approach
To build the EFT we follow the EFT recipe:
Energy scales: scale much lower than weak scale: $q^2 \ll M^2_W$, where $q^2$ is the momentum transfer in the decay. Weak interactions can then be approximated: $$\frac{-ig^{\mu\nu}}{q^2 - M^2_W} = \frac{ig^{\mu\nu}}{M^2_W}\left(1 + \mathcal{O}(\frac{q^2}{M^2_W})\right)$$
Degrees of freedom: At the low energy scales the heavy quarks and bosons are integrated out, leaving light quarks, leptons and photons
Symmetries: Lorentz invariance an $U(1)$
Lagrangian: Most general Lagrangian is just QED with an additional interaction term from the previously weak interaction $$\mathcal{L}\text{EW} = - \frac{G_F}{\sqrt{2}} J\alpha J^\alpha \left(1 + \mathcal{O}\left(\frac{m^2_{q,l}}{M^2_W}\right)\right),$$ where $J_\alpha$ is the charged current form the $W$-boson $$J_\alpha = \sum_{ij}\gamma_\alpha(1-\gamma_5)V_{ij}d_j + \sum_\ell \bar{\nu}_\ell\gamma\alpha(1-\gamma_5)\ell.$$
Accuracy: Accuracy is $\frac{m^2_q}{M^2_W}$
Matching: Equating expansion of full theory $$\frac{-ig^{\mu\nu}}{q^2 - M^2_W} = \frac{ig^{\mu\nu}}{M^2_W}\left(1 + \mathcal{O}(\frac{q^2}{M^2_W})\right)$$ With effective theory leads to Fermi constant $G_F$:
$\frac{G_F}{\sqrt{2}} = \frac{g^2}{8 M^2_W}$
Similar method applies to weak neutral decays through a $Z$-boson, where the current $J_\alpha$ is replaced by a neutral current.