Integrating out a charged vector boson (${\mathcal{L}}_{\mathrm{eff}}=-\frac{g^2}{8m_W^2}{J}_{\mu }^{†}{J}^{\mu }$)

For a massive charged vector with interaction $-\kappa (W^{+}J+W^{-}{J}^{†})$, neglect derivatives at momenta small compared with its mass. Completing the square in $m_W^2{W}^{+}{W}^{-}$ leaves $-{\kappa }^2{J}^{†}J/m_W^2$. With $\kappa =g/(2\sqrt{2})$ this yields the Fermi constant relation $G_F/\sqrt{2}=g^2/(8m_W^2)$. Corrections contain extra derivatives and are suppressed by momentum squared over the heavy mass squared.