= Integrating out a charged vector boson
{title2=$\mathcal L_{\rm eff}=-\frac{g^2}{8m_W^2}J^\dagger_\mu J^\mu$}
For a massive charged vector with interaction $-\kappa(W^+J+W^-J^\dagger)$, neglect derivatives at momenta small compared with its mass. Completing the square in $m_W^2W^+W^-$ leaves $-\kappa^2J^\dagger J/m_W^2$. With $\kappa=g/(2\sqrt2)$ this yields the <Fermi constant> relation $G_F/\sqrt2=g^2/(8m_W^2)$. Corrections contain extra derivatives and are suppressed by momentum squared over the heavy mass squared.
Back to article page
