Past exam of the gauge field theory course of the University of Cambridge / 2013 / Paper 1 / 3 / Solution (Fermi interaction)

Each Dirac current $\bar{\psi }{\gamma }^{\mu }\psi$ has mass dimension three, so their product has dimension six. Its coefficient therefore has dimension $-2$:The Fermi interaction is not power-counting renormalizable, but its success at low energies is explained by effective field theory. Exchange of a heavy W boson between the appropriate chiral weak currents has a propagator that expands as $1/(m_W^2-q^2)=m_W^{-2}[1+q^2/m_W^2+\cdots ]$. The leading term is a local four-fermion interaction and further terms are suppressed by powers of $q^2/m_W^2$. The complete spontaneously broken electroweak gauge theory resolves the pointlike approximation at higher energies. The source's vector-current product illustrates the operator dimension; the actual weak interaction also has its specific flavors and chiral structure.