= Solution
{disambiguate=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 $-2$:
$$
\boxed{[G_F]=-2.}
$$
The <Fermi interaction> is \b[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.
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