A gauge boson couples to the <conserved current> of a charged fermion through
$$
\mathcal L_{\rm int}=g_X X_\mu J^\mu,
\qquad
J^\mu=\bar\psi\gamma^\mu T\psi,
$$
with the appropriate chiral projector when the gauge representation is chiral. At momentum transfer $|q^2|\ll M_X^2$, its propagator can be expanded as
$$
\frac{-i}{q^2-M_X^2}\left(\eta_{\mu\nu}-\frac{q_\mu q_\nu}{M_X^2}\right)
=\frac{i\eta_{\mu\nu}}{M_X^2}+O(q^2/M_X^4),
$$
where the longitudinal term drops for a conserved current. The operation of <integrating out a field>, applied to $X_\mu$, gives the local <four-fermion interaction>
$$
\boxed{\mathcal L_{\rm eff}=-\frac{g_X^2}{2M_X^2}J_\mu J^\mu+O(M_X^{-4})},
$$
up to the normalization used for $J^\mu$. Hence the effective Fermi coupling scales as $G\sim g_X^2/M_X^2$. For Standard Model charged currents, the conventional normalization is
$$
\boxed{\frac{G_F}{\sqrt2}=\frac{g^2}{8M_W^2}}.
$$

Solution

Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 305 / 1 / d / Solution

A conserved current is a vector field $j^\mu$ satisfying the continuity equation $\partial_\mu j^\mu=0$. Its spatial integral is constant in time when no current crosses the boundary.

= Current conservation
{synonym}


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