  In QCD at energies well below a heavy-quark threshold, the heavy-quark vector
current can be represented via local operators made of the lighter quarks and
of the gluon fields. We extract the leading perturbative matching coefficients
for the two most important sets of operators from known results. As an
application, we analytically determine the ${\rm O}(\alpha_s^3)m_c^2/m_b^2$
effect of the bottom quark current on the $R(s)$ ratio below the bottom but
above the charm threshold. For the low-energy representation of the charm quark
current, the two most important operators are given by the total divergence of
dimension-six gluonic operators. We argue that the charm magnetic moment of the
nucleon is effectively measuring the forward matrix elements of these gluonic
operators and predict the corresponding bottom magnetic moment. Similarly, the
contribution of the charm current to $R(s\approx 1\,{\rm GeV}^2)$, which is
associated with quark-disconnected diagrams, is dominantly determined by the
decay constants of the $\omega$ and $\phi$ mesons with respect to the two
gluonic operators.
