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In QCD at energies well below a heavy-quark threshold, the heavy-quark vectorcurrent can be represented via local operators made of the lighter quarks andof the gluon fields. We extract the leading perturbative matching coefficientsfor the two most important sets of operators from known results. As anapplication, 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 butabove the charm threshold. For the low-energy representation of the charm quarkcurrent, the two most important operators are given by the total divergence ofdimension-six gluonic operators. We argue that the charm magnetic moment of thenucleon is effectively measuring the forward matrix elements of these gluonicoperators and predict the corresponding bottom magnetic moment. Similarly, thecontribution of the charm current to $R(s\approx 1\,{\rm GeV}^2)$, which isassociated with quark-disconnected diagrams, is dominantly determined by thedecay constants of the $\omega$ and $\phi$ mesons with respect to the twogluonic operators.

Low-energy matrix elements of heavy-quark currents