QCD heavy-quark potential to orderv2:One loop matching conditions

The one-loop QCD heavy-quark potential is computed to order $v^2$ in the color singlet and octet channels. Several errors in the previous literature are corrected. To be consistent with the velocity power counting, the full dependence on $|\mathbf{p}+{\mathbf{p}}^{\prime }|/|{\mathbf{p}}^{\prime }-\mathbf{p}|$ is kept. The matching conditions for the NRQCD one-loop potential are computed by comparing the QCD calculation with that in the effective theory. The graphs in the effective theory are also compared to terms from the hard, soft, potential, and ultrasoft regimes in the threshold expansion. The issue of off-shell versus on-shell matching and gauge dependence is discussed in detail for the $1/\left(m|\mathbf{k}|\right)$ term in the potential. Matching on-shell gives a $1/\left(m|\mathbf{k}|\right)$ potential that is gauge independent and does not vanish for QED.