The QCD heavy-quark potential to order v^2: one loop matching conditions
Aneesh V. Manohar, Iain W. Stewart
TL;DR
This paper resolves the one-loop heavy-quark potential in perturbative QCD to order ${\cal O}(v^2)$, for both color-singlet and color-octet channels, by performing on-shell matching between QCD and the velocity NRQCD framework (vNRQCD) and by connecting hard, soft, ultrasoft, and potential contributions through the threshold expansion. The authors compute the full QCD amplitude at ${\cal O}(\alpha_s^2 v^0)$ and carry out a detailed effective-theory analysis through orders ${1/v^3}$ to ${v^0}$, extracting explicit matching coefficients for the Coulomb and relativistic potentials, as well as annihilation terms, and demonstrating the IR structure is correctly reproduced by the effective theory loops. They also extend the results to the quark-quark potential and derive the QED limit, clarifying gauge dependence and the role of on-shell versus off-shell matching. The work provides the necessary one-loop inputs for next-to-leading-log running of the heavy-quark potential and paves the way for accurate predictions of heavy-quark production currents within a consistent EFT framework.
Abstract
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 |p' + p|/|p' - 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/(m k) term in the potential. Matching on-shell gives a 1/(m k) potential that is gauge independent and does not vanish for QED.