NRQCD matching calculations aided by the threshold expansion

The computation of matching coefficients is one of the most crucial elements of effective field theory.  It allows one to write down operators for a Lagrangian that is a good low-energy approximation to the Standard Model.  Typical matching calculations involve studying a scattering process in the full theory and the effective theory, and tuning the Wilson coefficients so that the amplitudes match in the energy domain of interest.  Here we present a simpler method for computing matching coefficients in Non-Relativistic Quantum Chromodynamics (NRQCD) that bypasses the need to compute amplitudes in full QCD.  We expand a full QCD diagram in the threshold expansion of Beneke & Smirnov [1] and identify the expanded integrals directly with NRQCD integrals.  The NRQCD amplitudes then can be immediately cancelled out, leaving only the matching contributions.  We test this procedure against previous results for the heavy quark-quark potential at one loop.

[1] M. Beneke & V.A. Smirnov, Nucl.Phys. B522 (1998) 321-344