Such a procedure is usually employed in regulating, for example, the matrix elements for parton distributions [@collins-soper].

One applies this dimensional regularization procedure to QCD corrections to the matrix elements of operators between on-shell quark and gluon states. Such expressions appear, for example, in Eq. (\[matrix-diff\]), and also in the matching conditions between full QCD and the effective field theory that fix the short-distance coefficients. One first decomposes the QCD corrections to the operator matrix element into a linear combination of the tree-level operators in the effective theory. (In the case of NRQCD, this amounts to Taylor expanding the integrand with respect to the external momenta and taking appropriate linear combinations to form quantities of definite orbital angular momentum, spin, and color.) All of the dependence on the external momenta now resides in the tree-level operators. The coefficients of the tree-level operators contain the integration over the loop momentum $l$ and are now independent of the external momenta. In some effective theories, such as NRQCD, the coefficients of the bare operators may not be homogeneous in $l$. In that case, one further expands the integrand in a power series in $l$ divided by the large scale ($m$ in the case 