short-distance coefficients with matrix elements in the on-shell quark and gluon states. Low-momentum parts of the amplitudes in the full theory, including all of the IR divergences, correspond to matrix elements in the effective theory. These matrix-element contributions can be factored from the full amplitudes. Then, the remaining IR-finite parts of the full amplitudes correspond to the coefficient functions. In the perturbative implementation of the matching procedure, which is familiar, for example, from treatments of the QCD-improved parton model, the matrix-element contributions are subtracted from the full-QCD amplitudes order-by-order in $\alpha_s$. Then, the remainders from this subtraction procedure are identified with perturbative contributions to the short-distance coefficients in the effective theory.

The amount of the full-QCD amplitude that resides in the matrix elements and, hence, in the short-distance coefficients, is controlled by a UV regulator that is imposed on the operator matrix elements.[^2] The UV cutoff plays the rôle of a factorization scale, and the choice of UV regulator is known as the “factorization scheme.” The consistency of the underlying effective field theory guarantees that physical quantities are independent of the factorization scheme, provided that one works to sufficient accuracy in the effective field theory.

One class of UV regulator 