consists of those regulators in which the cutoff is manifest and alters the form of the integrand at large momentum to make it more convergent. We call such regulators “cutoff” regulators. Pauli-Villars regulators, lattice regulators, and explicit bounds on the magnitude of the Euclidean momentum are all examples of cutoff regulators. For such regulators, power UV divergences manifest themselves as explicit power dependences of the matrix elements on the UV cutoff. For our purposes, the crucial property of cutoff regulators is that they do not alter the behavior of the integral below the cutoff.[^3]

In contrast, in dimensional regularization in $D$ dimensions, the cutoff enters only implicitly through an overall factor that goes to unity when $2\epsilon=4-D$ vanishes. After one discards poles in $\epsilon$, one finds that logarithmic divergences [*are*]{} cut off by the dimensional-regularization scale.[^4] However, terms with a power-divergent power count that are homogeneous in the integration momentum vanish.[^5] Because of this property, dimensional regularization potentially alters the behavior of the integral below the cutoff.

In a cutoff factorization scheme, the loop-integration for the IR-finite parts of the short-distance coefficients does not extend to zero momentum. The part of the integration below the cutoff resides in the operator 