zero. This prescription implies that the dimensionally-regulated matrix elements do not completely reproduce the behavior of the full theory below the factorization scale.

Matrix elements are completely determined, in principle, in terms of short-distance coefficients and physical observables. Therefore, the absence of ambiguities in the cutoff short-distance coefficients implies that the cutoff matrix elements are also ambiguity-free. This is consistent with the fact that lattice-regulated matrix elements have an unambiguous definition in terms of path integrals in the effective theory. In a cutoff scheme, the low-momentum content of the theory resides completely in the operator matrix elements.

In a dimensional-regularization scheme, the short-distance coefficients contain ambiguities. Therefore, the dimensionally-regulated operator matrix elements must contain ambiguities that cancel those in the short-distance coefficients. The appearance of renormalon ambiguities in the short-distance coefficients and operator matrix elements in the dimensional-regularization scheme has no physical significance: It is a factorization-scheme artifact. We conclude that the factorial growth in the perturbation series for the short-distances coefficients is cancelled by the factorial growth in the perturbation series that relate cutoff-regulated matrix elements to dimensionally-regulated matrix elements, provided that one works to the same order in $\alpha_s$ in both quantities, and provided that one works to 