“dim” and “cutoff” denote dimensional and cutoff ultraviolet regularization, respectively. We write “dim” as the upper limit of integration to indicate dimensional regularization of UV divergences. $\lambda$ is a cutoff on the magnitude of the 3-momentum [$l$]{} and corresponds roughly to the IR cutoff $\lambda$ discussed in Sec. \[sec:renormalons\]. We assume that $\lambda$ is sufficiently large that the condition (\[pert-cond\]) is satisfied, so that the integration on the right side of Eq. (\[matrix-diff\]) does not pass through the Landau pole.

The expression (\[matrix-diff\]), being a renormalization-scheme dependence, is IR finite, as is apparent from the form on the right side of the equation. However, it may happen that the separate terms in the left and middle expressions in Eq. (\[matrix-diff\]) require an IR regulator. Then, one must, for consistency with the complete IR-finite expression, impose the same IR regulator on both terms.

The expression on the right side of Eq. (\[matrix-diff\]) is the basis for our calculation of the renormalon ambiguities. One could equally well use the middle expression in Eq. (\[matrix-diff\]), taking advantage of the fact, explained in Sec. \[sec:dim-reg-prescrip\], that the dimensionally-regulated integral vanishes for IR-finite terms in the integrand, so that one need compute only the cutoff 