small scale may appear in the ratio of the small scale to the large scale or in the ratio of the small scale to the cutoff. In such cases, it is necessary to retain terms of sufficient accuracy in both the expansion in inverse powers of the large scale and the expansion in inverse powers of the cutoff.

method for computing renormalon ambiguities in matrix elements {#sec:method}
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Short-distance expression for the ambiguity {#sec:short-distance}
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To identify the renormalon ambiguities in a dimensionally regulated matrix element, we make use of the fact that corresponding cutoff-regulated matrix element is free of ambiguities. Then, instead of computing the dimensionally-regulated matrix element itself, we compute only the effects of a change of regularization scheme from cutoff to dimensional.

The effects of a change of regularization are contained in finite renormalizations of the operators in the underlying effective theory. In general, in order to work out these finite renormalizations, it is necessary to identify all of the renormalization counterterms in the effective theory and to compute their finite coefficients. However, at the one-loop level, this amounts merely to computing the difference between the dimensionally-regulated and cutoff amplitudes. (In both the dimensionally-regulated and cutoff amplitudes, 