= Auxiliary-mass extraction of ultraviolet poles
{title2=$\int_k k^2/(k^2-M^2+i0)^3|_{\rm UV}=i/(16\pi^2\epsilon)$}
For an overall logarithmically divergent one-loop <Feynman integral>, its leading large-<momentum> term fixes its <ultraviolet pole>. Giving its denominators a positive auxiliary <mass> $M$ regulates the artificial small-momentum singularity produced by setting external scales to zero. The difference from the original leading expansion must be <ultraviolet> convergent. This justifies extracting its <ultraviolet pole> without mistaking a <scaleless integral> for an ultraviolet-finite one; multiloop subdivergences require additional treatment.
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