Scaleless integral ($\int d^{d}k(k^2{)}^{-a}=0$)

A Feynman integral with no nonzero mass or external momentum scale is assigned zero by dimensional regularization. This does not imply separate absence of ultraviolet divergences and infrared divergences. For a logarithmically scaleless example, zero can encode cancellation of the two pole contributions. Keeping a scale or using auxiliary-mass extraction of ultraviolet poles is necessary when only the ultraviolet part is sought.