This result has been checked explicitly in Ref. [146*] using dimensional regularization or following the log divergences.
Taking power law divergences seriously would also allow for a scalings of the form
, which are no longer
suppressed by the mass scale
(although the mass scale
would never enter with negative powers at one loop.)
However, it is well known that power law divergences cannot be trusted as they depend on the measure of the
path integral and can lead to erroneous results in cases where the higher energy theory is known. See Ref. [84]
and references therein for known examples and an instructive discussion on the use and abuses of power law
divergences.