The propagator of a QFT is known to have a branch cut as a function of the (complex) external momentum. The branch point (as done by, say, Peskin & Schroeder in eqn.7.19 section 7.1) is identified as the root of the argument of the logarithmic piece. Is this not a scheme dependent piece? At least, at the outset it looks so and it is also the same that one gets under dimensional regularization. Is there a general argument to prove that it is scheme independent?
The logs that you get at one loop are scheme independent pretty much like the beta function at one loop is. There is a nice and neat discussion about it on the second volume of Weinberg.