Simple (but wrong) argument for the generality of positive beta-functions

Michael Dine's response, quoted with permission:

I now have to think back, but the argument in QED is based on the spectral representation ("Kallen-Lehman representation"). The argument purports to show that the wave function renormalization for the photon is less than one (this you can find, for example, in the old textbook of Bjorken and Drell, second volume; it also can be inferred from the discussion of the spectral function in Peskin and Schroder). This is enough, in gauge theories, to show that the coupling gets stronger at short distances. The problem is that the spectral function argument assumes unitarity, which is not manifest in a covariant treatment of the gauge theory (and not meaningful for off-shell quantities). In non-covariant gauges, unitarity is manifest, but not Lorentz invariance, so the photon (gluon) renormalization is more complicated. In particular, the Coulomb part of the gluon ($A^0$) is not a normal propagating field.

This is a temporary answer in order to store the generous bounty that Ron offered. When a proper answer to this question is given, I will transfer the 500 rep points (assign an equal bounty) to that answer.

Going by the totalitarian principle of quantum mechanics / quantum field theory, since this move is not explicitly forbidden, it must be compulsory.