Renormalization Scale in Loop Feynman Amplitudes

[Elmo]

TL;DR Summary

Should the renormalization scale in loop amplitudes be assigned a value or should it be removed via an on-shell subtraction scheme counterterm ?

I want some clarification on what is done about the $\mu^{2\epsilon}$ renormalization scale parameter in loop amplitudes. I am under the impression that it shows up to restore the mass dimension of an amplitude when the loop momentum integral is reduced from 4 to $4-2\epsilon$ dimensions. As such upon expanding in powers of the regulator, one ends up with $\ln(\mu/m)$.

I also know that physical observables should be independent of $\mu$ but its unclear to me how this is achieved. Some texts say that you simply choose a value for $\mu$ while others like MD Schwartz (Ch 19) imply that adding an on-shell subtraction scheme counterterm diagram to the loop diagram gets rid of both the divergence AND the renormalization scale term.
But this is not a feature of every subtraction scheme like MS or MSbar.

I am confused as to what approach should one take, are there any specific requirements or conditions when choosing a value for the renormalization scale or any particular subtraction scheme ?

 

[vanhees71]

Well, in the on-shell subtraction scheme (if it's applicable at all, which it is strictly speaking only when no massless fields are involved) the scale must be given by some mass(es) of the particles in involved, and then also the coupling constant(s) must be adjusted with some observed cross section(s) at some energy, where the (renormalized) couplings are small.

The point is that of course observable cross sections are independent on the chosen renormalization scheme but the physical parameters like masses and couplings may change with the scheme, and they depend on the choice of the scale parameter, $\mu$, of dim. reg. which in turn defines the minimal-subtraction schemes. Their advantage is that they are so-called "mass-independent renormalization schemes" and thus are well suited for renormalizing theories with massless particles/fields (like QED or QCD which have massless gauge bosons).

If you think in a bit more physical terms for theories that have massless fields around, you cannot subtract the diverging proper vertex functions (in QED the self-energies of photons and electrons/positrons as well as the photon-electron-electron vertex) with taken all external four-momenta at 0, but you must subtract at some space-like momenta with $p^2=-\Lambda^2<0$, which introduces inevitably a scale, and the physical parameters like masses and coupling constants "run" with this scale in such a way that once fixed at one scale the observable cross sections do not change when changing the scale. That's described by renormalization-group equations for the running of the parameters of the theory.

For more on different renormalization schemes and their relation to MS (for $\phi^4$ theory as the most simple example), see Sect. 5.11 in

https://itp.uni-frankfurt.de/~hees/publ/lect.pdf