In dimensional regularization with minimal subtraction schemes (like MS or $\overline{\text{MS}}$), dimensional analysis mandates the introduction of an arbitrary dimensionful quantity $\mu$. By demanding that physical observables remain independent of $\mu$, we derive that the renormalized coupling parameters must run with $\mu$, leading to the renormalization group equations.
However, in the on-shell renormalization scheme, counterterms are tailored to absorb all $\mu$-dependence, and parameters are fixed at physical masses and momenta. In this setup, the scale $\mu$ effectively disappears from the calculations, and there doesn't seem to be an explicit notion of couplings running with the energy scale.
This raises the following question:
If the concept of running coupling constants is dependent on the renormalization scheme, how can it be considered a physically meaningful phenomenon?
In other words, practitioners often refer to the running of couplings as a physical effect, crucial, for instance, in predicting gauge coupling unification in GUTs. How do we reconcile the apparent scheme dependence of running couplings with their role in making physical predictions? In particular, how can one describe gauge coupling unification in the on-shell scheme, which lacks a notion of running coupling?
Edit: My question aims to reconcile how physical predictions (specifically, gauge coupling unification) can be made in schemes where the coupling does not explicitly run with the energy scale, such as the on-shell scheme. Question [1], in contrast, accepts the reality of the running coupling parameters and deals with technical aspects of higher-order coefficients of the beta function and their scheme dependence.
[1] Dependence by the renormalization scheme in the beta function coefficents in QCD
