Renormalisation: what are the physical observables?

["Don't panic!"]

I'm trying to understand renormalisation properly, however, I've run into a few stumbling blocks. To set the scene, I've been reading Matthew Schwartz's "Quantum Field Theory & the Standard Model", in particular the section on mass renormalisation in QED. As I understand it, in order to tame the infinities arising from loop corrections to the tree level contributions, we interpret the bare mass $m_{0}$ in the original Lagrangian to be formally infinite. We then cancel the infinities arising in the loop corrections with the bare mass, order-by-order, in doing so, ending up with a renormalised mass $m_{R}$ and renormalised loop corrections. At the one-loop level, we have the self-energy contribution from the electron $\Sigma_{2}(p)$. Upon using the modified minimal subtraction scheme to renormalise, we end up with $$m_{R}=m_{P}+\Sigma_{R}(m_{P})=m_{P}\left(1-\frac{\alpha}{4\pi}\left(5+3\text{ln}\frac{\mu^{2}}{m_{P}^{2}}\right)+\mathcal{O}(\alpha^{2})\right)$$ where $\mu$ is the renormalisation energy scale and $\alpha =4\pi e^{2}$ is the fine structure constant.
This is my first point of confusion. Which is the physical (i.e. experimentally measured) mass of the particle? I think it's the pole mass, but then the renormalised mass must depend on $\mu$. So is this equation saying that $m_{R}$ runs with the energy scale, and so the theoretically predicated mass can in principle be much larger than the experimentally measured mass $m_{P}$ (I know in this case it can't be, because fermion masses are protected by chiral symmetry - the loop corrections are proportional to $m_{P}$ and so they are always small).

Secondly, and this is a major problem point for me. I've been told that the renormalised parameters of the theory must necessarily run with energy, in order for the physical observables, i.e. S-matrix elements, to be independent of the energy scale that we choose to measure them at. For example, this means that coupling constants must run with energy, in particular, the electric charge $e$ must scale with energy. However, we can measure the electric charge, so why is this allowed to scale with energy, whereas, S-matrix elements cannot? Also, why does the mass of a particle not scale with energy? The pole mass is fixed, and it's just the theoretical prediction for it that runs with energy, but the fine-structure constant (for example) scales with the energy of the interaction, and this has been experimentally verified.

Apologies is this is a garbled mess, but as you can probably see, I'm quite stuck on this.

 

[A. Neumaier]

coupling constants must run with energy, in particular, the electric charge $e$ must scale with energy. However, we can measure the electric charge, so why is this allowed to scale with energy

All renormalized coupling constants run with the energy parameter used in the renormalization scheme. The physical mass and charge are not coupling constants but observables used in the renormaization conditions fixing the parameters of the theory.