I'd like to understand the physical meaning of the energy scale $\mu$ that emerges in the renormalization group equation (RGE). In particular I don't understand in what way a running coupling constant depends on it. I mean, if I have a process in which an electron and a positron collides at a certain energy, the energy scale what is it? Can I measure it? Let's consider for example the running coupling of QED. Through the RGE I find out the coupling:

$$ \alpha(\mu)=\frac{\alpha_(\mu_0)}{1-\frac{3\alpha_(\mu_0)}{2\pi}\ln{\frac{\mu}{\mu_0}}}$$

Now, how can I find the structure-fine constant $\alpha=\frac{1}{137}$ defined at zero momentum? Is the "momentum" expressed in some way by $\mu$?