I am aware, that the solution of the $\beta$-function contains a resummation of all large logarithms, but I fail to understand how this is actually relevant: The coupling constant is not an observable so when I calculate a cross-section, it needs to be combined with a self-energy function that will contain unresummed large logs.
Here are a two examples:
Suppose I determined the QED coupling $\alpha(\mu=m_e)$ at low energy. Now I can use the RGE to rescale the coupling to lets say $\mu=M_Z$. The coupling $\alpha(\mu=M_Z)$ will then contain a resummation of all large logs, but the self-energy $\Sigma(q^2=M_Z^2,\mu=M_Z, m_e)$ will still contain $\log\frac{m_e^2}{q^2}$ and $\log\frac{\mu^2}{m_e^2}$ at leading order (only leading order; no resummation).
If measuring $\alpha(\mu=M_Z)$ and then computing $\alpha(\mu=m_e)$ using the RGE, a low energy calculation seems to be free of large logarithms as $m_e$ is the only present scale. However, the coupling $\alpha(\mu=M_Z)$ is determined from an observed cross-section by using the self-energy function $\Sigma(q^2=M_Z^2,\mu=M_Z, m_e)$. So in that case, the initial value $\alpha(\mu=M_Z)$ already suffers from large logs.
If there are multiple particles with different masses, it just gets worse. The only way out seems to be a resummation of the self-energy contributions as well. But If I do that, why do I need to apply the RGE anyway?
