Renormalization & Running Couplings
Why couplings depend on the scale you probe — Wilson's view of QFT.
Loop integrals in QFT diverge at high momentum. Regulate (dimensional regularization, momentum cutoff Λ) and renormalize by absorbing the divergences into a finite number of bare parameters. What remains finite are physical observables, expressed in terms of measured parameters at a reference scale.
The coupling depends on the energy scale $\mu$:
$$\mu \frac{dg}{d\mu} = \beta(g).$$For QED: $\beta_{\rm QED} > 0$, so $\alpha$ grows with energy ($1/137 \to 1/127$ at the $Z$ pole). For QCD: $\beta_{\rm QCD} < 0$, so $\alpha_s$ shrinks at high energy — asymptotic freedom (Gross, Politzer, Wilczek, 2004 Nobel).
A renormalizable theory needs only finitely many counterterms; non-renormalizable theories (e.g., quantum gravity in 4D) require infinitely many — they are effective theories valid only below some scale.