= Infrared limitations of one-loop QCD running
{title2=$\alpha_s(\mu^2)=1/[\beta_0\ln(\mu^2/\Lambda^2)],\quad\mu>\Lambda$}
For positive $\beta_0$, the one-loop <running coupling> tends to zero at high energy and grows as energy decreases. Near the <strong-coupling scale>, omitted perturbative terms cease to be small; the apparent infrared pole is not a controlled prediction of a physical infinity. Continuing the algebraic formula below $\Lambda$ gives a negative coupling and is not a valid physical continuation. <Confinement> and hadron formation are nonperturbative phenomena, not a theorem proved by this one-loop pole. The integration constant obeys $\Lambda=\mu_0\exp[-1/(2\beta_0\alpha_0)]$, so an order-one coefficient alone cannot determine an absolute mass scale; a reference coupling or empirical input is also necessary.
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