 Infrared limitations of one-loop QCD running

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.