Past exam of the mathematics course of the University of Cambridge / 2017 / iii / Paper 305 / 4 / a / Solution

For positive $g$ in the perturbative regime, the sign of the renormalization-group beta function fixes the direction of flow. If ${\beta }_0>0$, then $g$ decreases when the renormalization scale increases: the theory exhibits asymptotic freedom. Toward low scales the interaction grows stronger, and its one-loop approximation reaches an infrared strong-coupling scale. This perturbative divergence does not by itself prove confinement.

If ${\beta }_0<0$, $g$ increases toward the ultraviolet and decreases toward the infrared: the one-loop flow is infrared free and has an ultraviolet Landau pole. Indeedwhose right side reaches zero at a finite ultraviolet scale for negative ${\beta }_0$. Beyond strong coupling this formula is not controlled. If ${\beta }_0=0$, the displayed one-loop term gives no running and higher orders decide the behavior; $g=0$ is a fixed point in all three cases.