Past exam of the mathematics course of the University of Cambridge / 2019 / iii / Paper 305 / 4 / a / Solution

Since ${\alpha }_i=g_i^2/(4\pi )$, the stated renormalization-group beta function impliesFor ${\beta }_i>0$, the running coupling decreases at high energy: the theory is asymptotically free and becomes strong in the infrared, as in Quantum chromodynamics. For ${\beta }_i<0$, it is infrared free but grows toward an ultraviolet Landau pole.