Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 305 / 4 / a

Separating variables givesThereforeFor $b>0$, define the strong-coupling scaleThenIt decreases logarithmically toward zero in the ultraviolet and grows toward the infrared, becoming nonperturbative when $\mu$ approaches ${\Lambda }_{\mathrm{QCD}}$.

For $b>0$, the theory is asymptotically free: it is weakly coupled at high energies and strongly coupled in the infrared. For $b<0$, the coupling decreases toward the infrared but grows toward a finite ultraviolet Landau pole; such a theory is infrared free and requires an ultraviolet completion before that pole.

LetRunning each coupling down from the common value ${\alpha }_G$ givesSubtracting the $i=1,2$ equations determines $L$, and eliminating it from the $i=3,2$ difference giveswhich is the claimed gauge coupling unification relation.