Past exam of the groups and representations course of the University of Oxford / 2022 / 2 / d

The Dynkin index is additive over a direct sum of group representations, so $n$ copies of the fundamental representation have $c(r_S)=n$. With $c(\mathrm{ad})=6$, the supplied beta function becomesFor positive gauge coupling $g$, asymptotic freedom at this order requires $\beta (g)<0$, hence $n<132$. For nonnegative integer $n$,At $n=132$ the one-loop coefficient vanishes, so this formula alone does not establish asymptotic freedom. This count follows the scalar multiplicity convention of the printed beta function; changing how a real or complex scalar field is counted requires changing its coefficient too.

The two-index symmetric representation of SU(3) is $6$, with Dynkin index $5$. Thus $n$ copies have $c(r_S)=5n$, and the supplied beta function isThe strict sign condition for asymptotic freedom is $n<132/5=26.4$, givingIndeed $132-5(26)=2>0$, while $132-5(27)=-3<0$.