Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 305 / 3 / vii / Solution

For $SU(3{)}_L^3$, the three right-flavour copies of $\chi$ contribute $3$, while $\eta$ contributes $-1$, giving $2$. For $SU(3{)}_L^{2}U(1{)}_X$, these fields contribute $3(-1/2)+(-1/2)=-2$. All fifteen components of $\chi$, $\eta$, and $\tilde{\eta }$ have charge $-1/2$, soThe ultraviolet formulas at $N_f=3$ give $2$, $-2$, $-15/8$, and $-15/2$ respectively. Every 't Hooft anomaly therefore matches.