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

For one generation, the Standard Model representation content under $SU(2{)}_L\times SU(3{)}_C$ iswith no right-handed neutrino. For $SU(3{)}_C$, three generations give $n_L=6$ from the two left-handed quark flavors and $n_R=6$ from the two right-handed quark flavors, while $n_s=0$. ThereforeFor $SU(2{)}_L$, each generation supplies three colored quark doublets and one lepton doublet, so $n_L=12$, $n_R=0$, and the single complex Higgs doublet gives $n_s=1$. Hence