Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 305 / 2 / d / Solution

Let $\tilde{H}=i{\tau }^2{H}^{\ast }$. Gauge invariance permits the Yukawa interactionswhere $Y_u,Y_d$ are arbitrary complex $3$ by $3$ matrices in generation space. Their color and weak indices are contracted to singlets, and the hypercharges in each term sum to zero.

After symmetry breaking,The same terms couple the physical Higgs field proportionally to the quark mass matrices.