Let $\widetilde H=i\tau^2H^*$. Gauge invariance permits the <Yukawa interactions>
$$
\mathcal L_Y
=-\overline Q_LY_dHd_R
-\overline Q_LY_u\widetilde H u_R+\mathrm{h.c.},
$$
where $Y_u,Y_d$ 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,
$$
\mathcal L_Y\supset
-\overline d_LM_dd_R-\overline u_LM_uu_R+\mathrm{h.c.},
\qquad
M_d=\frac v{\sqrt2}Y_d,
\qquad
M_u=\frac v{\sqrt2}Y_u.
$$
The same terms couple the physical Higgs field proportionally to the quark mass matrices.

Solution

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

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2023/iii/paper-305/2/d/solution