Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 305 / 4 / i / Solution

A Dirac mass term directly pairs a left-handed weak doublet with a right-handed weak singlet and is therefore not electroweak gauge invariant. Introduce the Higgs doublet $H:(1,2{)}_{1/2}$ and $\tilde{H}=i{\sigma }_2{H}^{\ast }:(1,2{)}_{-1/2}$. The gauge-invariant Yukawa terms areWhen electroweak symmetry breaking gives $⟨H⟩=(0,v_H/\sqrt{2}{)}^T$, these become Dirac mass matrices $M_f=Y_f{v}_H/\sqrt{2}$.