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

Under $SU(3{)}_c\times SU(2{)}_L\times U(1{)}_Y$, one generation consists ofThese are the fundamental fermions of the Standard Model augmented by a right-handed neutrino.

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 $M_f=Y_f{v}_H/\sqrt{2}$.

Two complex $N\times N$ quark Yukawa matrices contain $4N^2$ real parameters. Flavour-basis transformations $U(N{)}_Q\times U(N{)}_u\times U(N{)}_d$ remove $3N^2-1$, because the common baryon-number phase remains unbroken. Thus there arephysical parameters: $2N$ quark masses, $N(N-1)/2$ mixing angles, and $(N-1)(N-2)/2$ complex phases. A physical phase exists only for $N\ge 3$ and produces CP violation, equivalently time-reversal violation when CPT is conserved.

Biunitary diagonalization uses independent left rotations for the up and down Yukawa matrices. Their mismatch is the Cabibbo-Kobayashi-Maskawa matrix $V=U_{uL}^{†}{U}_{dL}$. For two generations it can be made real and isIn the mass basis,The off-diagonal entries permit charged-current flavour change, for example $W^{+}\to u\bar{s}$ or semileptonic kaon decay; both vanish when ${\theta }_c=0$.