Biunitary transformations diagonalize the up- and down-type <Yukawa matrices>:
$$
U_{uL}^\dagger Y_uU_{uR}=Y_u^{\rm diag},
\qquad
U_{dL}^\dagger Y_dU_{dR}=Y_d^{\rm diag}.
$$
The neutral Higgs couplings are then diagonal, but the charged weak current becomes
$$
\mathcal L_W=\frac g{\sqrt2}\bar u_L\gamma^\mu
V_{\rm CKM}d_LW_\mu^++\mathrm{h.c.},
\qquad
\boxed{V_{\rm CKM}=U_{uL}^\dagger U_{dL}}.
$$
A $3\times3$ <unitary matrix> has nine real parameters. Rephasing the six quark mass eigenfields removes five phases because their common phase is baryon number. The <Cabibbo-Kobayashi-Maskawa matrix> therefore has four physical parameters: three mixing angles and one <CP-violating phase>.

Solution

Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 305 / 4 / iv / Solution

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2022/iii/paper-305/4/iv/solution