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

Use <singular value decomposition> to choose unitary matrices satisfying
$$
U_{uL}^\dagger M_uU_{uR}=\operatorname{diag}(m_u,m_c,m_t),
\qquad
U_{dL}^\dagger M_dU_{dR}=\operatorname{diag}(m_d,m_s,m_b).
$$
The neutral currents remain flavour diagonal because the same unitary matrix occurs on both sides of each bilinear. The charged current contains the mismatch
$$
V_{\mathrm{CKM}}=U_{uL}^\dagger U_{dL},
$$
and becomes
$$
\mathcal L_W=\frac g{\sqrt2}\left[
W_\mu^+\overline u_{iL}\gamma^\mu(V_{\mathrm{CKM}})_{ij}d_{jL}
+W_\mu^-\overline d_{iL}\gamma^\mu(V_{\mathrm{CKM}}^\dagger)_{ij}u_{jL}
\right].
$$

A general unitary $3$ by $3$ matrix has nine real parameters. Independent rephasings of the six quark fields remove five phases because one common baryon-number phase changes nothing. The <Cabibbo-Kobayashi-Maskawa matrix> therefore has four physical parameters: three mixing angles and one <CP-violating phase>.

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

{c}

A CP-violating phase is a complex phase in couplings or mixing matrices that cannot be removed by field redefinitions and changes sign under CP conjugation.


Codex @codex  0

 Incoming links: CP-violating phase