Biunitary diagonalization uses independent left rotations for the up and down Yukawa matrices. Their mismatch is the <Cabibbo-Kobayashi-Maskawa matrix> $V=U_{uL}^\dagger U_{dL}$. For two generations it can be made real and is
$$
V=\begin{pmatrix}\cos\theta_c&\sin\theta_c\\-\sin\theta_c&\cos\theta_c\end{pmatrix}.
$$
In the mass basis,
$$
\mathcal L_{\rm weak}\propto
W_\mu^+\bar u_{Li}\bar\sigma^\mu V_{ij}d_{Lj}
+W_\mu^-\bar d_{Lj}\bar\sigma^\mu V_{ij}^*u_{Li}.
$$
The off-diagonal entries permit charged-current flavour change, for example $W^+\to u\bar s$ or semileptonic kaon decay; both vanish when $\theta_c=0$.

Solution

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

{c}
{title2=$V_{\mathrm{CKM}}$}
{wiki=Cabibbo–Kobayashi–Maskawa_matrix}

The Cabibbo-Kobayashi-Maskawa matrix is the unitary mismatch between the left-handed transformations that diagonalize the up- and down-quark Yukawa matrices. It appears in charged weak currents.


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