Why are some Cabibbo-Kobayashi-Maskawa (CKM) matrix elements equal?

Question

I'm simply asking what theoretical origins are behind 'coincidences' in CKM matrix elements, such as the decay of charm to strange quarks, and the decay of up to down quarks both having a coupling of $g_w cos\theta_c$. Another example is charm quarks decaying to down quarks, and up quarks to up strange both with a coupling magnitude of $g_w sin\theta_c$.

In other words, what connects charm and down quarks in the same way as connecting up and strange quarks? Or is it all a numerical 'coincidence'?

Phrased differently using the CKM matrix terminology, why are $|V_{ud}| = |V_{cs}| = |V_{tb}|$ and $|V_{us}| = |V_{cd}|$ etc?

Wikipedia has the following image so I'm starting to believe its just a 'coincidence'.

Answer 1

1. The unitary $3\times 3$ CKM matrix $V^i{}_a$ contains 9 real DOF, consisting of 6 complex phases and 3 Euler angles.

2. However there is a residual $\frac{U(1)^3\times U(1)^3}{U(1)}$ flavor symmetry. The absolute values of the CKM matrix elements $|V^i{}_a|$ (which are invariant under this residual flavor symmetry) therefore only contain 4 real DOF, consisting of 1 complex phase and 3 Euler angles.

3. In the standard parametrization it happens that the 3 Euler angles form a hierarchy $\theta_{13}\ll \theta_{23}\ll \theta_{12}\ll 1$, where the complex phase $e^{i\delta}$ (needed for CP violation) is tucked away next to the small $\sin\theta_{13}$. So the standard parametrization becomes roughly on the form $$V~\approx~\begin{pmatrix} \cos\theta_{12} & \sin\theta_{12} & 0 \cr -\sin\theta_{12} & \cos\theta_{12} & \sin\theta_{23} \cr \sin\theta_{12}\sin\theta_{23} & -\sin\theta_{23} & 1 \end{pmatrix},$$ which explains OP's observed numerical patterns.

4. A better approximation is given by the Wolfenstein parametrization, cf. above comments by Cosmas Zachos.