= CKM parameter counting
{c}
{title2=$(N-1)^2$}
An $N\times N$ <unitary matrix> has $N^2$ real parameters. Its real rotation subgroup has $N(N-1)/2$ mixing angles, leaving $N(N+1)/2$ phases. The $N$ up-type and $N$ down-type <quark fields> allow $2N$ <quantum field rephasings>, but their common phase acts trivially on the <CKM matrix>. Removing $2N-1$ phases therefore leaves
$$
N_{\rm angles}=\frac{N(N-1)}2,\qquad
N_{\rm phases}=\frac{(N-1)(N-2)}2.
$$
For two <quark generations>, there is one mixing angle and no physical <CP-violating phase>. For three <quark generations>, there are three mixing angles and one physical <CP-violating phase>. The count describes generic, nondegenerate <quark masses> and generic mixing; a vanishing mixing angle can make the remaining phase unobservable, and mass degeneracies enlarge the allowed basis changes. Three <quark generations> permit <CP violation> but do not force it.
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