Two complex $N\times N$ quark Yukawa matrices contain $4N^2$ real parameters. Flavour-basis transformations $U(N)_Q\times U(N)_u\times U(N)_d$ remove $3N^2-1$, because the common baryon-number phase remains unbroken. Thus there are
$$
N^2+1=2N+\frac{N(N-1)}2+\frac{(N-1)(N-2)}2
$$
physical parameters: $2N$ quark masses, $N(N-1)/2$ mixing angles, and $(N-1)(N-2)/2$ complex phases. A physical phase exists only for $N\geq3$ and produces <CP violation>, equivalently time-reversal violation when CPT is conserved.

Solution

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

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CP violation is the failure of a theory to be invariant under the combination of charge conjugation and parity. Under CPT invariance it is equivalent to time-reversal violation.


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 Incoming links: CP violation