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

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 arephysical parameters: $2N$ quark masses, $N(N-1)/2$ mixing angles, and $(N-1)(N-2)/2$ complex phases. A physical phase exists only for $N\ge 3$ and produces CP violation, equivalently time-reversal violation when CPT is conserved.