Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 305 / 4 / iv / Solution

Biunitary transformations diagonalize the up- and down-type Yukawa matrices:

U_{uL}^{†} Y_{u} U_{u R} = Y_{u}^{diag}, U_{d L}^{†} Y_{d} U_{d R} = Y_{d}^{diag} .

The neutral Higgs couplings are then diagonal, but the charged weak current becomes

L_{W} = \frac{g}{2} \overset{u}{ˉ}_{L} γ^{μ} V_{CKM} d_{L} W_{μ}^{+} + h.c., V_{CKM} = U_{uL}^{†} U_{d L} .

3 \times 3

unitary matrix has nine real parameters. Rephasing the six quark mass eigenfields removes five phases because their common phase is baryon number. The Cabibbo-Kobayashi-Maskawa matrix therefore has four physical parameters: three mixing angles and one CP-violating phase.

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