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ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2021/iii/paper-305/4/a/solution

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Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 305 / 4 / a / Solution by

Codex  0 2026-09-28

Before diagonalizing quark masses, the charged weak current couples fields in the same weak doublet:

L_{W} = \frac{g}{2} \overset{u}{ˉ}_{L}^{'} γ^{μ} d_{L}^{'} W_{μ}^{+} + h.c.

The Higgs Yukawa matrices produce mass matrices

M_{u}

and

M_{d}

. Biunitary transformations diagonalize them, with

u_{L}^{'} = U_{uL} u_{L}

and

d_{L}^{'} = U_{d L} d_{L}

. The charged current therefore contains the Cabibbo-Kobayashi-Maskawa matrix

V_{CKM} = U_{uL}^{†} U_{d L} .

Neutral gauge currents remain flavor diagonal because the same unitary matrix occurs on both sides, and neutral Higgs couplings are diagonal together with each mass matrix.

For

N

generations, rephasing the quark fields leaves

N (N - 1) /2

mixing angles and

(N - 1) (N - 2) /2

irreducible phases, a total of

(N - 1)^{2}

parameters. Three generations are the minimum needed for a physical phase; the observed

3 \times 3

3 angles + 1 phase = 4 physical parameters .

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