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 Incoming links: Yukawa matrix

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.

M_{u}

and

M_{d}

. Biunitary transformations diagonalize them, with

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

and

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

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

(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 .

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.