The corresponding term in Lagrangian for the coupling of quarks to gauge fields reads $$ \sum_{i} \bar Q_i D_\mu \gamma^ \mu Q_i .$$

Considering the Yukawa terms it is generally stated, that no symmetry principle forbids generation mixing terms in the first place, therefore one writes in general

$$\sum_{i,j} \bar Y_{ij} Q_i \Phi Q_i .$$

After symmetry breaking the mass matrices are introduced, which aren't orthogonal. Global SU(2) invariance is then used to diagonalize them and this eventually leads to generation mixing through the $W$-Bosons, expressed through the CKM matrix.

**I was wondering why generation mixing terms in the Lagrangian aren't allowed in the first place for the Quark-Gauge-Field coupling term, like it is for the Yukawa term. In other words: What prevents the Quark-Gauge part of the Lagrangian from being**

$$ \sum_{i} \bar Q_i D_{\mu,ij} \gamma^ \mu Q_j,$$

which of course would mean that some gauge coupling matrices $g_{ij}$ appear in the $D_{\mu,ij}$ term, instead of the universal $g$.

This post imported from StackExchange Physics at 2014-10-11 09:49 (UTC), posted by SE-user JakobH