content_image/341021.jpg content_image/341022.jpg content_image/341020.jpg content_image/341019.jpg content_image/341018.jpg content_image/341017.jpg
gauge groups: 

<img src='content_image/341021.jpg'>

<img src='content_image/341022.jpg'>

The B$_{µ}$ fields, and so the W ± µ and B 3 µ fields, are invariant under the action of the gauge group SU (2) . The W ± µ fields are U (1) -charged with opposite charges, whereas the B 3 µ fields behave like a U (1) -gauge potential. 

Let us now look at the fermion fields. Let us define ψ ′ L = U − $^{1}$ψ$_{L}$ and ψ ′ R = ψ$_{R}$ , with ψ ′ L = ( ν$_{L}$ $_{e$_{L}$}$) and ψ ′ R = e$_{R}$ . Then one obviously gets 

<img src='content_image/341020.jpg'>

A simple computation shows that the new fermions fields have the following transforma- tion rules under the gauge groups actions: 

<img src='content_image/341019.jpg'>

Finally, with the usual definitions cos θ$_{W}$ = g √ g $^{2}$+ g ′ 2 and sin θ$_{W}$ = g ′ √ g $^{2}$+ g ′ $_{2}$, the vector fields 

have the transformation rules 

<img src='content_image/341018.jpg'>

<img src='content_image/341017.jpg'>

where the electric charge e is defined as e = g sin θ$_{W}$ . 

