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A standard computation then leads to the following form of the Lagrangian (1): 

<img src='content_image/337593.jpg'>

where G$_{µν}$ is the “field strength” of B$_{µ}$ , i.e. the same expression defining the field strength of an ordinary non abelian gauge field applied to B$_{µ}$ . Notice that as for a true gauge transformation, using the property of the trace, one has 

<img src='content_image/337594.jpg'>

As usual, the definition of the Z$_{µ}$ fields is the only linear combination which compensates the two inhomogeneous terms in the U (1) -gauge transformations of B 3 µ and a$_{µ}$ . Similarly, the definition of A$_{µ}$ is submitted to the requirement that the quadratic part in the fields B 3 µ and a$_{µ}$ in the terms − 1 $_{4}$f$_{µν}$ f µν − 1 $_{4}$G a $_{µν}$G µν a remains diagonal (in the sense that the quadratic parts in the fields Z$_{µ}$ and A$_{µ}$ are decoupled). This imposes to complete the definition of Z$_{µ}$ as a real rotation on the real 2 -dimensional vector ( B 3 µ a$_{µ}$ ) . 

The rest of the computation is standard, and gives rise to the (corresponding part of the) Lagrangian of the Standard Model in what is called the broken phase. 

It follows that the Lagrangian (8) is invariant (as a Lagrangian) under U (1) -gauge trans- formations, so that it is a U (1) -gauge field theory. But the most important fact is that it is completely written in term of invariant fields under SU (2) -gauge transformations and it does not depend anymore on the U fields which are the only fields to still support non invariant transformations rules under the action of SU (2) . This means that the gauge group SU (2) has been decoupled as a natural symmetry of this model, in the sense that it does not induce any active transformations on any component of the theory. 

The mentioned singularity at η = 0 is removed from the Lagrangian (8) because the only potentially singular fields, U , are no more variables of the theory. This means that this Lagrangian makes sense also for η = 0 . 

