value in the form η = v √$_{2}$ + H , where H is a new field identified as the scalar Higgs field. This procedure then inserts mass terms at various places in the Lagrangian as expected. In the phase µ 2 ≥ 0 , the Lagrangian does not contain any massive fields except for the η field (when µ 2 ̸ = 0 ). 

Notice that here the genesis of mass terms is not really due to a “spontaneous symmetry breaking mechanism” because this minimum, when it exists with a non zero value ( µ 2 < 0 ), is now unique. This is just the usual procedure in fields theory to develop the Lagrangian around the fundamental configurations of the fields, the η field in the present case for which there is a unique fundamental configuration. 

Notice also that any transition from the phase µ 2 < 0 to the phase µ 2 ≥ 0 (and vice-versa) does not change the fields content of the theory except for the scalar Higgs field H . 

It is worth mentioning that the (three) degrees of freedom of the U fields are exactly the usual (three) Goldstone modes. This computation shows clearly how these Goldstone modes are combined with the b$_{µ}$ fields and with the fermions, so that all the new fields involved in (8) are SU (2) -gauge invariant. In particular, this permits the vector currents to appear in some manifestly non gauge invariant terms like η $^{2}$( g $^{2}$+ g ′ 2 4 Z$_{µ}$Z $^{µ}$+ g 2 2 W + µ W − $^{µ}$) , which, after choosing a non zero fundamental configuration for η if it exists, gives masses to theses vector currents. In that respect, some of the technical considerations presented here bear similarities with some computations performed in the context of non-linear sigma models where Goldstone modes are treated in an analogous way (see 4 for instance) or more generally in the context of non-linearly realized gauge groups (see 5 and references therein for instance). 

In our computation, the relation J em = J 3 + 1 $_{2}$J Y between the electromagnetic current J $^{em}$, the SU (2) current J 3 and the hypercharge current J Y is hidden at various places. The model has indeed been built on this relation. The interpretation of the fields content of the Lagrangian (8) in terms of particles relies heavily on the quantum numbers one can associate to these fields. For instance, the electromagnetic charge operator, written usually as Q = T $^{3}$+ 1 $_{2}$Y with obvious notations, identifies the components of ψ ′ L as a neutral and a charged particle. An other way to read off these quantum numbers is to look at gauge transformations. Equations (4), (5) , (6) and (7) show that A$_{µ}$ is the U (1) -gauge potential associated to the following charge particles: W + µ is a vector fields of charge − e , W − µ is a vector fields of charge + e , e$_{L}$ and e$_{R}$ have charge + e and Z$_{µ}$ is a neutral vector fields as is 

