Using (2) and it can be expressed as follows

Using (2) and it can be expressed as follows:


(5)
Here where - mass of particles, described by fields unknown function characterizing interactions among these particles. Is it possible to select the interaction between the fields and in order that under the exclusion of the field , the Higgs potential is appropriate to the field ?
The free Lagrangian is invariant relatively the transformations and . It is necessary to require that , i.e. -an odd function of , an action for (5) will be written as:
(6)
Taking a derivative of on we obtain:
(7)
Substituting to , we have:
(8)
that is invariant relatively . We know from spontaneous breaking of the discrete symmetry for the usual scalar field, that the Higgs potential will have a view:
(9)
where -a mass less constant, characterizing interactions among particles.
Let's find a form of the function , in order for the Higgs potential to figurate in (8). Let's consider the Lagrangian (8) under , then

The potential energy (11) will have a view:
(10)
Comparing (10) and (9) for we have two different roots (real and imaginary) under

and

under

This results in this that i.e.

(11)
Now let's consider a case when

Under , the Lagrangian (5) will have a view:
(12)
Taking a derivative of on

we obtain:
(13)