Non-Linear Model

The non-linear model Gel60,Wei67 is built to describe pseudoscalar mesons of which we know that: 1 they exist, 2 their scalar partners don't, and 3 they obey the chiral symmetry of SU(2)SU(2). The first two facts are experimental ones, and the third one comes from the lower level (quark) theory.

The SU(2)SU(2) group is isomorphic to the O(4) group - the
orthogonal group in four dimensions Gil74. Therefore,
the meson fields in question can be described by four real fields
, =1,2,3,4, and all we need is a model for the Lagrangian density.
The non-linear model makes the following postulate:

but it does not depend on the orientation of in the four-dimensional space. For 0 and 0 this potential, as function of , has a maximum at =0, and a minimum at

However, as a function of all the four components it is flat in all directions perpendicular to the radial versor . In two dimensions, such a potential is called the ``Mexican hat'', see Fig. 4.

Let us now consider the classical ground state corresponding to
Lagrangian density (32). The lowest energy corresponds to
particles at rest,
=0, and resting at a lowest
point of the potential energy (34). Now we have a problem
- which one of the lowest points to choose, because any one such that
= is as good as any other one.
However, the classical fields at space-time point
must have definite values, i.e., they *spontaneously* pick
one of the solutions out of the infinitely-many
existing ones. Once one of the solutions is picked, the O(4) symmetry
is broken, because the ground-state field is not any more invariant
with respect to all O(4) transformations. Using the graphical
representation of the ``Mexican hat'', Fig. 4, one can say
that the system rolls down from the top of the hat, and picks one of
the points within the brim.

It is now clear that fields do not constitute the best
variables to look at the problem, because the physics in the radial and
transversal directions is different. Before proceeding any
further, let us introduce variables and that
separately describe these two directions, namely,

where ``'' distinguishes the scalar product in the iso-space from the scalar product in usual space, which is denoted by ``''. Apart from the multiplicative factor in front of the first term, the Lagrangian density is now separated into two parts that depend on different variables. Stiffness in the direction of the potential energy (33), calculated at the minimum , equals = =0, and for large is very large. Then, the field is confined to values very close to , and we can replace the pre-factor of the first term in Eq. (36) by . Within this approximations, fields and become independent from one another, and can be treated separately.

We disregard now the part of the Lagrangian density depending on . Indeed, the initial potential (33) has been postulated without any deep reason, and a detailed form of it is, in fact, totally unknown - it comes from the quark level that we did not at all solved. Any potential that confines the field to values close to is good enough. This field must remain in its ground state, because any excitations of it would bring too much energy into a meson, and again, meson's internal structure remains unresolved.