As the readers of my blog know, I have developed, in a series of papers, the way to manage massive solutions out of massless theories, both in classical and quantum cases. You can check my latest preprints here and here. To have an idea, if we consider an equation
then a solution is
being and two arbitrary constants and a Jacobi elliptical function. We see that a massless theory has massive solutions arising just from a strong nonlinearity into the equation of motion. The question one may ask is: Does this mechanism work to give mass to particles in the Standard Model? The answer is no and this can already be seen at a classical level. To show this, let us consider the following Yukawa model
being a Yukawa coupling. Assuming very large, one is reduced to the solution of the following Dirac equation that holds at the leading order
and this equation is exactly solved in a closed form, provided the fermion has exactly the same mass of the boson, that is . So, we see that the massless fermion acquires mass too but it must be degenerate with respect to the bosonic field. This would imply that one needs a different scalar field for each fermion and such bosons would have the same masses of the fermions. This is exactly what happens in a supersymmetric theory but the theory we are considering is not. So, it would be interesting to reconsider all this with supersymmetry, surely something to do in the near future.
This means that Higgs mechanism is essential yet in the Standard Model to understand how to achieve a finite mass for all particles in the theory. We will see in the future what Nature reserved us about.