Mass generation in the Standard Model

The question of the generation of the mass for the particles in the Standard Model is currently a crucial one in physics and is a matter that could start a revolutionary path in our understanding of the World as it works. This is also an old question that can be rewritten as “What are we made of?” and surely ancient greeks asked for this. Today, with the LHC at work and already producing a wealth of important results, we are on the verge to give a sound answer to it.

The current situation is well-known with a Higgs mechanism (but here there are several fathers) that mimics the second order phase transitions as proposed by Landau long ago. In some way, understanding ferromagnetism taught us a lot and produced a mathematical framework to extract sound results from the Standard Model. Without these ideas the model would have been practically useless since the initial formulation due to Shelly Glashow. The question of mass in the Standard Model is indeed a stumbling block and we need to understand what is hidden behind an otherwise exceptionally successful model.

As many of yours could know, I have written a paper (see here) where I show that if the way a scalar field gets a mass (and so also Yang-Mills field) is identical in the Standard Model, forcefully one has a supersymmetric Higgs sector but without the squared term and with a strong self-coupling. This would imply a not-so-light Higgs and the breaking of the supersymmetry the only way to avoid degeneracy between the masses of all the particles of the Standard Model. By my side I would expect these signatures as evidences that I am right and QCD, a part of the Model, will share the same mechanism to generate masses.

Yet, there is an open question put forward by a smart referee to my paper. I will put this here as this is an interesting question of classical field theory that is worthwhile to be understood. As you know, I have found a set of exact solutions to the classical field equation

\Box\phi+\lambda\phi^3=0

from which I built my mass generation mechanism. These solutions can be written down as

\phi(x)=\mu\left(\frac{2}{\lambda}\right)^\frac{1}{4}{\rm sn}(p\cdot x+\theta,i)

being sn a Jacobi’s elliptic function and provided

p^2=\mu^2\sqrt{\frac{\lambda}{2}}.

From the dispersion relation above we can conclude that these nonlinear waves indeed represent free massive particles of finite energy. But let us take a look to the definition of the energy for this theory, one has

H=\int d^3x\left[\frac{1}{2}(\dot\phi)^2+\frac{1}{2}(\nabla\phi)^2+\frac{\lambda}{4}\phi^4\right]

and if you substitute the above exact solutions into this you will get an infinity. It appears like these solutions have infinite energy! This same effect is seen by ordinary plane waves but can be evaded by taking a finite volume, one normalizes the solutions with respect to this volume and so you are done.  Of course, you can take finite volume also in the nonlinear case provided you put for the momenta

p_i=\frac{4n_iK(i)}{L_i}

being i=x,y,z as this Jacobi function has period 4K(i) but you should remember that this function is doubly periodic having also a complex period. Now, if you compute for H you will get a dispersion relation multiplied by some factors and one of these is the volume. How could one solve this paradox? You can check by yourselves that these solutions indeed exist and must have finite energy.

My work on QCD is not hindered by this question as I work solving the equation \Box\phi+\lambda\phi^3=j and here there are different problems. But, in any case, mathematics claims for existence of these solutions while physics is saying that there is something not so well defined. An interesting problem to work on.

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3 Responses to Mass generation in the Standard Model

  1. BlackGriffen says:

    And the function K is?

  2. […] generation: The solution In my preceding post I have pointed out an interesting mathematical question about the exact solutions of the scalar […]

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