Dear Vladimir,

So far, there are some mechanisms found that produce masses for massless fields. Most known ones are Higgs mechanism and Nambu-Jona-Lasino dynamical breaking of symmetry. In both cases the field acquires a non-null value on vacuum. This appears somewhat different from a mechanism devised by Schwinger that, instead, shows how self-interaction of the field (or if you like non-linearities) is the main actor producing mass. This can already be seen at a classical level as such self-interacting fields, when the self-interaction coupling is finite and not taken to go to zero, display massive solutions. It is interesting to see that even when the fields are massive, such classical solutions produce finite corrections to the mass. This is true also when you consider a Higgs-like Lagrangian and you have not even to check if there is a non-null vacuum expectation value. Rather this appears true a posteriori. It is interesting to note that when a scalar field like this interacts with a Fermion field, a Nambu-Jona-Lasinio model is immediately obtained. So, all the mechanisms cited so far are entangled in some way.

In the end, only experiment will say to us what is the proper one. Now, it is just a matter of personal taste and nothing else. Higgs mechanism is the most widely accepted even if no sound experimental verification exists at the moment. It is up to people at CERN to say if this is also the mechanism of choice for nature and presently they are those having more information about on Earth on this.

Classical solutions point to a situation where this Higgs-like particle found at LHC cannot be alone. This is what I discuss in my post. You can check such solutions by yourself using my paper and a tool for symbolic mathematics on computer. No much effort is needed.

I am eager to know from CERN if some interesting news is coming out in the next few months. But I will not tear my hairs off if things will not go this way.

Cheers,

Marco

]]>So, what is the true mechanism for particle masses? Self-action? Or, maybe, it is better to use phenomenological masses?

]]>Feel free to correct any misinterpretations or inaccuracies that you identify.

]]>Hi Adam,

I hope we agree that these classical solutions do satisfy the given equations of motion for any value of . But you can use your preferred symbolic mathematical tool and check this.

1) Sure.

2) The point is that, for (is this the case you are considering here?), the exact classical solution is given by a Jacobi function. In this case, if you perform a quantum analysis, assuming this is your zero mode to be singled out, the spectrum has the form being . I just called the excitation the Goldstone mode.

3) For the complex case, the classical exact solutions take the form , being the solution of the equation for the single component. You can try by yourself by substitution into the equations of motion but it is just a trivial application of the idea that all the components are taken to be equal.

I have not proved the uniqueness of these classical solutions but the interesting point is that they satisfy a massive dispersion relation. The corresponding quantum field theory displays also the interesting feature to be trivial at the leading order.

Marco

]]>Hi,

I don’t really get it…

-Because of the quantum fluctuations, you would expect that for \mu_0^2<0 but close enough to zero, the symmetry stays unbroken (it is broken at the mean-field level, but quantum fluctuations destroy it).

-Because it is a real field, you should not find a Goldstone mode, right ? And in any case the formula for $\epsilon_n shows that it is never 0, even fon n=0.

-For the complexe field, do you mean that the phase is not dynamical ? That's for sure not the case.

Cheers,

Adam

]]>Dear Adam,

Thank you for your interest about my paper. Eq.(3.12) is in the quantum treatment of these solutions and applies just at the case. That is the reason why I have a zero mode here.

For I have worked out this analysis elsewhere (see http://arxiv.org/abs/hep-th/0703203) and I have got the spectrum with a Goldstone mode.

For the complex field these solutions have the form of a complex phase factor multiplying such massive solutions. All that is seen there also applies in this case.

Marco

]]>I’m having a look at your J.Nonlin.Math.Phys paper, and there is something I don’t understand. If I’m right, you are studying a scalar field with one component (it’s a real field, and in the O(N) terminology, it is the case N=1). In euclidian space, this field theory is related to the Ising model. In particular, because the symmetry \phi \to -\phi is discrete, there is no Goldstone mode.

I don’t really understand why you say after equation 3.12 that you have a zero mode, as \epsilon_0 is different from zero…

We nevertheless expect a phase transition for this model at some finite (and negative) \mu_0^2, so you should definitely have a zero mode for some value of your parameters. Is that the case ?

Have you tried to solve the case of a complex field. Here you should expect a Goldstone mode.

Cheers,

Adam

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