Is Higgs alone?


I am back after the announcement by CERN of the restart of LHC. On May this year we will have also the first collisions. This is great news and we hope for the best and the best here is just the breaking of the Standard Model.

The Higgs in the title is not Professor Higgs but rather the particle carrying his name. The question is a recurring one since the first hints of existence made their appearance at the LHC. The point I would like to make is that the equations of the theory are always solved perturbatively, even if exact solutions exist that provide a mass also if the theory is massless or has a mass term with a wrong sign (Higgs model). All you need is a finite self-interaction term in the equation. So, you will have bad times to recover such exact solutions with perturbation techniques and one keeps on living in the ignorance. If you would like to see the technicalities involved just take a cursory look at Dispersive Wiki.

What is the point? The matter is rather simple. The classical theory has exact massive solutions for the potential in the form V(\phi)=a\phi^2+b\phi^4 and this is a general result implying that a scalar self-interacting field gets always a mass (see here and here). Are we entitled to ignore this? Of course no. But today exact solutions have lost their charm and we can get along with them.

For the quantum field theory side what could we say? The theory can be quantized starting with these solutions and I have shown that one gets in this way that these massive particles have higher excited states. These are not bound states (maybe could be correctly interpreted in string theory or in a proper technicolor formulation after bosonization) but rather internal degrees of freedom. It is always the same Higgs particle but with the capability to live in higher excited states. These states are very difficult to observe because higher excited states are also highly depressed and even more hard to see. In the first LHC run they could not be seen for sure. In a sense, it is like Higgs is alone but with the capability to get fatter and present himself in an infinite number of different ways. This is exactly the same for the formulation of the scalar field as originally proposed by Higgs, Englert, Brout, Kibble, Guralnik and Hagen. We just note that this formulation has the advantage to be exactly what one knows from second order phase transitions used by Anderson in his non-relativistic proposal of this same mechanism. The existence of these states appears inescapable whatever is your best choice for the quartic potential of the scalar field.

It is interesting to note that this is also true for the Yang-Mills field theory. The classical equations of this theory display similar solutions that are massive (see here) and whatever is the way you develop your quantum filed theory with such solutions the mass gap is there. The theory entails the existence of massive excitations exactly as the scalar field does. This have been seen in lattice computations (see here). Can we ignore them? Of course no but exact solutions are not our best choice as said above even if we will have hard time to recover them with perturbation theory. Better to wait.

Marco Frasca (2009). Exact solutions of classical scalar field equations J.Nonlin.Math.Phys.18:291-297,2011 arXiv: 0907.4053v2

Marco Frasca (2013). Scalar field theory in the strong self-interaction limit Eur. Phys. J. C (2014) 74:2929 arXiv: 1306.6530v5

Marco Frasca (2014). Exact solutions for classical Yang-Mills fields arXiv arXiv: 1409.2351v2

Biagio Lucini, & Marco Panero (2012). SU(N) gauge theories at large N Physics Reports 526 (2013) 93-163 arXiv: 1210.4997v2


Exact solutions go published!


My paper presenting exact solutions to classical scalar field theories, with a corresponding quantum formulation, has been accepted for publication in the Journal of Nonlinear Mathematical Physics.  The replacement on arxiv will appear tomorrow, the link is here. I would like to thank the Editor, Norbert Euler, and an anonymous referee that pointed out to me the existence of a zero mode in the quantum fluctuations.

Classical Yang-Mills theory and mass gap


One of the key ingredients to build up a quantum field theory is to have a set of solutions of classical equations of motion to start with. Then, given such solutions, we are able to perform perturbation theory and to extract results from the theory to be compared with experiment. I think that my readers are familiar with standard approach having free equations of the theory solved. When path integrals are used, we solve for the Green function of the free theory but we are talking about the same thing: we know how to solve our theory in some limit and then we build on it. So, to give an answer to the question of the mass gap for Yang-Mills theory, we have to know how to solve the theory in a limit we are not so familiar: strong coupling limit. So far, very few was known about this limit except knowledge acquired through lattice computations. Also in this latter case, for several years a lot of confusion pervaded the field: Does gluon propagator go to zero or not? Enlarging volumes produced an answer that is a reason for hot debate yet: Gluon propagator does not go to zero at very low energy but reaches a finite value. In literature this is known as the decoupling solution to be contrasted with the scaling solution having a propagator going to zero at very small momenta. If we know gluon propagator, we are able to compute the behavior of QCD at very low energies (see here) and this is a well-known fact since eighties.

The question of existence of a class of solutions for Yang-Mills theory to work with at low energies has been successfully answered quite recently. I have written a nice pair of papers that went published in respectful journals and permitted to solve all this matter (see here and here). Two papers were needed because Terry Tao showed that a proof in a key theorem (mapping theorem) was not correct. After this, I was able to give  an answer that both agreed. My aim in this post is to explain, with some simple mathematics, what is the content of this theorem that produces a set of classical solutions to build up a quantum field theory in the low-energy limit for Yang-Mills theory and so QCD.

The key element is a mapping theorem. We map two classical theories, one of this we are able to solve exactly. So, consider a massless scalar field theory


Contrarily to common wisdom, we are able to solve this exactly. Our solution can be written down as

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

provided that


Hera \mu is an integration constant with the dimension of energy, \theta is another integration constant and {\rm sn} is the snoidal Jacobi’s elliptic function. Why is this solution so interesting? The reason is that we started with a massless equation and the solution describes a wave with a massive dispersion solution of a free particle! This is the famous mass gap when we translate this result to quantum field theory. I have done this here. So, the classical theory already has the feature of a mass gap. Scalar theory proves to be trivial for the simple reason that we produce, in the low-energy limit, free massive excitations. This is a long awaited result that is going to get increasingly confirmed from other theoretical studies. I will discuss this issue in another post.

What is the relation, if any, between a massless scalar field theory and Yang-Mills theory? Indeed, there exists a deep relation in the low-energy limit, when the coupling becomes increasingly large, as the solutions of the two theories can be mapped. So, for SU(3), mapping theorem shows that


being \phi(x) our solution above provided the substitution \lambda\rightarrow\sqrt{3}g. This is a very beautiful result as this gives at once the following conclusions:

  • Strong coupling solutions of classical Yang-Mills theory are free massive waves.
  • Yang-Mills theory displays massive solutions already at classical level.
  • Quantum theory maintains such conclusions as I showed in my papers.

Lattice computations beautifully confirmed this mapping theorem in d=2+1 as showed by Rafael Frigori in a very nice paper (see here). Strong hints are also seen in d=3+1 by other authors and it would be very nice to see an extended computation in this case as the one Frigori did in d=2+1. For yourselves, you can check with Mathematica or Maple the equations given above. You will also see that gauge invariance is not hindered.

Classical solutions of Yang-Mills equations


So far, I have posted several posts in this blog about the question of classical solutions to Yang-Mills equations. This has produced some fuzz, mostly arisen from my published papers, as to such solutions may not be correct. Thanks to a wise intervention of Terry Tao, I was able to give a complete understanding of my solutions and a theorem was fully proved in a recent paper of mine to appear in Modern Physics Letters A (see here), agreed with Terry in a private communication. So, I think it is time to give a description of this result here as it appears really interesting showing how, already at a classical level, this theory can display massive solutions and a mass gap is already seen in this case. Then, it takes a really small step to get the corresponding proof in quantum field theory.

To understand how these solutions are obtained, let us consider the following equation for a scalar field


This is a massless self-interacting field. We can select a class of solutions by looking at the case of a rest reference frame. So, we put any dependence on spatial variables to zero and solve the equation


whose solutions are known and given by

\phi(t,0)=\mu\left(\frac{2}{\lambda}\right)^{\frac{1}{4}}{\rm sn}\left[\left(\frac{\lambda}{2}\right)^{\frac{1}{4}}\mu t+\theta,i\right]

being \mu and \theta two integration constants and sn a Jacobi elliptical function. Then, boosting this solution will produce an exact solution of the equation we started from given by

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

provided the following dispersion relation holds


and we see that, although we started with a massless field, self-interaction provided us massive solutions!

Now, the next question one should ask is if such a mechanism may be at work for classical Yang-Mills equations. These can be written down as

\partial^\mu\partial_\mu A^a_\nu-\left(1-\frac{1}{\alpha}\right)\partial_\nu(\partial^\mu A^a_\mu)+gf^{abc}A^{b\mu}(\partial_\mu A^c_\nu-\partial_\nu A^c_\mu)

+gf^{abc}\partial^\mu(A^b_\mu A^c_\nu)+g^2f^{abc}f^{cde}A^{b\mu}A^d_\mu A^e_\nu = 0

being \alpha chosen depending on the gauge choice, g the coupling and f^{abc} the structure constants of the gauge group taken to be SU(N). The theorem I proved in my paper above states that the solution given for the scalar field theory is an exact solution of Yang-Mills equations, provided it will not depend on spatial coordinates, for a given choice of Yang-Mills components (Smilga’s choice) and \lambda=Ng^2, otherwise the following identity holds


Here \eta_\mu^a is a set of constants arising with the Smilga’s choice. This theorem has the following implications: Firstly, when the coupling become increasingly large, a massless scalar field theory and Yang-Mills theory can be mapped each other. Secondly, already at the classical level, for a coupling large enough, a Yang-Mills theory gets massive solutions. We can see here that a mass gap arises already at a classical level for these theories. Finally, we emphasize that the above mapping appears to hold only in a strong coupling regime while, on the other side, these theories manifest different behaviors. Indeed, we know that Yang-Mills theory has asymptotic freedom while the scalar theory has not. The mapping theorem just mirrors this situation.

We note that these solutions are wave-like ones and describe free massive excitations. This means that these classical theories have to be considered trivial in some sense as these solutions seem to behave in the same way as the plane waves of a free theory.

One can build a quantum field theory on these classical solutions obtaining a theory manifesting a mass gap in some limit. This is has been done in several papers of mine and I will not repeat these arguments here.

Exact solutions on arxiv


As promised in my preceding post (see here), I have posted yesterday a preprint on this matter on arxiv (see here). I have presented all the exact solutions I was able to obtain at a classical level and I have given a formulation of the quantum field theory for a massless quartic theory. The key point in this case is the solution of the equation for the propagator


being \phi_c the given exact classical solution. As usual, I have used a gradient approximation and the solution of the equation


that I know when the phase in \phi_c(t,0) is quantized as (4n+1)K(i), being n an integer and K(i) an elliptic integral. This gives back a consistent result in the strong coupling limit, \lambda\rightarrow\infty, with my preceding paper on Physical Review D (see here).

The conclusion is rather interesting as quantum field theory, given from such subset of classical solutions, is trivial when the coupling becomes increasingly large as one has a Gaussian generating functional and the spectrum of a harmonic oscillator. This is in perfect agreement with common wisdom about this scalar theory. So, in some way, Jacobi elliptical functions that describe nonlinear waves behave as plane waves for a quantum field theory in a regime of a strong coupling.

Exact solutions of nonlinear equations


Recently, I have posted on the site of Terry Tao and Jim Colliander, Dispersive Wiki. I am a regular contributor to this beautiful effort to collect all available knowledge about differential equations and dispersive phenomena. Of course, I can give contributions as a solver of differential equations in the vein of a pure physicist. But mathematicians are able to give rigorous theorems on the behavior of the solutions without really solving them. I invite you to take some time to look at this site and, if you are an expert, to register and contribute to it.

My recent contribution is about exact solutions of nonlinear equations. This is a really interesting field and most of the relevant results come from soliton theory.  Terry posted on his blog about Liouville equation (see here). This equation is exactly solvable and is widely known to people working in string theory. But also one of the most known equations in physics literature can be solved exactly. My preprint shows this. Indeed I have to update it as, working on KAM theorem, I have obtained the exact solution to the following equation (check here on Dispersive Wiki):

\Box\phi +\mu_0^2\phi+\lambda\phi^3 = 0

that can be written as

\phi(x) = \pm\sqrt{\frac{2\mu^4}{\mu_0^2 + \sqrt{\mu_0^4 + 2\lambda\mu^4}}}{\rm sn}\left(p\cdot x+\theta,\sqrt{\frac{-\mu_0^2 + \sqrt{\mu_0^4 + 2\lambda\mu^4}}{-\mu_0^2 -  \sqrt{\mu_0^4 + 2\lambda\mu^4}}}\right)

being now the dispersion relation


As always \mu is an arbitrary parameter with the dimension of a mass. You can see here an example of mass renormalization due to interaction. Indeed, from the dispersion relation we can recognize the following renormalized mass


that depends on the coupling. This class of solutions clearly show how the nonlinearities produce contributions to mass. Either by modifying it or by generating it. So, it is not difficult to imagine that Nature may have adopted them to display mass wherever there is not.

As a by-product, I am now able to give a consistent quantum field theory in the infrared for the scalar field (always thank to my work on KAM theorem), obtaining the needed corrections to the propagator and the spectrum. I hope to find some time in the next days to add all this new material to my preprint. Meanwhile, enjoy Dispersive Wiki!

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