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

The question of the mass gap


Some years ago I proposed a set of solutions to the classical Yang-Mills equations displaying a massive behavior. For a massless theory this is somewhat unexpected. After a criticism by Terry Tao I had to admit that, for a generic gauge, such solutions are just asymptotic ones assuming the coupling runs to infinity (see here and here). Although my arguments on Yang-Mills theory were not changed by this, I have found such a conclusion somewhat unsatisfactory. The reason is that if you have classical solutions to Yang-Mills equations that display a mass gap, their quantization cannot change such a conclusion. Rather, one should eventually expect a superimposed quantum spectrum. But working with asymptotic classical solutions can make things somewhat involved. This forced me to choose the gauge to be always Lorenz because in such a case the solutions were exact. Besides, it is a great success for a physicist to find exact solutions to fundamental equations of physics as these yield an immediate idea of what is going on in a theory. Even in such case we would get a conclusive representation of the way the mass gap can form.

Finally, after some years of struggle, I was able to get such a set of exact solutions to the classical Yang-Mills theory displaying a mass gap (see here). Such solutions confirm both the Tao’s argument that an all equal component solution for Yang-Mills equations cannot hold in any gauge and also my original argument that an all equal component solution holds, in a general case, only asymptotically with the coupling running to infinity. But classically, there exist solutions displaying a mass gap that arises from the nonlinearity of the equations of motion. The mass gap goes to zero as the coupling does. Translating this in the quantum realm is straightforward as I showed for the Lorenz (Landau) gauge. I hope all this will help to better elucidate all the physics around strong interactions. My efforts since 2005 went in that direction and are still going on.

Marco Frasca (2009). Mapping a Massless Scalar Field Theory on a Yang-Mills Theory: Classical Case Mod. Phys. Lett. A 24, 2425-2432 (2009) arXiv: 0903.2357v4

Marco Frasca (2014). Exact solutions for classical Yang-Mills fields arXiv arXiv: 1409.2351v1

A beautiful paper on arxiv


Keeping on their way of producing sound work, Bogolubsky, Ilgenfritz, Mueller-Preussker and Sternbeck have got their paper (see here) published on Physics Letters B. This is a collaboration between people working in Russia, Germany and Australia. The main aim of this work is the computation on a lattice of the two-point functions and the running coupling of a pure Yang-Mills theory. They carry on lattice computations from (64)^4 to (96)^4 points entering into a deep enough infrared limit to get a meaningful behavior of the lattice theory in this case. I give below their main results

Gluon propagator

Gluon propagator

Dressing function of the ghost propagator

Dressing function of the ghost propagator

Running coupling

Running coupling

These results confirm completely the decoupling solution. The definition of the running coupling is the one proposed by Alkofer and von Smekal and it is my personal conviction that it conveys the right physical behavior of the theory. This is exactly the scenario I have derived in my paper (see here) that has been published on Physics Letters B too and has arisen a lot of rumors around. You will not find this paper cited in this work as these authors have concerns about gauge invariance in my computations. As you may know from my dispute with Terry Tao, gauge invariance is not a problem here. One could ask why a mathematical technique, like a gradient expansion is, should not work for Yang-Mills equations but it does for all other equations of mathematical physics. Anyhow, I am here ready to listen to whoever is able to prove this. With this proof in hand one should also warn all general relativists that use this technique and put it in their handbooks.

The authors conclude their paper by pointing out weaknesses in lattice computations that may bring in discussion their results. Finally, they ask if the other solution, the one with a scaling behavior, can emerge from lattice computations. The understanding of this question is surely of relevant interest. We stay tuned to hear news about.

Paper replacement


I have updated the paper with the answer to Terry Tao on arxiv (see here). No correction was needed, rather I have added a new result giving the next-to-leading order correction for the Yang-Mills field. This result is important as it shows the right approximate solution, in an expansion into the inverse of the coupling constant, for the mapping between the scalar and the Yang-Mills field. As we repeated a lot of times, Smilga’s solutions are all is needed to work out our argument as this relies on a gradient expansion. A gradient expansion at the leading order has a solution depending just on time variable. But, as this has been a reason for discussion, I have also shown to what extent my approach applies to the solution of the quartic scalar field given in the form

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

with p^2=\mu^2\left(\lambda/2\right)^{1\over 2} with \mu an integration constant and \lambda the coupling. But I would like to emphasize that the relevance of these solutions for the Yang-Mills case was just demanded by Tao’s criticism but it is not needed for my argument to work. So, the main result of this paper is that


As it has been noted elsewhere, higher order corrections are zero in the Lorenz gauge. This result is certainly not trivial and worth to be considered in a classical analysis of Yang-Mills equations.

Finally, we note as any concern about gauge invariance is just worthless. Smilga’s solutions are exact solutions of the Yang-Mills equations. Casting doubt on them using gauge invariance arguments should be put on the same ground as casting doubt on Kasner solution of Einstein equations using general covariance reasons. Nothing worth to spend time on but a poor excuse to ignore a good work.

Exact solutions of Yang-Mills theory: The situation


Some time passed by since Terry Tao was so kind to take a look to my work. His concern about a main theorem in my paper, the so called mapping theorem, was motivated by the fact that no proof exists that there are common solutions between Yang-Mills equations and the one of the quartic scalar field. This point is quite crucial as, if such solutions do not exist, I cannot do any claim about Yang-Mills theory.

Some people are in confusion yet about this matter and I find occasionally someone, e.g. the Czech guy, claiming that my paper is false also after I have proved that such solutions exist.

Of course, Terry meant to point out a weakness in the proof given in my paper as I gave no evidence whatsoever of the form of these solutions and so the proof is, at least, incomplete. My next preprint proved that such solutions indeed exist and my argument is true already at level of perturbation theory. The conclusion is straightforward: Smilga’s choice select a class of common solutions between Yang-Mills equations and a quartic scalar field. I have not presented them explicitly in my paper and this is the reason why all this arguing was started. Terry’s suggestion was to complete the proof  and this I have done.

Curiously enough, I was able to see such solutions only in the Smilga’s book. I think this was Smilga’s idea and was also my source of inspiration.  I was in need of these solutions to treat classical Yang-Mills equations with a gradient expansion against a lot of unmanageable chaotic solutions. I would like to remember here that this approach is quite common in physics. For interested readers, I invite them to look at this beautiful Wikipedia entry about BKL solution. This is the way this approach is used in general relativity with a widespread example as the Kasner solution. This is an exact solution of Einstein equations that depends solely on time. Exactly as happens to the solutions obtained by a Smilga’s choice from Yang-Mills equations. Indeed, I suspect that Kasner solution may be helpful to quantize Einstein equations in the infrared limit. Currently I have no time to exploit this but I have given a hint about here.

Dmitry Podolsky (see his blog here) hit correctly the point when asked for the fate of chaotic solutions in the infrared quantum field theory. Presently, the fact that they are not relevant has the status of a conjecture: No quantum field theory can be built out of classical chaotic solutions. I do not even know how to face this kind of question as no closed form chaotic solutions exist to start from.

Finally, this gives the current situation about this matter. My paper that started all this is correct and in agreement with current lattice results. People’s mood about lattice computations range from fully convinced to skeptical.  My view is that they represent correctly the infrared physics at hand but I am a supporter of these people working on lattice computations and so, my judgement should not be counted.



Today I have posted a paper on arxiv. It will appear on monday. This paper was required to my by Terry Tao to supplement the proof of the mapping theorem showing that indeed it holds.

If you cannot hold the paper is 09032357v1: preprint. Don’t trust that number as may change.

The argument may be put up with very simple words: If you trust Smilga’s solutions that depend only on time, a Lorentz boost will fit the bill.

I did it for you


It is very easy to show, from Yang-Mills equations, how to obtain a scalar field equation through the Smilga’s choice. Let us write down Yang-Mills equations

\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

using the choice A_1^1=A_2^2=A_3^3=\phi. This is really a great simplification. Smilga, in his book, already checked this for us but we give here the full computation. From above eqautions, the only critical term is the following

f^{abc}A^{b\mu}(\partial_\mu A^c_\nu-\partial_\nu A^c_\mu)

as this term would produce terms deviating from the known form of the scalar theory. For SU(2) we have f^{abc}=\epsilon^{abc} the fully-antisymmetric Levi-Civita tensor. This means that we will have

\epsilon^{a1c}A^{11}(\partial_1A_\nu^c-\partial_\nu A_1^c)+

\epsilon^{a2c}A^{22}(\partial_2A_\nu^c-\partial_\nu A_2^c)+

\epsilon^{a3c}A^{33}(\partial_3A_\nu^c-\partial_\nu A_3^c).

Where we have used largely Smilga’s choice. Now do the following. Take the following components to evolve \nu=1 a=1, \nu=2  a=2 and \nu=3, a=3. It easy to see that the possible harmful term is zero with the Smilgaì’s choice. Now, for the cubic term you should use the useful relation


and you will get back the quartic term.

The gauge fixing term can be easily disposed of through a rescaling of spatial variables while the kinematic term gives the right contribution. You will get three identical equations for the scalar field.

Of course, Smilga in his book already did this and I repeated his computations after the Editor of PLB asked for a revision having the referee already put out this problem. The Editorial work was done very well and two referees read the paper emphasizing errors where they were.

Finally, Tao’s critcism does not apply as I said. This does not mean that what he says is wrong. This means that does not apply to my case.

Update: As the question of the gauge fixing term appears so relevant, let me fix it once and for all. Firstly, I would like to point out that these solutions belong to a class of solutions in the Maximal Abelian Gauge (MAG). But let us forget about this and consider the question of gauge fixing. This term is arbitrarily introduced in the Lagrangian of the field in order to fix the gauge when a quantization procedure is applied. Due to gauge invariance and the fact that becomes an exact differential after partial integration, it useful to have it there for the above aims. The form that it  takes is

\frac{1}{\alpha}(\partial A)^2

and is put directly into the Lagrangian. How does this term become with the Smilga’s choice? One has


and the final effect is a pure rescaling into the space variables of the scalar field. In this way the argument is made consistent. One cannot take the other way around for the very nature of this term and claiming the result is wrong.

This particular class of solutions belongs to the subgroup of SU(N) given by the direct product of U(1). This is a property of MAG and all the matter is really consistent and works.

Finally, I invite people commenting this and other posts to limit herself to polite responses and in the realm of scientific discussion. Of course, doing something wrong happens and happened to anyone working in a scientifc endeavour for the simple reason that she is really doing things. People that only do useless criticisms boiling down to personal offenses are kindly invited to refrain from further interventions.

Update 2: I will get a paper published about this matter. Please, check here.


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