## Paper with a proof of confinement has been accepted

28/03/2018

Recently, I wrote a paper together with Masud Chaichian (see here) containing a mathematical proof of confinement of a non-Abelian gauge theory based on Kugo-Ojima criterion. This paper underwent an extended review by several colleagues well before its submission. One of them has been Taichiro Kugo, one of the discoverers of the confinement criterion, that helped a lot to improve the paper and clarify some points. Then, after a review round of about two months, the paper has been accepted in Physics Letters B, one of the most important journals in particle physics.

This paper contains the exact beta function of a Yang-Mills theory. This confirms that confinement arises by the combination of the running coupling and the propagator. This idea was around in some papers in these latter years. It emerged as soon as people realized that the propagator by itself was not enough to grant confinement, after extended studies on the lattice.

It is interesting to point out that confinement is rooted in the BRST invariance and asymptotic freedom. The Kugo-Ojima confinement criterion permits to close the argument in a rigorous way yielding the exact beta funtion of the theory.

## f0(500) and f0(980) are not tetraquarks

27/06/2014

Last week I have been in Giovinazzo, a really beautiful town near Bari in Italy. I participated at the QCD@Work conference. This conference series is now at the 7th edition and, for me, it was my second attendance. The most striking news I heard was put forward in the first day and represents a striking result indeed. The talk was given by Maurizio Martinelli on behalf of LHCb Collaboration. You can find the result on page 19 and on an arxiv paper . The question of the nature of f0(500) is a vexata quaestio since the first possible observation of this resonance. It entered in the Particle Data Group catalog as f0(600) but was eliminated in the following years. Today its existence is no more questioned and this particle is widely accepted. Also its properties as the mass and the width are known with reasonable precision starting from a fundamental work by Irinel Caprini, Gilberto Colangelo and Heinrich Leutwyler (see here). The longstanding question around this particle and its parent f0(980) was about their nature. It is generally difficult to fix the structure of a resonance in QCD and there is no exception here.

The problem arose from famous papers by Jaffe on 1977 (this one and this one) that using a quark-bag model introduced a low-energy nonet of states made of four quarks each. These papers set the stage for what has been the current understanding of the f0(500) and f0(980) resonances. The nonet is completely filled with all the QCD resonances below 1 GeV and so, it seems to fit the bill excellently.

Someone challenged this kind of paradigm and claimed that f0(500) could not be a tetraquark state (e.g. see here and here but also papers by Wolfgang Ochs and Peter Minkowski disagree with the tetraquark model for these resonances). The answer come out straightforwardly from LHCb collaboration: Both f0(500) and f0(980) are not tetraquark and the original view by Jaffe is no more supported. Indeed, people that know the Nambu-Jona-Lasinio model should know quite well where the f0(500) (or $\sigma$ ) comes from and I would also suggest that this model can also accommodate higher states like f0(980).

I should say that this is a further striking result coming from LHCb Collaboration. Hopefully, this should give important hints to a better understanding of low-energy QCD.

$\overline{B}^0\rightarrow J/ψπ^+π^-$ decays arXiv arXiv: 1404.5673v2
Irinel Caprini, Gilberto Colangelo, & Heinrich Leutwyler (2005). Mass and width of the lowest resonance in QCD Phys.Rev.Lett.96:132001,2006 arXiv: hep-ph/0512364v2
Jaffe, R. (1977). Multiquark hadrons. I. Phenomenology of Q^{2}Q[over ¯]^{2} mesons Physical Review D, 15 (1), 267-280 DOI: 10.1103/PhysRevD.15.267
Jaffe, R. (1977). Multiquark hadrons. II. Methods Physical Review D, 15 (1), 281-289 DOI: 10.1103/PhysRevD.15.281
G. Mennessier, S. Narison, & X. -G. Wang (2010). The sigma and f_0(980) from K_e4+pi-pi, gamma-gamma scatterings, J/psi,
phi to gamma sigma_B and D_s to l nu sigma_B Nucl.Phys.Proc.Suppl.207-208:177-180,2010 arXiv: 1009.3590v1

Marco Frasca (2010). Glueball spectrum and hadronic processes in low-energy QCD Nucl.Phys.Proc.Suppl.207-208:196-199,2010 arXiv: 1007.4479v2

## Kyoto, arXiv and all that

12/11/2012

Today, Kyoto conference HCP2012 has started. There is already an important news from LHCb that proves for the first time the existence of the decay $B_s\rightarrow\mu^+\mu^-$. They find close agreement with the Standard Model (see here). Another point scored by this model and waiting for new physics yet. You can find the program with all the talks to download here. There is a lot of expectations from the update on the Higgs search: The great day is Thursday. Meantime, there is Jester providing some rumors (see here on twitter side) and seem really interesting.

I have a couple of papers to put to the attention of my readers from arXiv. Firstly, Yuan-Sen Ting and Bryan Gin-ge Chen provided a further improved redaction of the Coleman’s lectures (see here). This people is doing a really deserving work and these lectures are a fundamental reading for any serious scholar on quantum field theory.

Axel Weber posted a contribution to a conference (see here) summing up his main conclusions on the infrared behavior of the running coupling and the two-point functions for a Yang-Mills theory. He makes use of renormalization group and the inescapable conclusion is that if one must have a decoupling solution, as lattice computations demand, then the running coupling reaches an infrared trivial fixed point. This is in close agreement with my conclusions on this matter and it is very pleasant to see them emerge from another approach.

Sidney Coleman (2011). Notes from Sidney Coleman’s Physics 253a arXiv arXiv: 1110.5013v4

Axel Weber (2012). The infrared fixed point of Landau gauge Yang-Mills theory arXiv arXiv: 1211.1473v1

## A new year full of promises

03/01/2012

We have left 2011 with a lot of exciting results from experiments. Neutrinos appear to move a bit faster than expected and Higgs provided some glimpses at CERN. Of course, this kind of Higgs appears somewhat boring at first being in the range of what Standard Model expected. But it is really too early to say something for sure. We expect definite answer for the next summer with a lot more data analyzed by people at CERN.

With the new year, I would like to point out to my readers a couple of nice papers that are really worthwhile reading. About CUDA and lattice QCD, my Portuguese friends, Pedro Bicudo and Nuno Cardoso,  made a relevant step beyond and made available their code for working for a generic SU(N) gauge group (see here, their code is here). As I have some time I will try their code. The work of these people is excellent and making their code worldwide available is really helpful for all our community.

Finally, Axel Maas put forward a revision of his very good review paper (see here). Axel gave important contributions to the current understanding of Yang-Mills theory and his paper yields a lucid description of these ideas that rely on a large effort on lattice computations and functional methods. Often, I complain about the fact that the community at large seems to not consider these lines of research reliable yet to work with. This is not true as the results they were able to get give since now sound results to work with and the most important of these are that Yang-Mill theory has indeed a mass gap and that this theory appears to display a running coupling reaching zero lowering momenta, a completely unexpected result that goes against common wisdom but this is just what lattice put out.

So, let me wish to you a great 2012 and I hope to share with you the excitement physics research is promising.

Nuno Cardoso, & Pedro Bicudo (2011). Generating SU(Nc) pure gauge lattice QCD configurations on GPUs with
CUDA and OpenMP arXiv arXiv: 1112.4533v1

Axel Maas (2011). Describing gauge bosons at zero and finite temperature arXiv arXiv: 1106.3942v2

## QCD at finite temperature

19/11/2011

The great news for me, in this week, has been the acceptance of my paper of QCD at finite temperature in Physical Review C (see here). This chance materialized after the excellent work of the referee that helped me to improve the paper in a significant way. For a good paper, such a way to review is a fundamental one and should be a rule. I have discussed this paper previously in my blog (see here and here) and I presented its content in a conference in Paris this year (see here). The contribution to proceedings is this.

I think that the main conclusions of this paper that should be emphasized is that the low-energy limit of QCD is a non-local Nambu-Jona-Lasinio model, a critical point exists at finite temperature with zero mass and zero chemical potential and that the instanton liquid picture of the vacuum of QCD is a very good one. These are fundamental questions in QCD that were waiting for an answer for so long.

I would like to thank all the people that, with their efforts and interest about my work, helped me to get these results published, in the end, in such an important journal.

Marco Frasca (2011). Chiral symmetry in the low-energy limit of QCD at finite temperature arXiv arXiv: 1105.5274v4

Marco Frasca (2011). Low-energy limit of QCD at finite temperature arXiv arXiv: 1110.0096v1

## An interesting review

14/09/2011

It is some time I am not writing posts but the good reason is that I was in Leipzig to IRS 2011 Conference, a very interesting event in a beautiful city.  It was inspiring to be in the city where Bach spent a great part of his life. Back to home, I checked as usual my dailies from arxiv and there was an important review by Boucaud, Leroy, Yaouanc, Micheli, Péne and Rodríguez-Quintero. This is the French group that produced striking results in the analysis of Green functions for Yang-Mills theory.

In this paper they do a great work by reviewing the current situation and clarifying  the main aspects of the analysis carried out using Dyson-Schwinger equations. These are a tower of equations for the n-point functions of a quantum field theory that can be generally solved by some truncation (with an exception, see here) that cannot be completely controlled. The reason is that the equation of lower order depends on n-point functions of higher orders and so, at some point, we have to decide the behavior of some of these higher order functions truncating the hierarchy. But this choice is generally not under control.

About these techniques there is a main date, Reigensburg 2007, when some kind of wall just went down. Since then, the common wisdom was a scenario with a gluon propagator going to zero when momenta go to zero while, in the same limit, the ghost propagator should go to infinity faster than the free case: So, the gluon propagator was suppressed and the ghost propagator enhanced at infrared. On the lattice, such a behavior was never explicitly observed but was commented that the main reason was the small volumes considered in these computations. On 2007, volumes reached a huge extension in lattice computations, till (27fm)^4, and so the inescapable conclusion was  that lattice produced another solution: A gluon propagator reaching a finite non-zero value and the ghost propagator behaving exactly as that of a free particle. This was also the prevision of the French group together with other researchers as Cornwall, Papavassiliou, Aguilar, Binosi and Natale. So, this new solution entered into the mainstream of the analysis of Yang-Mills theory in the infrared and was dubbed “decoupling solution” to distinguish it from the former one, called instead “scaling solution”.

In this review, the authors point out an important conclusion: The reason why authors missed the decoupling solution and just identified the scaling one was that their truncation forced the Schwinger-Dyson equation to a finite non-zero value of the strong coupling constant. This is a crucial point as this means that authors that found the scaling solution were admitting a non-trivial fixed point in the infrared for Yang-Mills equations. This was also the recurring idea in that days but, of course, while this is surely true for QCD, a world without quarks does not exist and, a priori, nothing can be said about Yang-Mills theory, a theory with only gluons and no quarks. Quarks change dramatically the situation as can also be seen for the asymptotic freedom. We are safe because there are only six flavors. But about Yang-Mills theory nothing can be said in the infrared as such a theory is not seen in the reality if not interacting with fermionic fields.

Indeed, as pointed out in the review, the running coupling was seen to behave as in the following figure (this was obtained by the German group, see here)

Running coupling of a pure Yang-Mills theory as computed on the lattice

This result is quite shocking and completely counterintuitive. It is pointing out, even if not yet confirming, that a pure Yang-Mills theory could have an infrared trivial fixed point! This is something that defies common wisdom and can explain why former researchers using the Dyson-Schwinger approach could have missed the decoupling solution. Indeed, this solution seems properly consistent with a trivial fixed point and this can also be inferred by the goodness of the fit of the gluon propagator with a Yukawa-like propagator if we content ourselves with the best agreement just in the deep infrared and the deep ultraviolet where asymptotic freedom sets in. In fact, with a trivial fixed point the theory is free in this limit but you cannot pretend agreement on all the range of energies with a free propagator.

Currently, the question of the right infrared behavior of the two-point functions for Yang-Mills theory is hotly debated yet and the matter that is at stake here is the correct understanding and management of low-energy QCD. This is one of the most fundamental physics problem and something I would like to know the answer.

Ph. Boucaud, J. P. Leroy, A. Le Yaouanc, J. Micheli, O. Péne, & J. Rodríguez-Quintero (2011). The Infrared Behaviour of the Pure Yang-Mills Green Functions arXiv arXiv: 1109.1936v1

Marco Frasca (2009). Exact solution of Dyson-Schwinger equations for a scalar field theory arXiv arXiv: 0909.2428v2

I. L. Bogolubsky, E. -M. Ilgenfritz, M. Müller-Preussker, & A. Sternbeck (2009). Lattice gluodynamics computation of Landau-gauge Green’s functions in
the deep infrared Phys.Lett.B676:69-73,2009 arXiv: 0901.0736v3

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## Paper on critical temperature revised

06/07/2011

Yesterday, I have uploaded a new version of my paper on the critical temperature of chiral symmetry breaking in QCD (see here). The reason for this was that there are some points in need for a better clarification. The main of these is the mapping theorem: I have added a sketch of a proof. The reason for this is that there is a common misunderstanding about it and that some people think  that this theorem is for quantum field theories. Indeed, it just establishes a map between classical solutions of a scalar field and a Yang-Mills field but in the asymptotic limit of a coupling going to infinity. Quantum theory does not enter at all here but these classical asymptotic solutions can be used to build up a perturbation theory for quantum field theory in the infrared, that is for low-energies, that is the range of interest for all the phenomenology we would like to understand.

Another recurring question is if this mapping breaks in some way gauge invariance. The answer is a resounding no as the proof does not select a gauge at the start but anyhow if one wants quantization a gauge must be selected.

Finally, I have better clarified the derivation of the critical temperature and added some more relevant references. I hope in this way that my arguments can be better understood. Indeed, presentation is one of the most difficult aspects of scientific communication and sometime it is a sound explanation of attrition between authors and referees.