## Nailing down the Yang-Mills problem

22/02/2014

Millennium problems represent a major challenge for physicists and mathematicians. So far, the only one that has been solved was the Poincaré conjecture (now a theorem) by Grisha Perelman. For people working in strong interactions and quantum chromodynamics, the most interesting of such problems is the Yang-Mills mass gap and existence problem. The solutions of this problem would imply a lot of consequences in physics and one of the most important of these is a deep understanding of confinement of quarks inside hadrons. So far, there seems to be no solution to it but things do not stay exactly in this way. A significant number of researchers has performed lattice computations to obtain the propagators of the theory in the full range of energy from infrared to ultraviolet providing us a deep understanding of what is going on here (see Yang-Mills article on Wikipedia). The propagators to be considered are those for  the gluon and the ghost. There has been a significant effort from theoretical physicists in the last twenty years to answer this question. It is not so widely known in the community but it should because the work of this people could be the starting point for a great innovation in physics. In these days, on arxiv a paper by Axel Maas gives a great recount of the situation of these lattice computations (see here). Axel has been an important contributor to this research area and the current understanding of the behavior of the Yang-Mills theory in two dimensions owes a lot to him. In this paper, Axel presents his computations on large volumes for Yang-Mills theory on the lattice in 2, 3 and 4 dimensions in the SU(2) case. These computations are generally performed in the Landau gauge (propagators are gauge dependent quantities) being the most favorable for them. In four dimensions the lattice is $(6\ fm)^4$, not the largest but surely enough for the aims of the paper. Of course, no surprise comes out with respect what people found starting from 2007. The scenario is well settled and is this:

1. The gluon propagator in 3 and 4 dimensions dos not go to zero with momenta but is just finite. In 3 dimensions has a maximum in the infrared reaching its finite value at 0  from below. No such maximum is seen in 4 dimensions. In 2 dimensions the gluon propagator goes to zero with momenta.
2. The ghost propagator behaves like the one of a free massless particle as the momenta are lowered. This is the dominant behavior in 3 and 4 dimensions. In 2 dimensions the ghost propagator is enhanced and goes to infinity faster than in 3 and 4 dimensions.
3. The running coupling in 3 and 4 dimensions is seen to reach zero as the momenta go to zero, reach a maximum at intermediate energies and goes asymptotically to 0 as momenta go to infinity (asymptotic freedom).

Here follows the figure for the gluon propagator

and for the running coupling

There is some concern for people about the running coupling. There is a recurring prejudice in Yang-Mills theory, without any support both theoretical or experimental, that the theory should be not trivial in the infrared. So, the running coupling should not go to zero lowering momenta but reach a finite non-zero value. Of course, a pure Yang-Mills theory in nature does not exist and it is very difficult to get an understanding here. But, in 2 and 3 dimensions, the point is that the gluon propagator is very similar to a free one, the ghost propagator is certainly a free one and then, using the duck test: If it looks like a duck, swims like a duck, and quacks like a duck, then it probably is a duck, the theory is really trivial also in the infrared limit. Currently, there are two people in the World that have recognized a duck here:  Axel Weber (see here and here) using renormalization group and me (see here, here and here). Now, claiming to see a duck where all others are pretending to tell a dinosaur does not make you the most popular guy  in the district. But so it goes.

These lattice computations are an important cornerstone in the search for the behavior of a Yang-Mills theory. Whoever aims to present to the World his petty theory for the solution of the Millennium prize must comply with these results showing that his theory is able to reproduce them. Otherwise what he has is just rubbish.

What appears in the sight is also the proof of existence of the theory. Having two trivial fixed points, the theory is Gaussian in these limits exactly as the scalar field theory. A Gaussian theory is the simplest example we know of a quantum field theory that is proven to exist. Could one recover the missing part between the two trivial fixed points as also happens for the scalar theory? In the end, it is possible that a Yang-Mills theory is just the vectorial counterpart of the well-known scalar field, the workhorse of all the scholars in quantum field theory.

Axel Maas (2014). Some more details of minimal-Landau-gauge Yang-Mills propagators arXiv arXiv: 1402.5050v1

Axel Weber (2012). Epsilon expansion for infrared Yang-Mills theory in Landau gauge Phys. Rev. D 85, 125005 arXiv: 1112.1157v2

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

Marco Frasca (2007). Infrared Gluon and Ghost Propagators Phys.Lett.B670:73-77,2008 arXiv: 0709.2042v6

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 (2010). Mapping theorem and Green functions in Yang-Mills theory PoS FacesQCD:039,2010 arXiv: 1011.3643v3

## Ending and consequences of Terry Tao’s criticism

21/09/2013

Summer days are gone and I am back to work. I thought that Terry Tao’s criticism to my work was finally settled and his intervention was a good one indeed. Of course, people just remember the criticism but not how the question evolved since then (it was 2009!). Terry’s point was that the mapping given here between the scalar field solutions and the Yang-Mills field in the classical limit cannot be exact as it is not granted that they represent an extreme for the Yang-Mills functional. In this way the conclusions given in the paper are not granted being based on this proof. The problem can be traced back to the gauge invariance of the Yang-Mills theory that is explicitly broken in this case.

Terry Tao, in a private communication, asked me to provide a paper, to be published on a refereed journal, that fixed the problem. In such a case the question would have been settled in a way or another. E.g., also a result disproving completely the mapping would have been good, disproving also my published paper.

This matter is rather curious as, if you fix the gauge to be Lorenz (Landau), the mapping is exact. But the possible gauge choices are infinite and so, there seems to be infinite cases where the mapping theorem appears to fail. The lucky case is that lattice computations are generally performed in Landau gauge and when you do quantum field theory a gauge must be chosen. So, is the mapping theorem really false or one can change it to fix it all?

In order to clarify this situation, I decided to solve the classical equations of the Yang-Mills theory perturbatively in the strong coupling limit. Please, note that today I am the only one in the World able to perform such a computation having completely invented the techniques to do perturbation theory when a perturbation is taken to go to infinity (sorry, no AdS/CFT here but I can surely support it). You will note that this is the opposite limit to standard perturbation theory when one is looking for a parameter that goes to zero. I succeeded in doing so and put a paper on arxiv (see here) that was finally published the same year, 2009.

The theorem changed in this way:

The mapping exists in the asymptotic limit of the coupling running to infinity (leading order), with the notable exception of the Lorenz (Landau) gauge where it is exact.

So, I sighed with relief. The reason was that the conclusions of my paper on propagators were correct. But these hold asymptotically in the limit of a strong coupling. This is just what one needs in the infrared limit where Yang-Mills theory becomes strongly coupled and this is the main reason to solve it on the lattice. I cited my work on Tao’s site, Dispersive Wiki. I am a contributor to this site. Terry Tao declared the question definitively settled with the mapping theorem holding asymptotically (see here).

In the end, we were both right. Tao’s criticism was deeply helpful while my conclusions on the propagators were correct. Indeed, my gluon propagator agrees perfectly well, in the infrared limit, with the data from the largest lattice used in computations so far  (see here)

As generally happens in these cases, the only fact that remains is the original criticism by a great mathematician (and Terry is) that invalidated my work (see here for a question on Physics Stackexchange). As you can see by the tenths of papers I published since then, my work stands and stands very well. Maybe, it would be time to ask the author.

Marco Frasca (2007). Infrared Gluon and Ghost Propagators Phys.Lett.B670:73-77,2008 arXiv: 0709.2042v6

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

Attilio Cucchieri, & Tereza Mendes (2007). What’s up with IR gluon and ghost propagators in Landau gauge? A puzzling answer from huge lattices PoS LAT2007:297,2007 arXiv: 0710.0412v1

## Return in Paris

15/06/2013

After two years since the last edition, I was back in Paris to participate to the Twelfth Workshop on Non-perturbative Quantum Chromodynamics. The conference is organized by high-energy group at Brown University and held at Institut d’Astrophysique de ParisProfessor Chung-I Tan and Professor Berndt Mueller from Duke University are the organizers. As it also happened in the precedent edition, the workshop was really interesting and rich of ideas for research. The first talk was given by Kostantinos Orginos and was about nuclear physics emerging from lattice computations. This is a matter that I am involved into as a “final user” and so, very near my interests. It is noteworthy to point out how current technology permits  to extract such results from lattice QCD making this a useful tool for the understanding of low-energy phenomenology. With Kostantinos,  his wife Vassiliki Panoussi and sons, we have had a nice social dinner on Tuesday night and I have had an interesting discussion about the current situation of lattice computations. The next speaker was Philippe de Forcrand that is well-known for his works on finite temperature QCD on the lattice.   He showed how the effective Yang-Mills theory at high temperature is surprisingly good with respect to lattice results also lowering temperatures at few times the critical temperature. Another interesting talk was the one by Peter Petreczy about the observables of QCD at finite temperature presenting also the most recent value for the critical temperature. As my readers may know, I computed this value in my recent paper on Physical Review C (see here) properly corrected by the mass gap of Yang-Mills theory. Norberto Scoccola and Daniel Gomez-Dumm showed similar results (see here).

On Tuesday it was the ultrarelativistic Heavy-ion collision session. This was particularly interesting and involved the talks of two friends of mine: Marco Ruggieri and Salvatore Plumari. In this area of research there is a really interesting and hot debated situation. On the other side, there is plenty of experimental results from RHIC and LHC. The session chair was Jean-Yves Ollitrault. He put the foundations to the current understanding of the quark-gluon plasma through a hydrodynamic approximation. What is observed in the experiments is the production of a flow of particles in a transverse direction named elliptic flow. This is a clear evidence of existence for the quark-gluon plasma. Marco and Salvatore work in the group of Vincenzo Greco at University of Catania in Italy. The idea they based their work on is to derive the hydrodynamic equations from a kinetic description as the one provided by the Boltzmann equation. This approach opens up the scene to the possibility to derive such an equation and the full description of the quark-gluon plasma starting directly from QCD and fixing the collisional integral of the kinetic equation. Of course, one should understand the applicability conditions but my take is that, being the running coupling going to zero due to asymptotic freedom, a quark-gluon plasma should have scarce multi-collision effects. On the other side, this is a charged plasma but lives for a very small time. This means that this approach can prove to be really successful. One of the open questions is if, going at higher energies, a state called “color glass condensate” should form and this is a matter of a hot debate in the community. This is creating some tension that is reminiscent of the story I recounted about Landau gauge propagators for pure Yang-Mills theory (see here). A color glass condensate gives an increasing lower bound on the viscosity to entropy ratio by a factor 2 with respect to $1/4\pi$, also computed from string theory, and appears less efficient with respect to observed elliptic flow at RHIC (see here). This kind of wars is often unproductive in physics and science at large as it slows down progress and good works could turn out unpublished. In situations like this, researchers should have eyes wide open and open minds granting all the contenders to be fairly listened waiting for experiments or careful lattice computations to say the last word. This should teach the history of Landau gauge propagators and also by looking back to history of physics. Otherwise we will stay on a silly forever war  where we are only able to prove to the rest of mankind that nothing has been learned from the past.

On Wednesday the session was dedicated to AdS/CFT, Holography, and Scattering. There was the talk of Carl Bender that is currently working on PT quantum mechanics. He is the pioneer of strong perturbation for quantum systems and quantum field theory. I often cited his work that has been a source of inspiration. David Dudal also spoke and discussed a holographic model for the analysis of strong ion collisions and the effect of the huge magnetic field generated. He gets results reminiscent of the Nambu-Jona-Lasinio model.  David is one of the proponents of the Refined Gribov-Zwanzinger model (see here). This is a real successful approach to the understanding of Landau gauge propagators and fits quite well with my results in the deep infrared behavior of a Yang-Mills theory as I also pointed out in my talk (see below).

It was a great workshop and I have been very happy to be there also this year. I hope people at Brown University will repeat this again. Thanks a lot!

Marco Frasca (2011). Chiral symmetry in the low-energy limit of QCD at finite temperature Phys. Rev. C 84, 055208 (2011) arXiv: 1105.5274v4

D. Gomez Dumm, & N. N. Scoccola (2004). Characteristics of the chiral phase transition in nonlocal quark models Phys.Rev. C72 (2005) 014909 arXiv: hep-ph/0410262v2

Ollitrault, J. (1992). Anisotropy as a signature of transverse collective flow Physical Review D, 46 (1), 229-245 DOI: 10.1103/PhysRevD.46.229

M. Ruggieri, F. Scardina, S. Plumari, & V. Greco (2013). Elliptic Flow from Nonequilibrium Color Glass Condensate Initial
Conditions arXiv arXiv: 1303.3178v1

David Dudal, John Gracey, Silvio Paolo Sorella, Nele Vandersickel, & Henri Verschelde (2008). A refinement of the Gribov-Zwanziger approach in the Landau gauge:
infrared propagators in harmony with the lattice results Phys.Rev.D78:065047,2008 arXiv: 0806.4348v2

Lieb, E., & Simon, B. (1973). Thomas-Fermi Theory Revisited Physical Review Letters, 31 (11), 681-683 DOI: 10.1103/PhysRevLett.31.681

Lieb, E., & Simon, B. (1977). The Thomas-Fermi theory of atoms, molecules and solids Advances in Mathematics, 23 (1), 22-116 DOI: 10.1016/0001-8708(77)90108-6

Marco Frasca (2006). Thermodynamic Limit and Decoherence: Rigorous Results Journal of Physics: Conference Series 67 (2007) 012026 arXiv: quant-ph/0611024v1

## Back to CUDA

11/02/2013

It is about two years ago when I wrote my last post about CUDA technology by NVIDIA (see here). At that time I added two new graphic cards to my PC, being on the verge to reach 3 Tflops in single precision for lattice computations.  Indeed, I have had an unlucky turn of events and these cards went back to the seller as they were not working properly and I was completely refunded. Meantime, also the motherboard failed and the hardware was largely changed  and so, I have been for a lot of time without the opportunity to work with CUDA and performing intensive computations as I planned. As it is well-known, one can find a lot of software exploiting this excellent technology provided by NVIDIA and, during these years, it has been spreading largely, both in academia and industry, making life of researchers a lot easier. Personally, I am using it also at my workplace and it is really exciting to have such a computational capability at your hand at a really affordable price.

Now, I am newly able to equip my personal computer at home with a powerful Tesla card. Some of these cards are currently dismissed as they are at the end of activity, due to upgrades of more modern ones, and so can be found at a really small price in bid sites like ebay. So, I bought a Tesla M1060 for about 200 euros. As the name says, this card has not been conceived for a personal computer but rather for servers produced by some OEMs. This can also be realized when we look at the card and see a passive cooler. This means that the card should have a proper physical dimension to enter into a server while the active dissipation through fans should be eventually provided by the server itself. Indeed, I added an 80mm Enermax fan to my chassis (also Enermax Enlobal)  to be granted that the motherboard temperature does not reach too high values. My motherboard is an ASUS P8P67 Deluxe. This is  a very good card, as usual for ASUS, providing three PCIe 2.0 slots and, in principle, one can add up to three video cards together. But if you have a couple of NVIDIA cards in SLI configuration, the slots work at x8. A single video card will work at x16.  Of course, if you plan to work with these configurations, you will need a proper PSU. I have a Cooler Master Silent Pro Gold 1000 W and I am well beyond my needs. This is what remains from my preceding configuration and is performing really well. I have also changed my CPU being this now an Intel i3-2125 with two cores at 3.30 GHz and 3Mb Cache. Finally, I added  16 Gb of Corsair Vengeance DDR3 RAM.

The installation of the card went really smooth and I have got it up and running in a few minutes on Windows 8 Pro 64 Bit,  after the installation of the proper drivers. I checked with Matlab 2011b and PGI compilers with CUDA Toolkit 5.0 properly installed. All worked fine. I would like to spend a few words about PGI compilers that are realized by The Portland Group. I have got a trial license at home and tested them while at my workplace we have a fully working license. These compilers make the realization of accelerated CUDA code absolutely easy. All you need is to insert into your C or Fortran code some preprocessing directives. I have executed some performance tests and the gain is really impressive without ever writing a single line of CUDA code. These compilers can be easily introduced into Matlab to yield mex-files or S-functions even if they are not yet supported by Mathworks (they should!) and also this I have verified without too much difficulty both for C and Fortran.

Finally, I would like to give you an idea on the way I will use CUDA technology for my aims. What I am doing right now is porting some good code for the scalar field and I would like to use it in the limit of large self-interaction to derive the spectrum of the theory. It is well-known that if you take the limit of the self-interaction going to infinity you recover the Ising model. But I would like to see what happens with intermediate but large values as I was not able to get any hint from literature on this, notwithstanding this is the workhorse for any people doing lattice computations. What seems to matter today is to show triviality at four dimensions, a well-acquired evidence. As soon as the accelerate code will run properly, I plan to share it here as it is very easy to get good code to do lattice QCD but it is very difficult to get good code for scalar field theory as well. Stay tuned!

## 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

## Large-N gauge theories on the lattice

22/10/2012

Today I have found on arXiv a very nice review about large-N gauge theories on the lattice (see here). The authors, Biagio Lucini and Marco Panero, are well-known experts on lattice gauge theories being this their main area of investigation. This review, to appear on Physics Report, gives a nice introduction to this approach to manage non-perturbative regimes in gauge theories. This is essential to understand the behavior of QCD, both at zero and finite temperatures, to catch the behavior of bound states commonly observed. Besides this, the question of confinement is an open problem yet. Indeed, a theoretical understanding is lacking and lattice computations, especially in the very simplifying limit of large number of colors N as devised in the ’70s by ‘t Hooft, can make the scenario clearer favoring a better analysis.

What is seen is that confinement is fully preserved, as one gets an exact linear increasing potential in the limit of N going to infinity, and also higher order corrections are obtained diminishing as N increases. They are able to estimate the string tension obtaining (Fig. 7 in their paper):

$\centering{\frac{\Lambda_{\bar{MS}}}{\sigma^\frac{1}{2}}\approx a+\frac{b}{N^2}}.$

This is a reference result for whoever aims to get a solution to the mass gap problem for a Yang-Mills theory as the string tension must be an output of such a result. The interquark potential has the form

$m(L)=\sigma L-\frac{\pi}{3L}+\ldots$

This ansatz agrees with numerical data to distances $3/\sqrt{\sigma}$! Two other fundamental results these authors cite for the four dimensional case is the glueball spectrum:

$\frac{m_{0^{++}}}{\sqrt{\sigma}}=3.28(8)+\frac{2.1(1.1)}{N^2},$
$\frac{m_{0^{++*}}}{\sqrt{\sigma}}=5.93(17)-\frac{2.7(2.0)}{N^2},$
$\frac{m_{2^{++}}}{\sqrt{\sigma}}=4.78(14)+\frac{0.3(1.7)}{N^2}.$

Again, these are reference values for the mass gap problem in a Yang-Mills theory. As my readers know, I was able to get them out from my computations (see here). More recently, I have also obtained higher order corrections and the linear rising potential (see here) with the string tension in a closed form very similar to the three-dimensional case. Finally, they give the critical temperature for the breaking of chiral symmetry. The result is

$\frac{T_c}{\sqrt{\sigma}}=0.5949(17)+\frac{0.458(18)}{N^2}.$

This result is rather interesting because the constant is about $\sqrt{3/\pi^2}$. This result has been obtained initially by Norberto Scoccola and Daniel Gómez Dumm (see here) and confirmed by me (see here). This result pertains a finite temperature theory and a mass gap analysis of Yang-Mills theory should recover it but here the question is somewhat more complex. I would add to these lattice results also the studies of propagators for a pure Yang-Mills theory in the Landau gauge, both at zero and finite temperatures. The scenario has reached a really significant level of maturity and it is time that some of the theoretical proposals put forward so far compare with it. I have just cited some of these works but the literature is now becoming increasingly vast with other really meaningful techniques beside the cited one.

As usual, I conclude this post on such a nice paper with the hope that maybe time is come to increase the level of awareness of the community about the theoretical achievements on the question of the mass gap in quantum field theories.

Biagio Lucini, & Marco Panero (2012). SU(N) gauge theories at large N arXiv arXiv: 1210.4997v1

Marco Frasca (2008). Yang-Mills Propagators and QCD Nuclear Physics B (Proc. Suppl.) 186 (2009) 260-263 arXiv: 0807.4299v2

Marco Frasca (2011). Beyond one-gluon exchange in the infrared limit of Yang-Mills theory arXiv arXiv: 1110.2297v4

D. Gomez Dumm, & N. N. Scoccola (2004). Characteristics of the chiral phase transition in nonlocal quark models Phys.Rev. C72 (2005) 014909 arXiv: hep-ph/0410262v2

Marco Frasca (2011). Chiral symmetry in the low-energy limit of QCD at finite temperature Phys. Rev. C 84, 055208 (2011) arXiv: 1105.5274v4

## Confinement revisited

27/09/2012

Today it is appeared a definitive updated version of my paper on confinement (see here). I wrote this paper last year after a question put out to me by Owe Philipsen at Bari. The point is, given a decoupling solution for the gluon propagator in the Landau gauge, how does confinement come out? I would like to remember that a decoupling solution at small momenta for the gluon propagator is given by a function reaching a finite non-zero value at zero. All the fits carried out so far using lattice data show that a sum of few Yukawa-like propagators gives an accurate representation of these data. To see an example see this paper. Sometime, this kind of propagator formula is dubbed Stingl-Gribov formula and has the property to have a fourth order polynomial in momenta at denominator and a second order one at the numerator. This was firstly postulated by Manfred Stingl on 1995 (see here). It is important to note that, given the presence of a fourth power of momenta, confinement is granted as a linear rising potential can be obtained in agreement with lattice evidence. This is also in agreement with the area law firstly put forward by Kenneth Wilson.

At that time I was convinced that a decoupling solution was enough and so I pursued my analysis arriving at the (wrong) conclusion, in a first version of the paper, that screening could be enough. So, strong force should have to saturate and that, maybe, moving to higher distances such a saturation would have been seen also on the lattice. This is not true as I know today and I learned this from a beautiful paper by Vicente Vento, Pedro González and Vincent Mathieu. They thought to solve Dyson-Schwinger equations in the deep infrared to obtain the interquark potential. The decoupling solution appears at a one-gluon exchange level and, with this approximation, they prove that the potential they get is just a screening one, in close agreement with mine and any other decoupling solution given in a close analytical form. So, the decoupling solution does not seem to agree with lattice evidence that shows a linearly rising potential, perfectly confining and in agreement with what Wilson pointed out in his classical work on 1974. My initial analysis about this problem was incorrect and Owe Philipsen was right to point out this difficulty in my approach.

This question never abandoned my mind and, with the opportunity to go to Montpellier this year to give a talk (see here), I presented for the first time a solution to this problem. The point is that one needs a fourth order term in the denominator of the propagator. This can happen if we would be able to get higher order corrections to the simplest one-gluon exchange approximation (see here). In my approach I can get loop corrections to the gluon propagator. The next-to-leading one is a two-loop term that gives rise to the right term in the denominator of the propagator. Besides, I am able to get the renormalization constant to the field and so, I also get a running mass and coupling. I gave an idea of the way this computation should be performed at Montpellier but in these days I completed it.

The result has been a shocking one. Not only one gets the linear rising potential but the string tension is proportional to the one obtained in d= 2+1 by V. Parameswaran Nair, Dimitra Karabali and Alexandr Yelnikov (see here)! This means that, apart from numerical factors and accounting for physical dimensions, the equation for the string tension in 3 and 4 dimensions is the same. But we would like to note that the result given by Nair, Karabali and Yelnikov is in close agreement with lattice data. In 3 dimensions the string tension is a pure number and can be computed explicitly on the lattice. So, we are supporting each other with our conclusions.

These results are really important as they give a strong support to the ideas emerging in these years about the behavior of the propagators of a Yang-Mills theory at low energies. We are even more near to a clear understanding of confinement and the way mass emerges at macroscopic level. It is important to point out that the string tension in a Yang-Mills theory is one of the parameters that any serious theoretical approach, pretending to go beyond a simple phenomenological one,  should be able to catch. We can say that the challenge is open.

Marco Frasca (2011). Beyond one-gluon exchange in the infrared limit of Yang-Mills theory arXiv arXiv: 1110.2297v4

Kenneth G. Wilson (1974). Confinement of quarks Phys. Rev. D 10, 2445–2459 (1974) DOI: 10.1103/PhysRevD.10.2445

Attilio Cucchieri, David Dudal, Tereza Mendes, & Nele Vandersickel (2011). Modeling the Gluon Propagator in Landau Gauge: Lattice Estimates of Pole Masses and Dimension-Two Condensates arXiv arXiv: 1111.2327v1

M. Stingl (1995). A Systematic Extended Iterative Solution for QCD Z.Phys. A353 (1996) 423-445 arXiv: hep-th/9502157v3

P. Gonzalez, V. Mathieu, & V. Vento (2011). Heavy meson interquark potential Physical Review D, 84, 114008 arXiv: 1108.2347v2

Marco Frasca (2012). Low energy limit of QCD and the emerging of confinement arXiv arXiv: 1208.3756v2

Dimitra Karabali, V. P. Nair, & Alexandr Yelnikov (2009). The Hamiltonian Approach to Yang-Mills (2+1): An Expansion Scheme and Corrections to String Tension Nucl.Phys.B824:387-414,2010 arXiv: 0906.0783v1

## Running coupling and Yang-Mills theory

30/07/2012

Forefront research, during its natural evolution, produces some potential cornerstones that, at the end of the game, can prove to be plainly wrong. When one of these cornerstones happens to form, even if no sound confirmation at hand is available, it can make life of researchers really hard. It can be hard time to get papers published when an opposite thesis is supported. All this without any certainty of this cornerstone being a truth. You can ask to all people that at the beginning proposed the now dubbed “decoupling solution” for propagators of Yang-Mills theory in the Landau gauge and all of them will tell you how difficult was to get their papers go through in the peer-review system. The solution that at that moment was generally believed the right one, the now dubbed “scaling solution”, convinced a large part of the community that it was the one of choice. All this without any strong support from experiment, lattice or a rigorous mathematical derivation. This kind of behavior is quite old in a scientific community and never changed since the very beginning of science. Generally, if one is lucky enough things go straight and scientific truth is rapidly acquired otherwise this behavior produces delays and impediments for respectable researchers and a serious difficulty to get an understanding of the solution of  a fundamental question.

Maybe, the most famous case of this kind of behavior was with the discovery by Tsung-Dao Lee and Chen-Ning Yang of parity violation in weak interactions on 1956. At that time, it was generally believed that parity should have been an untouchable principle of physics. Who believed so was proven wrong shortly after Lee and Yang’s paper. For the propagators in the Landau gauge in a Yang-Mills theory, recent lattice computations to huge volumes showed that the scaling solution never appears at dimensions greater than two. Rather, the right scenario seems to be provided by the decoupling solution. In this scenario, the gluon propagator is a Yukawa-like propagator in deep infrared or a sum of them. There is a very compelling reason to have such a kind of propagators in a strongly coupled regime and the reason is that the low energy limit recovers a Nambu-Jona-Lasinio model that provides a very fine description of strong interactions at lower energies.

From a physical standpoint, what does it mean a Yukawa or a sum of Yukawa propagators? This has a dramatic meaning for the running coupling: The theory is just trivial in the infrared limit. The decoupling solution just says this as emerged from lattice computations (see here)

What really matters here is the way one defines the running coupling in the deep infrared. This definition must be consistent. Indeed, one can think of a different definition (see here) working things out using instantons and one see the following

One can see that, independently from the definition, the coupling runs to zero in the deep infrared marking the property of a trivial theory. This idea appears currently difficult to digest by the community as a conventional wisdom formed that Yang-Mills theory should have a non-trivial fixed point in the infrared limit. There is no evidence whatsoever for this and Nature does not provide any example of pure Yang-Mills theory that appears always interacting with Fermions instead. Lattice data say the contrary as we have seen but a general belief  is enough to make hard the life of researchers trying to pursue such a view. It is interesting to note that some theoretical frameworks need a non-trivial infrared fixed point for Yang-Mills theory otherwise they will crumble down.

But from a theoretical standpoint, what is the right approach to derive the behavior of the running coupling for a Yang-Mills theory? The answer is quite straightforward: Any consistent theoretical framework for Yang-Mills theory should be able to get the beta function in the deep infrared. From beta function one has immediately the right behavior of the running coupling. But in order to get it, one should be able to work out the Callan-Symanzik equation for the gluon propagator. So far, this is explicitly given in my papers (see here and refs. therein) as I am able to obtain the behavior of the mass gap as a function of the coupling. The relation between the mass gap and the coupling produces the scaling of the beta function in the Callan-Symanzik equation. Any serious attempt to understand Yang-Mills theory in the low-energy limit should provide this connection. Otherwise it is not mathematics but just heuristic with a lot of parameters to be fixed.

The final consideration after this discussion is that conventional wisdom in science should be always challenged when no sound foundations are given for it to hold. In a review process, as an editorial practice, referees should be asked to check this before to kill good works on shaky grounds.

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

Ph. Boucaud, F. De Soto, A. Le Yaouanc, J. P. Leroy, J. Micheli, H. Moutarde, O. Pène, & J. Rodríguez-Quintero (2002). The strong coupling constant at small momentum as an instanton detector JHEP 0304:005,2003 arXiv: hep-ph/0212192v1

Marco Frasca (2010). Mapping theorem and Green functions in Yang-Mills theory PoS FacesQCD:039,2010 arXiv: 1011.3643v3

## QCD 12 and Higgs’ tears

08/07/2012

I have spent this week in Montpellier being a participant to QCD 12, a biannual conference organized by Stephan Narison. It is the third time that I go to Montpellier for this conference and there are always very good reasons for being there. Essentially, the quality of physics and beauty of the city are already worthwhile and sound arguments but also the excellent organization  by the host and the attention reserved to the guests are not the least. This year we have had the blessing of a historical event in physics: The discovery at CERN of the Higgs particle. Stephan organized the event with the webcast from CERN the first two hours on Wednesday and so we heard directly from Gianotti and Incandela what they were seeing at LHC.  The conference is a fair interplay between experiment and theory in a field, QCD, that is very active and with several important open problems. Maybe, we would like to emphasize that is QCD that gives mass to everyday things, and not the Higgs boson, and this means that the solution of the mass gap problem and the developing of proper methods to manage non-perturbative regimes are essential to the understanding of our common perception of reality. Indeed, Roberto Mussa of University of Turin remembered us an argument that  makes Higgs boson essential to everyday life: The stability of matter. Without the Higgs boson quarks would have equal masses and so, proton would decay into neutron. The difference in mass between u and d quarks is essential and this originates from Higgs boson.

In this conference several questions emerged that were absolutely exciting. Hadron spectrum is not so well understood both in the low and high part. There is a plenty of experimental results claiming for an explanation. Labs keep on finding resonances that have not an immediate explanation and make hard the life of us theoreticians. One should compare the situation with the case of electromagnetic interactions where a Rydberg formula was promptly found and understanding of bound states is now quite straightforward. For hadrons we have hard times already to catch what the structure of a resonance is. These difficulties arise from the missing of technique to manage non-perturbative problems in a way similar to the weak coupling limit. Indeed, on Wednesday, some approaches were given to manage this kind of situation and, besides my talk, the most common technique is AdS/QCD starting from Maldacena conjecture. This was also the argument of Stefano Nicotri and Floriana Giannuzzi. They are students of Pietro Colangelo and contributed to the organization of Lecce conference. I have spent a lot of good time with them and so we exchanged a lot of opinions about this matter. On this line, Hans Günter Dosch put all us down showing that the situation with this approach is not so fine. Simply, it appears like a proper model for the mapping between gravity and QCD is lacking yet but, of course, people is actively pursuing it.

A talk that gave me some interesting views was the one by Kenichi Konishi. He pointed out how the confinement can emerge looking at the behavior of the supersymmetric counterpart of Yang-Mills theory. He pointed out the problems with the idea of monopoles, already discussed by Kei-Ichi Kondo at Lecce. And you bet, when one looks at SYM one recover the condensation of a scalar field! Konishi works at University of Pisa where teaches quantum mechanics.

On the line of non-perturbative approaches were the talks by Matteo Giordano and Enrico Meggiolaro. They are trying to re-derive from first principles the Froissart bound. This is a bound on hadronic scattering that can be obtained just using unitarity and dispersion relations. This bound depends crucially on the mass gap of the theory and so, again, we are coping with all the problems given above. Meggiolaro showed that, using lattice computations, the limit can be recovered with the proper constant while Matteo is approaching this problem using AdS/QCD. With Matteo we meet again in Montpellier after four years. We remembered each other immediately and drunk a last beer before leaving on Friday night after the social dinner, with Montpellier streets full of people and pleasant noise.

A talk that I followed with a lot of interest was the one given by Pietro Falgari. He is working on the use of perturbation theory at high-energy in QCD to evaluate the production rate of pairs of top quarks. Even if in this limit perturbation theory can be applied in QCD, they have difficulties mostly related to resum a quite singular series with logarithmic contributions. So, also when perturbation theory applies, QCD does not save us from problems. With Pietro I have spent a lot of time in Montpellier and we left the city together on Saturday with the same flight.

An interesting talk was the one given by Eduardo de Rafael about the determination of the g factor of the muon. This is a truly relevant matter as this measurement can give a clue to new physics. But, as de Rafael pointed out, the critical point is the determination of the hadronic contribution. Presently, there is a 3.6 sigmas discrepancy between the theoretical computed value and the measured one. We cannot be confident that the evaluation of the hadronic part is not correctly accomplished.

Last but not least, the current work of Narison on heavy flavors with sum rules. This approach is now fairly well stable and provides results also better than other non-perturbative techniques. This has been shown in the talk by his collaborator Albuquerque from Sao Paolo. Of course, results like these should be a reference for experiments much in the same way are others as lattice computations. Finally, I would like to cite the talk by Robert Kaminski. He presented the fine work done in collaboration with R. Garcia-Martin J. R. Pelaez, J. Ruiz de Elvira aimed to a precise determination of the properties of f0(600) and f0(980). Their results are striking indeed as they fix very precise values to the mass and width of these resonances, in close agreement with preceding works. It is my personal conviction that a serious theoretical approach should be derive both the mass and the width of these resonances deriving at the same time their structure.

Wednesday was the great day. There was a lot of expectation and the great discovery was in the air predated by a lot of rumors here and there. Our organizers did a great work both providing the webcast from CERN and with a pair of talks on Friday from people of CMS and ATLAS. There has been a religious silence during the talks of Incandela and Gianotti just interrupted by applause at the announcements of the 5 sigmas discovery by the two groups. Following this, we discussed a lot about this matter and, besides it is very standard model-like this particle at the moment, we all were very cautious to claim supersymmetry dead. Rather we would like to know more about the rates in the various channels, results to be known in the near future in order to answer the question put forward by CERN director Rolf-Dieter Heuer: Which one? A girl at my conference asked for other four Higgs and we all know why. Talking with a colleague at ATLAS here in Montpellier, he told me a quite interesting figure for the WW channel but I will not disclose it. Work is in progress yet and data are really too fresh to be discussed. It is a matter of few months and we will know better about the nature of this new particle. Meanwhile, I would like to remember Higgs’ tears after the great announcement and the handshaking with Fabiola Gianotti, after the splendid talk by her, confirming the expectation of almost fifty years of waiting with hopes often not coming up. It is an achievement that very few scientists can claim in their lifetime. The same must apply identically to Englert, Brout, Guralnik, Hagen, and Kibble.

On Friday, the program was concluded by the talks of people from CERN, on behalf of ATLAS and CMS Collaborations. Pushpa Bhat from Fermilab talked on behalf of CMS Experiment while Robert Harrington from Particle Physics Experimental Group of University of Edinburgh talked on behalf of ATLAS Experiment. This was a great conclusion for the Conference, hearing directly from people at CERN, about the great achievement announced on Wednesday.

As a final remark, I would like to thank all people with whom I shared beautiful moments at this conference. Besides people I have already mentioned, I would like to thank Stefano Venditti, Antonio Cassese, Andrey Tayduganov, Federico Mescia, Benjamin Obherof. A great thank goes to Stephan Narison for giving me the chance to give a talk here, for giving me the chance to be chairman for the first time, and for the excellent and really enjoying organization in a beautiful city. See you again!

## QCD@Work 2012

23/06/2012

This week has been of great interest for me being one of the participants to QCD@Work 2012. I have had my contribution accepted by the organizers and so I gave a talk. The conference was held in a really beautiful city, Lecce here in Italy. This conference is organized jointly by University of Bari and University of Salento in Lecce.

The first day, Monday, at the very beginning, there were both ATLAS and CMS delivering their official results. Of course, gossip arrived also at the conference and people were very aware of it. But both CERN groups presented known results leaving some more room for future developments to be heard very soon. The talks in their pdf format are given here. At this conference I have had the good chance to meet Francesco Sannino. We have had interesting discussions sharing some time outside conference time. His talk was really interesting opening up some new venues to the understanding of what could be going on beyond the Standard Model. I would like to remember the proposal by Francesco and Ryttov on the exact beta function for Yang-Mills theory that appears really insightful (see here and refs therein). The afternoon parallel session concerned more strictly what I am currently doing in QCD. A striking talk was given by Kei-Ichi Kondo. Kondo is currently involved in research about infrared behavior of Yang-Mills theory and his conclusions are very similar to mine: There is an Abelian dominance in the low-energy limit. He was able to get Nambu-Jona-Lasinio model as low-energy limit of QCD much in the same way I did. In his talk he get the interquark potential starting from the idea that confinement arises from non-abelian monopoles in the theory. He also verified his approach through lattice computations. In the successive parallel session I heard the talk from Mirko Serino about a really innovative idea. Mirko is a PhD studend at University of Salento and together with Claudio Coriano, that is his professor, Luigi Delle Rose and Antonio Quintavalle are producing an analytical computation of Standard Model in presence of gravity. This kind of computation is highly non-trivial and quite complicated. The striking result they get is that appears a coupling between a scalar degree of freedom and the gauge field and this appears as a rather interesting new proposal for mass generation. I have talked with the students of Coriano and they were really excited by this result that is indeed really interesting and unexpected. The session ended with my talk.

The next day, Tuesday, it was the day for excursion and social dinner. The program was limited to two sessions in the morning. There were a couple of interesting talks, I mean with respect to my fields of interest, by Huang Mei and Antonio Vairo. The talk by Huang Mei managed to get consistent results about confinement in AdS/QCD. She shows that a way to reconcile confinement and holography is obtained  with the introduction of the condensate of the vector potential. This has been the starting point of a lot of discussions, also fired by an interesting comment by Antonio Vairo.  Vairo’s comment is about the fact that an operator product expansion (OPE) does not produce this condensate at large momenta.  By itself, this condensate is clearly not gauge invariant. We also know that a gluon mass is obtained through this condensate in some scenarios as refined Gribov-Zwanzinger theory that I was able to show is fairly consistent (see here). Of course, I would like to see OPE at small momenta, where the real thing happens, to conclude that Vairo’s comment applies as well. Vairo presented a talk on a non-trivial effect in QCD: Jet quenching. The idea is very similar to the one of a particle going through ordinary matter but in this case the matter is a quark-gluon plasma that has the effect to dump energetically the jet. In the afternoon we headed to Otranto, on the sea, for a guided visit and finally took the social dinner in a very pleasant place.

On Wednesday there was a number of talks very near my interests. The first one was delivered by Luigi Capozza on behalf of COMPASS collaboration. I always find the results of this group really striking. Their aim is to take a measurement of the components of the proton spin. These are usually divided in three parts: Quark contribution, gluon contribution and orbital contribution. The striking part of the measurement is that the gluon contribution is compatible with zero! From what we know in the high-energy limit, where the gluon concept is well-defined, these are spin-1 objects and so, it is not so straightforward to have zero contribution from them to the proton spin and indeed this is an important open problem in theoretical physics. The following talk was delivered by Mark Alford. With Mark we spent a lot of time going around Lecce and taking meats looking for very good restaurants.  I was very impressed both by his humor, sometime really sharp, and by his thorough knowledge of a lot of arguments and physics was surely not the last of them. The talk he delivered left me somewhat impressed and the reason is that he and some other few people is managing a really pioneering question: What is going on to nuclear matter inside collapsed stars? He modeled the matter inside a neutron star as a liquid. This liquid has essentially two components and one of this is a superfluid. Sometime, the fluid motion happens in such a way to produce a quadrupole configuration emitting gravitational waves and so the star loses energy. This mechanism is characterized by large amplitude waves. Mark said to me that they are in difficulty due to the strongly coupled situation of such a plasma that has the properties of a superfluid. He called them speculations but much of us know that also special relativity in 1905 was just speculation and there was no hope to see an experimental test in a short time. Next talk was delivered by Maxim Chernodub. Maxim was one of the companions, together with Mark, spending around time in Lecce. We have had several discussions on a lot of arguments. His talk was about his very important work on superconductor properties of the vacuum of QCD. He proved that $\rho$ mesons undergo condensation. This effect appears when a strong magnetic field is applied and this is a typical situation in collisions at RHIC or LHC with heavy ions even if this effect could be very well-hidden. On the other side, surely on lattice computations this could be seen quite straightforwardly. I have asked to Maxim if current approximations in lattice computations can give off the mark results but he said that the choice of quark masses, even if nonphysical,  are not a concern as it should also be for the choice of the lattice spacing. From a theoretical standpoint, he was able to show this effect with some smart computations recently appeared (see here). The final talk of this session was delivered by Marco Ruggieri. Marco is one of my best friends and he gave me a lot of really helpful comments as you can see from the acknowledgment in my talk. The argument was Yang-Mills thermodynamics well above the critical temperature that is 270 MeV. This work, done at University of Catania, is providing really important results. It should be stated that we are again in a strongly coupled regime. But, if you remember condensed matter questions, generally the effect of strongly coupled degrees of freedom goes to dressing particles, called quasi-particles, leaving you with a manageable description of the physical picture. What Marco and the group he works with have proven is that it appears that a quasi-particle description does work in this limit of very high-temperature. The order parameter they identify for the phases is the Polyakov loop and they provide support to the existence of a condensate of Z(3) lines. Z(3) symmetry arises from the fact that only non-colored objects can propagate in Yang-Mills theory and so, only objects with this symmetry are physical degrees of freedom (see here). These results provide an innovative view of what is found on lattice computations and are really challenging for theoreticians. In the afternoon there has been another session with some other interesting talks. The first of this was delivered by Hiroaki Abuki. Abuki’s approach follows an interesting research path to study QCD near the critical point. The idea is to do a Ginzburg-Landau ansatz expanding the Gibbs free energy in terms of chiral fields till fourth power. I think it would be interesting to see a complete justification of this approach starting from QCD as also happens for a superconductor and BCS theory. Hiroaki and his students were very nice companions during launches and dinners at Lecce. Following talk was delivered by Andreas Schmitt. Andreas is doing an important theoretical work in trying to justify phenomenological behavior of nuclear matter in extreme physical conditions starting from models of QCD. This work is essential to give an in-depth comprehension of the work of Mark Alford and collaborators. This is a very challenging activity working with a strongly coupled theory. The starting point for Andreas is Nambu-Jona-Lasinio model and I should say that this is very sound. He is able to show that chiral symmetry is recovered under large magnetic fields. Finally, I have listened the talk of Motoi Tachibana. Spending time with Motoi in Lecce was really enjoying. His talk was about a collaboration with Marco Ruggieri and people at INFN Gran Sasso. This work is aimed to a deeper understanding of the ultradense nuclear matter on the same line of the preceding ones. This new path of research I was not aware of entails the same difficulties of other standard approaches to QCD: One has to cope with a strongly coupled theory. Of course, recurring to Nambu-Jona-Lasinio model is the right way but all the parameters should be properly fixed using QCD and here lies the main question. As my readers probably know, I have approached this problem in one of my most recent papers (see here) and I have presented part of it to this conference. It would be interesting to extend possible applications of it further in such very extreme conditions.

I have to say that this proved to be a very interesting conference, well-organized and I have had the chance to say this personally to Pietro Colangelo, one of the organizers. Knowing how Italian universities are managed by our political class should make clearer yet how great the job has been done by the organizers to get all this machine properly work. Finally, Lecce is a very beautiful city and is worthwhile a staying there. It is another part of this conference that I enjoyed a lot.

Marco Frasca (2012). Condensates in the refined Gribov-Zwanziger scenario arXiv arXiv: 1202.4105v2

M. N. Chernodub (2010). Superconductivity of QCD vacuum in strong magnetic field Phys.Rev.D82:085011,2010 arXiv: 1008.1055v2

M. Ruggieri, P. Alba, P. Castorina, S. Plumari, C. Ratti, & V. Greco (2012). Polyakov Loop and Gluon Quasiparticles in Yang-Mills Thermodynamics arXiv arXiv: 1204.5995v1

Marco Frasca (2011). Chiral symmetry in the low-energy limit of QCD at finite temperature Phys. Rev. C 84, 055208 (2011) arXiv: 1105.5274v4