Yang-Mills theory paper gets published!



Exact solutions of quantum field theories are very rare and, normally, refer to toy models and pathological cases. Quite recently, I put on arxiv a pair of papers presenting exact solutions both of the Higgs sector of the Standard Model and the Yang-Mills theory made just of gluons. The former appeared a few month ago (see here) while the latter has been accepted for publication a few days ago (see here). I have updated the latter just today and the accepted version will appear on arxiv on 2 January next year.

What does it mean to solve exactly a quantum field theory? A quantum field theory is exactly solved when we know all its correlation functions. From them, thanks to LSZ reduction formula, we are able to compute whatever observable in principle being these cross sections or decay times. The shortest way to correlation functions are the Dyson-Schwinger equations. These equations form a set with the former equation depending on the higher order correlators and so, they are generally very difficult to solve. They were largely used in studies of Yang-Mills theory provided some truncation scheme is given or by numerical studies. Their exact solutions are generally not known and expected too difficult to find.

The problem can be faced when some solutions to the classical equations of motion of a theory are known. In this way there is a possibility to treat the Dyson-Schwinger set. Anyhow, before to enter into their treatment, it should be emphasized that in literature the Dyson-Schwinger equations where managed just in one way: Carl BenderUsing their integral form and expressing all the correlation functions by momenta. It was an original view by Carl Bender that opened up the way (see here). The idea is to write the Dyson-Schwinger equations into their differential form in the coordinate space. So, when you have exact solutions of the classical theory, a possibility opens up to treat also the quantum case!

This shows unequivocally that a Yang-Mills theory can display a mass gap and an infinite spectrum of excitations. Of course, if nature would have chosen the particular ground state depicted by such classical solutions we would have made bingo. This is a possibility but the proof is strongly related to what is going on for the Higgs sector of the Standard Model that I solved exactly but without other matter interacting. If the decay rates of the Higgs particle should agree with our computations we will be on the right track also for Yang-Mills theory. Nature tends to repeat working mechanisms.

Marco Frasca (2015). A theorem on the Higgs sector of the Standard Model Eur. Phys. J. Plus (2016) 131: 199 arXiv: 1504.02299v3

Marco Frasca (2015). Quantum Yang-Mills field theory arXiv arXiv: 1509.05292v1

Carl M. Bender, Kimball A. Milton, & Van M. Savage (1999). Solution of Schwinger-Dyson Equations for ${\cal PT}$-Symmetric Quantum Field Theory Phys.Rev.D62:085001,2000 arXiv: hep-th/9907045v1


That strange behavior of supersymmetry…



I am a careful reader of scientific literature and an avid searcher for already published material in peer reviewed journals. Of course, arxiv is essential to accomplish this task and to satisfy my needs for reading. In these days, I am working on Dyson-Schwinger equations. I have written on this a paper (see here) a few years ago but this work is in strong need to be revised. Maybe, some of these days I will take the challenge. Googling around and looking for the Dyson-Schwinger equations applied to the well-known supersymmetric model due to Wess and Zumino, I have uncovered a very exciting track of research that uses Dyson-Schwinger equations to produce exact results in quantum field theory. The paper I have got was authored by Marc Bellon, Gustavo Lozano and Fidel Schaposnik and can be found here. These authors get the Dyson-Schwinger equations for the Wess-Zumino model at one loop and manage to compute the self-energies of the involved fields: A scalar, a fermion and an auxiliary bosonic field. Their equations are yielded for three different self-energies, different for each field. Self-energies are essential in quantum field theory as they introduce corrections to masses in a propagator and so enters into the physical part of an object that is not an observable.

Now, if you are in a symmetric theory like the Wess-Zumino model, such a symmetry, if it is not broken, will yield equal masses to all the components of the multiplet entering into the theory. This means that if you start with the assumption that in this case all the self-energies are equal, you are doing a consistent approximation. This is what Bellon, Lozano and Schaposnik just did. They assumed from the start that all the self-energies are equal for the Dyson Schwinger equations they get and go on with their computations. This choice leaves an open question: What if do I choose different self-energies from the start? Will the Dyson-Schwiner equations drive the solution toward the symmetric one?

This question is really interesting as the model considered is not exactly the one that Witten analysed in his famous paper  on 1982 on breaking of a supersymmetry (you can download his paper here). Supersymmetric model generates non-linear terms and could be amenable to spontaneous symmetry breaking, provided the Witten index has the proper values. The question I asked is strongly related to the idea of a supersymmetry breaking at the bootstrap: Supersymmetry is responsible for its breaking.

So, I managed to numerically solve Dyson-Schwinger equations for the Wess-Zumino model as yielded by Bellon, Lozano and Schaposnik and presented the results in a paper (see here). If you solve them assuming from the start all the self-energies are equal you get the following figure for coupling running from 0.25 to 100 (weak to strong):

All equal self-energies for the Wess-Zumino model

It does not matter the way you modify your parameters in the Dyson-Schwinger equations. Choosing them all equal from the start makes them equal forever. This is a consistent choice and this solution exists. But now, try to choose all different self-energies. You will get the following figure for the same couplings:

Not all equal self-energies for the Wess-Zumino model

This is really nice. You see that exist also solutions with all different self-energies and supersymmetry may be broken in this model. This kind of solutions has been missed by the authors. What one can see here is that supersymmetry is preserved for small couplings, even if we started with all different self-energies, but is broken as the coupling becomes stronger. This result is really striking and unexpected. It is in agreement with the results presented here.

I hope to extend this analysis to more mundane theories to analyse behaviours that are currently discussed in literature but never checked for. For these aims there are very powerful tools developed for Mathematica by Markus Huber, Jens Braun and Mario Mitter to get and numerically solve Dyson-Schwinger equations: DoFun anc CrasyDSE (thanks to Markus Huber for help). I suggest to play with them for numerical explorations.

Marc Bellon, Gustavo S. Lozano, & Fidel A. Schaposnik (2007). Higher loop renormalization of a supersymmetric field theory Phys.Lett.B650:293-297,2007 arXiv: hep-th/0703185v1

Edward Witten (1982). Constraints on Supersymmetry Breaking Nuclear Physics B, 202, 253-316 DOI: 10.1016/0550-3213(82)90071-2

Marco Frasca (2013). Numerical study of the Dyson-Schwinger equations for the Wess-Zumino
model arXiv arXiv: 1311.7376v1

Marco Frasca (2012). Chiral Wess-Zumino model and breaking of supersymmetry arXiv arXiv: 1211.1039v1

Markus Q. Huber, & Jens Braun (2011). Algorithmic derivation of functional renormalization group equations and
Dyson-Schwinger equations Computer Physics Communications, 183 (6), 1290-1320 arXiv: 1102.5307v2

Markus Q. Huber, & Mario Mitter (2011). CrasyDSE: A framework for solving Dyson-Schwinger equations arXiv arXiv: 1112.5622v2

An interesting review



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


No scaling solution with massive gluons



Some time ago, while I was just at the beginning of my current understanding of low-energy Yang-Mills theory, I wrote to Christian Fischer to know if from the scaling solution, the one with the gluon propagator going to zero lowering momenta and the ghost propagator running to infinity faster than the free particle in the same limit,  a mass gap could be derived. Christian has always been very kind to answer my requests for clarification and did the same also for this so particular question telling to me that this indeed was not possible. This is a rather disappointing truth as we are accustomed with the idea that short ranged forces need some kind of massive carriers. But physics taught that a first intuition could be wrong and so I decided not to take this as an argument against the scaling solution. Since today.

Looking at arxiv, I follow with a lot of interest the works of the group of people collaborating with Philippe Boucaud.   They are supporting the decoupling solution as this is coming out from their numerical computations through the Dyson-Schwinger equations. A person working with them, Jose Rodríguez-Quintero, is producing several interesting results in this direction and the most recent ones appear really striking (see here and here). The question Jose is asking is when and how does a scaling solution appear in solving the Dyson-Schwinger equations? I would like to remember that this kind of solution was found with a truncation technique from these equations and so it is really important to understand better its emerging. Jose solves the equations with a method recently devised by Joannis Papavassiliou and Daniele Binosi (see here) to get a sensible truncation of the Dyson-Schwinger hierarchy of equations. What is different in Jose’s approach is to try an ansatz with a massive propagator (this just means Yukawa-like) and to see under what conditions a scaling solution can emerge. A quite shocking result is that there exists a critical value of the strong coupling that can produce it but at the price to have the Schwinger-Dyson equations no more converging toward a consistent solution with a massive propagator and the scaling solution representing just an unattainable limiting case. So, scaling solution implies no mass gap as already Christian told me a few years ago.

The point is that now we have a lot of evidence that the massive solution is the right one and there is no physical reason whatsoever to presume that the scaling solution should be the true solution at the critical scaling found by Jose. So, all this mounting evidence is there to say that the old idea of Hideki Yukawa is working yet:  Massive carriers imply limited range forces.

J. Rodríguez-Quintero (2011). The scaling infrared DSE solution as a critical end-point for the family
of decoupling ones arxiv arXiv: 1103.0904v1

J. Rodríguez-Quintero (2010). On the massive gluon propagator, the PT-BFM scheme and the low-momentum
behaviour of decoupling and scaling DSE solutions JHEP 1101:105,2011 arXiv: 1005.4598v2

Daniele Binosi, & Joannis Papavassiliou (2007). Gauge-invariant truncation scheme for the Schwinger-Dyson equations of
QCD Phys.Rev.D77:061702,2008 arXiv: 0712.2707v1

The Saga of Landau-Gauge Propagators: A Short History



I have never discussed too much in-depth the history of the matter of Yang-Mills propagators in Landau gauge even if I often expressed a clearcut position. This is a kind of disclaimer when I say that I would not like to offend the work of anyone but my results agree excellently well with lattice computations that my point of view cannot be much different. But a recent paper on arxiv by Attilio Cucchieri and Tereza Mendes and an email exchange with Attilio motivated the idea to put down these rows to give my audience an idea of the stake we are playing for and why no peace treaty has been signed yet by people working in this area of research.

Firstly, I would like to give an idea of why this part of our scientific community is pursuing such a task to identify the gluon and ghost propagators of a pure Yang-Mills theory. The most obvious reason is to understand confinement. The idea that confinement is coded into these propagators dates back to the works of Vladimir Gribov taken to their natural extension by Daniel Zwanziger. Daniel, that I have had the luck to meet and hear in Ghent last year, did a great job in this direction and proved that, for Yang-Mills theory to be confining, the gluon propagator must go to zero with momenta. This scenario was then named Gribov-Zwanzinger arising from the contributions of these authors. It implies that positivity is maximally violated by the propagator and the real space propagator should be seen to cut the time axis. I would like to emphasize that the propagator is a gauge-dependent quantity, even if the spectrum one could obtain from it is not, and here we aim to talk about Landau-gauge propagators both for gluon and ghost fields that are generally easier to manage both on a lattice and theoretically.

The next and also relevant reason to get such propagators is to understand how Yang-Mills theory behaves at lower energies, as we know quite well its behavior at higher ones, and if a mass gap indeed forms. This could have impact on a lot of activities in high-energy physics and nuclear physics. In accelerator facilities one needs to have an exact idea of what is the background arising from QCD and this is not quite well controlled. We have seen a clear example with the charge asymmetry seen by CDF at Tevatron at 3.4 sigma. We cannot be sure this is new physics yet and if a mass gap exists at lower energies, what happens to such massive particles going to higher energies? So, my conclusion here is that we cannot live forever ignoring low energy behavior of QCD as its complete understanding could have impact at unexpected large scales.

After the work of Zwanzinger, people was motivated to get an explicit form of these propagators. Two approaches were clearly at hand. The first one is the use of large computer facilities to solve the theory on the lattice. The other is to attack the problem theoretically through a non-perturbative hierarchy of equations: Dyson-Schwinger equations. The first technique has the drawback that increasing resources are needed to approach meaningful volumes to get a proper understanding of the the theory. At the start of nineties the computing facilities today available were just a dream. On the side of Dyson-Schwinger equations the problem is mathematically very easy to state but very difficult to solve: How to truncate the hierarchy to get the proper results at lower energies? On 1997 an important paper by Lorenz von Smekal, Andreas Hauck and Reinhard Alkofer made its appearance on Physical Review Letters (see here). The authors of this paper claimed to have found a proper truncation of the Dyson-Schwinger hierarchy of equations showing that the gluon propagator should go to zero at lower momenta while the ghost propagator should go to infinity faster than the free particle case. Also the running coupling should reach a finite non-zero fixed point in the same limit. The so-called “scaling solution” was born. The importance of this paper relies on the fact that it strongly supports the Gribov-Zwanzinger scenario and so the theoretical results of these authors appeared vindicated! The idea of the scaling solution and the school built on it by von Smekal and Alkofer with a lot of students after is one of the main aspects of the history we are telling here.

After this paper, a lot of them followed on high impact journals and all the community working on Landau-gauge propagators was genuinely convinced that this indeed should have been the right behavior of propagators. Also the common wisdom that there was a fixed point at infrared was supported by these results. Indeed, people doing lattice computations seemed to confirm these findings even if the gluon propagator was never seen to converge toward zero. But this was said to depend just on the small volumes used by them with the inherent limitations of the computer facilities at that time. Anyhow, data fits seemed to agree quite well with the scaling solution. At this point, till the begin of the new century, the scaling solution become a paradigm for all the community working on the computation of propagators both theoretically and numerically.

At the beginning of the new century things started to change. People started to solve Dyson-Schwinger equations with computers and the results did not appear to agree with the scaling solution. The advantage to solve Dyson-Schwinger equations numerically is that the limitations due to the volume that were plaguing lattice computations, here are absent. It is worthwhile to cite a couple of papers (here and here) that have their culmination in a work by Joannis Papavassiliou, Daniele Binosi and Arlene Aguilar (see here). It should be noticed that the second one of these three papers went unpublished  and the first one met severe difficulties to get published. Indeed, it should be remembered that a paradigm was already formed while these papers contain completely opposite results. What was found was really shocking: the gluon propagator was seen to reach a finite non-zero value and the ghost propagator was going to infinite like that of a free particle! This could be said to be the discover of the “decoupling solution” but it is not completely true. Such a solution was obtained by John Cornwall back in the eighties and since then was waiting for a confirmation (see here). I would like to emphasize that Joannis Papavassiliou worked about this with Cornwall and this work merged in some way with that of solving numerically Dyson-Schwinger equations. The really striking part of these results was that the gluon acquires mass dynamically with a mechanism that is alike the one Schwinger devised long ago and this is an essential starting point to understand confinement.

In parallel to this research line, computers improved and more powerful ones were available during these years to attack the problem on the lattice. Increasing volumes did not seem to change the situation. The scaling solution appeared more and more distant from numerical results. The crushing event happened with the Regensburg conference, Lattice 2007. Large volume computations were finally available. Three contributions appeared by a Brazilian group (Attilio Cucchieri and Tereza Mendes), a Russian-German group (I. Bogolubsky, E.M. Ilgenfritz, M. Muller-Preussker and A. Sternbeck) and a Portuguese-German group (O. Oliveira, P. J. Silva, E.-M.Ilgenfritz, A. Sternbeck). Attilio and Tereza considered huge volumes as 27fm! There was no doubt that the solution with massive gluons, the one contradicting the initial paradigm introduced by Alkofer, von Smekal and Hauck, was the one seen on lattice at large volumes.The propagator was not going to zero but at a finite non-zero value reaching a plateau at lower energies. Also on the lattice the gluon appeared to get a mass at least in four dimensions.

These results were shocking but the question is not settled yet and a lot I would have more to say. People supporting the scaling solution is still there alive and kicking and the propagators war is still on. Nobody wants to cast armies down and surrender. This means that I will have a lot to write yet and the reasons to keep alive this blog are several. For the moment I hope to have kept your attention alive and if you have something more to add or to precise we are open to comments.

von Smekal, L., Hauck, A., & Alkofer, R. (1997). Infrared Behavior of Gluon and Ghost Propagators in Landau Gauge QCD Physical Review Letters, 79 (19), 3591-3594 DOI: 10.1103/PhysRevLett.79.3591

Aguilar, A., & Natale, A. (2004). A dynamical gluon mass solution in a coupled system of the Schwinger-Dyson equations Journal of High Energy Physics, 2004 (08), 57-57 DOI: 10.1088/1126-6708/2004/08/057

Ph. Boucaud, J. P. Leroy, A. Le Yaouanc, A. Y. Lokhov, J. Micheli, O. Pene, J. Rodriguez-Quintero, & C. Roiesnel (2005). The Infrared Behaviour of the Pure Yang-Mills Green Functions arxiv arXiv: hep-ph/0507104v4

Aguilar, A., Binosi, D., & Papavassiliou, J. (2008). Gluon and ghost propagators in the Landau gauge: Deriving lattice results from Schwinger-Dyson equations Physical Review D, 78 (2) DOI: 10.1103/PhysRevD.78.025010

Cornwall, J. (1982). Dynamical mass generation in continuum quantum chromodynamics Physical Review D, 26 (6), 1453-1478 DOI: 10.1103/PhysRevD.26.1453

Attilio Cucchieri, & Tereza Mendes (2007). What’s up with IR gluon and ghost propagators in Landau gauge? A
puzzling answer from huge lattices PoSLAT2007:297,2007 arXiv: 0710.0412v1

I. L. Bogolubsky, E. -M. Ilgenfritz, M. Müller-Preussker, & A. Sternbeck (2007). The Landau gauge gluon and ghost propagators in 4D SU(3) gluodynamics in
large lattice volumes PoSLAT2007:290,2007 arXiv: 0710.1968v2

O. Oliveira, P. J. Silva, E. -M. Ilgenfritz, & A. Sternbeck (2007). The gluon propagator from large asymmetric lattices PoSLAT2007:323,2007 arXiv: 0710.1424v1

What about ghosts?


As most of my readers know, in order to quantize Yang-Mills field one has to introduce a ghost. This result is due to Fadeev and Popov and since then, technology to work out high-energy behavior of QCD has been widely known. When you have a field you will also have a propagator as also unphysical degrees of freedom, as the ghost is, do propagate. But, in the end of your computations, all these contributions magically disappear giving a meaningful result. When you do your high energy computations you will have as a bare ghost propagator the one of a free particle but this field behaves quite strangely violating spin-statistic theorem. The main question we are concerned here is: How does ghost propagator behave in the infrared (low-energy) limit? Some researchers proposed that this propagator should go to infinity faster than that of a free particle (enhancement): This has been dubbed scaling solution . You  can read a nice paper about by Alkofer and von Smekal describing this kind of solution (see here). Lattice simulations went otherwise. In order to have an idea you can read a paper by Cucchieri and Mendes (see here) that shows, at a leading order of small momenta, that the ghost propagator, in the low-energy limit, is that of a free particle! This kind of solution has been dubbed decoupling solution.

Alike Alkofer and von Smekal, other authors thought to use Dyson-Schwinger equations to get the infrared behavior of such quantities like the ghost propagator. A French group, Boucaud,  Gomez ,  Leroy,  Le Yaouanca,  Micheli, Pene, Rodriguez-Quintero, has produced a lot of important papers showing how decoupling solution indeed comes out. I should say that I am an enthusiastic fan of this group as their results coincide perfectly with my findings. They work mostly numerically on lattice and solving Dyson-Schwinger equations also using interesting theoretical approaches. Quite recently, they put out a beautiful paper (see here) where they solve Dyson-Schwinger equation for the ghost propagator but using a smart trick to make it independent from the one of the gluon propagator. They just take the simplest hypothesis for a gluon propagator


(this is exactly the first term of my propagator!), than they show that the solution for the ghost propagator goes like a free propagator plus a logarithmic correction at higher momenta that they are able to compute.  This solution coincides quite perfectly with lattice computations. Gluon mass is seen to be around 500 MeV as it must be (that is also my case). So, a massive propagator (or a massive gluon) implies necessarily a decoupling solution as is seen on lattice computations. This conclusion is quite striking but is not enough. To have a clear idea of this finding one needs to understand what happens, with such an ansatz, to the scaling solution. This has been obtained in a paper appeared today by Rodriguez-Quintero (see here). The conclusion is again striking: A scaling solution emerges only for a critical coupling when enhancement is asked for in the ghost propagator. This, at best, means that this solution is atypical and this gives also a hint why is not seen on lattice computations for 3d and 4d. I would like to remember that the scaling solution appears in lattice computations in 2d when Yang-Mills theory is trivial and has not dynamics. It would be interesting to add similar terms to their ansatz for the gluon propagator: They should be able to recover my gluon propagator with the right spectrum to be compared with quenched lattice computations for QCD.

These results are really shocking but I should say that most has yet to be done on the way to get a complete understanding of Yang-Mills theory. Papers analyzing both scaling and decoupling solutions are fundamental to learn the relevance of such solutions and how they can come out. Presently, decoupling solution is strongly supported by lattice computations and several theoretical works, not last my papers, and I hope that future analysis could hopefully decide for the right scenario.

Solving Dyson-Schwinger equations


Sunday I posted a paper of mine on arxiv (see here). I was interested on managing a simple interacting theory with the technique of Dyson-Schwinger equations. These are a set of exact equations that permit to compute all the n-point functions of a given theory. The critical point is that a lower order equation depends on higher order n-point functions making the solution of all set quite difficult. The most common approach is to try a truncation at some order relying on some physical insight. Of course, to have a control on such a truncation could be a difficult task and the results of a given computation should be carefully checked. The beauty of these equations relies on their non-perturbative nature to be contrasted with the severe difficulty in solving them.

In my paper I consider a massless \phi^4 theory and I solve exactly all the set of Dyson-Schwinger equations. I am able to do this as I know a set of exact solutions of the classical equation of the theory and I am able to solve an apparently difficult equation for the two point function. At the end of the day,  one gets the exact propagator, the spectrum and the beta function. It is seen that this theory has only trivial fixed points. I was able to get these results on another paper of mine. So, it is surely comforting to get identical results with different approaches.

Finally,  I can apply  the mapping theorem with Yang-Mills theories, recently proved thanks also to Terry Tao intervention, to draw conclusions on them in the limit of a very large coupling. In the paper you can find a formulation of this theorem as agreed with Terry, a direct consequence of my latest accepted paper on this matter (see here).

I think this paper adds an important contribution to our understanding of Dyson-Schwinger equations presenting an exact non-trivial solution of them.

A first attempt for the Coulomb Gauge


Coulomb gauge is quite peculiar for the behavior of the Green functions of a Yang-Mills theory. It makes computations more involved and recent lattice computations seem to point toward some different behavior with respect to the case of Landau gauge. Indeed, the propagator is dependent on the gauge choice and could seem not so much useful. We know that things do not stay this way from our understanding of simpler theories as quantum electrodynamics. Some physics can be extracted from them and, if analytically obtained, we can build up a quantum field theory. This explains the effort of a good part of the scientific community for their understanding in the low-energy limit of Yang-Mills theory.

While in the Landau gauge the gluon propagator on the lattice  is seen to reach a finite non-null value at zero momenta, in the Coulomb gauge there are strong indications that the gluon propagator is strongly suppressed bending toward zero at lower momenta. This was firstly seen by Attilio Cucchieri (see here for a more recent analysis). This situation has been somehow improved in a recent paper (see here) in a Tokyo-Berlin-Adelaide collaboration. An understanding of the underlying physics would be improved through the analysis of the ghost propagator. This should be infrared enhanced. But from the paper of Cucchieri, Mendes and Maas that I cited above, it is not seen to change from a free behavior with changing the gauge. This is a relevant indicator and should be exploited wherever one tries to see the behavior of the propagators in different gauges.

Of course, the best way to have an insight on such Green functions is through analytical means. This is overdue in the case of the Coulomb gauge. Today on arxiv appeared a notable attempt by Alkofer, Maas and Zwanziger (see here). Alkofer and Zwanziger are pioneers in the use of Dyson-Schwinger equations for gauge theories and have given important contributions to quantum field theory in this area. So, their conclusions are relevant. They show that a more demanding truncation scheme is needed for this case while it appears not trivial at all the emerging of a linear rising potential. This paper puts the foundations for further work in this direction whose understanding is essential to Yang-Mills theory in the low-energy limit.

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