## Well below 1%

14/04/2017

When a theory is too hard to solve people try to consider lower dimensional cases. This also happened for Yang-Mills theory. The four dimensional case is notoriously difficult to manage due to the large coupling and the three dimensional case has been treated both theoretically and by lattice computations. In this latter case, the ground state energy of the theory is known very precisely (see here). So, a sound theoretical approach from first principles should be able to get that number at the same level of precision. We know that this is the situation for Standard Model with respect to some experimental results but a pure Yang-Mills theory has not been seen in nature and we have to content ourselves with computer data. The reason is that a Yang-Mills theory is realized in nature just in interaction with other kind of fields being these scalars, fermions or vector-like.

In these days, I have received the news that my paper on three dimensional Yang-Mills theory has been accepted for publication in the European Physical Journal C. Here is tha table for the ground state for SU(N) at different values of N compared to lattice data

N Lattice     Theoretical Error

2 4.7367(55) 4.744262871 0.16%

3 4.3683(73) 4.357883714 0.2%

4 4.242(9)     4.243397712 0.03%

4.116(6)    4.108652166 0.18%

These results are strikingly good and the agreement is well below 1%. This in turn implies that the underlying theoretical derivation is sound. Besides, the approach proves to be successful both also in four dimensions (see here). My hope is that this means the beginning of the era of high precision theoretical computations in strong interactions.

Andreas Athenodorou, & Michael Teper (2017). SU(N) gauge theories in 2+1 dimensions: glueball spectra and k-string tensions J. High Energ. Phys. (2017) 2017: 15 arXiv: 1609.03873v1

Marco Frasca (2016). Confinement in a three-dimensional Yang-Mills theory arXiv arXiv: 1611.08182v2

Marco Frasca (2015). Quantum Yang-Mills field theory Eur. Phys. J. Plus (2017) 132: 38 arXiv: 1509.05292v2

## Yang-Mills theory paper gets published!

30/12/2016

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: Using 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

## Do quarks grant confinement?

21/07/2014

In 2010 I went to Ghent in Belgium for a very nice Conference on QCD. My contribution was accepted and I had the chance to describe my view about this matter. The result was this contribution to the proceedings. The content of this paper was really revolutionary at that time as my view about Yang-Mills theory, mass gap and the role of quarks was almost completely out of track with respect to the rest of the community. So, I am deeply grateful to the Organizers for this opportunity. The main ideas I put forward were

• Yang-Mills theory has an infrared trivial fixed point. The theory is trivial exactly as the scalar field theory is.
• Due to this, gluon propagator is well-represented by a sum of weighted Yukawa propagators.
• The theory acquires a mass gap that is just the ground state of a tower of states with the spectrum of a harmonic oscillator.
• The reason why Yang-Mills theory is trivial and QCD is not in the infrared limit is the presence of quarks. Their existence moves the theory from being trivial to asymptotic safety.

These results that I have got published on respectable journals become the reason for rejection of most of my successive papers from several referees notwithstanding there were no serious reasons motivating it. But this is routine in our activity. Indeed, what annoyed me a lot was a refeee’s report claiming that my work was incorrect because the last of my statement was incorrect: Quark existence is not a correct motivation to claim asymptotic safety, and so confinement, for QCD. Another offending point was the strong support my approach was giving to the idea of a decoupling solution as was emerging from lattice computations on extended volumes. There was a widespread idea that the gluon propagator should go to zero in a pure Yang-Mills theory to grant confinement and, if not so, an infrared non-trivial fixed point must exist.

Recently, my last point has been vindicated by a group that was instrumental in the modelling of the history of this corner of research in physics. I have seen a couple of papers on arxiv, this and this, strongly supporting my view. They are Markus Höpfer, Christian Fischer and Reinhard Alkofer. These authors work in the conformal window, this means that, for them, lightest quarks are massless and chiral symmetry is exact. Indeed, in their study quarks not even get mass dynamically. But the question they answer is somewhat different: Acquired the fact that the theory is infrared trivial (they do not state this explicitly as this is not yet recognized even if this is a “duck” indeed), how does the trivial infrared fixed point move increasing the number of quarks? The answer is in the following wonderful graph with $N_f$ the number of quarks (flavours):

From this picture it is evident that there exists a critical number of quarks for which the theory becomes asymptotically safe and confining. So, quarks are critical to grant confinement and Yang-Mills theory can happily be trivial. The authors took great care about all the involved approximations as they solved Dyson-Schwinger equations as usual, this is always been their main tool, with a proper truncation. From the picture it is seen that if the number of flavours is below a threshold the theory is generally trivial, so also for the number of quarks being zero. Otherwise, a non-trivial infrared fixed point is reached granting confinement. Then, the gluon propagator is seen to move from a Yukawa form to a scaling form.

This result is really exciting and moves us a significant step forward toward the understanding of confinement. By my side, I am happy that another one of my ideas gets such a substantial confirmation.

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

Markus Hopfer, Christian S. Fischer, & Reinhard Alkofer (2014). Running coupling in the conformal window of large-Nf QCD arXiv arXiv: 1405.7031v1

Markus Hopfer, Christian S. Fischer, & Reinhard Alkofer (2014). Infrared behaviour of propagators and running coupling in the conformal
window of QCD arXiv arXiv: 1405.7340v1

## 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

## 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