Lorenz von Smekal has been one of the proponents of the functional approach to the understanding of infrared Yang-Mills theory. He is currently working at University of Adelaide where a lot of important work on lattice computations is performed. Today on arxiv appeared a paper by him (see here). I would like to report here his words in the introduction
“Without infrared enhancement of the ghosts in Landau gauge, the global gauge charges of covariant gauge theory are spontaneously broken. Within the language of local quantum field theory the decoupling solution can thus only be realised if and only if it comes along with a Higgs mechanism and massive physical gauge bosons. The Schwinger mechanism can in fact be described in this way, and it can furthermore be shown that a non-vanishing gauge boson mass, by whatever mechanism it is generated, necessarily implies the spontaneous breakdown of global symmetries.”
The “decoupling solution” cited here is the one currently seen on lattice computations having a finite gluon propagator at zero momenta and a ghost propagator behaving like a free particle without any fixed point in the running coupling. The point here is that, in this paragraph, the truth about the real situation of Yang-Mills theory in the infrared is simply exposed. Classical solutions exist that display such dynamical generation of mass for the massless scalar field theory and Yang-Mills theory (see here) and a quantum field theory can be built with them making the above argument truly consistent.
But the point to be emphasized here is the proposal of von Smekal arriving to present a modification of lattice computations. His proposal relies on a recent work done with Andre Sternbeck (see here) where they study the limit of the Yang-Mills theory. Indeed, in this limit they recover the results obtained by functional methods that disagree with lattice computations. Again, they simply freeze the dynamics and get meaningless results as also happens when one compares D=1+1 Yang-Mills theory with no dynamics and the D=4 case. Besides, recent QCD computations on the lattice, relying on known formulations of the Yang-Mills side, give too striking results to go to look for reformulations (see my post).
My view is that functional methods are generally useless. Also when the right solution is hit, of course numerically, one is not able to do any kind of real calculation in QCD. In physics this means that no true understanding is reached. One of the points that should have warned people working with functional methods is that no mass gap is ever obtained and there is no way to recover the low-energy phenomenology of QCD. But having a mass gap produces immediately a Nambu-Jona-Lasinio model from QCD from the ratio to the square of the mass gap itself and this is a real understanding as Nambu-Jona-Lasinio model gives a lot of comprehension of low energy phenomenology.
I think this paper is worth an in depth reading as it contains several pieces of true awareness. My criticisms should not be of any concern for such a good work.
In our preceding posts we have largely discussed what are the results emerging from lattice about the gluon and ghost propagators and the running coupling and how functional methods, in the way they are currently adopted, fail to reach agreement with lattice computations at very large volumes. But we want to resume here what are the main conclusions that are obtained from such applications of functional methods. People working in this way fix the gluon propagator as
and for the ghost
then the claim is made that the relation does hold while the functions are taken to be regular as momenta go to zero. From the relation between the exponents we can conclude that, excluding the trivial solution, if the gluon propagator goes to zero at lower momenta, that is , than we must have that means that the ghost propagator must go to infinity at lower momenta. What they get is that the ghost propagator should go to infinity faster than a free particle. If this would be true all the confining scenarios (Zwanzinger-Gribov and Kugo-Ojima) hold true. The ghost holds a prominent role and, last but not least, a proper defined running coupling goes to a fixed point to lower momenta.
Lattice computations say that all this is blatantly wrong. Indeed, we have learned from them that
- Gluon propagator reaches a finite non-null value at lower momenta.
- Ghost propagator is that of a free particle and so ghosts play no role at lower momenta.
- Running coupling is seen to approach zero at lower momenta.
From this we can easily derive our exponents as defined by people working with functional methods: and so that and no relation between exponents is seen to exist. So we have got a clear cut criterion to say when a published paper about infrared behavior of Yang-Mills theory is blatantly wrong independently on the prestige of the journal that publishes it. This happens all the times the relation is assumed to hold. I can grant that there are a lot around of these wrong papers published on the highest ranked journals. If you have time and you need fun try to search for them.
I would like to say that this belongs to dynamics of science. We are presently in a transition situation about our matter, a situation similar at that happened after the discovery of the resonance that took some time before people agreed on its nature. So, there is nothing to say to editors or referee and also to authors as mistakes are the most common facts in physics and very few people hit the right track after a wide cemetery of mistakes and wrong theories.
Functional methods are techniques used in these years to manage Yang-Mills theory. This name arose from the various methods people invented to solve Dyson-Schwinger equations. These are a tower of equations, meaning by this that the equation for the two-point function will depend on the three point function and so on. These are exact equations: When you solve them you get all the hierarchy of n-point functions of the theory. So, the only way to manage them to understand the behavior of Yang-Mills theory at lower momenta is by devising a proper truncation of the hierarchy. A similar situation can be found in statistical mechanics with kinetic equations. For a gas we know that collisions with a higher number of particles give smaller and smaller contributions and we are able to provide a meaningful truncation of the hierarchy. For the Dyson-Schwinger equations, generally, we are not that lucky and the choice of a proper truncation can be verified only through lattice computations. This means that the choice of a given truncation scheme may imply an uncontrolled approximation with all the consequences of the case. A beatiful paper about this approach is due to Alkofer and von Smekal (see here). This paper has been published on Physics Report and describes in depth all the elements of functional methods for Yang-Mills theory. Alkofer and von Smekal proposed a truncation scheme for Dyson-Schwinger equations that provided the following scenario:
- Gluon propagator should go to zero at lower momenta.
- Ghost propagator should go to infinity faster than a free particle propagator at lower momenta.
- A proper defined running coupling should reach a fixed point in the infrared.
The reason why this view reached success is due to the fact that gives consistent support to currently accepted confinement scenarios. Today we know as the history has gone. Lattice computations showed instead that
- Gluon propagator reaches a non-zero value at lower momenta.
- Ghost propagator is practically the same of that of a free particle.
- Running coupling as defined by Alkofer and von Smekal goes to zero at lower momenta.
So, after years where people worked to support the scenario coming from functional methods, now the community is trying to understand why the truncation scheme proposed by Alkofer and von Smekal seems to fail. On this line of research, Axel Maas showed recently, with lattice computations, that for D=1+1 the scenario is exactly those Alkofer and von Smekal proposed (see here and here). So, now people is try to understand why for D=1+1 functional method seems to work and for higher dimensions this does not happen.
I think that these are not good news for functional methods. The reason of this is that a pure Yang-Mills theory in D=1+1 is trivial. Trivial here means that this theory has not dynamics at all! This result was obtained some years ago by ‘t Hooft (see here) and published on Nuclear Physics B. He showed this using light cone coordinates. Then, by eliminating gluonic degrees of freedom he obtained a two-dimensional formulation of QCD with non-trivial solutions. In our case this means that the truncation scheme adopted by Alkofer and von Smekal simply does not work because removes all the dynamics of the Yang-Mills field and these are also the implications for the confinement scheme this approach should support. Indeed, a proper numerical solution of Dyson-Schwinger equations proves that the right scenario can also be obtained (see here). These authors met difficulties to get their paper accepted by an archival journal. Today, we should consider this work an important step beyond in our understanding of Yang-Mills theory.
My view is that we have to improve on the work of Alkofer and von Smekal to make it properly work at higher dimensionalities. This without forgetting all other works that gave the right solution straightforwardly.