## Gluon and ghost propagators on the lattice

For a true understanding of QCDat low energies we need to know in a clear way the behavior of the gluon and ghost propagators in this limit. This should permit to elucidate the general confinement mechanism of a Yang-Mills theory in the strong coupling limit. Things are made still more interesting if this comprehension should imply a dynamical mechanism for the generation of mass. The appearance of a mass in a otherwise massless theory gives rise to the so-called “mass gap” that should appear in a Yang-Mills theory to explain the short range behavior of the strong force.

A lot of computational effort has been devoted to this task mainly due to three research groups in Brazil, Germany and Australia. These authors aimed to reach larger volumes on the lattice as some theoreticians claimed loudly that finite volume effects was entering in these lattice results. This because all the previous theoretical research using the so called “functional methods”, meaning in this way the solution of Dyson-Schwinger equations in some approximation, pointed toward a gluon propagator going to zero at low momenta, a ghost propagator going to infinity faster than the free case and the running coupling (defined in some way) reaching a fixed point in the same limit. These theoretical results support views about confinement named Gribov-Zwanziger and Kugo-Ojima scenarios after the authors that proposed them.

Two points should be emphasized. The idea that the running coupling in the infrared should go to a fixed point is a recurrent prejudice in literature that has no theoretical support. The introduction of a new approach to solve the Dyson-Schwinger equations by Alkofer and von Smekal supported this view and  the confinement scenarios that were implied in this way. These scenarios need the gluon propagator going to zero. But these authors are unable to derive any mass gap in their theory and so we have no glueball spectrum to compare with experiment and other lattice computations. Anyhow, this view has been behind all the research on the lattice that was accomplished in the last ten years and whatever computation was done on the lattice, the gluon propagator refused to go to zero and the running coupling never reached a fixed point.

It should be said that some voices out of the chorus indeed there are. These are mainly the group of Philippe Boucaud in France and some numerical works on Dyson-Schwinger equations due to Natale and Aguilar in Brazil. These authors were able to describe the proper physical situation as appears today on lattice but they met difficulties due to skepticism and prejudices about their work due to the mainstream view described above. Since 2005 I proposed a theoretical approach that fully accounts for these results and so I just entered into the club of some heretical view! I will talk of this in my future posts.

After the Lattice 2007 Conference (for the proceedings see here) people working on lattice decided to not pursue further increasing volumes as now an agreement is reached on the behavior of the propagators. This view is at odds with functional methods and no precise understanding exists about why things are so. Some results in two dimensions agree with functionalmethods but the theory is known to be trivial in this case (there is a classical paper by ‘t Hooft about). To give you an idea of the situation I have put the following lattice results at $(13.2fm)^4$ here, $(19.2fm)^4$ here and $(27fm)^4$  here respectively:

and finally

Then we can sum up the understanding obtained from the lattice about Yang-Mills theory:

• Gluon propagator reaches to finite value at low momenta.
• Ghost propagator is that of a free particle.
• Running coupling, using the definition of Alkofer and von Smekal, goes to zero at low momenta.

These results agree perfectly with the ones obtained by Boucaud et al., Aguilar and Natale and myself. In a future post we will discuss the way these lattice results seem to give a consistent view with current phenomenology of light scalar mesons.

### 6 Responses to Gluon and ghost propagators on the lattice

1. […] We have shown the results of the lattice computations of the gluon propagator here. It would be important to get an understanding of what is going on in the low momentum limit. […]

2. Alejandro Rivero says:

Figures 1 and 2 of _340 are impressive. What is going on?

Also, I assume the results are for pure QCD, is it? No fermions there? It is even more fantastic, because it stresses a beyond-standard-model fact: that the mass values of the quarks conspire to live near the peak of the gluon dress. So what will be happen if/when we allow quark loops to contribute to the dressing?

3. mfrasca says:

Yes, these results apply just when no quark is present, just gluons. It is pure Yang-Mills. Anyhow, recent computations by MILC Collaboration show that presence of quarks does not change this picture dramatically and this means that quark contribution is there but the behavior of gluon propagator in the infrared is preserved. In order to have an idea of what is going on just read my post

https://marcofrasca.wordpress.com/2008/06/27/meaning-of-lattice-results-for-the-gluon-propagator/

The beautiful thing is that, once the gluon propagator is known one also knows interquark potential.

Marco

4. […] computations on huge lattices. We would like to remember here that huge lattices arrived at (see here) and now there is no more interest to increase volumes. So, we have here two researchers, just two, […]

5. […] aware of them. This is the situation seen on the computations of gluon and ghost propagators (see here). The situation in this case if far more simpler as there are no quarks. This is pure Yang-Mills […]

6. […] and the propagators in this way. The propagators have been obtained on very large lattice (see here). We have often commented about them and we can give a summary […]