## Gluon and ghost propagators on the lattice

25/06/2008

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.

## When is a physical system strongly perturbed?

25/06/2008

Generally speaking, a physical system is described by a set of differential equations whose solution is given by a state variable $u$. So, one has $L_0(u)=0$. The solution of this set of differential equations gives all one needs to obtain observables, that is numbers, to be compared with experiments. When the physical system undergoes the effect of a perturbation $L_1(u)$ the problem to be solved is

$L_0(u)+\lambda L_1(u) = 0$

and we say that perturbation is small (or weak) as we consider the limit $\lambda\rightarrow 0$.  In this case we look for a solution series in the form

$u=u_0+\lambda u_1+O(\lambda^2)$.

The case of a strong perturbation is then easily obtained. We say that a physical system is strongly perturbed when the limit $\lambda\rightarrow\infty$ is considered and the solution series we look for has the form

$u=u'_0+\frac{1}{\lambda}u'_1+O\left(\frac{1}{\lambda^2}\right)$.

We will see that this solution series can generally be built and can be obtained by a “duality principle” in perturbation theory arising from the freedom that exists in the choice of what is the perturbation and what is left unperturbed.