Yang-Mills theory in D=1+1

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.

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