Running coupling is an important quantity in quantum field theory. It identifies the behavior of an interacting theory in the given limit. Indeed, its importance emerged in a full glory after Gross, Wilczek and Politzer showed that QCD becomes a free theory at high energies (asymptotic freedom). For this reason, they were awarded a well deserved Nobel prize in 2004 (see here). Their result implies that the strong interaction coupling goes to zero as the energy involved in the particle interactions becomes larger. Significant experimental confirmations emerged in the course of time but this result already explained the success of Bjorken scaling. The work of Gross, Wilczek and Politzer served also to convince the community that QCD was the right theory to explain strong interactions. This was a long sought result. This matter is so well-acquired today that people is able to work out higher order corrections to this result and found it in close agreement with experimental evidence. So, you can see that this is the story of a great success in physics.

Then, one may ask what happens in the low energy limit. Here we are in troubles as there are no acquired techniques to manage QCD than working with a computer on a lattice. But the situation is not yet at our hands also using computers. The reason relies on the fact that no generally accepted definition of a running coupling constant exists for the low energy limit. But a lot of prejudices exist about. A firm conviction of a large part of the community is that a meaningful running coupling for strong interactions should reach a fixed point as the energy becomes smaller. The reason for this is that, being these interactions so strong, a coupling cannot be zero. Of course, this is wishful thinking and we have no proof whatsoever that things stay that way. So, people work the other way round looking for a definition that grants the existence of a fixed point. An example of this can be found in this paper.

When I read something like this “I reach for my gun” as Hawking would say. The reason is that, in the course of time, we have gathered a lot of expectations and would-be about Yang-Mills theory and QCD in the infrared that we are no more able to distinguish between what is proved and what is not and, mostly, when we are just trying to give a chance to our wishful thinking to be reality. Till now, nobody was able to let us know what are the right excitations of the theory in the low energy limit and so, there is no reason on Earth to believe that the running coupling do not reach zero value lowering the energy. So, your theory could admit bound states with zero charge but surely you do not need a fixed point running coupling here.

But if you look back to all that people believing that glueballs are just bound states of gluons, they are in strong need for an infrared fixed point of the theory. Clearly, they hope that describing the theory with two wrong concepts may turn it into a rigth one. Things do not stay that way and evidence is coming out that such bound states that diagonalize the theory are indeed with zero charge. Bound states anyway.