## How far are we from a proof of existence of the mass gap?

29/06/2008

A well acquired fact is that there are seven problems that will be awarded with a rich prize by the Clay Institute (see here). One of these problems touch us that work on QCD very near. This is the question of the existence of a mass gap for a Yang-Mills theory (see here). I would like to emphasize that Clay Institute is a mathematical institution and, as such, it is acquired that a proof given by a physicist could not be enough to satisfy the criteria of professional mathematicians to be called a proof. Anyhow, one can always suppose that a sound mathematical idea due to some physicists can be made rigorous enough by the proper intervention of a mathematician. But this last passage is generally neither that simple nor obtainable in a short time.

As we have discussed here, present lattice results are already enough to have given to physicists a sound proof of the existence of a mass gap for a Yang-Mills theory in D=3+1. In this post I will avoid to discuss about theoretical work in D=3+1 but rather I would like to point out some relevant work appeared for the case D=2+1. In this case there are two categories of papers to be considered. The first category corresponds to the works of Bruce McKellar and Jesse Carlsson. These works are largely pioneering. In these papers the authors consider the theory on the lattice and try to solve it through analytical means (here, here and here appeared in archival journals). They reached a relevant conclusion:

The spectrum of the Yang-Mills theory in D=2+1 is that of an harmonic oscillator.

This conclusion should be compared with the results in D=3+1 on the lattice. We have seen here that looking in a straightforward way to these computations one arrives easily to the conclusion that the theory is trivial. Yes, it has a mass gap but is trivial. The only missing block is the spectrum. So, Carlsson and McKellar give us the missing step. Also the spectrum is consistent with the view that the theory is trivial.

Then we look at the second category of papers. These papers arose from an ingenious idea due to Kim, Karabali and Nair (e.g. see here) that introduced the right variables to manage the theory. In this way one reduces the problem to the one of diagonalizing a Hamiltonian obtaining eigenstates and eigenvalues. Building on this work, Leigh, Minic and Yelnikov were able to postulate a new nontrivial form of the ground state wavefunctional producing the spectrum of the theory in D=2+1 in closed analytical form (see here and here). The spectrum was given as the zeros of Bessel functions that in some approximation can be written as that of an harmonic oscillator. The open problem with this latter approach relies on the proof of existence of the postulated wavefunctional. This may not be easy.

The conclusion to be drawn from this is that we have already sound evidences that a mass gap for Yang-Mills theory exists. These proofs could not be satisfactory for a mathematician but surely for us physicists give a solid ground to work on.