## What is a glueball?

Recently I have read a post in Dmitry’s blog by Fabien Buisseret claiming the following conclusion:

“In the present post were summarized various arguments showing that the glueballs and gluelumps currently observed in lattice QCD can be understood in terms of bound states of a few transverse constituent gluons. In this scheme, the lowest-lying glueballs can be identified with two-gluon states, while the lightest negative-C glueballs are compatible with three-gluon states.”

Indeed he considers free gluons interacting each other through a given potential forming bound states. Of course, as all of you may be aware, nobody in the Earth was able to prove that, in the low energy limit, gluons are the right states entering into a quantum Yang-Mills theory. So, this view appears as a well rooted prejudice in the community.

Let me explain what I mean with a classical example. I take the following quartic theory

$\partial^2\phi+\lambda\phi^3=0.$

In the small coupling limit you will get plane waves plus higher order corrections. Assume these plane waves are gluons as we all of us is aware from high-energy QCD. Indeed, these plane waves describe massless excitations. Now I claim that these solutions should hold also when the coupling $\lambda$ becomes increasingly large. But here I have the exact solution

$\phi(x)=\mu\left(\frac{2}{\lambda}\right)^{1\over 4}{\rm sn}(p\cdot x,i)$

being sn a Jacobi snoidal function and $\mu$ an arbitrary constant. But now

$p^2=\mu^2\left(\frac{\lambda}{2}\right)^{1\over 2}$

and I am describing massive excitations that are not resembling at all my plane wave solutions given above. The claim is blatantly wrong already at a classical level with this very simple example.

This proves without any doubt that the view of glueballs as bound states of gluons is plainly wrong as nobody knows the behavior of a Yang-Mills theory in the infrared limit and so, nobody knows what are the right glue excitations for the theory here. As you may have realized, if you would know this you will be just  filed for a Millenium Prize. This means that, unless we learn how to treat the theory at low energies, all this kind of approaches are doomed.