What is the current understanding of Yang-Mills theory (pure non-Abelian gauge theory without matter field) in the infrared limit? (To avoid the subtlety of renormalizability, we may restrict our scope to lattice gauge theories.) Is the spectrum gapped or not? I am aware of the beta function of its coupling constant in dimension four, what about higher dimensions? Is it always confining in all dimensions?
I believe the answer is yes for any dimension $d\ge 2$. However, this is on a fixed lattice. As Solenodon said for $d>4$ you can't perform the continuum limit. The result I mentioned is in the article by Osterwalder and Seiler "Gauge field theories on a lattice".