# Is Yang-Mills theory confining in any dimensions?

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?

• "inferred"$\to$"infra-red"? Sep 26, 2017 at 9:59
• Do you mean lattice gauge theory? The ordinary perturbative approach is nonrenormalizable in $d > 4$. Sep 26, 2017 at 15:47

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".

• Sorry, this seems wrong. In $d > 4$ dimensions Yang-Mills has a nontrivial UV fixed point at $g = g_c$ and it becomes trivial at large distances if $g < g_c$.
– user159249
Sep 26, 2017 at 17:16
• @Marty: any references for this nontrivial UV fixed point? And please don't just say "This seems wrong" without being more specific about this "this" you are talking about. Otherwise it means everything I said is wrong. Sep 26, 2017 at 17:19
• The beta function for $SU(N)$ Yang-Mills in $d=4+\epsilon$ is roughly speaking $\beta(g) = \epsilon g - C g^2 + O(g^3)$ with $C > 0$ (depending on $N$ in the usual way). The first term is just dimensional analysis, the second reflects asymptotic freedom in $d=4$. So for sufficiently small $g$, you get this phase diagram.
– user159249
Sep 26, 2017 at 20:51
• @Marty: OK but do you have references about this mystery fixed point? Also, the result I mentioned is a rigorous mathematical theorem which applies in the large $g$ regime. So assuming $g_c$ exists, it would have to be repulsive if one runs the RG the way one should run it, i.e., from UV to IR. If the initial condition satisfies $0<g<g_c$ then you go to $0$. But if $g>g_c$ you go to $\infty$. So there is no contradiction with what you said. Sep 26, 2017 at 21:03
• I don't know any papers for the non-SUSY case. Since this fixed point has $O(1)$ coupling at $d=5$ it's essentially impossible to construct it in practice. There are SUSY YM theories with similar phase diagrams in $d=5$ that have received attention in the 1990s-2000s, thinking of some Seiberg and Intrilligator papers.
– user159249
Sep 26, 2017 at 21:30