# Perturbative vs. non-perturbative approaches to a well-defined Yang-Mills theory in 4 dimensions

Another question regarding the Yang-Mills Existence and Mass Gap problem (http://www.claymath.org/sites/default/files/yangmills.pdf). Does the problem require that the "construction" of a four dimensional quantum Yang-Mills be non-perturbative? I get the feeling that this problem is to make notions such as the renormalization group rigorous, and thus is perturbative, but isn't lattice gauge theory already mathematically well-defined? If so, why can this not be used as an approach to this problem? Essentially, which is the preferable approach as specified by the problem: perturbative or non-perturbative?

• It is believed that the mass gap is a purely non-pertubative effect hence a pertubative approach does not work. Jun 26, 2014 at 10:45
• @TobiasDiez Why is this? Jun 26, 2014 at 14:33
• The reason is, that the mass gap is proportional to $exp(- g^2)$, where $g$ is the coupling constant. So in perturbation theory you send $g \rightarrow 0$ and hence the mass gap also vanishes. See for example, page 29 in media.scgp.stonybrook.edu/presentations/20120117_3_Witten.pdf Jun 26, 2014 at 21:02
• Yes, of course @TobiasDiez meant $e^{-1/g^2}$. Jun 28, 2014 at 20:26