Apparently (?), a line operator over a very large loop with length $L$ can obey either perimeter law or area law, $-\log\langle U\rangle\sim L^a$ with $a=1,2$, respectively. We call these options "deconfimenent" and "confinement", and this seems to be of crucial importance in QCD.
Is there an argument as to why these two are the only options? Can we have fractional exponent $a$? For example, $a=1/2$ would be an even milder decay than perimeter law, would we still call $U$ deconfined in such case? Similarly, $a=3/2$ is faster than perimeter but slower than area, is such a line confined?
I could perhaps convince myself that only integral $a$ can appear in perturbation theory, but the large-loop behaviour is in general highly non-perturbative, with lots of non-trivial anomalous dimensions, etc. So I don't find a perturbative argument reliable.
Also: I am interested in general QFTs, not necessarily 4d gauge theories with $U$ a Wilson line. I would also like to understand other types of line operators, in other types of QFTs, such as e.g. something like a Verlinde line in 2d. Is there a similar classification of area vs perimeter in such cases? Or does this only work for Wilson loops in 4d? I believe it also makes sense to discuss confimenent in gauge theories in other dimensions, so there should exist some similar criterion.
