Wilson-'t Hooft classification and the perimeter law for dyonic loops in 3d monopole semiclassics

We investigate the long-distance behavior of dyonic loop operators in 4d $SU(N)$ gauge theories on $\mathbb{R}^3 \times S^1$ using the 3d monopole semiclassics. If we employ the naive definition of the 't Hooft loop in the Abelianized regime, the dyonic loop operators do not admit the well-defined computations within the effective field theory. Moreover, if one forcibly proceeds with the computations of their expectation values, all the dyonic loops turn out to show the area law, which contradicts the prediction of the Wilson-'t Hooft classification. In this paper, we resolve this puzzle by employing the notion of screening for line operators, and we argue that the dyonic loops are screened by a defect known as the twist vortex, which is non-dynamical in the infrared effective theory but is dynamical in the original ultraviolet theory. The dyonic loops properly dressed by twist vortices admit the well-defined computations within the effective field theory, and we reproduce the kinematic prediction of the Wilson-'t~Hooft classification using the $3$d monopole semiclassics. Furthermore, we apply our framework to the thermal deconfined phase to evaluate the dual string tension, elucidating the topological nature of $\mathbb{Z}_N$ domain walls. We confirm that the domain-wall state has the phase transition at $θ=π$ in the thermal deconfined phase despite the fact that the bulk state is smooth there.