Phases of Wilson Lines in Conformal Field Theories

Abstract

We study the low-energy limit of Wilson lines (charged impurities) in conformal gauge theories in 2+1 and 3+1 dimensions. As a function of the representation of the Wilson line, certain defect operators can become marginal, leading to interesting renormalization group flows and for sufficiently large representations to complete or partial screening by charged fields. This result is universal: in large enough representations, Wilson lines are screened by the charged matter fields. We observe that the onset of the screening instability is associated with fixed-point mergers. We study this phenomenon in a variety of applications. In some cases, the screening of the Wilson lines takes place by dimensional transmutation and the generation of an exponentially large scale. We identify the space of infrared conformal Wilson lines in weakly coupled gauge theories in 3+1 dimensions and determine the screening cloud due to bosons or fermions. We also study QED in 2+1 dimensions in the large $N_f$ limit and identify the nontrivial conformal Wilson lines. We briefly discuss 't Hooft lines in 3+1-dimensional gauge theories and find that they are screened in weakly coupled gauge theories with simply connected gauge groups. In non-Abelian gauge theories with S-duality, this together with our analysis of the Wilson lines gives a compelling picture for the screening of the line operators as a function of the coupling.

Figures (2)

• Figure 1: An illustration of the $\beta$-function associated with the parameter $g$ in equation \ref{['eq_MoreGeneralLine']}.
• Figure 2: Plots of the scalar profile (blue) and the electric field (orange) as functions of the distance from the probe charge, all normalized to be dimensionless. The analysis was carried out for $\frac{e^2q}{2\pi}=1.02$ and $\lambda/e^2=\frac{1}{2}$ by numerically solving the classical equations of motion that follow from \ref{['eq_Action_SQED4']}, with boundary conditions such that the fields decay at infinity, while at a minimal radial position $r=r_0$ we have $\left.F_{0r}\right|_{r=r_0}=\frac{e^2q}{4\pi r^2}$, and $\left.|\phi|^2\right|_{r=r_0}=0$, with $\phi \neq 0$ for all $r>r_0$. Different boundary conditions for the scalar field lead to a qualitatively similar plot.