Branes screening quarks and defect operators

Here we generalize a well-known computation and uncover a phase-transition, showing that Wilson lines do not necessarily exhibit Coulomb scaling laws in AdS/BCFT at zero temperature. The area difference between a surface that returns to the boundary, and one that plunges into the bulk, determines the potential between two quarks. This classic AdS/CFT calculation is naturally extended to Wilson surfaces associated to general p-form symmetries in boundary conformal field theories (BCFTs) by embedding a Karch-Randall (KR) brane in the geometry. We find (generalized) Coulomb law scaling in subregion size $Γ$ is recovered only above the critical angle for the brane, $θ_{c,p}$. The potential between the two quarks (or defect operators) vanishes precisely when the surface connecting them ceases to exist at $θ_{c,p}$. This screening effect, where the operators are fully screened below the critical angle, is a phase transition from Coulomb law to perimeter law with the brane angle $θ_b$ acting as an order parameter. This effect is also explored at finite temperature where we introduce a new regularization procedure to obtain closed-form results.