Branes Screening Quarks and Defect Operators

TL;DR

The paper investigates how Karch-Randall branes screen the potential between quarks and extended defect operators in AdS/BCFT, revealing a brane-angle–driven phase transition at a critical angle $\theta_{c,p}$ between Coulomb-like and perimeter-law behavior. By analyzing minimal $p$-dimensional area surfaces in AdS with KR branes, the authors show that Coulomb scaling $V_{q\bar q}\sim \Gamma^{1-p}$ holds only above $\theta_{c,p}$ and that the potential vanishes at the transition as the brane-endpoint surfaces detach. At finite temperature, they develop a shadow-based regularization to obtain closed-form area differences, yielding explicit results such as $\Delta A_c = \frac{\sqrt{\pi}}{1-d}\frac{\Gamma\left(\frac{1-p}{d-1}\right)}{\Gamma\left(\frac{1-p}{d-1}+\frac{1}{2}\right)}$, illustrating that screening persists in thermal settings. The findings point to a universal, island-like mechanism governing defect observables across dimensions, with ties to top-down IIB constructions and potential broader principles for holographic BCFTs.

Abstract

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.

Figures (6)

• Figure 1: It is well-known that for $p>1$ the $p$-dimensional extremal surface dual to the closed Wilson surface $W(C)$ exhibits Coulomb law as $\Gamma \rightarrow 0$, both for empty AdS and at finite temperature. However, after a KR brane is introduced with angle $\theta_b$, the behavior as $\Gamma\rightarrow0$ depends on whether $\theta_b$ is greater than or less than the critical angle $\theta_c$. Here we depict an ICFT rather than a BCFT to emphasize the close connection with the quark-antiquark potential.
• Figure 2: The physical interpretation of $f_p(\theta)$ is that it determines whether the extremal surface undershoots $(f_p(\theta)<0)$ or overshoots $(f_p(\theta)>0)$ the defect at $\Gamma=0$. On the LHS, above the critical angle, the surface overshoots the defect when $f_p(\theta)>0$, which gives $\Gamma>0$ and $A_p<0$. In the middle, at the critical angle $\theta_{c,p}$, $f_p(\theta)= A_p = \Gamma = 0$. Note the scale invariance at the critical angle. On the RHS we are below the critical angle; we have $f_p(\theta)<0$ and $\Gamma<0$, so the brane-anchored surfaces cease to exist.
• Figure 3: Here we illustrate the behavior at the critical angle $\theta_{c,p}$ at finite temperature for $p$-dimensional extremal surfaces dual to $p-1$ dimensional defect operators. The background geometry is the $d=7$ dimensional black string. Spatial Wilson surfaces anchored to the defect ($\Gamma=0$) touch the brane at the critical anchor $u_b$. This brane angle is slightly below the critical angle for $p=3$ surfaces: $\theta_{c,3} \approx .98687$, and above $\theta_{c,1}$ and $\theta_{c,2}$, which means the $p=1$ and $p=2$ dimensional surfaces have vanishing critical anchors. Only their area differences diverge with (generalized) Coulomb law scaling.
• Figure 4: Critical anchors $u_b$ are shown for the ($d=10$) dimensional black string geometry as a function of brane angle $\theta_b$ and surface dimension $p$. The $u_b$ vanish above the critical angle $\theta_{c,p}$. The same limiting behavior at $\theta_{c,p}$ was found for $p-$dimensional surfaces, with $p<d$, for each bulk dimension $d>2$. Similar comments apply to Figures \ref{['fig:constantd']} and \ref{['fig:constantp']}; $u_b$ is monotonically increasing in both $p$ and $d$, and monotonically decreasing in $\theta_b$. This phenomenon occurs in every asymptotically AdS geometry with an embedded Karch-Randall brane. The surface for $p=1$ is not pictured because $\theta_{c,1}=0$.
• Figure 5: The critical anchors $u_b$ are displayed as a function of brane angle $\theta_b$ and bulk dimension $d$ for surface dimension $p=2$ in the black string geometry. The $u_b$ vanish at the critical angle $\theta_{c,2}$. The same limiting behavior was found at $\theta_{c,p}$ for $p$-dimensional extremal surfaces, with $p<d$, for each bulk dimension $d$.