Position dependence of the holographic entanglement entropy for an accelerating quark-antiquark pair

TL;DR

This work computes the holographic entanglement entropy (EE) of the gluonic field sourced by a quark-antiquark pair undergoing uniform back-to-back acceleration, generalizing from a symmetric entangling surface to a displaced one described by $h$. Using the replica trick in the gravity dual and a probe-brane (string) description, the authors derive a three-term decomposition of the EE: a contact term from the intersection of the string worldsheet with the Ryu-Takayanagi surface, a worldsheet stress-energy contribution, and a vanishing counterterm, yielding a closed expression that reduces to the known $S = \frac{\sqrt{\lambda}}{3}$ in the symmetric limit and diverges as $h/b \to 1$. They express the result in AdS$_5$ with a dimensionally extended formula to AdS$_{d+1}$/CFT$_d$, and discuss implications for worldsheet interpretations and potential connections to JT gravity-like structures in low dimensions. The analysis crucially leverages conformal maps that send the entangling surface to a sphere and translate the quark trajectory to a shifted circle, enabling a tractable replica calculation even when the configuration is neither static nor thermally interpretable.

Abstract

Through the holographic correspondence, we compute the entanglement entropy of the gluonic field sourced by a quark-antiquark pair undergoing uniform back-to-back acceleration. Previous calculations had obtained this only for the case where the entanglement surface is located midway between the quark and antiquark. Here, we consider the more general case with a relative lateral displacement, and determine the entanglement entropy as a function of the distance between the quark and the entanglement surface. This setup is of interest because it departs from the usual simplifying conditions of staticity and thermality, and because it yields more information about the entanglement pattern in the gluonic field and about the possibility of eventually developing a purely worldsheet interpretation for said entanglement.

Figures (6)

• Figure 1: Setup for the computation of entanglement entropy in the presence of a quark (filled red circle) and antiquark (unfilled red circle) that accelerate back-to-back with uniform acceleration $b^{-1}$ and closest distance of approach $2b$. The red curves are the $q$-$\bar{q}$ worldlines, and the horizontal line is the time slice where the entanglement entropy of the gluonic and other fields is examined. The blue dot indicates the location of the entangling surface $\partial A$, which is a 2-dimensional plane coming out of the picture, with $A$ the spatial region depicted by the blue semi-infinite segment to the right of $\partial A$. (a) The symmetric configuration, analyzed previously in the literature, where each of the particles is at the same distance from $\partial A$. (b) The asymmetric configuration of interest in the present paper, where $\partial A$ is displaced a distance $h\le b$ to the left of the midpoint between the quark and antiquark.
• Figure 2: The $q$-$\bar{q}$ trajectories are shown in red. At $x^0=0$ the quark is at a distance $b-h$ and the antiquark at a distance $b+h$ from the entangling surface denoted as $\partial A$ (blue point).
• Figure 3: (a) These are the trajectories of the $q$-$\bar{q}$ pair after a Wick rotation. They are described by a shifted circle with its origin located at $(0,h)$. (b) This is the result of applying a set of conformal transformations to the configuration in (a). The ES is mapped from a hyperplane to a sphere (blue circle) of radius $r'=b$ and the $q$-$\bar{q}$ trajectory is mapped to a shifted circle in the $x'^0$-$x'^1$ plane with a new radius $b'$ (red lines).
• Figure 4: The red lines represent the $q$-$\bar{q}$ trajectories in the primed coordinates. We have plotted the trajectories for different values of $h$ (taking $b=1$). The blue points indicate the locations of the spherical ES. The $h=0$ plot corresponds to the vertical line, while the $h=1$ plot corresponds to the circumference that intersects the ES.
• Figure 5: The figure shows the $x^1$–$x^2$ plane at time $t=0$. Region $A$ (shaded in light blue) is defined by $x^1 \geq 0$, while region $B$ is its complement. The entanglement surface lies along $x^1 = 0$ (indicated by the blue line). The bounded operators $O_A$ and $O_B$ are inserted at $x^1 = \delta$ and $x^1 = -\delta$, respectively, and are represented by crosses ($\times$). The Wilson loop intersects the $t=0$ plane at $x^1_q = -b + h$ (red circle) and $x^1_{\bar{q}} = b + h$ (white circle).