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

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.

💡 Research Summary

The paper investigates the entanglement entropy (EE) associated with the gluonic field generated by a uniformly accelerated quark–antiquark (q–={q}) pair in a (3+1)-dimensional conformal field theory (CFT), extending previous work that only treated the symmetric configuration where the entangling surface (ES) lies exactly midway between the two particles. By allowing a lateral displacement of the ES (or equivalently of the q–={q} pair) by a distance h, the authors study a more general setup in which the two particles are at different distances from the ES. This breaks the special conformal symmetry that rendered the reduced density matrix thermal in the symmetric case, and thus the computation must be performed without relying on a thermal interpretation.

From the CFT side, the quark and antiquark follow hyperbolic worldlines that, in Euclidean signature, trace a circle of radius b with its centre displaced by h along the spatial axis. The dual bulk description is a U‑shaped Nambu–Goto string in pure AdS₅ whose endpoints sit on the boundary and accelerate with proper acceleration a = b⁻¹. The induced metric on the string worldsheet is that of an eternal AdS₂ black hole, possessing two horizons and an Einstein–Rosen bridge that encodes the EPR entanglement of the colour‑singlet pair.

The authors first perform a series of conformal transformations that map the planar ES (x₁ = 0) to a sphere of radius b in a new set of coordinates. Under these transformations the displaced circular trajectory of the q–={q} pair is mapped to another circle with new radius b′ and centre h′, given explicitly by
b′ = 2b³h/(2b + h), h′ = b(2b² − h²)h/(2b + h).
Corresponding bulk isometries are then applied, showing that the original hemispherical cap (the Euclidean continuation of the string worldsheet) is mapped to a new cap with parameters b′, h′.

To compute the EE, the paper employs the replica trick in its gravitational formulation (the generalized gravitational entropy method of Lewkowycz and Maldacena). Crucially, the authors use the result of