Exact results for the entanglement entropy and the energy radiated by a quark

TL;DR

The paper derives exact, UV-finite expressions for the entanglement entropy of a spherical region containing a heavy quark in both N=4 SYM and ABJM, exploiting a mapping to thermal entropy on S^1 × H^{d−1} and the known Wilson loop results from localization. It presents a general relation S_W = log⟨W⟩ + ∫⟨T_{ττ}⟩_W, with ⟨T_{μν}⟩_W fixed by symmetry up to a coefficient h_w, and provides explicit formulas for h_w in multiple theories. In N=4 SYM, S_W is given by (1 − 4λ∂_λ/3) log⟨W⟩ for the 1/2 BPS circular loop, with exact weak/strong coupling interpolations and checks against string theory. In ABJM, a similar approach yields S_W = (1 − 1/2 ∂_b) log⟨W_b⟩|_{b=1} and an exact expression for the 1/6 BPS Bremsstrahlung function B^{1/6}, computed from the derivative of the Wilson loop matrix model; a proposed relation between h_w and B is explored via an improved stress tensor. Collectively, the results bridge Wilson-loop data, entanglement, and radiation in strongly coupled gauge theories, offering concrete, testable predictions and a framework for extending to other supersymmetric theories.

Abstract

We consider a spherical region with a heavy quark in the middle. We compute the extra entanglement entropy due to the presence of a heavy quark both in ${\cal N}=4 $ Super Yang Mills and in the ${\cal N}=6$ Chern-Simons matter theory (ABJM). This is done by relating the computation to the expectation value of a circular Wilson loop and a stress tensor insertion. We also give an exact expression for the Bremsstrahlung function that determines the energy radiated by a quark in the ABJM theory.

Figures (8)

• Figure 1: $(a)$ is the entanglement entropy of a disk in the presence of a Wilson line$^{\ref{['FootNoteForFigureCaption']}}$, $(b)$ is the thermal entropy in thermal hyperbolic space in the presence of a Polyakov loop and $(c)$ is the entropy of a plane in the presence of a circular loop, which can also be interpreted as the entropy induced by a quark/anti-quark pair undergoing hyperbolic motion .
• Figure 2: $(a)$ String coming from the boundary and ending at the horizon. $(b)$ In Euclidean space we have a worlsheet wrapping the radial and time directions with a disk topology. We can use it to compute the entropy.
• Figure 3: Entropy (solid blue) vs $\lambda$. We compare it with the weak coupling expansion (dashed purple) to order $O(\lambda^{8})$ and with the strong coupling result up to three loops (dashed red). Note that the weak coupling expansion has a finite radius of convergence ($|\lambda| \sim 14.6$).
• Figure 4: The $\tau$ circle shrinks at $\theta=0$ while $\phi$ shrinks $\theta=\frac{\pi}{2}$. We are going to put a loop at $\theta=\frac{\pi}{2}$. The geometry is not singular, we just draw the cones to denote that the circles shrink.
• Figure 5: Entropy (solid blue) of the 1/6 BPS loop in the planar limit. Weak coupling (dashed purple) up to $O(\lambda^{10})$ and strong coupling (red purple) at three loops.