Generalized quark-antiquark potential at weak and strong coupling

TL;DR

This work introduces a two-parameter family of Wilson loops in ${\mathcal N}=4$ SYM on ${\mathbb S}^3\times\mathbb R$ that interpolate between a 1/2-BPS line/circle and antiparallel lines, defining a generalized quark–antiquark potential $V(\phi,\theta,\lambda)$. It analyzes the observable at weak coupling up to two loops and at strong coupling via AdS/CFT, obtaining explicit one- and two-loop results and a one-loop string determinant expressed as an integral, with analytic expansions around the 1/2-BPS configuration. In the antiparallel-lines limit $\phi\to\pi$, both weak- and strong-coupling analyses reproduce known results, providing a nontrivial cross-check across regimes. The near straight-line expansion reveals a simple, largely two-parameter structure controlled by $\theta^2-\phi^2$ and suggests a potential route to all-loop insights through insertions along the Wilson loop and a focus on the most connected graphs. Altogether, the paper lays a framework for interpolating observables across coupling and for exploring all-loop dynamics in a controlled setting.

Abstract

We study a two-parameter family of Wilson loop operators in N=4 supersymmetric Yang-Mills theory which interpolates smoothly between the 1/2 BPS line or circle and a pair of antiparallel lines. These observables capture a natural generalization of the quark-antiquark potential. We calculate these loops on the gauge theory side to second order in perturbation theory and in a semiclassical expansion in string theory to one-loop order. The resulting determinants are given in integral form and can be evaluated numerically for general values of the parameters or analytically in a systematic expansion around the 1/2 BPS configuration. We comment about the feasibility of deriving all-loop results for these Wilson loops.

Figures (3)

• Figure 1: Antiparallel lines on $\mathbb{S}^3\times\mathbb{R}$ with Lorentzian signature can be mapped by different conformal transformations to hyperbolas in Minkowski space, arranged so that they all pass through the points $\pm1$ on the horisontal axis. The thin lines on the cylinder map to the boundary of Minkowski space.
• Figure 2: Pairs of rays intersecting at angles $\phi=0,\pi/4, \pi/2, 3\pi/4$ get mapped by different conformal transformations to pairs of intersecting arcs, interpolating between the circle and a pair of antiparallel lines.
• Figure 3: Curves showing $V^{(1)}(\phi,0)$ (blue, wide dashes), $V^{(2)}(\phi,0)$ (green, dash-dot), $V^{(0)}_{AdS}(\phi,0)$ (red, short dashes) And $V^{(1)}_{AdS}(\phi,0)$ (purple, short dash-dot). Note that all have a simple pole at $\phi=\pi$, with different residues. The dotted lines show the perturbative expansions of $V^{(0)}_{AdS}$ and $V^{(1)}_{AdS}$ around $\phi=0$ to order $\phi^8$, (\ref{['V0ads-expand']}), (\ref{['V1ads-expand']}), which furnish a good approximation up to $\phi\sim2$.