The quark anti-quark potential and the cusp anomalous dimension from a TBA equation

TL;DR

This work constructs a boundary Thermodynamic Bethe Ansatz framework to compute the quark–antiquark potential (cusp anomalous dimension) in planar N=4 SYM as a function of two angles, φ and θ, and the coupling λ. By modeling local Z insertions on a Wilson line as an open spin chain with two boundaries, the authors fix the boundary reflection matrix via symmetry and boundary crossing, introduce the cusp through a relative boundary rotation, and apply the BTBA to obtain exact, L-dependent equations; L=0 yields the cusp, while a small-angle reduction yields a simplified system that can be solved perturbatively up to three loops. They perform consistency checks against Luscher corrections in both weak and strong coupling and connect the near-BPS small-angle limit to localization results through the Bremsstrahlung function B(λ), including a three-loop weak-coupling match. The results bridge integrability-based BTBA techniques with localization and classical-string insights, providing a unified framework for the cusp anomalous dimension in AdS/CFT. Potential directions include numerical BTBA studies, strong-coupling expansions, and extensions to related Wilson-loop setups and ABJM-like theories.

Abstract

We derive a set of integral equations of the TBA type for the generalized cusp anomalous dimension, or the quark antiquark potential on the three sphere, as a function of the angles. We do this by considering a family of local operators on a Wilson loop with charge L. In the large L limit the problem can be solved in terms of a certain boundary reflection matrix. We determine this reflection matrix by using the symmetries and the boundary crossing equation. The cusp is introduced through a relative rotation between the two boundaries. Then the TBA trick of exchanging space and time leads to an exact equation for all values of L. The L=0 case corresponds to the cusped Wilson loop with no operators inserted. We then derive a slightly simplified integral equation which describes the small angle limit. We solve this equation up to three loops in perturbation theory and match the results that were obtained with more direct approaches.

Figures (12)

• Figure 1: ( a) A Wilson line with a cusp angle $\phi$. ( b) Under the plane to cylinder map the two half lines in (a) are mapped to a quark anti-quark pair sitting at two points on $S^3$ at a relative angle of $\pi -\phi$. The quark anti-quark lines are extended along the time direction.
• Figure 2: The BTBA trick. The same partition function can be viewed in two ways (\ref{['openclosed']}). In the open string channel it is a trace over all states in the open string Hilbert space. In this case Euclidean time runs along the $T$ arrow. Alternatively we can view it as the propagation of a closed string along the $L$ arrow. The closed string has length $T$ and propagates over a Euclidean time $L$. The two boundary conditions, now lead to two boundary states that create the closed strings that propagate along the closed string channel.
• Figure 3: Unfolding of $R(p)$ into $S(p,-p)$. There is a non-trivial map between dotted and checked indices. See appendix \ref{['Rmatrix']} for details.
• Figure 4: Computation of the reflection phase at strong coupling. We have a soliton at the boundary, which is at rest at $\sigma=0$. There is also an image soliton coming from the right. Then the soliton with momentum $p$ scatters through the soliton at rest and the one with momentum $-p$, leading to a certain time delay. From the time delay we can compute the derivative of the reflection phase with respect to the energy.
• Figure 5: ( a) We have a strip with pairs of particles being exchanged. The two colors represent the two types of indices. In ( b) we unfolded this into a cylinder computation. The $K$ matrices became $S$ matrices for a single $\widetilde{su}(2|2)$. ( c) Using crossing we have moved the lines. The red circles indicates the action of the matrix $m$.