Analytic Solution of Bremsstrahlung TBA II: Turning on the Sphere Angle

TL;DR

This work delivers an exact analytic solution for the Bremsstrahlung cusp TBA in the near-BPS limit of planar N=4 SYM, extending previous θ=0 results to arbitrary twist θ and operator length L. By reducing the Bremsstrahlung TBA to FiNLIE and solving with a twisted Q-function Ansatz, the authors derive a compact determinant form for the cusp energy Γ_L(g, φ, θ) that interpolates between weak and strong coupling and passes nontrivial checks against localization and classical string theory. The solution hinges on an analytic structure built from deformed Bessel functions, Baxter equations, and Bethe-like relations, and is shown to be compatible with the modern Pμ-system formulation. These results provide a powerful nonperturbative handle on Wilson-line observables in AdS/CFT and hint at matrix-model connections for the underlying integrable structure.

Abstract

We find an exact analytical solution of the Y-system describing a cusped Wilson line in the planar limit of N=4 SYM. Our explicit solution describes anomalous dimensions of this family of observables for any value of the `t Hooft coupling and arbitrary R-charge L of the local operator inserted on the cusp in a near-BPS limit. Our finding generalizes the previous results of one of the authors & Sever and passes several nontrivial tests. First, for a particular case L=0 we reproduce the predictions of localization techniques. Second, we show that in the classical limit our result perfectly reproduces the existing prediction from classical string theory. In addition, we made a comparison with all existing weak coupling results and we found that our result interpolates smoothly between these two very different regimes of AdS/CFT. As a byproduct we found a generalization of the essential parts of the FiNLIE construction for the gamma-deformed case and discuss our results in the framework of the novel ${\bf P}μ$-formulation of the spectral problem.