Analytic Solution of Bremsstrahlung TBA

TL;DR

This work analytically solves the Bremsstrahlung TBA for the quark–antiquark potential on $S^3$ in the near-BPS limit with $L$ units of R-charge. By reducing an infinite TBA to a Finite set of Nonlinear Integral Equations (FiNLIE) and then solving the FiNLIE analytically, the authors derive a simple expression for the Bremsstrahlung function $B_L(g)$ and reveal two equivalent descriptions: an effective Baxter equation and a matrix-model formulation. The $L=0$ result agrees with localization, while finite $L$ results provide new predictions; in the large-$L$ limit the analysis yields a classical string algebraic curve whose matrix-model saddle reproduces the same curve, and zeros of a polynomial $P_L$ quantize this curve. Overall, the paper unveils a deep link between FiNLIE, Baxter-type structures, matrix models, and classical string geometry in AdS/CFT, offering a new route to exact results in planar ${ m N}=4$ SYM.

Abstract

We consider the quark--anti-quark potential on the three sphere or the generalized cusp anomalous dimension in planar N=4 SYM. We concentrate on the vacuum potential in the near BPS limit with $L$ units of R-charge. Equivalently, we study the anomalous dimension of a super-Wilson loop with L local fields inserted at a cusp. The system is described by a recently proposed infinite set of non-linear integral equations of the Thermodynamic Bethe Ansatz (TBA) type. That system of TBA equations is very similar to the one of the spectral problem but simplifies a bit in the near BPS limit. Using techniques based on the Y-system of functional equations we first reduced the infinite system of TBA equations to a Finite set of Nonlinear Integral Equations (FiNLIE). Then we solve the FiNLIE system analytically, obtaining a simple analytic result for the potential! Surprisingly, we find that the system has equivalent descriptions in terms of an effective Baxter equation and in terms of a matrix model. At L=0, our result matches the one obtained before using localization techniques. At all other L's, the result is new. Having a new parameter, L, allows us to take the large L classical limit. We use the matrix model description to solve the classical limit and match the result with a string theory computation. Moreover, we find that the classical string algebraic curve matches the algebraic curve arising from the matrix model.

Figures (5)

• Figure 1: ( a) A quark--anti-quark pair sitting at two points on $S^3$ at a relative angle $\pi -\phi$. The quark--anti-quark lines are extended along the (Euclidian) time direction. ( b) Under the cylinder to plane conformal map, the quark and anti-quark lines in ( a) are mapped to the two half lines of a Wilson line with a cusp angle $\phi$.
• Figure 2: Plots of $\eta,\rho$ and $\varrho$ at $L=0$ for various values of coupling from ${g=0.05}$ (bottom/blue) to ${g=0.25}$ (top/red) with a step $\Delta g=0.02$.
• Figure 3: Plots of $\eta,\rho$ and $\varrho$ for various values of coupling from $g=0.06$ (bottom/blue) to $g=0.5$ (top/red) with a step $0.04$ with $L=1$.
• Figure 4: Plots of energies for various $L$'s as a function of coupling. The dots (from bottom to top) correspond to $L=0$ (bottom/blue), $L=1$ (red), $L=2$ (yellow), $L=3$ (top/green).
• Figure 5: Algebraic curve $\frac{x^2-1}{x}p(x)$ (real part is plotted). The cuts are discretized by zeros of $P_L(x)$ (red dots).