$QQ$-system and non-linear integral equations for scattering amplitudes at strong coupling

Abstract

We provide the two fundamental sets of functional relations which describe the strong coupling limit of scattering amplitudes in $\mathcal{N}=4$ SYM dual to Wilson loops in $Ad{S}_3$: the basic $QQ$-system and the derived $TQ$-system. We use the $TQ$ relations and the knowledge of the main properties of the $Q$-function (eigenvalue of some $Q$-operator) to write the Bethe Ansatz equations, {\it viz.} a set of ('complex') non-linear-integral equations, whose solutions give exact values to the strong coupling amplitudes/Wilson loops. Moreover, they have some advantages with respect to the ('real') non-linear-integral equations of Thermodynamic Bethe Ansatz and still reproduce, both analytically and numerically, the findings coming from the latter. In any case, these new functional and integral equations give a larger perspective on the topic.

Latex, 22 pages