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

Abstract

Abstract We provide the two fundamental sets of functional relations which describe the strong coupling limit in AdS3 of scattering amplitudes in $$ \mathcal{N} $$ N = 4 SYM dual to Wilson loops (possibly extended by a non-zero twist l): 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, 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 also applicable to the realm of $$ \mathcal{N} $$ N = 2 SYM BPS spectra.