Beyond cusp anomalous dimension from integrability

We study the first sub-leading correction O((lns)0) to the cusp (minimal) anomalous dimension in the high spin expansion of finite twist operators in N=4 SYM theory. Since this approximation is still governed by a linear integral equation (derived already from the Bethe ansatz equations in a previous paper), we finalise it better in order to study the weak and strong coupling regimes. In fact, we emphasise how easily the weak coupling expansion can be obtained, confirms the known four loop results and predicts the higher orders. Eventually, we pay particular attention to the strong coupling regime showing agreement and predictions in comparison with the string expansion; speculations on the 'universal' part (upon subtracting the collinear anomalous dimension) are brought forward.