The five-gluon amplitude in the high-energy limit

We consider the high-energy limit of the colour-ordered one-loop five-gluon amplitude in the planar maximally supersymmetric N = 4 Yang-Mills theory in multi-Regge kinematics where all of the gluons are strongly ordered in rapidity. We apply the calculation of the one-loop pentagon in D = 6-2 epsilon performed in a companion paper [1] to compute the one-loop five-gluon amplitude through to O(epsilon(2)). Using the factorisation properties of the amplitude in the high-energy limit, we extract the one-loop gluon-production vertex to the same accuracy, and, by exploiting the iterative structure of the gluon-production vertex implied by the BDS ansatz, we perform the first computation of the two-loop gluon-production vertex up to and including finite terms.