The one-loop six-dimensional hexagon integral and its relation to MHV amplitudes in N=4 SYM

We provide an analytic formula for the (rescaled) one-loop six-dimensional scalar hexagon integral (Phi) over tilde (6) with all external legs massless, in terms of classical polylogarithms. We show that this integral is closely connected to two integrals appearing in one- and two-loop amplitudes in planar N = 4 super-Yang-Mills theory, Omega((1)) and Omega((2)). The derivative of Omega((2)) with respect to one of the conformal invariants yields (Phi) over tilde (6), while another first-order differential operator applied to (Phi) over tilde (6) yields Omega((1)). We also introduce some kinematic variables that rationalize the arguments of the polylogarithms, making it easy to verify the latter differential equation. We also give a further example of a six-dimensional integral relevant for amplitudes in N = 4 super-Yang-Mills theory.