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 Φ̃6 with all external legs massless, in terms of classical polylogarithms. We show that this integral is closely connected to two integrals appearing in oneand two-loop amplitudes in planar N = 4 super-Yang-Mills theory, Ω(1) and Ω(2). The derivative of Ω(2) with respect to one of the conformal invariants yields Φ̃6, while another first-order differential operator applied to Φ̃6 yields Ω (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.