We provide an analytic formula for the (rescaled) one-loop six-dimensional scalar hexagon integral Φ ̃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, Ω(1) and Ω(2). The derivative of Ω(2) with respect to one of the conformal invariants yields Φ ̃₆ , while another first-order differential operator applied to Φ ̃₆ 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.
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Dixon et al. (2011) studied this question.