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We provide an analytic formula for the (rescaled) one-loop scalar hexagonintegral $\tilde\Phi_6$ with all external legs massless, in terms of classicalpolylogarithms. We show that this integral is closely connected to twointegrals appearing in one- and two-loop amplitudes in planar $\\mathcal{N}=4$super-Yang-Mills theory, $\Omega^{(1)}$ and $\Omega^{(2)}$. The derivative of$\Omega^{(2)}$ with respect to one of the conformal invariants yields$\tilde\Phi_6$, while another first-order differential operator applied to$\tilde\Phi_6$ yields $\Omega^{(1)}$. We also introduce some kinematicvariables that rationalize the arguments of the polylogarithms, making it easyto verify the latter differential equation. We also give a further example of asix-dimensional integral relevant for amplitudes in $\\mathcal{N}=4$super-Yang-Mills.

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