Abstract
We provide an analytic formula for the (rescaled) one-loop six-dimensional scalar hexagon integral \( {\tilde{\Phi }_6} \) 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 \( \mathcal{N} = 4 \) super-Yang-Mills theory, Ω(1) and Ω(2). The derivative of Ω(2) with respect to one of the conformal invariants yields \( {\tilde{\Phi }_6} \), while another first-order differential operator applied to \( {\tilde{\Phi }_6} \) 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 \( \mathcal{N} = 4 \) super-Yang-Mills theory.
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Dixon, L.J., Drummond, J.M. & Henn, J.M. The one-loop six-dimensional hexagon integral and its relation to MHV amplitudes in \( \mathcal{N} = 4 \) SYM. J. High Energ. Phys. 2011, 100 (2011). https://doi.org/10.1007/JHEP06(2011)100
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DOI: https://doi.org/10.1007/JHEP06(2011)100