Two-loop six-point planar massless feynman integrals to higher ϵ orders

A bstract In this work, we calculate two-loop six-point planar massless Feynman integrals at higher orders in the dimensional regulator ϵ , corresponding to higher transcendental weights. In previous works, these integrals were calculated up to weight four for the purpose of two-loop gauge theory amplitudes. Using modern rational reconstruction methods, we identify the complete alphabet with 269 letters relevant to all weights, derive the analytic canonical differential equation and obtain the symbols up to weight six. As a proof of concept, using a new method with Chebyshev pseudospectral transport, we show that the corresponding pure basis can be efficiently evaluated up to weight six, i.e., to $$ \mathcal{O}\left({\epsilon}^2\right) $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>O</mml:mi> <mml:mfenced> <mml:msup> <mml:mi>ϵ</mml:mi> <mml:mn>2</mml:mn> </mml:msup> </mml:mfenced> </mml:math> in a physical scattering region. The results of this work can be applied to future three-loop amplitudes and provide new data for the formal study of symbols and cluster algebras.