Two-loop Six-point Planar Massless Feynman Integrals to Higher $\epsilon$ Orders

TL;DR

This paper advances the calculation of two-loop six-point planar massless Feynman integrals by extending the epsilon expansion to higher weights, deriving the complete alphabet, and demonstrating efficient evaluation methods.

Contribution

It introduces a complete alphabet with 269 letters, derives the canonical differential equation, and evaluates the integrals up to weight six using a new pseudospectral method.

Findings

Complete alphabet with 269 letters identified for all weights.

Derived the analytic canonical differential equation for the integrals.

Efficient evaluation of pure basis up to weight six using Chebyshev pseudospectral transport.

Abstract

In this work, we calculate two-loop six-point planar massless Feynman integrals at higher orders in the dimensional regulator $\epsilon$, 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}(\epsilon^2)$ 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.