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

Abstract

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 for the all-weight orders, 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}(ε^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 algebra.

Figures (2)

• Figure 1: The block structure of the differential equation for the DP family. The light green block represents the differential equation for the integrals with fewer than $9$ propagators, which were analytically reconstructed in Abreu:2024fei and Henn:2025xrc. Furthermore, the light blue area represents the first, second, third and fifth row of the off-diagonal block, which were analytically reconstructed in Henn:2025xrc. The deep blue block represents the cut differential equation for the top-sector integrals with $9$ propagators, which were reconstructed in Henn:2021cyv. In this work, we analytically reconstruct the fourth row of the off-diagonal part, represented by the carnelian red block. The numbers in the figure show the corresponding block sizes.
• Figure 2: The origin of the genuinely new letters $W_{308}$ and $W_{311}$. The differential equation matrix entry $(4,6)$ is the coefficient of derivative of $I_4^\text{DP}$ over $I_6^\text{DP}$, which provides a new letter $W_{308}$. Note that $I_4^\text{DP}$ is reflection invariant and the coefficient on reflected integral of $I_6^\text{DP}$, $I_{23}^\text{DP}$, should be $d\log(\mathcal{R}(W_{308}))$. It is determined that $\mathcal{R}(W_{308}) =W_{299} W_{311}$, where $W_{311}$ is also a new letter. The numerators for the pure integrals $I_6^\text{DP}$ and $I_{23}^\text{DP}$ are listed.