An Analytic Computation of Three-Loop Five-Point Feynman Integrals

TL;DR

This paper presents an analytic method for computing complex three-loop five-point Feynman integrals using differential equations and algebraic geometry, advancing multi-loop amplitude calculations.

Contribution

It introduces a novel approach combining differential equations and algebraic geometry for three-loop five-point integrals, with a new representation of weight six functions.

Findings

Successfully evaluated the three-loop five-point integral family.

Developed a new representation for weight six functions.

Enabled potential future analytic computations of multi-leg Feynman integrals.

Abstract

We evaluate the three-loop five-point pentagon-box-box massless integral family in the dimensional regularization scheme, via canonical differential equation. We use tools from computational algebraic geometry to enable the necessary integral reductions. The boundary values of the differential equation are determined analytically in the Euclidean region. To express the final result, we introduce a new representation of weight six functions in terms of one-fold integrals over the product of weight-three functions with weight-two kernels that are derived from the differential equation. Our work paves the way to the analytic computation of three-loop multi-leg Feynman integrals.