Complete computation of all three-loop five-point massless planar integrals

TL;DR

This work completes the analytic computation of all massless planar three-loop five-point Feynman integrals across four topologies. By constructing pure integral bases and canonical differential equations, and by deriving analytic boundary values up to weight six, the authors obtain a complete three-loop planar pentagon alphabet and the corresponding symbol space, enabling fast and precise numerical evaluation. The results provide essential building blocks for N^3LO predictions in 2→3 processes and offer a foundation for future non-planar and gauge-theory bootstrap studies. The methodology integrates advanced IBP reduction via NeatIBP, Baikov-based numerator construction, and rigorous alphabet/symbol analysis, with thorough numerical validation against independent approaches.

Abstract

We calculate all three-loop, five-point, massless planar Feynman integral families in the dimensional regularization scheme. This is a new milestone in Feynman integral computations. The analysis covers four distinct families of Feynman integrals for this configuration, for all of which we derive the canonical differential equations. Our results also confirm a prediction on the three-loop five-point alphabet. The boundary values are analytically determined. Using these differential equations, the integrals can be evaluated to high precision efficiently. Our work establishes the foundation for next-to-next-to-next-to-leading-order (N$^3$LO) calculation of the production of three massless final states, as well as corresponding bootstrap studies in gauge theories.

Figures (2)

• Figure 1: Genuine three-loop five-point integral families.
• Figure 2: New sub-topologies in PBP family.