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

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.