Nine-Propagator Master Integrals for Massless Three-Loop Form Factors

TL;DR

This paper completes the calculation of three-loop massless form factor integrals by deriving analytic and high-precision numerical results for previously unknown nine-propagator diagrams, advancing precision in quantum field theory computations.

Contribution

It introduces the first analytic results for certain three-loop master integrals with nine propagators, expanding the set of known integrals for massless form factors.

Findings

Analytic results for one integral using Riemann zeta functions.

High-accuracy numerical coefficients for remaining integrals.

Completion of the set of master integrals needed for three-loop form factors.

Abstract

We complete the calculation of master integrals for massless three-loop form factors by computing the previously-unknown three diagrams with nine propagators in dimensional regularisation. Each of the integrals yields a six-fold Mellin-Barnes representation which we use to compute the coefficients of the Laurent expansion in epsilon. Using Riemann zeta functions of up to weight six, we give fully analytic results for one integral; for a second, analytic results for all but the finite term; for the third, analytic results for all but the last two coefficients in the Laurent expansion. The remaining coefficients are given numerically to sufficiently high accuracy for phenomenological applications.