Master Integrals for Massless Three-Loop Form Factors: One-Loop and Two-Loop Insertions

TL;DR

This work identifies and computes the master integrals needed for massless three-loop quark and gluon form factors, focusing on insertion topologies with bubble insertions. Master integrals are categorized into three classes, with six insertion topologies solved analytically to all orders in ε and cross-validated by sector decomposition. Explicit ε-expansions are provided for A_{5,1}, A_{5,2}, A_{6,1}, A_{6,3}, A_{7,1}, and A_{7,2}, including high-order poles and transcendental constants, enabling construction of the complete three-loop form factors. The results illuminate the infrared structure at this order and establish a solid basis for further analytic and numerical techniques, such as Mellin-Barnes representations, to complete the master integral set.

Abstract

The three-loop form factors in massless QCD can be expressed as a linear combination of master integrals. Besides a number of master integrals which factorise into products of one-loop and two-loop integrals, one finds 16 genuine three-loop integrals. Of these, six have the form of a bubble insertion inside a one-loop or two-loop vertex integral. We compute all master integrals with these insertion topologies.

Figures (1)

• Figure 1: Three-loop master integrals with massless propagators. The incoming momentum is $q=p_1+p_2$. Outgoing lines are considered on-shell and massless, i.e. $p_1^2=p_2^2=0$.