Abstract
We present the complete set of planar master integrals relevant to the calculation of three-point functions in four-loop massless Quantum Chromodynamics. Employing direct parametric integrations for a basis of finite integrals, we give analytic results for the Laurent expansion of conventional integrals in the parameter of dimensional regularization through to terms of weight eight.
Highlights
We establish some notation and describe our enumeration of the planar four-loop form factor master integrals
We present the complete set of planar master integrals relevant to the calculation of three-point functions in four-loop massless Quantum Chromodynamics
After carrying out integral reductions for all Feynman integrals with our inhouse reduction code, we find just ninety-nine master integrals in ninety-seven sectors
Summary
We establish some notation and describe our enumeration of the planar four-loop form factor master integrals. As a first non-trivial step, we construct a single Reduze 2 [28,29,30] integral family (see table 1) which covers all planar sectors (or topologies). We make highlysymmetric choices for the auxiliary propagators of the four-loop planar ladder form factor integral topology. At this stage, Reduze 2 allows for the construction of a compact sector selection encoding the minimal number of sectors for which integration by parts reductions are required. Only two of our master integral topologies are of the multi-component type For these topologies, we prefer to work with squared propagators, marked with dotted edges at the level of graphs (see figure 2). A D1 (k1) D2 (k2) D3 (k3) D4 (k4) D5 (p1 − k1) D6 (p1 − k1 + k2) D7 (p1 − k1 + k2 − k3) D8 (p1 − k1 + k2 − k3 + k4) D9 (p2 + k1) D10 (p2 + k1 − k2) D11 (p2 + k1 − k2 + k3) D12 (p2 + k1 − k2 + k3 − k4) D13 (k1 − k2) D14 (k2 − k3) D15 (k3 − k4) D16 (k1 − k2 + k3) D17 (k2 − k3 + k4) D18 (k1 − k2 + k3 − k4)
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