The Hopf Algebra Structure of the Two Loop Three Mass Non-Planar Feynman Diagram

TL;DR

This paper extends the Hopf algebra approach to compute two-loop non-planar Feynman diagrams with three mass scales, confirming the symbol alphabet matches the planar case and reconstructing the full integral result.

Contribution

It demonstrates that the Hopf algebra method and symbol alphabet used for planar diagrams also apply to non-planar diagrams with multiple mass scales, providing a systematic analysis and reconstruction.

Findings

The non-planar symbol alphabet is identical to the planar case.

The Hopf algebra method applies to non-planar two-loop diagrams.

Full integral results can be reconstructed from the symbol.

Abstract

The method of using Hopf algebras for calculating Feynman integrals developed by Abreu et al. is applied to the two-loop non-planar on-shell diagram with massless propagators and three external mass scales. We show that the existence of the method of cut Feynman diagrams comprising of the coproduct, the first entry condition and integrability condition that was found to be true for the planar case also holds for the non-planar case; furthermore, the non-planar symbol alphabet is the same as for the planar case. This is one of the main results of this work, and they have been obtained by a systematic analysis of the relevant cuts, using the symbolic manipulation codes HypExp and PolyLogTools. The obtained result for the symbol is cross-checked by an analysis of the known two-loop original Feynman integral result. In addition, we also reconstruct the full result from the symbol. This is the other main result in this paper.