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.
Highlights
In a series of publications [1,2,3,4], Abreu et al have developed the study of cut Feynman diagrams with the intention to exploit generalized unitarity, as encoded in the corresponding Hopf algebras, to calculate Feynman integrals in those cases where the final results and the corresponding cuts can be expressed by multiple polylogarithms
In [1], inspired by the work of Duhr [5], Abreu et al have initiated a new method of calculating Feynman integrals which is based on cut diagrams and Hopf algebras
We have considered a nonplanar two-loop diagram to study the new method in this case as well
Summary
In a series of publications [1,2,3,4], Abreu et al have developed the study of cut Feynman diagrams with the intention to exploit generalized unitarity, as encoded in the corresponding Hopf algebras, to calculate Feynman integrals in those cases where the final results and the corresponding cuts can be expressed by multiple polylogarithms. While at the one-loop level, the calculations are straightforward, the corresponding diagrams exhibit important and useful mathematical properties such as the integrability condition and the first-entry condition made evident by the new formalism These are related to the discovery that Feynman integrals obey Hopf Algebras, which plays an important role in the new method. The calculation using the new method involves several steps: First of all, the total cut diagram for any one channel (of the external momenta), which is a sum of all the possible cuts (the cut propagator momenta add up to the channel momentum), needs to be evaluated for that channel
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