Hopf algebra structure of the two loop three mass nonplanar Feynman diagram

The method of using Hopf algebras for calculating Feynman integrals developed by Abreu et al. is applied to the two-loop nonplanar 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 nonplanar case; furthermore, the nonplanar symbol alphabet is the same as for the planar case. This is one of the main results of this work which 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 second main result of this paper.

Figure 1

The planar and nonplanar two-loop diagrams discussed in this paper.

Figure 2

$C(p_1^2,p_2^2,p_3^3)$.

Figure 3

$\mathrm{Cut}[(k+l-p_2{)}^2,l^2,k^2]$