Some investigations into algebraic and geometric properties of Feynman integrals and related topics

Abstract

We describe our investigations into various formal properties of Feynman integrals and of scattering amplitudes obtained from studying algebraic structures. There are variety of ways to study these properties. Out of which, we use method of regions and method of using Hopf algebra for studying Feynman integrals and a geometry based approach for scattering amplitudes. In the method of regions based approach, we use the ASPIRE program, which is based on the Landau singularities and the method of Power geometry to unveil the regions required for the evaluation of a given Feynman diagram asymptotically in a given limit. It also allows for the evaluation of scaling coming from the top facets. In this work, we relate the scaling having equal components of the top facets of the Newton polytope to the maximal cut of given Feynman integrals. We have therefore connected two independent approaches to the analysis of Feynman diagrams. In the second approach, 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 crosschecked 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. Finally, inspired by the recent work of Nima Arkani Hamed and collaborators who introduced the notion of positive geometry to account for the structure of tree-level scattering amplitudes in bi-adjoint $\phi^3$ theory, which led to one-loop descriptions of the integrands. Here we consider the one-loop integrals themselves in $\phi^3$ theory. In order to achieve this end, the geometrical construction offered by Schnetz for Feynman diagrams is hereby extended, and the results are presented. The extension relies on masking the loop momentum variable with a constant and proceeding with the calculations. The results appear as a construction given in a diagrammatic manner. The significance of the resulting triangular diagrams is that they have a common side amongst themselves for the corresponding Feynman diagrams they pertain to. Further extensions to this mathematical construction can lead to additional insights into higher loops. A mathematica code has been provided in order to generate the final results given the initial parameters of the theory.