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.