Graphical method for computing an infinite family of Feynman periods graphical method for computing an infinite family of Feynman periods

The goal of the present study is to demonstrate the possibility to analyze and evaluate certain multiple integrals that give residues of Feynman amplitudes by means of graph-theoretical methods. The method is applied to calculate the 'wheel with N spokes' family of 'Feynman periods' in a massless scalar quantum field theory. All calculations are performed in position space. We show that this specific integration problem can be mapped to a problem of enumerating weighted paths on an associated graph with weighted edges and vertices. The problem can be further reduced to the known problem of enumerating 2-Motzkin paths with N-2 steps, which gives the Catalan number CN-1. Relevant decompositions of Catalan numbers are also derived