2017/04/20 by Iain Crump, Crump, Iain
Mathematics · Physics and Astronomy · #05C21 #05C31 #81T18 #Black Holes and Theoretical Physics #Combinatorics (math.CO) #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Noncommutative and Quantum Gravity Theories #advanced mathematical theories
paper · pdf · doi:10.48550/arxiv.1704.06350
openalex publication_date 2017/04/20 · openalex created_date 2024/04/11 · openalex updated_date 2026/07/28
Feynman diagrams in ϕ4 theory have as their underlying structure 4-regular graphs. In particular, any 4-point ϕ4 graph can be uniquely derived from a 4-regular graph by deleting a vertex. The Feynman period is a simplified version of the Feynman integral, and is of special interest, as it maintains much of the important number theoretic information from the Feynman integral. It is also of structural interest, as it is known to be preserved by a number of graph theoretic operations. In particular, the vertex deleted in constructing the 4-point graph does not affect the Feynman period, and it is invariant under planar duality and the Schnetz twist, an operation that redirects edges incident to a 4-vertex cut. Further, a 4-regular graph may be produced by a 3-sum operation on triangles in two 4-regular graphs. The Feynman period of this graph with a vertex deleted is equal to the product of the Feynman periods of the two smaller graphs with one vertex deleted each. These operations currently explain all known instances of non-isomorphic 4-point ϕ4 graphs with equal periods. With this in mind, other graph invariants that are preserved by these operations for 4-point ϕ4 graphs are of interest, as they may provide insight into the Feynman period. In this thesis the extended graph permanent is introduced; an infinite sequence of residues from prime order finite fields. It is shown that this sequence is preserved by these three operations, and has a product property. Additionally, computational techniques will be established, and an alternate interpretation will be presented as the point count of a novel graph polynomial. Further, the previously existing c2 invariant and Hepp bound are examined, two graph invariants that are conjectured to be preserved by these graph operations. A combinatorial approach to the c2 invariant is introduced.