Graph Invariants with Connections to the Feynman Period in $\phi^4$ Theory
Iain Crump
TL;DR
This paper explores graph invariants related to Feynman periods in $ $ theory, introducing the extended graph permanent and analyzing its invariance properties, with implications for understanding the structure of Feynman graphs.
Contribution
The paper introduces the extended graph permanent, demonstrating its invariance under key graph operations and providing new computational and interpretative methods.
Findings
Extended graph permanent is preserved by key graph operations.
The sequence of residues from the extended graph permanent has a product property.
Alternative interpretation as point count of a novel graph polynomial.
Abstract
Feynman diagrams in theory have as their underlying structure 4-regular graphs. In particular, any 4-point 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…
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Taxonomy
Topicsadvanced mathematical theories · Black Holes and Theoretical Physics · Noncommutative and Quantum Gravity Theories