A 4-point Feynman diagram in scalar φ 4 theory is represented by a graph G which is obtained from a connected 4-regular graph by deleting a vertex. The associated Feynman integral gives a quantity called the period of G which is invariant under a number of graph operationsnamely, planar duality, the Schnetz twist, and it also does not depend on the choice of vertex deleted to form G.In this article we study a graph invariant we call the graph permanent, which was implicitly introduced in a paper by Alon, Linial and Meshulam [1]. The graph permanent applies to any graph G = (V, E) for which |E| is a multiple of |V | − 1 (so in particular to graphs obtained from a 4-regular graph by removing a vertex). We prove that the graph permanent, like the period, is invariant under planar duality and the Schnetz twist when these are valid operations, and we show that when G is obtained from a 2k-regular graph by deleting a vertex, the graph permanent does not depend on the choice of deleted vertex.
We give a characterization of 3-connected graphs which are planar and forbid cube, octahedron, and H minors, where H is the graph which is one ∆ − Y away from each of the cube and the octahedron. Next we say a graph is Feynman 5-split if no choice of edge ordering gives an obstruction to parametric Feynman integration at the fifth step. The 3-connected Feynman 5-split graphs turn out to be precisely those characterized above. Finally we derive the full list of forbidden minors for Feynman 5-split graphs of any connectivity.
We create for all graphs a new invariant, an infinite sequence of residues from prime order finite fields, constructed from the permanent of a reduced incidence matrix. Motivated by a desire to better understand the Feynman period in φ 4 theory, we show that this invariant is preserved by all graph operations known to preserve the period. We further establish properties of this sequence, including computation techniques and alternate interpretations as the point count of a novel polynomial.
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