Feynman symmetries of the Martin and $c_2$ invariants of regular graphs

TL;DR

This paper introduces a new sequence derived from regular graphs that captures symmetries of Feynman period integrals and relates to known invariants, providing new insights into graph invariants in quantum field theory.

Contribution

The paper defines a novel sequence for regular graphs based on the Martin polynomial that encodes symmetries of Feynman periods and relates to the $c_2$ invariant and extended graph permanent.

Findings

The sequence respects all known symmetries of Feynman period integrals.

It determines the $c_2$ invariant and extended graph permanent.

It proves the completion conjecture for the $c_2$ invariant at all primes.

Abstract

For every regular graph, we define a sequence of integers, using the recursion of the Martin polynomial. This sequence counts spanning tree partitions and constitutes the diagonal coefficients of powers of the Kirchhoff polynomial. We prove that this sequence respects all known symmetries of Feynman period integrals in quantum field theory. We show that other quantities with this property, the $c_2$ invariant and the extended graph permanent, are essentially determined by our new sequence. This proves the completion conjecture for the $c_2$ invariant at all primes, and also that it is fixed under twists. We conjecture that our invariant is perfect: Two Feynman periods are equal, if and only if, their Martin sequences are equal.