On $c_2$ invariants of 4-regular Feynman graphs

TL;DR

This paper derives a new formula for computing the $c_2$ invariant of 4-regular Feynman graphs, overcoming previous limitations due to the absence of 3-valent vertices, enabling analysis of more complex graphs.

Contribution

A novel formula for the $c_2$ invariant of 4-valent vertices in Feynman graphs is introduced, expanding computational techniques beyond 3-valent cases.

Findings

Enables computation of $c_2$ invariants for 4-regular graphs.

Facilitates analysis of graphs with small loop numbers.

Overcomes previous technical obstacles in Feynman graph invariants.

Abstract

The obstruction for application of effective techniques like denominator reduction for the computation of the $c_2$ invariant of Feynman graphs in general is the absence of a 3-valent vertex for the initial steps. In this paper such a formula for a 4-valent vertex is derived. The formula allows to compute the $c_2$ invariant of new graphs, for instance, 4-regular graphs with small loop number.