Some remarks on Feynman rules for non-commutative gauge theories based on groups $G\neq U(N)$

TL;DR

This paper investigates the structure of Feynman rules in non-commutative gauge theories for subgroups of U(N), revealing limitations in partial summations and disproving some conjectured rules for SO(N).

Contribution

It provides a detailed analysis of partial summations in non-commutative gauge theories for subgroups of U(N), highlighting the absence of certain simplifications and disproving specific conjectures.

Findings

Partial summations do not produce new Feynman rules for G⊂U(N), G≠U(M)

A cancellation mechanism is crucial in the U(N) case but absent for subgroups

Certain SO(N) Feynman rules cannot be derived from the enveloping algebra approach.

Abstract

We study for subgroups $G\subseteq U(N)$ partial summations of the $\theta$-expanded perturbation theory. On diagrammatic level a summation procedure is established, which in the U(N) case delivers the full star-product induced rules. Thereby we uncover a cancellation mechanism between certain diagrams, which is crucial in the U(N) case, but set out of work for $G\subset U(N)$. In addition, an explicit proof is given that for $G\subset U(N), G\neq U(M), M<N$ there is no partial summation of the $\theta $-expanded rules resulting in new Feynman rules using the U(N) star-product vertices and besides suitable modified propagators at most a $finite$ number of additional building blocks. Finally, we show that certain SO(N) Feynman rules conjectured in the literature cannot be derived from the enveloping algebra approach.