Color Grosse-Wulkenhaar models: One-loop $\beta$-functions

TL;DR

This paper computes the one-loop beta functions of color Grosse-Wulkenhaar models, revealing asymptotic freedom for N>1 and triviality issues for N<1, with special behavior at N=1.

Contribution

It provides the first detailed one-loop beta function analysis for color Grosse-Wulkenhaar models, highlighting their asymptotic properties based on N.

Findings

Asymptotic freedom for N>1

Triviality or Landau ghost for N<1

Beta function vanishes at N=1

Abstract

The $\beta$-functions of O(N) and U(N) invariant Grosse-Wulkenhaar models are computed at one loop using the matrix basis. In particular, for ``parallel interactions", the model is proved asymptotically free in the UV limit for $N >1$, and has a triviality problem or Landau ghost for $N<1$. The vanishing $\beta$-function is recovered solely at N=1. We discuss various possible consequences of these results.