Why are tensor field theories asymptotically free?

TL;DR

This paper explains the combinatorial reasons behind the asymptotic freedom of tensor field theories, contrasting them with vector and matrix models, and clarifies how different symmetries influence their renormalization behavior.

Contribution

It provides a pedagogic analysis of the combinatorial structures that lead to asymptotic freedom in tensor field theories, highlighting the role of crossing symmetry.

Findings

Tensor models exhibit asymptotic freedom due to their combinatorial structure.

Matrix models transition from asymptotic freedom to safety because of crossing symmetry.

Vector models lack asymptotic freedom due to absence of wave function renormalization.

Abstract

In this pedagogic letter we explain the combinatorics underlying the generic asymptotic freedom of tensor field theories. We focus on simple combinatorial models with a $1/p^2$ propagator and quartic interactions and on the comparison between the intermediate field representations of the vector, matrix and tensor cases. The transition from asymptotic freedom (tensor case) to asymptotic safety (matrix case) is related to the crossing symmetry of the matrix vertex whereas in the vector case, the lack of asymptotic freedom ("Landau ghost"), as in the ordinary scalar case, is simply due to the absence of any wave function renormalization at one loop.