New Fixed Points from Melonic Interactions

TL;DR

This paper explores the complex phase space of tensor field theories using the functional renormalization group, revealing new fixed points and rich dynamics that could lead to novel continuum geometries.

Contribution

It introduces the first analysis of the full tensorial phase space beyond isotropic parts, identifying non-Gaussian fixed points and new anisotropic phases.

Findings

Discovery of non-Gaussian fixed points in all regimes of $0<s\le r$

Identification of both isotropic and anisotropic fixed points

Potential candidates for asymptotic safe fixed points at critical dimensions

Abstract

Generalizations of vector field theories to tensors allow to similarly apply large-$N$ techniques but find a richer though often still tractable structure. However, the potential of such tensor theories has not been fully exploited since only a symmetry-reduced ``isotropic'' part of their phase space has been studied so far. Here we present for the first time the richness of the tensorial phase space applying the functional renormalization group to tensor fields of rank $r$ in the cyclic-melonic potential approximation including the flow of anomalous dimensions. Due to a decoupling of the flow equations of the $r$ couplings at given order, we find non-Gaussian fixed points in any regime of $0<s\le r$ non-vanishing coupling types. Each of these regimes contains isotropic fixed points of Wilson-Fisher type as well as new anisotropic fixed points. This new classification reveals a rich structure of renormalization group dynamics including candidates for asymptotic safe fixed points even at critical dimension. Considering the tensorial interactions as generating discrete geometries, the various fixed points correspond to continuum limits of distinguished ensembles of triangulations raising hope to find new classes of continuum geometry in this way.