Prismatic Large $N$ Models for Bosonic Tensors

TL;DR

This paper introduces a large N bosonic tensor model with prism-like index structure, solving it via Schwinger-Dyson equations, and explores its fixed points and operator spectrum across various dimensions.

Contribution

It provides the first large N solution of a bosonic tensor model with prism topology and analyzes its fixed points and spectrum using both Schwinger-Dyson equations and epsilon expansion.

Findings

Spectrum of bilinear operators has no complex dimensions in certain dimensions.

Identifies a prismatic fixed point with real couplings for large N.

Fixed point persists down to N≈54 before merging and becoming complex.

Abstract

We study the $O(N)^3$ symmetric quantum field theory of a bosonic tensor $\phi^{abc}$ with sextic interactions. Its large $N$ limit is dominated by a positive-definite operator, whose index structure has the topology of a prism. We present a large $N$ solution of the model using Schwinger-Dyson equations to sum the leading diagrams, finding that for $2.81 < d < 3$ and for $d<1.68$ the spectrum of bilinear operators has no complex scaling dimensions. We also develop perturbation theory in $3-\epsilon$ dimensions including eight $O(N)^3$ invariant operators necessary for the renormalizability. For sufficiently large $N$, we find a "prismatic" fixed point of the renormalization group, where all eight coupling constants are real. The large $N$ limit of the resulting $\epsilon$ expansions of various operator dimensions agrees with the Schwinger-Dyson equations. Furthermore, the $\epsilon$ expansion allows us to calculate the $1/N$ corrections to operator dimensions. The prismatic fixed point in $3-\epsilon$ dimensions survives down to $N\approx 53.65$, where it merges with another fixed point and becomes complex. We also discuss the $d=1$ model where our approach gives a slightly negative scaling dimension for $\phi$, while the spectrum of bilinear operators is free of complex dimensions.