Flow equation for the scalar model in the large $N$ expansion and its applications

TL;DR

This paper investigates the flow equations of the O(N) scalar model at next-to-leading order in 1/N expansion, revealing fixed points and corrections to the emergent AdS geometry, with implications for understanding quantum field theories and holography.

Contribution

It derives NLO flow equations for the O(N) model, analyzes fixed points of the running coupling, and computes NLO corrections to the emergent AdS metric.

Findings

The running coupling exhibits both UV and IR fixed points.

NLO corrections alter the asymptotic AdS structure of the emergent geometry.

The AdS radius decreases at NLO in the IR limit, indicating quantum corrections to holographic duals.

Abstract

We study the flow equation of the O($N$) $\varphi^4$ model in $d$ dimensions at the next-to-leading order (NLO) in the $1/N$ expansion. Using the Schwinger-Dyson equation, we derive 2-pt and 4-pt functions of flowed fields. As the first application of the NLO calculations, we study the running coupling defined from the connected 4-pt function of flowed fields in the $d+1$ dimensional theory. We show in particular that this running coupling has not only the UV fixed point but also an IR fixed point (Wilson-Fisher fixed point) in the 3 dimensional massless scalar theory. As the second application, we calculate the NLO correction to the induced metric in $d+1$ dimensions with $d=3$ in the massless limit. While the induced metric describes a 4-dimensional Euclidean Anti-de-Sitter (AdS) space at the leading order as shown in the previous paper, the NLO corrections make the space asymptotically AdS only in UV and IR limits. Remarkably, while the AdS radius does not receive a NLO correction in the UV limit, the AdS radius decreases at the NLO in the IR limit, which corresponds to the Wilson-Fisher fixed point in the original scalar model in 3 dimensions.