Concerning the Double Scaling Limit in the $O(N)$ Vector Model in Four-Dimensions

TL;DR

This paper reevaluates the 1/N expansion in the four-dimensional O(N) vector model, highlighting the instability at the critical point and questioning the feasibility of a double-scaling limit in this context.

Contribution

It demonstrates that the effective potential becomes complex at the critical point, indicating an unstable vacuum and challenging the existence of a double-scaling limit in four dimensions.

Findings

Effective potential is complex at the critical point.

Unstable vacuum signals issues with the 1/N expansion.

Double-scaling limit likely does not exist in this model.

Abstract

The 1/N expansion for the O(N) vector model in four dimensions is reconsidered. It is emphasized that the effective potential for this model becomes everywhere complex just at the critical point, which signals an unstable vacuum. Thus a critical O(N) vector model cannot be consistently defined in the 1/N expansion for four-dimensions, which makes the existence of a double-scaling limit for this theory doubtful.