Are there scaling solutions in the O(N)-models for large N in d>4?

TL;DR

This paper investigates the existence of scaling solutions in large N O(N) models in dimensions greater than four, using the functional renormalization group, and finds that such solutions are either unbounded or singular.

Contribution

It demonstrates that in the large N limit and within a specific approximation, critical potentials in these models are either unbounded or singular, challenging the existence of stable solutions.

Findings

Critical potentials are unbounded from below.

Critical potentials are singular at finite field values.

No stable scaling solutions found in the studied regime.

Abstract

There have been some speculations about the existence of critical unitary O(N)-invariant scalar field theories in dimensions 4<d<6 and for large N. Using the functional renormalization group equation, we show that in the lowest order of the derivative expansion, and assuming that the anomalous dimension vanishes for large $N$, the corresponding critical potentials are either unbounded from below or singular for some finite value of the field.