Comment on A. Patrascioiu and E. Seiler's paper ``Nonuniformity of the $1/N$ Expansion for Two-Dimensional $O(N)$ Models''

TL;DR

This paper clarifies that the non-uniformity in the $1/N$ expansion for 2D $O(N)$ models is consistent with asymptotic freedom and occurs in both continuum and lattice formulations, while the expansion for dimensionless quantities remains uniform.

Contribution

It demonstrates that the non-uniformity observed is predicted by asymptotic freedom and clarifies its presence in different formulations.

Findings

Non-uniformity in temperature of the $1/N$ expansion is compatible with asymptotic freedom.

The $1/N$ expansion for dimensionless quantities converges uniformly.

Non-uniformity occurs both in continuum and on the lattice.

Abstract

We remind that the non-uniformity in the temperature of the $1/N$ expansion for dimensionful quantities pointed out in Patrascioiu and Seiler's paper is not only compatible with but is predicted by asymptotic freedom, and it is present in the continuum as well as on the lattice. The convergence of the $1/N$ expansion for adimensional quantities is perfectly uniform.