Comment on the ``$\theta$-term renormalization in the (2+1)-dimensional $CP^{N-1}$ model with $\theta$ term''

TL;DR

This paper corrects a numerical inaccuracy in the previously published coefficient of the $eta$-function for the Chern-Simons term in the $CP^{N-1}$ model, affirming that the main conclusions remain valid.

Contribution

It provides the correct numerical value for the $eta$-function coefficient, clarifying an error in prior work on the $CP^{N-1}$ model with a $ heta$ term.

Findings

Corrected the $eta$-function coefficient value.

Confirmed that main conclusions of prior work are unaffected.

Clarified the numerical discrepancy in earlier results.

Abstract

It is found that the recently published first coefficient of nonzero $\beta$-function for the Chern-Simons term in the $1/N$ expansion of the $CP^{N-1}$ model is untrue numerically. The correct result is given. The main conclusions of Park's paper are not changed.