Comment on "Remark on the renormalization group equation for the Penner model"

TL;DR

This paper critically examines and corrects a previous derivation of the renormalization group equation for the Penner model's free energy derivatives, clarifying the proper form and extending its applicability.

Contribution

It explicitly shows the incorrectness of Johnston's sum over punctures and corrects the derivation of the RG equation for the Penner model's free energy derivatives.

Findings

Johnston's sum over punctures for the third derivative is incorrect.

The differentiated RG equation previously proposed is invalid.

The corrected differentiated RG equation is extended to any derivative order.

Abstract

We show explicitly that the sum over punctures for the three times derivative for the Penner free energy $F_{0}^{3}$, given by D.A. Johnston, Phys.Rev.D 51 (1995) is not correct. As a consequence, Eq.(21), the differentiated version for the renormalization group (RG) equation, is wrong. Also, his conclusion that the differentiated version of the (RG) equation for the three-times derivative of the free energy can be obtained from the higher genus (RG) equation can not be true. Finally, the differentiated version of the (RG) equation is extended to any $s$ derivative of the free energy $F_{0}$.