Flow Equations for U_k and Z_k

TL;DR

This paper clarifies the correct derivation of flow equations for the potential and kinetic term in scalar field theory, resolving previous misconceptions about their dependence on fluctuation determinant expansions.

Contribution

It provides the correct derivation of coupled RG flow equations for U_k and Z_k, correcting prior claims about their dependence on fluctuation determinant expansion.

Findings

Correct set of coupled differential equations for U_k and Z_k established

Previous claims about expansion dependence are shown to be incorrect

Clarification of the derivation procedure for flow equations

Abstract

By considering the gradient expansion for the wilsonian effective action S_k of a single component scalar field theory truncated to the first two terms, the potential U_k and the kinetic term Z_k, I show that the recent claim that different expansion of the fluctuation determinant give rise to different renormalization group equations for Z_k is incorrect. The correct procedure to derive this equation is presented and the set of coupled differential equations for U_k and Z_k is definitely established.