Inhomogeneous and Homogeneous Renormalization Group Equations for the Effective Potential

TL;DR

This paper rederives the renormalization group equation for the effective potential, showing it becomes homogeneous when normalized, and explores differences in vacuum energy density in spontaneous symmetry breaking.

Contribution

It demonstrates that the normalized effective potential satisfies a homogeneous RGE and distinguishes vacuum energy in SSB from the symmetric case.

Findings

Normalized effective potential obeys a homogeneous RGE.

Vacuum energy density appears only in spontaneous symmetry breaking.

Normalized potentials differ significantly between symmetric and SSB cases.

Abstract

The inhomogeneous renormalization group equation for the effective potential is rederived. It is shown that when the effective potential is normalized by the normalization condition on the generating functional, its renormalization group equation is homogeneous. This is demonstrated in the case of massive $\phi^4$-theory. We also show that for the case of spontaneous symmetry breaking, the normalized effective potential is completely different from the symmetric case, though the two cases satisfy the same RGE with the same RG-functions. It is concluded that the vacuum energy density arises only in the case of SSB.