Vacuum Energy and Renormalization of the Field-Independent Term

TL;DR

This paper investigates the nonperturbative renormalization of the constant, field-independent term in the functional renormalization group, highlighting its physical significance in various dimensions and proposing a subtraction method for proper renormalization.

Contribution

It introduces a subtraction method for the RG flow of the constant term in four dimensions when the Gaussian fixed point is absent, addressing a key issue in FRG analysis.

Findings

The constant term relates to ground-state energy in quantum mechanics.

In four dimensions, it corresponds to the cosmological constant.

A subtraction method is proposed for UV scaling when the Gaussian fixed point is missing.

Abstract

Due to its construction, the nonperturbative renormalization group (RG) evolution of the constant, field-independent term (which is constant with respect to field variations but depends on the RG scale $k$) requires special care within the Functional Renormalization Group (FRG) approach. In several instances, the constant term of the potential has no physical meaning. However, there are special cases where it receives important applications. In low dimensions ($d=1$), in a quantum mechanical model, this term is associated with the ground-state energy of the anharmonic oscillator. In higher dimensions ($d=4$), it is identical to the $\Lambda$ term of the Einstein equations and it plays a role in cosmic inflation. Thus, in statistical field theory, in flat space, the constant term could be associated with the free energy, while in curved space, it could be naturally associated with the cosmological constant. It is known that one has to use a subtraction method for the quantum anharmonic oscillator in $d=1$ to remove the $k^2$ term that appears in the RG flow in its high-energy (UV) limit in order to recover the correct results for the ground-state energy. The subtraction is needed because the Gaussian fixed point is missing in the RG flow once the constant term is included. However, if the Gaussian fixed point is there, no further subtraction is required. Here, we propose a subtraction method for $k^4$ and $k^2$ terms of the UV scaling of the RG equations for $d=4$ dimensions if the Gaussian fixed point is missing in the RG flow with the constant term. Finally, comments on the application of our results to cosmological models are provided.