Vacuum energy and renormalization of the field-independent term

Abstract

Due to its construction, the nonperturbative renormalizationgroup (RG) evolution of the constant, field-independent term (which is constantwith respect to field variations but depends on the RG scale k) requiresspecial care within the Functional Renormalization Group (FRG) approach.In several instances, the constant term of the potential has nophysical meaning. However, there are special cases where it receives importantapplications. In low dimensions (d = 1), in a quantum mechanical model, thisterm is associated with the ground-state energy of the anharmonic oscillator.In higher dimensions (d = 4), it is identical to the Λ termof the Einstein equationsand it plays a role in cosmic inflation. Thus, instatistical field theory, in flat space, the constant term could be associatedwith the free energy, while in curved space, it could be naturally associatedwith the cosmological constant. It is known that one has to use a subtractionmethod for the quantum anharmonic oscillator in d = 1 to remove the k 2term that appears in the RGflow in its high-energy (UV) limit in order to recover thecorrect results for the ground-state energy. The subtraction is needed becausethe Gaussian fixed point is missing in the RG flow once the constant term isincluded. However, if the Gaussian fixed point is there, no furthersubtraction is required. Here, we propose a subtraction method for k 4 andk 2 terms of the UV scaling of the RG equations for d = 4 dimensions if theGaussian 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.