Vacuum Energy and Renormalization of the Field-Independent Term. (arXiv:2107.06069v2 [hep-th] UPDATED)
<a href="http://arxiv.org/find/hep-th/1/au:+Marian_I/0/1/0/all/0/1">I. G. Marian</a>, <a href="http://arxiv.org/find/hep-th/1/au:+Jentschura_U/0/1/0/all/0/1">U. D. Jentschura</a>, <a href="http://arxiv.org/find/hep-th/1/au:+Defenu_N/0/1/0/all/0/1">N. Defenu</a>, <a href="http://arxiv.org/find/hep-th/1/au:+Trombettoni_A/0/1/0/all/0/1">A. Trombettoni</a>, <a href="http://arxiv.org/find/hep-th/1/au:+Nandori_I/0/1/0/all/0/1">I. Nandori</a>
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.
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.
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