Abstract
We revisit the decoupling effects associated with heavy particles in the renormalization group running of the vacuum energy in a mass-dependent renormalization scheme. We find the running of the vacuum energy stemming from the Higgs condensate in the entire energy range and show that it behaves as expected from the simple dimensional arguments meaning that it exhibits the quadratic sensitivity to the mass of the heavy particles in the infrared regime. The consequence of such a running to the fine-tuning problem with the measured value of the Cosmological Constant is analyzed and the constraint on the mass spectrum of a given model is derived. We show that in the Standard Model (SM) this fine-tuning constraint is not satisfied while in the massless theories this constraint formally coincides with the well known Veltman condition. We also provide a remarkably simple extension of the SM where saturation of this constraint enables us to predict the radiative Higgs mass correctly. Generalization to constant curvature spaces is also given.
Highlights
It is widely accepted that our today’s universe is undergoing the phase of the accelerated expansion which is commonly explained by the presence of the Cosmological Constant (CC)Svac = d4x√−g LSM + ξ φ†φ R + a1 Rμ2 ναβ + a2 Rμ2 ν + a3 R2 + a4R − 1 (R + 2 16π Gvac vac) . (1)The renormalization procedure for the theory (1) consists of the renormalization of the Standard Model (SM) matter fields, couplings and masses, non-minimal coupling ξ and the gravitational couplings a1,2,3,4, Gvac and vac
We are going to work in the low energy domain of the gravitational physics and, for that reason, the short distance effects from the higher derivative terms a1,2,3,4, in (1) are not important for our considerations, and so we start with the usual bare Hilbert–Einstein action with coupling constants Gvac, vac supplemented with nonminimal coupling ξ : SH E =
We will comment on the inconsistencies of the similar derivation presented in [11], as we demonstrate the importance of considering the Renormalization Group (RG) running of the total vacuum energy ρvac + ρind since, ρvac and ρind run separately, it is only the sum that exhibits behavior consistent with the Appelquist– Carazzone decoupling theorem [4]
Summary
It is widely accepted that our today’s universe is undergoing the phase of the accelerated expansion which is commonly explained by the presence of the Cosmological Constant (CC). We are going to work in the low energy domain of the gravitational physics and, for that reason, the short distance effects from the higher derivative terms a1,2,3,4, in (1) are not important for our considerations, and so we start with the usual bare Hilbert–Einstein action with coupling constants Gvac, vac supplemented with nonminimal coupling ξ : SH E =. The physical vacuum energy ρphys consists of several additional parts. The physical value is measured at the cosmological RG scale μc, which is experimentally given by μc = O(10−3) eV, as ρphys = ρvac(μc) + ρind(μc) + . We deal with the time-independent classical curved background and will derive the RG evolution of ρvac and ρind of the form (5) taking into account the decoupling effects due to massive particles by using the mass-dependent RG formalism. In “Appendices” we provide the technical details, as well as generalize the flat spacetime results to the spaces with constant curvature
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