Give the neo-Archimedes a non-minimal coupling to spacetime curvature and he could....

...  protect the Higgs from quantum fluctuations...

The present paper will point out an exception to {the} inevitable destabilization {of the electroweak scale by additive power-law quantum corrections [4]} noting that the Higgs field, being a doublet of fundamental scalars, necessarily develops the non-minimal Higgs-curvature interaction [7]

 ∆V(H, R)=ζRH^†H   ...

with which the Higgs vacuum expectation value (VEV) {v²=−m_H²/λ_H} changes to

v²=(−m_H²− 4ζV₀/M_Pl²) / (λ_H + ζm_H²/M²_Pl)  ...

and this new VEV can be stabilized by fine-tuning ζ to counterbalance the quadratic divergences δm_H²∝Λ_UV² with the quartic divergences δV₀∝Λ_UV⁴. Quantum corrections to the SM parameters are independent of ζ if gravity is classical, and thus ζ acts as a gyroscope that stabilizes the electroweak scale against violent UV contributions. This novel fine-tuning scheme is in accord with Sakharov’s induced gravity approach, and continues to hold also in extensions of the SM involving extra Higgs fields (additional Higgs doublets or singlet scalars or scalar multiplets belonging to larger gauge groups)... 

 The workings of the fine-tuning ... is best exemplified by the special value of ζ

 ζ₀=1/(n_F−n_B)×(6h_t²−6λ_H−3g_Y²/4−9g₂²/4)×(M_Pl²/Λ_UV²)  ...

{where h_t is top quark Yukawa coupling, g_Y (g₂) is the hypercharge (isospin) gauge coupling, and n_F (n_B) is the total number of fermions (bosons) in the SM. The}... numerical value is ζ₀≈1/15 for Λ_UV≈M_Pl. It is smaller than the conformal value 1/6 [7] and much much smaller than the Higgs inflation value 10⁴ [11]. As a function of Λ_UV, ζ₀ completely eradicates the power-law UV contribution ... and the concealed logarithmic corrections give the usual renormalization properties of the Higgs VEV. Obviously, smaller the Λ_UV larger the ζ₀ though there remains lesser and lesser need to fine-tuning if Λ_UV gets closer and closer to the Fermi scale... 

To discuss further, we state that ζ fine-tuning can have a variety of implications for model building and phenomenology. Below we highlight some of them briefly: ...

• The matter sector does not have to be precisely the SM. The finetuning mechanism here works also in extensions of the SM which include extra scalar fields provided that each scalar assumes a non-minimal coupling to curvature...  The scalar fields can be additional Higgs doublets, singlet scalars or multiplets of scalars belonging to larger gauge groups. The VEV of each scalar is of the form in (4), and can be fine-tuned individually without interfering with the VEVs of the remaining scalars.

• The classical gravity assumption in the present work can be lifted to include quantum gravitational effects [22]. In this case, non-minimal coupling spreads into the SM parameters through graviton loops. Moreover, this quantum gravitational setup is inherently non-renormalizable [15]. These factors can obscure the process of fine-tuning ζ.

• There have been various attempts [23] to nullify the quadratic divergence in Higgs VEV by introducing singlet scalars. This is now known to be not possible at all, even when vector-like fermions are included [24]. Nevertheless, non-minimal coupling between curvature scalar and some scalar fields can help stabilize both electroweak and hidden scales ... and then masses of the particles in the SM and hidden sector get automatically stabilized.

Effects of Curvature-Higgs Coupling on Electroweak Fine-Tuning

Durmus A. Demir

(Submitted on 1 May 2014 (v1), last revised 10 May 2014 (this version, v2))

... and connect the electroweak scale with the cosmological inflation one

Recently, by Demir [10], it has been shown that the one-loop quadratic divergences can be suppressed completely if Higgs coupling to spacetime curvature is finely tuned. The most interesting aspect of this fine-tuning is that it is phantasmal if gravity is classical. The reason for this phantom behaviour is that the Higgs-curvature coupling does not appear in quantum corrections to the SM parameters. Moreover, particle masses are sensitive only to the Higgs vacuum expectation value (VEV); they are completely immune to what mechanism has set the Higgs VEV to that specific value appropriate for electroweak interactions. In this sense, one is able to stabilize the Higgs boson through a “soft fine-tuning” that does not interfere with workings of the SM [10] (see Refs. [11, 12] for quantum corrections in curved background).  

In the present work, we discuss implications of the quartic divergences. More specifically, we show that the quartic divergences induce an enormous vacuum energy which can inflate the Universe. The scale of inflation sets the UV scale and determines the degree of soft finetuning. In fact, quartic contributions give the plateau section of the slow-roll inflaton potential and fully governs the inflationary epoch for parameter ranges preferred by the softly fine-tuned Higgs mass. As a matter of fact, an analysis of the inflationary phase is rather timely since recent measurements of the tensor-to-scalar ratio r in CMB polarization have the potential to fix the scale of inflation...  

In this paper, exposed is one such model in which inflationary dynamics and electroweak stability are directly correlated. Extended SM scenarios keeping the Higgs vacuum stable while yielding the high-scale inflation successfully exist in the literature by incorporating either an additional U(1)B−L symmetry [22], with non-minimal coupling of the Higgs kinetic term with the Higgs field [23], with the Einstein tensor [24], with both the Higgs field and the Einstein tensor [25]... 

As a matter of fact, quantum corrections shift the scalar curvature by δR=(Λ_UV⁴/M_Pl²)×(n_F−n_B)/(4π)²... after neglecting subleading quadratic and logarithmic corrections... This curvature correction, proportional to n_F−n_B, takes enormous values if there is no fermion-boson degeneracy in the theory. The SM exhibits no such symmetry... the background de Sitter spacetime is found to have the Hubble constant H²=δR/12... This gives numerically H≃0.18Λ_UV²/M_Pl ... for the SM spectrum where the Hubble parameter acts like a moment arm in a see-saw creating a balance between Λ_UV and M_Pl. The CMB observations such as BICEP2 and Planck can measure H when foreground is small. This then fixes Λ_UV, directly. For H=10¹⁶GeV, as reported by BICEP2, one finds Λ_UV=3.7×10¹⁷GeV. This means that a measurement of the Hubble constant H determines the upper validity limit Λ_UV of the SM and, in general, smaller the H smaller the Λ_UV. The ongoing and upcoming experiments on CMB polarization place upper limits on tensor to scalar ratio r... 

In consequence, quartic quantum corrections from matter loops inflate the Universe with a Hubble constant determined by the UV scale Λ_UV. The flatness, homogeneity and isotropy of the observable Universe can be understood by some 60 e-foldings in a rather short time interval. The crucial question concerns exit of the Universe from this exponential expansion phase. This is not possible with a constant vacuum energy. The resolution comes from the fact that, the vacuum energy does actually change in time due to phase transitions occurring as the Universe expands. In fact, a decaying cosmological constant was proposed decades ago by Dolgov [28]. In this sense, as an inherent assumption in inflationary cosmology, the vacuum energy can be ascribed as the energy density of a slowly-varying real scalar field. It could be modelled in various ways, and slow-roll of the scalar field along the model potential can give a graceful exit from inflationary epoch such that the vacuum energy at the beginning has effectively decayed into matter and radiation during reheating. 

Given the regularized action in Ref. [10], there arise {two viable}... scenarios to be considered as emerging from sliding cutoff scale: ... {one is a} chaotic inflation with non-minimal coupling {ζ_φ=1/[6(4π)²√(n_F−n_B)}... previously studied by Refs. [30, 31] and it is found that slow-roll conditions ... cannot be realized unless ζ_φ ≤ 10^-3. As  ζ_φ≃10^-4 in our scenario, inflationary dynamics could be successfully driven by the inflaton field...

Softly Fine-Tuned Standard Model and the Scale of Inflation

Beste Korutlu

(Submitted on 29 Oct 2015)