If a Higgs field is conformally coupled to gravity, then it can give rise to the scale invariant density perturbations. We make use of this result in a realistic inert Higgs doublet model, where we have a pair of Higgs doublets conformally coupled to the gravity in the early universe. The perturbation of the inert Higgs is shown to be the scale invariant. This gives rise to the density perturbation observed through CMB by its couplings to the standard model Higgs and the subsequent decay. Loop corrections of this conformally coupled system gives rise to electroweak symmetry breaking. We constrain the couplings of the scalar potential by comparing with the amplitude and spectrum of CMB anisotropy measured by WMAP and this model leads to a prediction for the masses of the lightest Higgs and the other scalars.
It is well known that to generate the density perturbation of the CMB of the magnitude observed by COBE and WMAP, we need an inflationary period generated by the flat potential of a scalar field with coupling λ ∼ 10 -10 in a λφ 4 theory. For standard model Higgs, λ is approximately ∼ 1 and the Higgs can not be used as inflaton. A way out was proposed by Bezrukov and Shaposhnikov [1] who coupled the standard model Higgs with the Ricci scalar with a large coupling constant ξ ∼ 10 4 . This large coupling leads to problem with unitarity [2][3][4] of graviton-scalar scattering. Some attempts to solve the unitarity problem associated with the large Higgs curvature coupling are in [5][6][7] In this paper we follow a different approach for the generation of scale invariant density perturbations. It was shown by Rubakov and collaborators [8][9][10] that a conformally coupled field rolling down a quartic potential can generate scale invariant density perturbation. These perturbations can become superhorizon in an inflationary era or in a ekpyrotic scenario [11].
We work with the inert Higgs doublet (IDM) model [12,13] with conformal couplings to the Ricci scalar. The mass terms which give rise to electroweak symmetry breaking are generated by the Coleman-Weinberg method [14].
The requirement of scale invariance at high energy scale and electro-weak symmetry breaking at low energies fixes the coupling constants of the theory. Specifically we find that the quartic coupling of the inert doublet, predicts a spectral index of the power spectrum of the perturbations to be consistent with observations. The amplitude of the power spectrum P ζ can be tuned to be consistent with the observations by choosing a suitable curvaton mechanism. This model specifies the mass of the Higgs boson to be m h = 291 GeV and the mass of the dark matter m A 0 = 550 GeV which can be tested respectively at the LHC and in cosmic ray observations. The main aim of this paper has been to show that a Higgs potential with not too small couplings can be a viable source of the observed scale invariant density perturbations. The scale invariant density perturbations become superhorizon during a phase of inflation at the electroweak scale. However other cosmological scenarios like a bounce models [11] of making the density perturbations superhorizon may be equally viable with our model.
In Section-2 we describes the basics of the Inert Doublet Model (IDM). The one-loop correction to the potential and the calculation for running of coupling constant are briefly discussed in Section-3. We study the generation of the scale invariant density perturbation from the inert higgs doublet in Section-4. In Section-5 we list the scalar mass spectrum predicted by this model and identify the dark matter candidate.
Inert Doublet Model (IDM) is a economical extension of Standard Model which solves the problem of naturalness [12] and it can also explain the electroweak symmetry breaking [13]. The lagrangian of this model respect the Z 2 symmetry, under which all Standard model particles including the SM Higgs H 1 are even and an extra scalar doublet H 2 is odd.
Due to Z 2 symmetry, the cubic term and yukawa term for H 2 doublet are forbidden. This makes the inert doublet stable and its neutral component can be a candidate for dark matter. The two Higgs doublets H 1 and H 2 can be written in terms of their component fields as,
The most general renormalisable potential will be,
We consider the conformal case where µ 1 = µ 2 = 0. V c is the constant potential, which acts as cosmological constant and can be formed from the vev of different Higgs fields. We have chosen V c = 3.66 × 10 8 GeV 4 such that the minimum of the total potential becomes zero at present era. In the early universe the cosmological constant gives rise to an exponential expansion during which the scale invariant perturbations of the phase of the neutral component of H 2 become super-horizon. To achieve this we need that the potential is such that in the early universe, V ∼ -|λ 2 ||H 2 | 4 and the neural component of H 2 rolls down this quartic potential while the minimum of H 1 is at H 1 = 0. In the present era the potential should be such that the minima occurs at H 2 = 0 and H 1 = v = 246GeV which gives rise to the electro-weak symmetry breaking. We show in the next section how this is achieved by radiative corrections starting from a scale invariant tree level potential.
We derive the one-loop correction to the potential (1) following Coleman-Weinberg formalism [14]. The generic one-loop correction to the potential can be written as [15],
where J i is the spin of the fields and m i are the tree level masses, function of the Higgs field.
The double derivative of the tree level potential (1) with respect to the fields give the the tree level masses, which are,
where λ L,S ≡ λ 3 + λ 4 ± λ 5 . We regularize the divergent terms in Eq. ( 2) using the cut-off scale Λ and obtain
The divergence in Eq
This content is AI-processed based on open access ArXiv data.
