Higgs inflation in a radiative seesaw model

We investigate a simple model to explain inflation, neutrino masses and dark matter simultaneously. This is based on the so-called radiative seesaw model proposed by Ma in order to explain neutrino masses and dark matter by introducing a $Z_2$-odd isospin doublet scalar field and $Z_2$-odd right-handed neutrinos. We study the possibility that the Higgs boson as well as neutral components of the $Z_2$-odd scalar doublet field can satisfy conditions from slow-roll inflation and vacuum stability up to the inflation scale. We find that a part of parameter regions where these scalar fields can play a role of an inflaton is compatible with the current data from neutrino experiments and those of the dark matter abundance as well as the direct search results. A phenomenological consequence of this scenario results in a specific mass spectrum of scalar bosons, which can be tested at the LHC, the International Linear Collider and the Compact Linear Collider.


I. INTRODUCTION
The new particle with the mass of 126 GeV which has been found at the LHC [1,2] is showing various properties that the Higgs boson must have. It is likely that the particle is the Higgs boson. If this is the case, the Standard Model (SM) of elementary particles is confirmed its correctness not only in the gauge interaction sector but also in the sector of electroweak symmetry breaking. By the discovery of the Higgs boson, all the particle contents in the SM are completed. This means that we are standing on the new stage to search for new physics beyond the SM. There are several empirical reasons why we consider the new physics. Phenomena such as neutrino oscillation [3][4][5][6][7][8], existence of dark matter [9] and baryon asymmetry of the Universe [9][10][11] cannot be explained in the SM.
Cosmic inflation at the very early era of the Universe [12], which is a promising candidate to solve cosmological problems such as the horizon problem and the flatness problem, also requires the additional scalar boson, the inflaton.
The determination of the Higgs boson mass at the LHC opens the door to directly explore the physics at very high scales. Assuming the SM with one Higgs doublet, the vacuum stability argument indicates that the model can be well defined only below the energy scale where the running coupling of the Higgs self-coupling becomes zero. For the Higgs boson mass to be 126 GeV with the top quark mass to be 173.1 GeV and the coupling for the strong force to be α s = 0.1184, the critical energy scale is estimated to be around 10 10 GeV by the NNLO calculation, although the uncertainty due to the values of the top quark mass and α s is not small [13]. The vacuum seems to be metastable when we assume that the model holds up to the Planck scale. This kind of analysis gives a strong constraint on the scenario of the Higgs inflation [14] where the Higgs boson works as an inflaton, because the inflation occurs at the energy scale where the vacuum stability is not guaranteed in the SM.
Recently, a viable model for the Higgs inflation has been proposed, in which the Higgs sector is extended including an additional scalar doublet field [15].
In order to generate tiny masses of neutrinos, various kinds of models have been proposed.
The simplest scenario is so called the seesaw mechanism, where the tiny neutrino masses are generated at the tree level by introducing very heavy particles, such as right-handed neutrinos [16], a complex triplet scalar field [17], or a complex triplet fermion field [18]. The radiative seesaw scenario is an alternative way to explain tiny neutrino masses, where they are radiatively induced at the one loop level or at the three loop level by introducing Z 2 -odd scalar fields and Z 2 -odd right-handed neutrinos [19][20][21]. An interesting characteristic feature in these radiative seesaw models is that dark matter candidates automatically enter into the model because of the Z 2 parity.
In this Letter, we discuss a simple model to explain inflation, neutrino masses and dark matter simultaneously, which is based on the simplest radiative seesaw model [20]. Both the Higgs boson and neutral components of the Z 2 -odd scalar doublet can satisfy conditions for slow-roll inflation [22] and vacuum stability up to the inflation scale. We find that a part of the parameter region where these scalar fields can play a role of the inflaton is compatible with the current data from neutrino experiments and those of the dark matter abundance as well as the direct search results [23]. A phenomenological consequence of scenario results in a specific mass spectrum of scalar fields, which can be tested at the LHC, the International Linear Collider (ILC) [24] and the Compact Linear Collider (CLIC) [25].

II. LAGRANGIAN
We consider the model, which is invariant under the unbroken discrete Z 2 symmetry, with the Z 2 -odd scalar doublet field Φ 2 and right-handed neutrino ν R to the SM with the SM Higgs doublet field Φ 1 [20]. Quantum charges of particles in the model are shown in Table I. Dirac Yukawa couplings of neutrinos are forbidden by the Z 2 symmetry. The Yukawa interaction for leptons is given by where φ, r and θ are defined as with taking a large field limit ξ 1 h 2 1 /M 2 P + ξ 2 h 2 2 /M 2 P ≫ 1. For stabilizing r as a finite value, we need to impose following conditions [15]; Parameters in the scalar potential should satisfy the constraint from the power spectrum [9,15]; where a is given as a ≡ ξ 1 /ξ 2 . When the scalar potential satisfies the conditions in Eqs.

B. Dark Matter
We assume that the CP-odd boson A is the lightest Z 2 odd particle. (By changing the sign of the coupling constant λ 5 , the similar discussion can be applied with the CP-even boson H to be the lightest.) When λ 5 is very small such as O(10 −7 ), A is difficult to act as the dark matter because the scattering process AN → HN opens, where N is a nucleon.
The cross section is too large to be consistent with the current direct search results for dark matter [27][28][29]. In Ref. [15], the authors claim that both the Higgs boson and Z 2 -odd neutral scalar bosons can work as the inflatons when the dark matter (H or A) has the mass of 600 GeV if λ 5 10 −7 . However, as recently discussed in Ref. [28], the bound from direct search results are getting stronger, and such a dark matter is not allowed anymore in this model without a fine tuning among the scalar self-coupling constants. We here take λ 5 ≃ 10 −6 and at the inflation scale. With this choice, the process AN → HN can be avoided kinematically.
Still masses of A and H are almost the same value. The coannihilation process AH → XX via the Z boson is important to explain the abundance of the dark matter where X is a particle in the SM, because the pair annihilation process AA → XX via the h boson is suppressed due to the constraint from the inflation. The cross section of AH → XX depends only on the mass of the dark matter. Therefore, the mass of the dark matter A is constrained from the abundance of the dark matter as where we have used the nine years WMAP data [9] . The scattering process AN → AN then comes mainly from the diagram of the SM-like Higgs boson mediation. The cross section is given by [29,30] where [31] and f T G = 0.944 [32]. The mass m A should be approximately a half of m h [33] in order for the dark matter to be consistent with the abundance from the WMAP experiment [9] and the upper bound on the scattering cross section for AN → AN from the XENON100 experiment [23]. The coupling constant λ hAA should satisfy at the low energy scale for consistency to satisfy the data from the XENON100 experiment.

C. Tiny Neutrino Masses
In this model, tiny neutrino masses are generated by the one loop diagram in Fig. 1 [20].
The neutrino mass (m ν ) ij are given by The neutrino mixing matrix is explained by neutrino Yukawa coupling constants (Y ν ) k i . The magnitude of tiny neutrino masses can be explained when GeV −1 because λ 5 and masses of scalar bosons, m H and m A , are constrained from the conditions of inflation and dark matter. Our model then can be consistent with current experimental data for neutrinos [3][4][5][6][7][8]

D. Running of Scalar Coupling Constants
In the SM, the energy scale cannot reach to the inflation scale because the quartic coupling constant of the Higgs boson is inconsistent with the unbounded-from-below condition at 10 10 GeV scale when m t = 173.1 GeV and α s = 0.1184 [13]. On the other hand, if we consider extended Higgs sectors such as the two Higgs doublet model, the vacuum stability condition on the quartic coupling constant for the SM-like Higgs boson can be relaxed due to the effect of the additional quartic coupling constants [34]. Therefore, these models can be stable up to the inflation scale 1 We calculate these coupling constants by using the renormalization group equations with the following beta functions [35]; β(λ 1 ) = 1 16π 2 12λ 2 1 + 4λ 2 3 + 2λ 2 4 + 2λ 2 5 + 4λ 3 λ 4 − 12y 4 t + 12y 2 t λ 1 We here impose the conditions of triviality and vacuum stability (the unbounded-below-condition) up to the inflation scale. In Fig 2, running of the scalar coupling constants are shown between the electroweak scale and the inflation scale. The vacuum instability due to λ 1 is avoided by the effect of the Higgs selfcoupling constants with Z 2 -odd scalar bosons [34]. In Table II,

E. Mass Spectrum
Let us evaluate the mass spectrum of the model under the constraint from inflation, the neutrino data and the dark matter data as well as the vacuum stability condition. In our model, there are nine parameters in the scalar sector; i.e., ξ 1 , ξ 2 , µ 2 1 , µ 2 2 , λ 1 , λ 2 , λ 3 , λ 4 and λ 5 .
First of all, as the numerical inputs, we take v = 246 GeV and m h = 126 GeV. Second, we use the conditions to explain the thermal fluctuation; i.e., the allowed region for the mass of the dark matter A is determined from the constraint of the dark matter abundance from the WMAP data in Eq. (11). We here take m A = 130 GeV as a reference value.
Further numerical input comes from the perturbativity of λ 2 up to the inflation scale; i.e., λ 2 (µ inf ) = 2π, where µ inf is the inflation scale 10 17 GeV. The parameter set in Table II can be consistent with these numerical inputs and the constraints are given in Eqs. (3), (7), (8), (9), (10) , (13) and (22). The mass spectrum of the scalar bosons is determined as where the mass difference between A and H is about 500 KeV.
The mass spectrum is not largely changed even if m A is varied with in its allowed region.
Consequently, in our scenario, the following relation for the mass is obtained; The bounds on m H ± is obtained in order to satisfy the conditions from Eqs. (3), (7) and (22). Therefore, we can test the model by using the mass spectrum at collider experiments.

IV. PHENOMENOLOGY
Masses of Z 2 -odd scalar bosons have been constrained by the LEP experiment. In our scenario, m H ± should be around 170 GeV, which is above the lower bound given by the LEP experiment [38,39]. From the Z boson width measurement, m H + m A should be larger than m Z [38,40]. In addition, there is a bound on HA production at the LEP. However, when m H − m A < 8 GeV, masses of neutral Z 2 -odd scalar bosons are not constrained by the LEP [38,40]. The contributions to the electroweak parameters [41] from additional scalar bosons loops are given by [42,43] where In all of our parameters, it is consistent with current electroweak precision data with 90% Confidence Level (C.L.) [43].
The detectability of H, A and H ± at the LHC has been studied in Ref. [44][45][46]. They conclude that it could be difficult to test pp → AH + /HH + /H + H − processes because cross sections of the background processes are very large. The process of pp → AH could be tested with about the 3σ C.L. with the various benchmark points for masses for A and H However, it would be difficult to test pp → AH in our model. In our parameter set, m H and m A are about 130 GeV. In this case, after imposing the basic cuts [44][45][46], event number of pp → AH is negligibly small. Furthermore, the total decay width of H is about 10 −29 GeV.
In this case, H would pass through the detector. Therefore, this signal is difficult to be detected at the LHC.
We now discuss signals of H, A and H ± at the ILC with √ s = 500 GeV. In the following, we use Calchep 2.5.6 for numerical evaluation [47]. First, we focus on the H ± pair production process e + e − → Z * (γ * ) → H + H − → W +( * ) W −( * ) AA → jjℓνAA, where j denotes a hadron jet [48]. The final state of this process is a charged lepton and two jets with a missing momentum. The energy of the two-jet system E jj satisfies the following equation because of the kinematical reason; E jj is evaluated by using our parameter set as 15 GeV < E jj < 94 GeV.
The distribution of E jj of the cross section for e + e − → Z * (γ * ) → H + H − → W +( * ) W −( * ) AA → jjℓνAA is shown in Fig. 3. The important background processes against e + e − → Z * (γ * ) → H + H − → W +( * ) W −( * ) AA → jjℓνAA are e + e − → W + W − → jjℓν and e + e − → Z(γ)Z → jjℓℓ with a missing ℓ event. In these processes, the missing invariant mass is zero. These backgrounds could be well reduced by imposing an appropriate kinematic cuts. We expect that m H ± and m A can be measured by using the endpoints of E jj at the ILC after the background reduction.
When the mass difference between A and H is sizable, it could also be detected by using the endpoint of E jj . However, in our mass spectrum, it is predicted that masses of A and H are almost degenerated. When we detect H ± but we cannot detect the clue of this process at the ILC, it seems that masses of A and H are almost same value.
Finally, we discuss prediction on the diphoton decay of the Higgs boson h. BR(h → γγ) in the model, which the SM with Z 2 -odd scalar doublet, has been studied in Ref. [49]. The deviation in our model from the SM is given by where N c and Q f are the color and electromagnetic charges of the top quark, respectively.
where τ x and f (x) are given by When we use our parameter set in Eq. (23), the ratio is calculated as In our model, BR(h → γγ) is smaller than the SM results due to constraints from the conditions of the inflation and the dark matter. These ratio is at most 10 % because our model contains only one charged scalar field.

V. DISCUSSION AND CONCLUSION
In this Letter, we have not explicitly discussed baryogenesis. It is likely not difficult to complement the mechanism for baryogenesis to our model via leptogenesis [50]. In Ref. [28], the possibility of the leptogenesis in the Ma model [20] has been studied in details under the constraint of current neutrino and dark matter data. By using the typical value for λ 5 in our model λ 5 ≃ 10 −6 ), the scenario of baryogenesis through the leptogenesis would be difficult if masses of the right-handed neutrinos are about 1 TeV.
On the other hand, the possibility of electroweak baryogenesis would also be interesting [51]. The condition of strong first order phase transition is compatible with m h = 126 GeV in the framework of two Higgs doublet models [52] including the inert doublet model [53]. In such a case, an important phenomenological consequence is a large deviation in the loop-corrected prediction on the hhh coupling [54], by which the scenario can be tested when the hhh coupling is measured at future colliders such as the ILC or the CLIC.
an extension has to be needed in order to get additional CP violating phases, which are required for successful baryogenesis.
We have studied the simple scenario to explain inflation, neutrino masses and dark matter simultaneously based on the radiative seesaw model with the Higgs inflation mechanism.
We find that the parameter region where Z 2 -odd scalar fields can play a role of the inflaton is compatible with the current data from neutrino experiments and those of the dark matter abundance as well as the direct search results. This scenario predicts a specific mass spectrum for the scalar fields, which can be measured at the LHC and the ILC with √ s = 500 GeV. Our model is a viable example for the TeV scale model for inflation (and neutrino with dark matter) which is testable at collider experiments.