Higgs inflation and Higgs portal dark matter with right-handed neutrinos

We investigate the Higgs inflation and the Higgs portal dark matter with the right-handed neutrino. The dark matter and the right-handed neutrino in the Higgs inflation play important roles in explaining the recent experimental results of the Higgs and top masses, and the cosmic microwave background by BICEP2 at the same time. This inflation model predicts $805~{\rm GeV} \lesssim m_{\rm DM} \lesssim1220~{\rm GeV}$ for the DM mass, $1.05 \times10^{14}~{\rm GeV} \lesssim M_R \lesssim 2.04 \times10^{14}~{\rm GeV}$ for the right-handed neutrino mass, and $8.42 \lesssim \xi \lesssim 12.4$ for the non-minimal coupling within $m_H=125.6 \pm 0.35~{\rm GeV}$ for the Higgs and $M_t=173.34 \pm 0.76~{\rm GeV}$ for the top masses.


Introduction
The standard model (SM) has achieved great success in the last few decades. In particular, the discovery of the Higgs boson at the LHC experiment [1,2] was the last key ingredient to finalize the SM. However, there are several unsolved problems in the SM, e.g., explanations of the origin of dark matter (DM), the tiny active neutrino masses, and the inflation, etc. Regarding the inflation, the BICEP2 experiment has recently reported the non-vanishing tensor-to-scalar ratio [3] #1 : This result has not been confirmed yet, but a number of explanations for the result have since been presented. One interesting attempt is given in the context of the Higgs inflation [5]- [16] (see Refs. [17]- [31] for recent discussions in the ordinary Higgs inflation and related works after the BICEP2 result). In particular, Ref. [19] showed that a Higgs potential with ξ = 7 of the non-minimal Higgs coupling and a suitable plateau for the inflation can explain the above BICEP2 result. But the plateau is realized by a slightly smaller top mass than the experimental range as M t = 173.34 ± 0.76 GeV [32]. Then, Ref. [29] pointed out that the Higgs inflation with the Higgs portal DM (see, e.g., Refs. [33]- [44]) and the right-handed neutrino can explain the result with the experimental central values of the Higgs and top masses, and ξ ≃ 10.1. In this scenario, DM (a real singlet scalar) and the right-handed neutrino play a crucial role in the realization of a suitable Higgs potential, i.e. these particles are important for high-energy behavior in the evolution of the Higgs self-coupling λ(µ) obeying the renormalization group equation (RGE). In addition, the right-handed neutrino can generate the tiny active neutrino mass through the type-I seesaw mechanism. Reference [29] found one solution, in which the central values were taken for the Higgs and top masses, for realistic inflation. For the quantum corrections to the coupling constants (β-functions) in the model, the numerical analyses included the next-to-leading-order (NLO) contributions of the SM particles and the leading-order (LO) ones from DM and the righthanded neutrino. In this work, we will analyze the model based on the NLO computations for both the SM and the singlet particles. In addition, the latest experimental errors for the Higgs and top masses will be taken into account in the calculations.
As a result, we will point out that this inflation model can explain the results of cosmological observations within regions for 805 GeV m DM 1220 GeV for the DM mass, 1.05 × 10 14 GeV M R 2.04 × 10 14 GeV for the right-handed neutrino mass, and 8.42 ξ 12.4 for the non-minimal Higgs coupling to the Ricci scalar with m H = 125.6 ± 0.35 GeV [45] for the Higgs and M t = 173.34±0.76 GeV [32] for the top masses. A strong correlation between #1 Actually, the joint analysis by BICEP, the Keck Array, and Planck concludes that this signal can be explained by dust [4]. However, this setup is still attractive from phenomenological points of view. Furthermore, we will show that the main results do not change much even if we take the tensor-to-scalar ratio as r = 0.048 which is the central value reported in Ref. [4].
the DM and the right-handed neutrino masses will also be shown and the allowed region of a DM mass will be checked by future DM detection experiments.
The paper is organized as follows: In section 2, we will explain our model. In section 3, we will briefly review the context of the Higgs inflation. Our numerical results will be given in section 4. Section 5 is devoted to the conclusions. We will also present the relevant β-functions up to 2-loop level in the model in the Appendix.

Extension of the Standard Model
In this letter, we consider the extended SM with a real singlet scalar and right-handed neutrinos. The adequate Lagrangians can be written as where L SM is the SM Lagrangian in which the Higgs potential is included as shown in Eq. (3).
H is the Higgs doublet field and v EW is its vacuum expectation value. L, S, and N are the lepton doublet, an SM gauge singlet real scalar, and the right-handed neutrino fields, respectively. The coupling constants k, λ S , and y N are the portal coupling of H and S, the quartic self-coupling of S, and the neutrino Yukawa coupling constant. M R is the right-handed neutrino mass. We assume that the singlet scalar has odd parity under an additional Z 2 symmetry. Hence, we can have a candidate for DM when the mass and the portal coupling k of S are taken to be appropriate values. The observed tiny neutrino masses can be obtained by the conventional type-I seesaw mechanism #2 . We have fixed the active neutrino mass as 0.1 eV here.
Here, we show the RGEs to solve: where X symbolizes the SM gauge couplings, the top and the neutrino Yukawa couplings, and the scalar quartic couplings in our model, and we define t ≡ ln(µ/1 GeV) with the renormalization scale µ. In this analysis, we divide the energy region into three parts between the Z #2 The right-handed neutrino with the mass M R generates one active neutrino mass. Others can be obtained by lighter right-handed neutrinos with smaller Yukawa coupling. When the neutrino Yukawa couplings are smaller than the bottom Yukawa coupling, the contributions from the neutrino Yukawa couplings to the βfunctions are negligible. In this work, we consider a case in which the neutrino Yukawa coupling of only one generation of the right-handed neutrino is effective in the β-functions. boson mass scale, M Z , and the Planck scale, M pl = 2.435 × 10 18 GeV. Each part can be con- The behavior of λ(µ) can be roughly supposed as follows: At first, the evolution of k becomes small as k(M Z ) becomes small, because the β-function for k is proportional to k itself. In this case, the evolution of λ(µ) is really close to the SM one. Second, the contribution from the additional scalar pushes up the evolution of λ(µ). Thus, we can make λ(µ) positive up to M pl with sufficient magnitude of the additional scalar contribution. On the other hand, the contribution from the right-haded neutrinos pulls down λ(µ) in the region of M R ≤ µ ≤ M pl . A typical behavior of the running of the Higgs quartic coupling is shown in Fig.1. As a result, we can have a suitable Higgs potential with a plateau for the inflation around O(10 18 ) GeV [29]. These are the important features for Higgs inflation with a real scalar field and right-handed neutrinos #3 . In our estimation, the free parameters are the values of the right-handed neutrino and DM masses and non-minimal coupling ξ.

Higgs inflation with singlets
First, we briefly review the ordinary Higgs inflation [6] with the action in the so-called Jordan frame, #3 We assume that λ S (M Z ) = 0.1 just as a sample point. where ξ is the non-minimal coupling of the Higgs to the Ricci scalar R, H = (0 , h) T / √ 2 is given in the unitary gauge, and L SM includes the Higgs potential of Eq. (3). With the conformal transformation (ĝ µν ≡ Ω 2 g µν with Ω 2 ≡ 1 + ξh 2 /M 2 pl ) which denotes the transformation from the Jordan frame to the Einstein one, one can rewrite the action as whereR is the Ricci scalar in the Einstein frame given byĝ µν , and χ is a canonically renormalized field as One can calculate the slow-roll parameters as where U(χ) is obtained #4 as #4 In this analysis, we take only the quartic term of the Higgs field into account because the quadratic coupling can be negligible at the inflationary scale. Moreover, in our calculation, we use the β-functions of coupling constants including the self-coupling of the Higgs up to 2-loop level. This means that our analysis also contains quantum effects, which partially include effects from the loop level potential.
A typical behavior of the Higgs potential U(χ) is shown in Fig.2. Using slow-roll parameters, the spectral index and the tensor-to-scalar ratio are evaluated as n s = 1 − 6ǫ + 2η and r = 16ǫ, respectively. Finally, the number of e-foldings is given by where h 0 (h end ) is the initial (final) value of h corresponding to the beginning (end) of the inflation. At the point h end , the slow-roll conditions (ǫ , |η| ≪ 1) are broken. The Higgs inflation can be realized even in the SM if the top mass is fine-tuned as M t = 171.079 GeV for m H = 125.6 GeV [16,19,24]. With these values, the Higgs potentials have a plateau and r ≃ 0.2 can be achieved by taking ξ = 7 [19]. Even though this framework can accomplish a relevant amount of e-foldings, the required top mass, M t ≃ 171.1 GeV, is outside M t = 173.34 ± 0.76 GeV [32]. On the other hand, without the plateau, a sufficient amount of e-foldings can be obtained by assuming ξ ∼ O(10 4 ). However, this case is plagued with too tiny a tensor-to-scalar ratio on the order of 10 −3 , which is inconsistent with the recent BICEP2 result. Consequently, we extend the SM with a real scalar and right-handed neutrinos, in which the evolution of λ (equivalent to the Higgs potential) is changed, in order to reproduce the values of cosmological parameters within the experimental range M t .

Numerical analysis at 2-loop order
In this section, we give the results of our numerical analysis. We solve RGEs #5 at 2-loop level for the β-functions of the relevant couplings in the model #6 . As we discussed above, we analyze within the experimental ranges of the Higgs mass m h = 125.6 ± 0.35 GeV [45] and the top mass M t = 173.34 ± 0.76 GeV [32].
As we mentioned in Sec.2, we assume that the additional real scalar is DM. In the case that the DM mass is greater than the Higgs mass (m S > m h ), the portal coupling k(M Z ) is well approximated as [46,30] log 10 k(M Z ) ≃ −3.63 + 1.04 log 10 m DM GeV , where m DM denotes the mass of DM given by m 2 DM = m 2 S + kv 2 EW /2. Our results are shown in Fig.3. One can see that the required values of M R , m DM , and ξ for reproducing suitable cosmological parameters are on a line in the M R -m DM plane. This is because the form of the Higgs potential is strictly constrained and it should be uniquely realized by taking suitable values of M R , m DM , and ξ for given m H and M t . A lighter (heavier) top (Higgs) mass gives lower bounds on M R , m DM , and ξ, while a heavier (lighter) top (Higgs) mass leads to upper bounds on these parameters. #5 Here, we consider that this renormalization scale is the same as h/Ω. #6 There are theoretical uncertainties between the low-and high-energy parameters as discussed in Refs. [9,22]. But we assume such uncertainties are small enough and can be neglected.
k(M Z ) is determined by a condition that the singlet scalar gives a sufficient amount of the relic density of the DM #8 , i.e. Ω DMh 2 = 0.119 (see, e.g., Refs. [41,46]) where Ω DM is the #7 There are theoretical uncertainties in matching the high-and low-energy parameters [9]. For example, they change by about 0.5 GeV for the 126 GeV Higgs mass. This leads to uncertainties of about ±30 GeV for the DM mass, ±5 × 10 13 GeV for the right-handed neutrino mass, and ±0.6 for the non-minimal coupling in our model. #8 In this work, we have neglected the non-thermal production of DM via portal interaction due to the smallness of the cross section. Therefore, we can estimate the total amount of DM abundance by thermal production.  [4]. We show only three points here and the meaning of each point is same as in Fig.3. density parameter of the DM andh is the Hubble constant. From this condition, k(M Z ) is in the range 0.25 k(M Z ) 0.38. The region is consistent with the current experimental constraints so far, and future experiments, e.g., XENON1T or LUX for the direct detection, or the combined analysis of the Fermi+CTA+Planck observations might make them clear [46]. Finally, we also comment on the validity of our β-functions. Since the value of ξ is around 10 as in the above results and it is much smaller than the previous work [8], we can utilize our β-functions up to the inflation scale.

After the joint analysis of BICEP, Keck Array and Planck
The joint analysis of BICEP, the Keck Array and Planck showed only an upper bound on the tensor-to-scalar ratio [4] #9 of We show our result respecting r = 0.048 +0.035 −0.032 in Fig.4. One can easily see that the mass scales of right-handed neutrino and DM do not drastically change compared with the r = 0.2 case. Therefore, the future direct-detection experiments of DM can reach even if the mass range of DM shown in Fig.4. The difference appears in the magnitude of non-minimal coupling, ξ. Its magnitude should be slightly larger in the range 9.23 ξ 13.5.

#9
The latest independent result from the Planck Collaboration gives a consistent result [48].

Conclusions
We have investigated the Higgs inflation model with the Higgs portal DM and the right-handed neutrino by the use of β-functions up to 2-loop level. In addition, the latest experimental errors of the top and Higgs masses have been taken into account in the calculations. As a result, we pointed out that this inflation model can explain the results of cosmological observations n s = 0.9600 ± 0.0071, r = 0.20 +0.07 −0.06 , and 52.3 N 59.7 within regions of 805 GeV m DM 1220 GeV for the DM mass, 1.05 × 10 14 GeV M R 2.04 × 10 14 GeV for the righthanded neutrino mass, and 8.42 ξ 12.4 for the non-minimal Higgs coupling to the Ricci scalar with m H = 125.6 ± 0.35 GeV for the Higgs and M t = 173.34 ± 0.76 GeV for the top masses. Furthermore, we have shown that the scales of the right-handed neutrino and DM are not significantly changed even though the tensor-to-scalar ratio decreases as r < 0.12. On the other hand, the non-minimal coupling should be slightly larger than the r = 0.2 case as 9.23 ξ 13.5. There is a strong correlation between the DM and the right-handed neutrino masses because the form of the Higgs potential is strictly constrained; it should be uniquely realized by taking suitable values of M R , m DM , and ξ for given m H and M t . The DM mass region in our analysis will be confirmed by future DM detections. Next, are for the top and neutrino Yukawa couplings. Lastly, β λ = 24λ 2 − 2(3y 4 t + y 4 N ) + 4λ 3y 2 t + y 2 N − 3λ g ′2 + 3g 2 + 3 8 2g 4 + (g ′2 + g 2 ) 2 + 1 2 k 2 + 1 16π 2 − 312λ 3 + 36λ 2 g ′2 + 3g 2 − λ − β k = k 4k + 12λ + λ S + 6y 2 t + 2y 2 N − β λ S = 3λ 2 S + 12k 2 + 1 16π 2 − 17 3 λ 3 S − 20k 2 λ S − 48k 3 + 24k 2 (g ′2 + 3g 2 ) − (3y 2 t + y 2 N ) , (25) are for the SM Higgs quartic, the portal between the SM Higgs and the singlet scalar, and the singlet scalar quartic couplings.