Multiple Point Principle of the Standard Model with Scalar Singlet Dark Matter and Right Handed Neutrinos

We consider the multiple point principle (MPP) of the Standard Model (SM) with the scalar singlet Dark Matter (DM) and three heavy right-handed neutrinos at the scale where the beta function $\beta_{\lambda}$ of the effective Higgs self coupling $\lambda_{\text{eff}}$ becomes zero. We make the two-loop analysis and find that the top quark mass $M_{t}$ and the Higgs portal coupling $\kappa$ are strongly related each other. One of the good points in this model is that the larger $M_{t}{1mm}(\gtrsim 171\text{GeV})$ is allowed. This fact is consistent with the recent experimental value \cite{ATLAS:2014wva} $M_{t}=173.34\pm0.76$ GeV, which corresponds to the DM mass $769{1mm}\text{GeV}\leq m_{\text{DM}}\leq 1053 {1mm}\text{GeV}$.


Introduction
The discovery of the Higgs like particle and its mass [1,2] is a very meaningful result for the Standard Model (SM). It suggests that the Higgs potential can be stable up to the Planck scale M pl and also that both of the Higgs self coupling λ and its beta function β λ become very small around the Planck scale. This fact attracts much attention, and there are many works which try to find its physical meaning [3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20].
One of the interesting and meaningful studies is to analyze how the physics beyond the SM affects such a criticality. For example, recently there has been a two loop analysis about the Higgs portal Z 2 scalar model [21]. In this model, the SM singlet scalar is a Dark Matter (DM) candidate, and it is found that its mass can be predicted to be 400GeV < m DM < 470GeV from the requirement that λ and β λ simultaneously become zero at 10 17 GeV, which is usually called the multiple point principle (MPP) [3,4,5].
In this paper, we study the MPP of the next minimal extension of the SM, namely, besides the Higgs portal Z 2 scalar, there are SM singlet heavy right-handed neutrinos [22,23]. We calculate the two-loop beta functions and the one-loop effective potential of the Higgs field, and see whether the Higgs potential can become flat at the scale where β λ becomes zero. Although, within the renormalizable Lagrangian, there are also two scalar couplings in this model (see Eq.(11)), we focus on λ (and β λ ) in this paper 1 . The existence of heavy right-handed neutrinos is naturally needed if we try to explain the light neutrino masses by the seesaw mechanism. Thus, this model is phenomenologically very interesting because it can explain both of DM and the light neutrino masses. In this paper, we consider three right-handed neutrinos whose Majorana mass is given by This paper is organized as follows. In Section2, we review the MPP of the pure SM for the later discussion. In Section3, we give the two-loop analysis of the SM with the scalar singlet DM and three right handed neutrinos. In Section4, the summary is given.

Preliminary -Multiple Point Principle of SM -
In the SM, the one loop effective potential in Landau gauge is given by where Here, µ is the renormalization scale, Γ(φ) is the wave function renormalization and λ(µ), y t (µ), g 2 (µ) and g Y (µ) are the renormalized couplings 2 . By using those results, the effective Higgs self coupling λ eff (φ, µ) can be defined as Because the contribution from the one loop effective potential V tree (φ, µ) is minimized when φ ≃ µ, we put φ = µ in the following discussion.
The left panel of Fig.1 shows λ eff (φ) as a function of φ. For the initial values, we have used the numerical results of [24], and the Higgs mass is fixed at We use Eq.(7) as a typical value in the following discussion. The band corresponds 95% CL deviation of the top quark pole mass M t . For the 1σ level, this is given by [25] M t = 171.2 ± 2.4GeV.
If we assume that all the other parameters of the SM except for M t are fixed, we can find the scale Λ β where β λ eff becomes zero as a function of M t . Here, β λ eff means The right panel of Fig.1 shows Λ β as a function of M t . The MPP requires that λ eff (Λ β ) should become zero, and predicts This is the MPP of the pure SM. In the next section, we discuss the MPP of the SM with the scalar singlet DM and three right-handed neutrinos.

MPP of the SM with Scalar Singlet Dark Matter and Right Handed Neutrinos
We consider the following renormalizable Lagrangian: Here, H is the Higgs field, S is the SM singlet real scalar field, m DM is its mass, ν Ri are right-handed neutrinos, M Rij are their Majorana masses, and (Y Rij , y νij ) are the Yukawa couplings. For simplicity, we assume that M Rij , Y Rij and y νij are diagonalized, and also that they are equal respectively for the three generations. In this case, Eq.(11) becomes Thus, there are six unknown parameters in this model. In the following discussion, to distinguish the initial values of these parameters at µ = M t from their running couplings, we put the subscript 0 for their initial values, like κ 0 . Because S is the candidate of the DM, m DM and κ must satisfy some relation such that they can explain the observed energy density [27] For m DM M h , this relation is approximately given by [28] log 10 κ ≃ −3.63 + 1.04 log 10 m DM GeV .
Moreover, if we assume that the neutrino mass is 0.1eV, y ν and M R must satisfy where v h is the Higgs expectation value. This is the usual relation of the seesaw mechanism. In the following discussion, we choose M R = 10 13 GeV, and y ν is fixed by Eq. (16).
To discuss how the effective couplings behave at the high energy scale, we must know the renormalization group equations (RGEs) of this model. Their results are presented in Appendix A. Here, note that the contributions from the heavy righthanded neutrinos should be taken into account at the scale where µ ≥ M R . The 1-loop effective potential of the Higgs field is given by where  In these expressions, we have put S = 0 because we now focus on the MPP of the Higgs sector 3 . Furthermore, we can neglect m DM in Eq.(18) because its effect is very small when φ ≫ m DM . As well as Section2, we put φ = µ, and define the effective Higgs self coupling λ eff as One can see that λ eff depends mainly on M t and κ 0 , and hardly on λ DM0 and Y R0 . This is because Y R appears in β λ at the two-loop level, and λ DM does not appear (see Eq. (30) in Appendix A). Therefore, by fixing λ DM and Y R , we can relate M t and κ 0 from the MPP. By doing the same procedure of Section2, we can calculate the scale Λ β where β λ eff becomes zero, and obtain λ eff (Λ β ) as a function of M t and κ 0 . Fig.3 shows the results. For comparison, the tree level result (left) is also shown. Here, "tree" means we consider the tree level potential, but the RGEs are two-loop level. The parameter region where λ eff (λ DM )(Λ β ) > 0 is filled by blue (red), and especially, the λ eff (Λ β ) = 0 line is drawn by green. The intersecting region is allowed from the stability of the potential, and the MPP predicts that M t and κ 0 should exist on the green line. One of the good points of this model is that the larger M t is allowed unlike the SM, and this is consistent with the recent experimental values [25,26] M t = 173.34 ± 0.76GeV. (21) A few comments are needed. First, the curves which represent Λ β = 10 16 GeV, 10 17 GeV and 10 18 GeV are also shown in Fig.3 respectively by red, blue and orange lines. Second, the lower bound of κ 0 , which corresponds to λ DM (Λ β ) = 0, depends on Y R0 because both of them come into β λ DM at the one-loop level. For example, when we decrease Y R , a smaller value of κ 0 is allowed. Finally, as is seen from Fig.3, we can also demand λ DM (λ β ) = 0 as well as λ eff (λ β ) = 0. However, it is difficult to realize β λ DM (Λ β ) = 0 simultaneously. See Appendix B for more details.

Summary
We have discussed the MPP of the SM with the scalar singlet DM and right-handed neutrinos. We have found that λ eff and β λ eff can simultaneously become zero within the reasonable parameter region. The MPP predicts the strong relation between the portal coupling κ and the top mass M t . Unlike the pure SM, the large M t is allowed in this case, which is favorable for the recent experimental values [25,26] M t = 173.34 ± 0.76GeV.
Although we have found that the MPP can be satisfied for the Higgs potential in this paper, it is difficult to realize the exact flatness of the scalar potential at some high energy scale Λ; See Appendix B for the details. It would be interesting to consider a generalization of this model in such a way that the MPP is realized for the whole scalar fields.