Multiple-point principle with a scalar singlet extension of the Standard Model

We suggest a scalar singlet extension of the standard model, in which the multiple-point principle (MPP) condition of a vanishing Higgs potential at the Planck scale is realized. Although there have been lots of attempts to realize the MPP at the Planck scale, the realization with keeping naturalness is quite difficult. Our model can easily achieve the MPP at the Planck scale without large Higgs mass corrections. It is worth noting that the electroweak symmetry can be radiatively broken in our model. In the naturalness point of view, the singlet scalar mass should be of ${\cal O}(1)\,{\rm TeV}$ or less. We also consider right-handed neutrino extension of the model for neutrino mass generation. The model does not affect the MPP scenario, and might keep the naturalness with the new particle mass scale beyond TeV, thanks to accidental cancellation of Higgs mass corrections.


Introduction
The observed mass of the Higgs boson may imply that Higgs self-coupling vanishes at a high energy scale in the framework of the standard model (SM). About twenty years ago, Ref. [1] suggested the multiple-point principle (MPP) at the Planck scale, and predicted a Higgs boson mass of 135±9 GeV with 173 ± 5 GeV for the top quark mass. The MPP means that there are two degenerate vacua in the SM Higgs potential, V (v H ) = V (M Pl ) = 0 and V (v H ) = V (M Pl ) = 0, where V is the effective Higgs potential, v H = 246 GeV is the vacuum expectation value (VEV) of the Higgs doublet, and M Pl = 2.44 × 10 18 GeV is the reduced Planck scale. One is our vacuum at the electroweak (EW) scale, and the other vacuum lies at the Planck scale, which can be realized by the Planck-scale boundary conditions of vanishing effective Higgs self-coupling, λ H (M Pl ) = 0, and its beta function, β λ H (M Pl ) = 0. Furthermore, an asymptotic safety scenario of gravity [2] predicted a 125 GeV Higgs boson mass with a few GeV uncertainty. This scenario also pointed out that λ H (M Pl ) ∼ 0 and β λ H (M Pl ) ∼ 0 (see also Refs. [3][4][5][6][7][8][9][10][11][12][13][14] for more recent analyses).
Although Ref. [1] was able to predict the approximate Higgs boson mass, the MPP condition cannot fit the observed 125 GeV Higgs boson mass with the recent data inputs. In fact, within the context of the SM, the MPP condition at the Planck scale leads to a Higgs boson mass of 129.1 ± 1.5 GeV by using 173.10 ± 0.59 exp ± 0.3 th GeV for the world-averaged top quark mass [12]. There have been lots of attempts to realize the MPP at the Planck scale so far [15][16][17][18][19][20][21][22][23][24][25][26][27][28][29][30][31][32][33]. For example, in Ref. [18] the MPP at the Planck scale is achieved by introducing a scalar dark matter and a large Majorana mass of the right-handed neutrino. In this case, the masses of dark matter and the right-handed neutrino PTEP 2017, 013B03 N. Haba et al. can be predicted. However, there is a tension from the viewpoint of naturalness, since the Higgs mass corrections via the heavy particles well exceeds the EW scale. Actually, it turns out to be quite difficult to realize the MPP at the Planck scale while keeping naturalness.
The difficulty is related with the renormalization group (RG) running of the Higgs self-coupling. In order to satisfy λ H (M Pl ) = 0 and β λ H (M Pl ) = 0 simultaneously, there should exist one or more new particles which change β λ H adequately from the SM case. In almost all cases, such new particles need to be much heavier than the EW scale, as long as the Higgs self-coupling is "continuous" during the RG running. However, when a new scalar field couples with the Higgs doublet and develops nonzero VEV, the Higgs self-coupling has a tree-level threshold correction [34][35][36][37]. 1 The correction causes a gap between the Higgs self-coupling in the extended model and the one in the effective theory, which is identified as the SM one. It has been shown that using the gap, the EW vacuum can be stabilized in a scalar singlet extended model [35] and type-II seesaw model [37]. Most importantly, even if the new scalar particle is as light as a TeV scale, the gap can appear. Then, the model does not affect the naturalness in the sense of Bardeen [38].
Here, we comment on the naturalness. According to Bardeen's argument, in quantum corrections quadratic divergences can be treated as an unphysical quantity, so that only logarithmic divergences should be concerned. In this sense, there is no hierarchy problem within the SM, which possesses an approximate scale invariance and its stability is guaranteed by the smallness of the logarithmic corrections. Since the logarithmic corrections can be taken into account as a beta function of the Higgs mass parameter, the naturalness can be evaluated with the solution of its RG equation. Namely, it is natural if the Higgs mass parameter does not significantly change during the RG running. We will apply this sense of naturalness to our model.
In this paper, we will investigate the MPP condition in a scalar singlet extended model, which can be consistent with the 125 GeV Higgs boson mass. Our model is explained in the next section, in which we show the gap explicitly. Numerical analyses of the MPP scenario are given in Sect. 3. We will find that the EW symmetry can be radiatively broken in our model. We also discuss the naturalness of the Higgs mass. In Sect. 4, we will introduce right-handed neutrinos into the scalar singlet extended model to incorporate active neutrino masses. We will show that in the presence of the right-handed neutrinos, the MPP scenario can be realized. It will be pointed out that even if the singlet scalar and the right-handed neutrinos are much heavier than the EW scale, the model might keep the naturalness thanks to an accidental cancellation of Higgs mass corrections coming from them. Finally, we will summarize our results in Sect. 5.

Scalar singlet extension
We consider a simple extension of the SM with a real singlet scalar field. The scalar potential is given by [36] where H and S are the Higgs doublet and the scalar singlet fields, respectively. In this paper, we consider the case with m 2 S > |m 2 H | and μ HS > 0, and omit a linear term of the singlet scalar field, which can vanish by a shift of the field. Note that we do not assume an ad hoc Z 2 symmetry, and PTEP 2017, 013B03 N. Haba et al. then, we will find that μ HS plays an important role for the vacuum stability and the EW symmetry breaking. In the unitary gauge, the scalar fields are written by where v H and v S are vacuum expectation values. The Higgs VEV is v H = 246 GeV, and v S has a negative small value in our setup, as will be discussed below.
The minimization conditions of the potential are given by From Eq. (3), the Higgs VEV is obtained by To realize the EW symmetry breaking, the Higgs mass term m 2 H is negative at the EW scale, and 2(−m 2 H ) > λ HS v 2 S + 2μ HS v S should be satisfied. Without any fine-tuning, we can expect μ HS m S by a naive dimensional analysis. Thus, |v S | should be much smaller than v H for m 2 S |m 2 H |. The nonzero Higgs VEV induces a tadpole for the singlet scalar due to the μ HS term. If we neglect the cubic term of S, Eq. (4) is approximated by and its order of magnitude is O(v 2 H /m S ) for μ HS m S . In the no tadpole limit μ HS → 0, v S vanishes. The assumption of μ S = 0 seems to be unnatural, but it is necessarily required by the MPP condition, as discussed later. Actually, we will find that λ S and λ HS also vanish by the MPP condition.
The mass matrix for the scalar fields is expressed by the second derivatives of the potential at the VEVs: with and We identify the lighter eigenstate φ 1 with the SM-like Higgs, and its mass eigenvalue m φ 1 corresponds to the observed Higgs boson mass M h = 125 GeV. In our numerical calculation, we will take into account a renormalization group effect for the Higgs mass. The scalar-mixing matrix is defined by For |m 2 H | m 2 S μ 2 S , the mixing coupling is obtained by sin α ≈ μ HS v H /m 2 S , and it must be lower than the experimental bound | sin α| ≤ 0.36 given by the LHC Run 1 data [39]. This constraint induces m S μ HS 685 GeV, and also |v S | 45 GeV from Eq. (6).
In the low energy effective theory, the tree-level effective Higgs potential is given by [36] with Note that the Higgs self-coupling has a nontrivial gap λ ≡ μ 2 HS /m 2 S , which can play a crucial role to make the EW vacuum stable as in a scenario in Refs. [35,37]. In particular, the Higgs self-coupling λ H can vanish at the UV scale, e.g. the Planck scale, as well as the effective Higgs self-coupling λ SM explaining the observed Higgs boson mass, which has been studied in a type-II seesaw model [34]. This scenario indicates that We show the RG running of the Higgs self-coupling in Fig. 1, where we have used the beta functions given in Appendix A. The vertical and horizontal axes show the Higgs self-coupling and renormalization scale μ, respectively. Here, we have considered m S as the cutoff of the SM, and taken the boundary condition λ SM = λ H − λ at μ = m S = 1 TeV. Figure 1 shows that the Higgs self-coupling remains positive up to the Planck scale, and thus the EW vacuum can be stabilized.

Multiple-point principle
The   scalar (μ S ), because the highest term of S must be an even function to realize the degenerate vacua. Once μ S vanishes, it also remains zero. In the rest of this paper, we can take away λ S , λ HS , and μ S from our discussion. Note that for the vacuum around the Planck scale, v H ∼ M Pl , the stationary condition (4)  It is worth noting that λ is uniquely determined for a given m S , once the MPP condition and Eq. (16) are required. Then, μ HS is determined by μ 2 HS = λ m 2 S . In addition, v S is exactly obtained by Eq. (6) because of λ S = λ HS = 0 and μ S = 0. As a result, our model is controlled by only one free parameter. In the following, we choose m S as the free parameter.
The left panel of Fig. 2 shows the m S dependence of λ as the blue line. The red line and black dashed line show λ H (m S ) and λ SM (m S ), respectively. We find that λ is almost constant, and thus μ HS (m S ) is roughly proportional to m S as shown in the right panel of Fig. 2. To stabilize the EW vacuum, the Higgs self-coupling should remain positive up to the Planck scale. Thus, m S has to be smaller than 10 10 GeV, and we do not consider the heavier case.
Since λ is almost constant, v S is almost inversely proportional to m S . We find that −35 GeV v S < 0 GeV, and in particular |v S | < 1 GeV for m S > 4 TeV. The scalar-mixing angle is obtained by Thus, we can estimate sin α < 0.01 for m S > 4 TeV. Note that the whole parameter region is safe from the LHC Run 1 constraint | sin α| ≤ 0.36 [39]. This result is different from the estimation discussed below Eq. (13). The reason is that the estimation comes from μ HS m S , while the MPP condition requires μ HS 0.1 m S . It is remarkable that the EW symmetry is radiatively broken in our model.  At the end of this section, we mention the naturalness of the Higgs mass. When m S is much higher than the EW scale, it induces |m 2 , that is, the RG running of m 2 H is highly tuned to realize the observed Higgs mass. Here, we define the fine-tuning level as δ ≡ m 2 GeV. For example, δ = 10 indicates that we need to fine-tune the Higgs mass squared at an accuracy level of 10%. Figure 5 shows the m S dependence of δ, and we find δ = 1, 10, and 100 correspond to m S 1.3 TeV, 3.0 TeV, and 9.0 TeV, respectively. Therefore, from the naturalness point of view, there should exist the singlet scalar at O(1) TeV scale. We have found that m 2 H (M Pl ) vanishes for m S 950 GeV, and becomes negative in the lower m S region, in which the radiative EW symmetry breaking does not occur. For a tadpole diagram which contributes Higgs mass correction, it is tiny due to the heavy mass of m s . 2

Additional extension with right-handed neutrinos
In addition to the singlet scalar, we can introduce right-handed neutrinos to explain the active neutrino masses. The interaction parts of the Lagrangian including right-handed neutrinos are given by where L and N are lepton doublet and right-handed neutrino fields, respectively. Imposing the MPP condition at the Planck scale, β λ S (M Pl ) = 0 is required, and then Y N vanishes at all energy scales (see Appendix B). Therefore, the new parameters are only Y ν and M N , the same as the usual type-I seesaw model [42][43][44][45]. These parameters should satisfy the seesaw relation where m ν is the active neutrino mass matrix calculated by mass eigenvalues and the PMNS matrix [46,47].
When we consider the   (20) or where, using the seesaw relation, we have defined Tr(Y † (on the righthand side M N is a number not a matrix). The effective neutrino mass m eff is typically given by the heaviest active neutrino mass, and N ν means the relevant number of right-handed neutrinos. Since the singlet scalar and the right-handed neutrinos oppositely contribute to m 2 H , the Higgs mass corrections might be accidentally canceled (at the one-loop level).
We show a contour plot of δ in Fig. 6, where the horizontal and vertical axes show m S and M N , respectively. For the calculation of Eqs. (20) and (21), we have taken N ν = 1 and m eff = 0.05 eV as reference values. The positive δ region, in which the singlet scalar contribution is dominant, can drive the radiative EW symmetry breaking as mentioned above. When the right-handed neutrino mass becomes larger, the value of δ becomes smaller and vanishes at a specific point. From Eqs. (20) and (21), the point is estimated by If this relation is realized, δ can be small and hence our scenario can be natural even for the masses of singlet scalar and right-handed neutrinos 1 TeV.

Summary
We have investigated the scalar singlet extension of the SM with the MPP condition, in which the scalar potential has two degenerate vacua at the EW and UV scales.  [41]. The MPP conditions strongly restrict our model parameters, and there is only one free parameter left in our analysis, which we have taken as the singlet mass m S . We have shown the m S dependence of some model predictions, and found that our model is consistent with the LHC Run 1 results for the SM Higgs boson properties.
To simultaneously realize the MPP condition and the observed Higgs mass, the singlet-Higgs-Higgs coupling μ HS plays an important role. Furthermore, this coupling induces the radiative EW symmetry breaking. When the singlet mass is much larger than the Higgs mass, the μ 2 HS term dominates the beta function of the Higgs mass squared β m 2 H . Then, the sign of m 2 H can flip during the RG running, that is, m 2 H becomes negative toward the EW scale while positive at the Planck scale. We have found that this behavior can occur for m S > 950 GeV. On the other hand, too large m S causes the fine-tuning problem of the Higgs mass. To avoid the problem, there should exist the singlet scalar at O(1) TeV scale.
In order to incorporate the neutrino masses and flavor mixings to the singlet scalar extended model, we have introduced right-handed neutrinos and investigated the MPP scenario. Here, new parameters Y ν and M N are introduced, which are the neutrino Dirac-Yukawa coupling and the righthanded neutrino Majorana mass matrices, respectively, leading to the type-I seesaw mechanism. For Y ν O(1) (or equivalently M N O(10 14 ) GeV), the running of all couplings except m 2 H are almost the same as before. Therefore, the model can realize the MPP scenario as well as explaining the active neutrino masses.
It might be possible to solve the fine-tuning problem of the Higgs mass by an accidental cancellation of Higgs mass corrections coming from the singlet scalar and the right-handed neutrinos. We have found its approximate condition as Eq. (22). If the condition is satisfied, the masses of the singlet scalar and right-handed neutrinos can exceed O(1) TeV.