Vanishing Higgs Potential in Minimal Dark Matter Models

We consider the Standard Model with a new particle which is charged under $SU(2)_{L}$ with the hypercharge being zero. Such a particle is known as one of the dark matter (DM) candidates. We examine the realization of the multiple point criticality principle (MPP) in this class of models. Namely, we investigate whether the one-loop effective Higgs potential $V_{\text{eff}}(\phi)$ and its derivative $dV_{\text{eff}}(\phi)/d\phi$ can become simultaneously zero at around the string/Planck scale, based on the one/two-loop renormalization group equations. As a result, we find that only the $SU(2)_L$ triplet extensions can realize the MPP. More concretely, in the case of the triplet Majorana fermion, the MPP is realized at the scale $\phi={\cal{O}}(10^{16}\text{GeV})$ if the top mass $M_{t}$ is around $172$ GeV. On the other hand, for the real triplet scalar, the MPP can be satisfied for $10^{16}\text{ GeV}\lesssim\phi\lesssim10^{17}$GeV and $172\text{ GeV}\gtrsim M_{t}\gtrsim171$ GeV, depending on the coupling between the Higgs and DM.

In [3,4], the Higgs mass was predicted to be around 130GeV by the requirement that λ(µ) and β λ (µ) simultaneously become zero around M pl . 1 Namely, the minimum of the Higgs potential V (φ) around M pl vanishes. Such a requirement is called the multiple point criticality principle (MPP), and there have been many suggestions [56,57,58,59,60,61,46,39,62,64] that this principle might be closely related to physics at the Planck scale. One of the good points of the principle is its predictability: The low-energy effective couplings are fixed so that the minimum of the potential takes zero around M pl . See [39,62,63,64,65] for examples of the prediction.
By taking the fact that the MPP is realized in the SM into consideration, a natural question is whether the MPP can be also realized in the models beyond the SM. It is meaningful to consider the MPP of these models because we can understand whether the SM is actually special among them. One of the interesting extensions is adding a new weakly interacting fermion χ or scalar X, which is a n χ(X) representation of SU (2) L with the hypercharge Y χ(X) . Such extensions are phenomenologically well studied because they have dark matter (DM) candidates when Y χ(X) = 0 [66,67,68]. In this paper, we focus on Y χ(X) = 0, that is, Majorana fermions and real scalars. We examine the realization of the MPP of these models, based on the one/two-loop renormalization group equations (RGEs). We use the effective Higgs self coupling λ eff and its beta function β λ eff defined from the one-loop effective Higgs potential V eff (φ). Their definitions and the two-loop RGEs when we add a new fermion are presented in Appendix A. In the case of the new scalar (fermion), we only have to consider n X = 3 (n χ = 3, 5) since the scalar couplings (SU (2) L coupling g 2 ) rapidly blow(s) up when n X ≥ 4 [69] (n χ ≥ 7 [66]), and the theory does not valid up to M pl . For the septet and nonet fermion cases, we discuss this point in Appendix B.
In the following discussion, we regard the top mass M t as a free parameter, and the Higgs mass is varied within [70] M h = 125.09 ± 0.32GeV.
As for the initial values of the MS SM couplings, we use the results of [19]. For illustration, the Y χ = 0 cases are also discussed in Appendix C.
As a result, M t and Λ MPP are uniquely predicted because there is no additional free parameter. The results are depending on 124.77GeV ≤ M h ≤ 125.41GeV. 2 The upper panels of Fig.1 show the runnings of the SM parameters where M h = 125.09GeV, and M t is correspondingly fixed so that the MPP is realized. Here, we also show the SM running of g 2 by the dashed green line for comparison. Furthermore, in the middle and lower panels, we show the corresponding λ eff (left) and V eff (φ) (right). In these figures, the one-loop results are also shown. One can actually see that the potential and its derivative simultaneously become zero at a high energy scale, and that the only triplet can have the other vacuum near the string/Planck scale. We note that the two-loop effects are small. Now let us consider a new scalar. As mentioned before, the remaining possibility is n X = 3 [69]. The potential of the scalar fields is Here, H is the SM Higgs doublet. The one-loop RGEs which are different from those of the SM are as follows 3 : Furthermore, there is an additional contribution to V eff (φ): where   In order to discuss the Higgs potential around the cutoff scale Λ, it is meaningful to consider how the existence of a new particle changes the behavior of the bare Higgs mass m B as a function of Λ. 5 This is because m B would appear in the Higgs potential above Λ [31]. We now examine whether m B vanishes around the string scale or not. 6 See [13] for the evaluation of m B in the SM. 4 The mass of a new scalar suffers from fine-tuning problem. However, because our motivation in this paper is to distinguish the minimal dark matter models in the context of the MPP, we take Eq.(11) as the dark matter mass. 5 Within field theory, the quadratic divergence does not appear after the renormalization. However, it can have the physical meaning if we consider the scale around the Planck/string one, because the SM couples with the gravity. In this paper, we assume that the physics around the Planck scale is described by string theory, which is the cutoff theory whose universal cutoff scale is given by the string scale. This is why we take the universal cutoff in the calculation of m 2 B . See [13] for the detail. 6 The vanishing bare mass is so-called Veltman condition [74]. From the point of view of low energy field theory, Here, let us focus on (n χ(X) , Y χ(X) ) = (3, 0) at one-loop level. For χ, m B is given by where the couplings are evaluated at µ = Λ. On the other hand, for X, m 2 B | 1-loop becomes The left (right) panel of Fig.3 shows m B as a function of Λ when a new particle is fermion (scalar). Here, the green contour is the SM prediction when M t = 171.2 GeV, and blue bands correspond to the 2σ deviation from it [73]: In the right panel, we change κ at µ = M X from 0 to 0.4, and they are represented by a red band. Depending on the values of M t and κ, one can see that the scale at which m B becomes zero quite changes. In both of cases, m B can take zero around the string scale 7 . In addition to the vanishing λ at around the string scale, this fact may suggest the MPP is realized at this scale.
In conclusion, we have studied the MPP of the SM with a weakly interacting new particle with its hypercharge being zero. When a new particle is a fermion, we have found that the top mass M t and Λ MPP can be uniquely predicted. On the other hand, when a new particle is scalar, there exists a new scalar coupling κ. Due to this coupling, we have found that Λ MPP and M t drastically change. In both of cases, only the triplets survive from the point of view that the other vacuum should exist around the string/Planck scale and that the theory is valid up to this scale. The analysis of this paper suggests that the SM and its triplet extensions are special in that the MPP can be realized around the string/Planck scale. m 2 B = 0 is accidental and seems to require the fine-tuning at the Planck scale. We hope that m 2 B = 0 comes from some mechanism related to the physics at the Planck scale. 7 In order to obtain the correct electroweak symmetry breaking, we need to add small negative mass term to the Higgs potential, which is much small than Λ 2 . However, in the case of the SU (2) L triplet scalar, it may be possible to realize the electroweak symmetry breaking by the Coleman-Weinberg mechanism. See Appendix D. We thank the referee for pointing this out.

Appendix A Two-loop renormalization group equations and one-loop effective Higgs potential
The two-loop RGEs of the SM with a new fermion which is a n χ representation of SU (2) L with the hypercharge Y χ are as follows 8 : Here, t = ln µ with µ being the renormalization scale, Γ is the wave function renormalization of the Higgs, C n and S n are the Casimir and Dynkin index, and η = 1, 1 2 for Dirac and Weyl fermion. The two-loop RGEs of g Y and g 2 are agreement with [51] by putting y t = 0.
The one-loop effective Higgs potential is where and In Eq. (21), we have neglected the contribution from the Higgs quartic term because it is small when we consider the MPP. In principle, µ should be determined as a function of φ so that V 1-loop is minimized. However, in this paper, µ is taken to be φ for simplicity. It is known that this is a good approximation [44]. From V eff , we define λ eff and β λ eff as follows: Appendix B Landau pole in septet and nonet fermion As mentioned in the introduction, in cases of n χ = 7 and 9, there exists a scale Λ LP at which g 2 becomes infinity below M pl , which is well known as the Landau Pole. Therefore, these theories are not favored from the point of view of perturbativity (triviality) up to the string/Planck scale. For completeness, we give numerical results of the Landau pole in Fig.4.
Here, the two-loop results are shown by dashed lines. As is known, the one-loop Landau pole can be analytically solved: where S n is the Dynkin index. From Fig.4, one can see that the two-loop effect is relatively important.  Appendix C New Fermion with Y χ = 0 Here, we consider a new fermion with Y χ = 0. As well as the real n χ = 7 and 9 cases, the Landau pole of g 2 exists below M pl when n χ ≥ 5 [51]. So, let us here focus on n χ ≤ 4 9 .
Here, we leave M χ as a free parameter 10 . The left (right) panel of Fig.5 shows Λ MPP (M t ) as a function of M χ for each (n χ , Y χ ). 9 For n χ = 3 and 4, the LP of the U (1) Y gauge coupling g Y also appears below M pl respectively when Y χ = 2 and = 3/2. This is why we only show Y χ = 1 when n χ = 3 in Fig.5. 10 Furthermore, when n χ =1, 2 and 3, there are additional Yukawa couplings among the SM leptons (L i , E Ri ), the Higgs H and χ. However, we can neglect these effects because the lepton masses are small.

Appendix D Electroweak symmetry breaking by Coleman-Weinberg mechanism
Here, we discuss a possibility to realize the electroweak symmetry breaking by the Coleman-Weinberg mechanism in the case of the SU(2) triplet scalar. The one-loop effective Higgs potential is where m DM (φ) and ∆V 1-loop (µ, φ) are given by Eq.(10) and Eq.(21) respectively, and we have assumed that the quadratic term vanishes at the tree-level. In the following, we choose µ = m DM (φ). Then, φ develops the vacuum expectation value v because the negative quadratic term appears from the second term in Eq. (25). The resultant vacuum expectation value is where we have neglected the 1-loop correction to the quartic term. It is interesting that the successful electroweak symmetry breaking is realized for κ ≃ 0.04 which is also favored by the MPP around the Planck scale.