Cosmological implications of Standard Model criticality and Higgs inflation

The observed Higgs mass indicates that the Standard Model can be valid up to near the Planck scale $M_\text{P}$. Within this framework, it is important to examine how little modification is necessary to fit the recent experimental results in particle physics and cosmology. As a minimal extension, we consider the possibility that the Higgs field plays the role of inflaton and that the dark matter is the Higgs-portal scalar field. We assume that the extended Standard Model is valid up to the string scale $10^{17}\text{GeV}$. (This translates to the assumption that all the non-minimal couplings are not particularly large, $\xi\lesssim 10^2$, as in the critical Higgs inflation, since $M_\text{P}/\sqrt{10^2}\sim 10^{17}\text{GeV}$.) We find a correlated theoretical bound on the tensor-to-scalar ratio $r$ and the dark matter mass $m_\text{DM}$. As a result, the Planck bound $r<0.09$ implies that the dark-matter mass must be smaller than 1.1TeV, while the PandaX-II bound on the dark-matter mass $m_\text{DM}>0.7\pm0.2\text{TeV}$ leads to $r\gtrsim 2\times10^{-3}$. Both are within the range of near-future detection. When we include the right-handed neutrinos of mass $M_\text{R}\sim 10^{14}$GeV, the allowed region becomes wider, but we still predict $r\gtrsim 10^{-3}$ in the most of the parameter space. The most conservative bound becomes $r>10^{-5}$ if we allow three-parameter tuning of $m_\text{DM}$, $M_\text{R}$, and the top-quark mass.


Introduction
The Higgs field is the only elementary scalar whose existence is experimentally confirmed. The observed Higgs mass indicates that the Standard Model (SM) can be valid up to near the Planck scale M P = 1/ √ 8πG 2.4 × 10 18 GeV, that is, all the couplings remain perturbative and the Higgs potential is stable; see e.g. Refs. [1,2,3,4,5,6,7,8,9,10,11,12,13,14]. From string-theory point of view, this fact suggests that string theory is directly connected to the SM at the string scale Λ ∼ 10 17 GeV.
Furthermore, the Higgs potential can be very small and flat around the Planck scale by tuning the top quark mass within the experimental error. This fact, the so-called criticality of the SM, suggests that something non-trivial is happening around the Planck scale and that the SM remains valid without much modification up to the scale. 1 Within this paradigm, it is important to examine how little modification is necessary to fit the recent experimental results in particle physics and cosmology, especially the inflation and the dark matter.
The flatness of the Higgs potential suggests that the Higgs field can play the role of inflaton. Indeed, if we trust the SM even at the Planck scale, we can realize a phenomenologically viable inflation, namely the critical Higgs inflation, by introducing a non-minimal coupling of order 10-10 2 [50,51,52]; see also Ref. [53]. 2 In this paper, we do not assume any particular form of the Higgs potential at the Planck scale. 3 Instead, we study consequences of a general postulate that the Higgs field plays the role of inflaton above the string scale, assuming that the SM (extended with dark matter) is reliable below it; see Fig. 1. 4 After the end of the inflation, the slow-roll condition on the Higgs field is violated. In order for the fields to roll down to the electroweak (EW) scale, the potential height must be smaller than the inflation energy V inf in the whole region ϕ ≤ Λ. Note that even if there exists a local maximum with its height smaller than V inf , it does not interrupt the rolling down to the EW scale because the slow-roll condition is already violated. As we do not specify the shape of the inflaton potential above Λ, we cannot predict precisely the cosmological parameters such as the spectral index n s and the tensor-to-scalar ratio r. However, we may still put a lower bound on V inf from the highest value of the Higgs potential in the region ϕ < Λ, which can be converted into the lower bound on r.
It is certain that there exists a dark matter (DM). As one of the simplest realization, we employ the Higgs portal Z 2 scalar dark matter model [58,59,60,61,62,63,64,65,66], 5 though our analysis itself is applicable for any other model that modifies the running of the Higgs quartic coupling. We consider the generic region of the DM mass m DM being larger than the Higgs mass. Then its thermal abundance fixes the relation between the DM mass and the Higgs-DM coupling κ to be m DM κ × 3.2 TeV, and the spin-independent DM-

Slow-roll
Higgs field value Effective Higgs potential ' ⇤ V inf Figure 1: Schematic figure for the Higgs field as an inflaton nucleon elastic cross section is determined to be σ SI ∼ 10 −45 cm 2 [68,69]; see Fig. 2 for more discussion. The latest 1.6 σ bound from the PandaX-II experiment [70] reads m DM 0.7 TeV. When we do not include the right-handed neutrinos, we find that the current observational bounds r < 0.09 and m DM 0.7 TeV lead to the theoretical bounds m DM 1.1 TeV and r 2 × 10 −3 , respectively, which are well within the range of near future detection.
We have also studied the case with the right-handed neutrinos that account for the observed neutrino oscillations through the seesaw mechanism [71,72]. We find that when their mass is in the range M R 10 13 GeV, the results are the same as in the case without them. As we increase M R , the bound becomes milder up to the scale 10 14 GeV, and then becomes tighter up to 10 15 GeV at which the right-handed neutrino contribution makes the Higgs potential unstable. Combining these results, we find the absolute theoretical bound r > 10 −5 . If we restrict m DM 1.3 TeV, we obtain a stronger bound r 10 −3 for a reasonable top-quark mass range, as we will see in Fig. 7.
This paper is organized as follows. In Sec. 2, we show our basic strategy how to put a lower bound on r without the knowledge of higher scales. In Sec. 3, we review the Higgs-portal Z 2 scalar dark matter. In Sec. 4, we show our results without the effects from the heavy right-handed neutrinos. In Sec. 5, we show the results with the right-handed neutrinos for several representative values of their mass M R . In Sec. 6, we show the allowed region when we vary both M R and the top-quark mass m t . In Sec. 7, we summarize and discuss our results. In Appendix A, we show the renormalization group equations (RGEs) that we employ in the computation of the effective potential. In Appendix B, we discuss that the shape of allowed region, shown in Secs. 4-6, can be understood in terms of the difference of potential shape.

Lower bound on tensor-to-scalar ratio
We present our basic strategy how to put a lower bound on r without the knowledge of the physics at the higher Higgs-field value ϕ > Λ, extending the analysis in Ref. [56]. In the slow-roll inflation, the observable A s and r are written in terms of V and V inf : where is the slow-roll parameter. Eliminating V , we obtain This gives a linear relation between r and V inf since A s is fixed by the CMB observation to be A s 2.2 × 10 −9 [73]. During the inflation, the inflaton field value is larger than Λ; see Fig. 1. After the end of inflation, the field continues to roll down the potential hill and becomes the low-energy Higgs field that we know in the SM. In order not to prevent the rolling down to the EW scale, the maximum value of the effective potential in the region ϕ ≤ Λ, which we call V max ϕ≤Λ , must be smaller than the energy density during the inflation V inf : From Eqs. (3) and (4), Thus, we obtain the lower bound on r from V max ϕ≤Λ only.

Z 2 Higgs-portal scalar model
In this paper, we employ the Higgs-portal Z 2 scalar dark matter. Below the scale Λ our Lagrangian is where Φ is the SM Higgs doublet and S is the Higgs portal scalar field which has Z 2 symmetry. Hereafter we use ϕ := √ 2Φ † Φ. The singlet S is identified as the DM, whose mass is where v 246 GeV is the Higgs vacuum expectation value (VEV). The parameters λ S and κ affects V ϕ≤Λ through the renormalization group (RG) running of the Higgs quartic coupling, while m S does not. We assume that S does not acquire a Planck scale VEV and thus does not affect the inflation. In this model, the thermal abundance of the DM fixes the relation between κ and m DM in the non-resonant region [68]: 6 log 10 κ −3.63 + 1.04 log 10 m DM GeV .
This relation allows us to convert m DM into κ and vice versa. On the other hand, the spinindependent cross section and the dark-matter mass are related by [68] σ SI =   [70]. The upper side of each curve is excluded with 1.6 σ C.L. We superimpose the purple line representing Eq. (9) and the light purple region denoting the error of σ SI coming from the uncertainty of the overall coupling f N = 0.30±0.03; see Ref. [68].
The resultant 1.6 σ constraint on the DM mass reads, as said in Introduction, The uncertainty ±0.2 TeV comes from that of the f N ; see the caption of Fig. 2. Similarly, the 1.6 σ results from the LUX [74] and XENON1T [75] experiments imply m DM > 0.5 ± 0.1 TeV and m DM > 0.6 ± 0.1 TeV, respectively. The corresponding lower bound on κ is κ > 0.2 for all the three experiments. We will employ the pole mass of the top quark m t as an input parameter for the RG analysis below. The Monte-Carlo mass of the top quark has been precisely measured to be m MC t = 173.1±0.6 GeV [76]. However, the relation between m MC t and the pole mass m t is still unclear, and there remains uncertainty at least of 1 GeV; see e.g. Ref. [77] for a recent review. A theorist's combination of the pole mass, derived from the cross-section measurements, reads m t = 173.5 ± 1.1 GeV [76]. Hereafter we take conservatively two ranges: 169 GeV < m t < 178 GeV, which roughly corresponds to 2σ and 4σ ranges of the above combination, respectively.

Analysis without heavy right-handed neutrinos
As shown in the previous section, we need to compute V ϕ≤Λ in order to put the lower bound on r. We may find the excluded region in the r-m DM plane by obtaining V max ϕ≤Λ as a function of κ for each fixed set of (λ S , m t ) and converting κ to m DM via Eq. (8). We first present our method of analysis in Sec. 4.1. Then we show our results in Sec. 4.2.

Method of analysis
We employ the two-loop RGEs, which are summarized in Appendix A. The input parameters for them are λ S , κ, and the pole mass of top quark m t . 7 From the obtained running couplings, we determine V ϕ≤Λ . Then we exclude the parameter region in which V becomes negative in ϕ ≤ Λ or the perturbativity of couplings is violated. For the perturbativity, we demand that all the couplings are smaller than √ 4π 3.5 in all the region ϕ ≤ Λ. This condition chooses the region κ ≤ 0.5 (m DM 1.6 TeV) for λ S = 0. 8 In this paper, we restrict to the case λ S = 0 except for the right of Fig. 3 in which we instead take λ S = 0.6 for comparison. 9 For numerical computation, we first neglected the wave-function renormalization Γ(ϕ) given in Eq. (20). We have also estimated the largest possible deviation due to Γ(ϕ) by setting ϕ = Λ; see Appendix A for details.

Results
We plot the allowed region in r-m DM plane for λ S = 0 in the upper panel of Fig. 3. A solid and dashed lines denote the results with and without the effects of Γ, respectively, mentioned above. The region below each line is excluded, with its rainbow-color corresponding to each m t value. In the right, we superimpose the excluded region for λ S = 0.6 on the envelope of the left for a comparison with larger value of λ S . To understand the form of envelope, we need to know how the Higgs potential changes its shape with parameters, see Appendix B. 10 Fig. 3 shows that the Planck constraint r < 0.09 [78] leads to bounds on m t and m DM : This bound on m DM is stricter than the above-mentioned perturbativity bound m DM 1.6 TeV. When we take the lower bound by the PandaX-II experiment [70], m DM 0.7 TeV (κ 0.2), we obtain the lower bound on tensor-to-scalar ratio We can explore this possibility in near-future experiments such as the POLARBEAR-2 [79], LiteBIRD [80] and CORE [81]. Table 1: Neutrino masses obtained from the absolute values of mass-squared differences in the notation of Ref. [82].

Analysis with right-handed neutrinos
We introduce the heavy right-handed neutrinos that account for the observed neutrino masses through the seesaw mechanism, and obtain the lower bound on r for each DM mass.

Method of analysis
The observational constraints on mass of left-handed neutrinos are the upper bound on the sum of masses and their squared differences; see e.g. Ref. [82]. Under this condition, we consider the following three typical patterns of mass relations: The mass pattern is most hierarchical when the lightest one is 0. In Degenerate case, the upper bound on the sum of neutrino masses reads m i 0.1 eV. 11 In Table 1, we show the 11 The left-handed neutrino mass 0.1 eV for the three degenerate neutrinos corresponds to i mi = 0.3 eV. The 2σ upper bound from the TT-only analysis is i mi < 0.715 eV, while that from the TT+lensing+ext gives i mi < 0.234 eV [78]. See also Refs. [83,84] for more recent analyses that give a tighter bound i mi < 0.12 eV.  Table 3: Constraints that will be obtained from future observations of m DM and r for Normal Hierarchy.
mass pattern by setting the lightest one to be zero (0.1 eV) for the cases of normal/Inverted Hierarchy (Degenerate), using the mass-squared differences in Ref. [82]. For the three cases, we approximate the heaviest n ν neutrinos as having a common mass m ν and the remaining 3 − n ν ones as being massless as shown in Table 2. 12 Under the existence of heavy right-handed neutrino, the remaining input parameters to determine V ϕ≤Λ are λ S , κ, m t , and the right-handed neutrino mass M R,i . For simplicity, we assume that M R,i (i = 1, 2, 3) are identical: M R,i = M R . The Yukawa coupling of neutrino is given by the seesaw mechanism: We show the β-functions in this case in Appendix A.

Results for Normal Hierarchy
We show the results for Normal Hierarchy, n ν = 1, in Fig. 4. The right-handed neutrino mass M R is fixed in each panel: 10 13 , 10 14 , 10 14.4 ( 2.5 × 10 14 ), 10 14.5 ( 3.2 × 10 14 ), 10 14.6 ( 4.0 × 10 14 ), and 10 14.7 ( 5.0 × 10 14 ) in units of GeV. The color of envelope in each panel, denoted by the thick line, corresponds to the color in the plots in Sec. 6, in which the discussion for more general values of M R will be given. Note that the thick line is obtained by tuning one parameter m t for fixed M R , and its minimum corresponds to the two parameter tuning of m t and m DM .
With the right-handed neutrino, we have one more theoretical parameter M R in addition to m t to determine from the observational constraints of r and m DM . From Fig. 4, we see that the larger the M R is, the smaller the allowed region becomes, for a given lower bound on m t . This is because the right-handed neutrinos and the top quark have a similar effect on the Higgs potential, namely, they drives the Higgs quartic coupling smaller through the RG running towards high scales, and therefore they tend to make the Higgs potential negative if they both are heavy. 13 If we e.g. set m t > 169 GeV, we have the constraint M R 10 14.8 GeV ( 6.3 × 10 14 GeV); see also Fig. 8 in Sec. 6.
As we increase M R and switch panels in Fig. 4, we see that the value of m DM at the minimum point of the envelope becomes larger: 600 GeV, 870 GeV, etc. In particular, it goes beyond the perturbativity bound, indicated by the gray band, when M R = 10 14.7 GeV. Therefore, if the right-handed neutrino mass is larger than that, we have a stringent lower bound: r 10 −2 . On the other hand, the plot with M R 10 13 GeV is almost the same as the case without right-handed neutrinos shown in the left of Fig. 3.
Let us see implications of future discoveries of the DM and r: 14 • Suppose that m DM = 1 TeV (κ 0.31) and r = 0.01 are found. Then the right-handed neutrino mass is predicted to be in the narrow range 10 14 GeV M R 10 14.6 GeV and the top-quark mass is constrained from above: m t < 174 GeV.
• If we discover m DM = 1.5 TeV (κ 0.47) and r = 0.01, we obtain the theoretical lower bound M R 10 14.6 GeV, while the top quark mass is less constrained: 171 GeV < m t < 178 GeV. However, M R and m t are highly correlated in this case. Therefore if one of them is fixed, the other is precisely predicted.
• See Table 3 for other pairs of m DM and r. Generically the heavy DM mass tends to predict the heavy top-quark mass and M R . The smaller the r is, the tighter the range of m t . Especially, if we discover m DM = 1.5 TeV and r = 0.001, m t and M R are accurately predicted.
We can predict r bound or m DM to some extent by considering typical input parameters. When we choose m t = 173 GeV and M R = 10 14 GeV, we obtain the bound m DM ∼ 860 GeV-970 GeV for r < 0.09.

Results for Inverted Hierarchy
We show the results for the case of Inverted Hierarchy (n ν = 2) in Fig. 5. In this case, the right-handed neutrinos lighter than ∼ 10 13 GeV do not affect the analysis, similarly as in the case of Normal Hierarchy. However, the upper bound on M R is slightly different: M R 10 14.7 GeV, see also Fig. 7. Let us summarize implications of future discoveries of the DM and r again: • If we discover m DM = 1 TeV and r = 0.01, we obtain 10 13.9 GeV M R 10 14.4 GeV and m t < 174 GeV.
• If we discover m DM = 1.5 TeV and r = 0.01, M R must be larger than ∼ 10 14.6 GeV and 170 GeV < m t < 178 GeV. Although we cannot obtain the global narrow bounds on M R and m t , they are highly correlated as in the case of Normal Hierarchy.
See Table 4 for other pairs of m DM and r.

Results for Degenerate case
We show the results for Degenerated case (n ν = 3) in Fig. 6. The right-handed neutrinos lighter than ∼ 10 13 GeV do not affect the analysis, similarly as other cases. The upper bound on M R is smaller than in other cases: M R 10 14.2 GeV 1.6 × 10 14 GeV. We summarize implications of future discoveries m DM and r in Table 5. The right-handed neutrino mass tend to be lighter than hierarchical cases due to the heavy m ν . However, the prediction of m t is similar to the other cases.

Allowed region for all parameter space
In the previous section, we have plotted the bounds for various m t in each panel of fixed M R .
In each panel, we have also shown the envelope of different m t lines. This envelope is our theoretical lower bound on r for a given M R . In Sec. 6.1, we show these envelopes altogether in the same plot, and give the absolute lower bound on r for varying m t and M R . In Sec. 6.2, we see the same absolute lower bound in a different way, by changing the order of fixing m t and M R .

Lower bound on r for each M R
In Figs. 7 and 8, we plot our theoretical lower bounds on r for various M R when we allow the top-quark pole mass within roughly 2σ and 4σ ranges shown in Eqs. (11) and (12), respectively. We also give the envelope of these lines, which gives the allowed region for varying m t and M R . In the plot, each colored line represents the lower bound on r, and corresponds to the envelope denoted by the thick colored line in Secs. 5. 2-5.4. We also show the absolute lower bound by the black line. 15 We explain the envelope denoted by the black line in Fig. 7: • We see that the allowed region is enlarged to from the n ν = 0 case in Eq. (15), which is read from the black line in Fig. 3 (being close the orange M R = 10 13 GeV line in Fig. 7). This is because the loop corrections of heavy right-handed neutrinos reduce V max ϕ≤Λ . 15 We have plotted the envelope, denoted by the black line, as follows: 1) Each MR-fixed line has a minimum. Make an interpolating function which linearly join all these minimum points. 2) Each mt-fixed line has a minimum. Make another interpolating function which linearly join all these minimum points. 3) Make a function that chooses the smaller value of these two for each mDM. 4) In large mDM region, we replace the interpolated bound with the lower bound determined by the maximal mt; see the caption of Fig. 9 to see how mt gives the bound. Note that these interpolating functions are evaluated only for 600 GeV < mDM < 1600 GeV, and hence they are untrustworthy in the extrapolated regions mDM < 600 GeV and mDM > 1600 GeV. This does not do any harm because these regions are already excluded by the direct DM search and by the perturbativity, respectively.
• The lower bound on r increases rapidly in the region m DM 1.3 TeV due to the upper end of the parameter m t < 176 GeV.
• In the region near the envelope denoted by the black line, the two input parameters m t and M R are simultaneously tuned to minimize the potential height V max ϕ≤Λ .
• If one allows to adjust the three parameters, m t , M R , and m DM simultaneously, then the lowest point of the black line, r ∼ 10 −5 , is realized. This might be the case if some logic that demands the fine tuning, such as the multiple-point principle [15,16,17], is indeed applicable.
In Fig. 8, we plot for a wider range of the top-quark mass (12). The lower bound on r, denoted by the black line, increases in the region m DM 1.3 TeV in the cases of n ν = 1 and 2 because of the difference of potential shapes explained in Appendix B, while its rise in the region m DM 1.5 TeV is due to the upper end of the parameter m t < 178 GeV.

Lower bound on r for each m t
In Fig. 9, we show the lower bound on r for each fixed m t with M R being varied. The envelope, denoted by the black line, is the same as the global lower bound on r in Sec. 6.1 except for the range explained in the caption.
The lowest possible value of m DM for each m t does not depend on the number of neutrinos n ν . Therefore, for any given lower bound on m t , we obtain the corresponding lower bound of m DM without any assumptions on the other parameters in the neutrino sector. We see that the lowest point for a given m t moves right as we increase m t . This leads to a strong correlation between m t and m DM regardless of n ν if r < 10 −3 .  Fig. 3 for the shaded region.  Fig. 7 for 169 GeV < m t < 178 GeV.  Fig. 3 for the corresponding plot without right-handed neutrinos and for the explanation of the shaded region.

Summary and discussion
We have calculated the lower bound on the tensor-to-scalar ratio, r, for each given mass of the Higgs-portal Z 2 scalar dark matter, m DM , under the simple assumption that the extrapolation of the Higgs-field direction plays the role of inflaton at ϕ > Λ. The advantage of our approach is that we may obtain the lower bound on r without knowing any detail of the high-scale physics.
In the case without the heavy right-handed neutrinos, we have obtained the theoretical bounds on (i) the DM mass m DM 1.1 TeV, (ii) the tensor-to-scalar ratio r 2 × 10 −3 , and (iii) the pole mass of the top quark 171 GeV < m t < 175 GeV from the current observational constraints r < 0.09 and m DM 0.7 TeV. We see that (i) and (ii) are rather stringent and are well within the near-future detection.
With the heavy right-handed neutrinos, we obtain the wider allowed region in the r-m DM plane, r 10 −5 and m DM 1.6 TeV, if we allow a three-parameter tuning. Altough the region r 10 −3 is hard for the planned near-future observations, we may still explore it in combination with the HL-LHC and future neutrino experiments because of the strong correlations between m t , m DM and the right-handed neutrino mass M R .
The lower bound on r may slightly be affected when we relax the positivity condition on the Higgs potential by e.g. taking into account the thermal correction or by replacing it with the vacuum meta-stability. Because our bound is coming from the maximum value of effective Higgs potential, rather than the minimum, the lower bound on r would be reduced only by a factor of few even if we allow the negative value of the potential minimum of the order of the height of the potential maximum. Of course we should make sure that finally the EW vacuum is chosen in the late time in such a case.
In our RG analysis, we have assumed that all the fields are massless. 16 In general, the non-minimal couplings ξϕ 2 R and ξ S S 2 R cause the mass terms of the order of ξH 2 ϕ 2 and ξ S H 2 S 2 , respectively. Under a classical field value ϕ cl , we get Such effective masses √ ξλϕ 2 cl /M P and √ ξ S λϕ 2 cl /M P are smaller enough than µ ∼ ϕ cl since we consider the non-minimal couplings ξ, ξ S 10 2 and the small quartic coupling λ 10 −1 in the region ϕ cl 10 17 GeV.
It would be interesting to investigate the cosmology of our scenario after the inflation. For example, the Higgs field may be trapped at a false vacuum, and a mini inflation may results from it, depending on the initial condition at the end of the main inflation. 17 Afterwards, the true-vacuum bubbles should be created by the tunneling process, and the first order phase transition be completed by the bubble collision, which generates primordial gravitational waves and black holes. In addition, if the Higgs potential has an inflection point, the scalar perturbations could be enhanced, depending on how the slow-roll condition is well satisfied there, and could lead to another mechanism of the formation of primordial black holes. We hope to return to these issues in the future.

A Renormalization group equations
We have calculated the lower bound on r as follows: 1. Solve the RGEs (21)-(28) (shown in the end of this section) for given parameters. 18 The effects from right-handed neutrino is introduced only at high energy scale ϕ ≥ M R : We set n ν = 0 and M R = 0 in ϕ < M R . As the boundary condition to solve the RGEs, we have used Eqs. (11)- (15) in [69] and the values in Table 6.
2. Calculate the one-loop effective Higgs potential where is the effective Higgs-self coupling. 19 We set µ = ϕ when we calculate λ eff ; see [52,86] for the details about the renormalization scale µ of λ eff . The one-loop wave-function 18 In this section, we omit the argument of the couplings but it should be understood that they are not the values at µ = mt; see footnote 7. 19 The last term in the braces is introduced to naively take into account the effect of the neutrino loop on the effective potential. We have checked that its effect is at most few percent.
B Explaining the form of envelope by potential shape In this section, we explain the shape of envelopes denoted by the black or colored-thick line in the figures. Let us start with the case without heavy right-handed neutrino (the left of Fig. 3). At the minimum point of envelope, the maximum value of the Higgs potential becomes smallest. In Fig. 10, we show the shape of potential near this point. At this minimum point, the height at the local maximum (at ϕ 5 × 10 16 GeV in the case of Fig. 10) becomes identical to the height at ϕ = Λ. If the local potential minimum (at ϕ = 9 × 10 16 GeV in the case of Fig. 10) is moved left, then the height at ϕ = Λ becomes larger, while moved right, the height at the local maximum becomes larger. The left of the minimum point of the envelope is governed by the local maximum of the potential, while the right by the value at ϕ = Λ. Now we turn to the case with right-handed neutrinos. The envelope is denoted by the colored thick line in Figs. 4-6. We note that each envelope in Fig. 8 has, in addition to the cusp, one more (hardly seeable) non-smooth point, which is located, e.g. in the n ν = 1 panel, at (m DM , log 10 r) ∼ (1050 GeV, −1.5) for M R = 10 14 GeV, (1100 GeV, −1.7) for M R = 10 14.1 GeV, (1150 GeV, −2) for M R = 10 14.2 GeV, and (1220 GeV, −2.6) for M R = 10 14.3 GeV. This is because the Higgs potential has two local minima in general: The one at higher (lower) Figure 10: The shape of Higgs potential with the values of m t and m DM (κ) that corresponds to a point near the minimum of the envelope in the left of Fig. 3. Figure 11: Left: Typical potential shape when its minimum is at ϕ > Λ. Center: Typical potential shape when the minimum at larger ϕ gives lower height. Right: Typical potential shape when the minimum at smaller ϕ gives lower height.
ϕ is due to the neutrino (top quark) contribution. 20 There are the following three kinds of potential shapes: (i) On the left side of the cusp of each envelope, the potential minimum at higher ϕ is located at ϕ > Λ; see the left of Fig. 11.
(ii) In between the cusp and the non-smooth point of each envelope, there are two potential minima and the height of the one at larger ϕ is smaller; see the center of Fig. 11.
(iii) On the right side of the non-smooth point of each envelope, there are two potential minima and the height of the one at lower ϕ is smaller; see the right of Fig. 11.
These two potential minima are degenerate on the black line in Figs. 7 and 8, and the cases (ii) and (iii) become identical.
We can see this non-smooth point from another point of view in Fig. 12. The left end of each rainbow-colored line for m t 174 GeV is the M R → 0 limit, and is the same as the black line in Fig. 3 of n ν = 0. However, they do not touch the black n ν = 0 line for m t 175 GeV. This is because there arises the two local minima of the potential as in the case (iii): The minimum at lower ϕ can be tuned to be zero by the top-quark contribution, and the neutrino contribution may produce the minimum at higher ϕ, which allows to reduce the maximum potential height freely. The dotted magenta line in Fig. 12 links the left ends of the (dashed) rainbow-colored lines for the region m t 175 GeV, and is the same as the line joining the non-smooth points mentioned above.
Finally, we explain why the envelope in Fig. 12 (black, lower) is bent in the large m DM region. At the point where the dotted magenta line touches the envelope (black, lower), the potential height of the two degenerate minima becomes zero. 21 On its right side, the case (i) is realized, and the maximum potential height becomes much larger than the cases (ii) and (iii) with the two minima.
mt=178 GeV Figure 12: The same plot as in Fig. 9. The black n ν = 0 line in Fig. 3 is also superimposed. The dotted magenta line show the location of the non-smooth points explained in the text, and links the left ends of the rainbow-colored lines for m t 175 GeV.