#  OU-HET/948 Cosmological implications of Standard Model criticality and
Higgs inflation

Yuta Hamada, 1,2 Hikaru Kawai, 3 Yukari Nakanishi, 4 and Kin-ya Oda 4
1Department of Physics, University of Wisconsin, Madison, WI 53706, USA
2KEK Theory Center, IPNS, KEK, Tsukuba, Ibaraki 305-0801, Japan
3Department of Physics, Kyoto University, Kyoto 606-8502, Japan
4Department of Physics, Osaka University, Osaka 560-0043, Japan
yhamada<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

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 1.1 TeV, while the PandaX-II
bound on the dark-matter mass $m_{\text{DM}}>0.7\pm 0.2\,\text{TeV}$ leads to
$r\gtrsim 2\times 10^{-3}$. 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.

###### Contents

  1. 1 Introduction
  2. 2 Lower bound on tensor-to-scalar ratio
  3. 3 $Z_{2}$ Higgs-portal scalar model
  4. 4 Analysis without heavy right-handed neutrinos
    1. 4.1 Method of analysis
    2. 4.2 Results
  5. 5 Analysis with right-handed neutrinos
    1. 5.1 Method of analysis
    2. 5.2 Results for Normal Hierarchy
    3. 5.3 Results for Inverted Hierarchy
    4. 5.4 Results for Degenerate case
  6. 6 Allowed region for all parameter space
    1. 6.1 Lower bound on $r$ for each $M_{\text{R}}$
    2. 6.2 Lower bound on $r$ for each $m_{t}$
  7. 7 Summary and discussion
  8. A Renormalization group equations
  9. B Explaining the form of envelope by potential shape

##  1 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 $M_{\text{P}}=1/\sqrt{8\pi
G}\simeq 2.4\times 10^{18}\,\text{GeV}$, 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 $\Lambda\sim 10^{17}\,\text{GeV}$.

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.111 This paradigm includes e.g. the multiple
point criticality principle (MPP) [15, 16, 17, 18, 19], the (classical)
conformality [20, 21, 20, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34,
35, 36, 37, 38, 39], the asymptotic safety [40, 41, 42], the hidden duality
and symmetry [43, 44], and the maximum entropy principle [45, 46, 47, 48, 49].
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].222 In the original Higgs inflation model [54, 55],
the flatness of the Higgs potential has not been assumed and a large non-
minimal coupling $\xi\sim 10^{4\text{--}5}$ has been required.

In this paper, we do not assume any particular form of the Higgs potential at
the Planck scale.333 If one assumes e.g. that the Higgs field fits in the
massless state in string theory, one may compute the potential at the trans-
Planckian scale and discuss the possibility of its mixing with other
directions such as the radius of the extra dimension at the high scales [56,
57].  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. 1.444 This is
consistent with the natural assumption that all the non-minimal couplings are
not particularly large, namely, at most $10^{2}$, as in the critical Higgs
inflation.  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 $V_{\text{inf}}$ in the whole region $\varphi\leq\Lambda$.
Note that even if there exists a local maximum with its height smaller than
$V_{\text{inf}}$, 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 $\Lambda$, 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
$V_{\text{inf}}$ from the highest value of the Higgs potential in the region
$\varphi<\Lambda$, which can be converted into the lower bound on $r$.

Figure 1: Schematic figure for the Higgs field as an inflaton

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],555 See Ref. [67] for a model with extra dark
Higgs field.  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 $m_{\text{DM}}$ being larger than the Higgs
mass. Then its thermal abundance fixes the relation between the DM mass and
the Higgs-DM coupling $\kappa$ to be $m_{\text{DM}}\simeq\kappa\times
3.2\,\text{TeV}$, and the spin-independent DM-nucleon elastic cross section is
determined to be $\sigma_{\text{SI}}\sim 10^{-45}\,\text{cm}^{2}$ [68, 69];
see Fig. 2 for more discussion. The latest 1.6 $\sigma$ bound from the PandaX-
II experiment [70] reads $m_{\text{DM}}\gtrsim 0.7\,\text{TeV}$. When we do
not include the right-handed neutrinos, we find that the current observational
bounds $r<0.09$ and $m_{\text{DM}}\gtrsim 0.7\,\text{TeV}$ lead to the
theoretical bounds $m_{\text{DM}}\lesssim 1.1\,\text{TeV}$ and $r\gtrsim
2\times 10^{-3}$, 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 $M_{\text{R}}\lesssim
10^{13}\,\text{GeV}$, the results are the same as in the case without them. As
we increase $M_{\text{R}}$, the bound becomes milder up to the scale
$10^{14}\,\text{GeV}$, and then becomes tighter up to $10^{15}\,\text{GeV}$ 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 $m_{\text{DM}}\gtrsim 1.3\,\text{TeV}$, we obtain
a stronger bound $r\gtrsim 10^{-3}$ 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 $M_{\text{R}}$. In Sec. 6, we show the allowed region
when we vary both $M_{\text{R}}$ 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.

##  2 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 $\varphi>\Lambda$,
extending the analysis in Ref. [56]. In the slow-roll inflation, the
observable $A_{s}$ and $r$ are written in terms of $\epsilon_{V}$ and
$V_{\text{inf}}$:

$\displaystyle A_{s}$ $\displaystyle={1\over
24\pi^{2}}{1\over\epsilon_{V}}{V_{\text{inf}}\over M_{\text{P}}^{4}},$
$\displaystyle r$ $\displaystyle=16\epsilon_{V},$ (1)

where

$\displaystyle\epsilon_{V}:={M_{\text{P}}^{2}\over 2}\left(V_{,\varphi}\over
V\right)^{2},$ (2)

is the slow-roll parameter. Eliminating $\epsilon_{V}$, we obtain

$\displaystyle r$ $\displaystyle={2\over 3\pi^{2}}{1\over
A_{s}}{V_{\text{inf}}\over M_{\text{P}}^{4}}.$ (3)

This gives a linear relation between $r$ and $V_{\text{inf}}$ since $A_{s}$ is
fixed by the CMB observation to be $A_{s}\simeq 2.2\times 10^{-9}$ [73].

During the inflation, the inflaton field value is larger than $\Lambda$; 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 $\varphi\leq\Lambda$, which we call
$V_{\varphi\leq\Lambda}^{\text{max}}$, must be smaller than the energy density
during the inflation $V_{\text{inf}}$:

$\displaystyle V_{\varphi\leq\Lambda}^{\text{max}}<V_{\text{inf}}.$ (4)

From Eqs. (3) and (4),

$\displaystyle r>r_{\text{bound}}:={2\over 3\pi^{2}}{1\over
A_{s}}{V_{\varphi\leq\Lambda}^{\text{max}}\over M_{\text{P}}^{4}}.$ (5)

Thus, we obtain the lower bound on $r$ from
$V_{\varphi\leq\Lambda}^{\text{max}}$ only.

##  3 $Z_{2}$ Higgs-portal scalar model

In this paper, we employ the Higgs-portal $Z_{2}$ scalar dark matter. Below
the scale $\Lambda$ our Lagrangian is

$\displaystyle\mathcal{L}=\mathcal{L}_{\text{SM}}+{1\over
2}(\partial_{\mu}S)^{2}-{1\over 2}m_{S}^{2}S^{2}-{\lambda_{S}\over
4!}S^{4}-{\kappa\over 2}S^{2}\Phi^{\dagger}\Phi,$ (6)

where $\Phi$ is the SM Higgs doublet and $S$ is the Higgs portal scalar field
which has $Z_{2}$ symmetry. Hereafter we use
$\varphi:=\sqrt{2\Phi^{\dagger}\Phi}$. The singlet $S$ is identified as the
DM, whose mass is

$\displaystyle m_{\text{DM}}^{2}=m_{S}^{2}+{\kappa v^{2}\over 2},$ (7)

where $v\simeq 246\,\text{GeV}$ is the Higgs vacuum expectation value (VEV).
The parameters $\lambda_{S}$ and $\kappa$ affects $V_{\varphi\leq\Lambda}$
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
$\kappa$ and $m_{\text{DM}}$ in the non-resonant region [68]:666 In the region
of our interest, $0.1\lesssim\kappa\lesssim 0.5$, this roughly translates as
$m_{\text{DM}}\simeq\kappa\times 3.2\,\text{TeV}$ [69].

$\displaystyle\log_{10}\kappa\simeq-3.63+1.04\log_{10}{m_{\text{DM}}\over\text{GeV}}.$
(8)

This relation allows us to convert $m_{\text{DM}}$ into $\kappa$ and vice
versa. On the other hand, the spin-independent cross section and the dark-
matter mass are related by [68]

$\displaystyle\sigma_{\text{SI}}$
$\displaystyle=\frac{\kappa^{2}f_{N}^{2}}{4\pi}\left(\frac{m_{n}m_{\text{DM}}}{m_{n}+m_{\text{DM}}}\right)^{2}\frac{m_{n}^{2}}{m_{H}^{4}m_{\text{DM}}^{2}}\simeq
1.1\times
10^{-45}\left(m_{\text{DM}}\over\text{TeV}\right)^{0.08}\,\text{cm}^{2}.$ (9)

The resultant $1.6\,\sigma$ constraint on the DM mass reads, as said in
Introduction,

$\displaystyle m_{\text{DM}}>0.7\pm 0.2\,\text{TeV}.$ (10)

The uncertainty $\pm 0.2\,\,\text{TeV}$ comes from that of the $f_{N}$; see
the caption of Fig. 2. Similarly, the $1.6\,\sigma$ results from the LUX [74]
and XENON1T [75] experiments imply $m_{\text{DM}}>0.5\pm 0.1\,\text{TeV}$ and
$m_{\text{DM}}>0.6\pm 0.1\,\text{TeV}$, respectively. The corresponding lower
bound on $\kappa$ is $\kappa>0.2$ for all the three experiments.

Figure 2: Fig. 5 (a) in Ref. [70]. The upper side of each curve is excluded
with $1.6\,\sigma$ C.L. We superimpose the purple line representing Eq. (9)
and the light purple region denoting the error of $\sigma_{\text{SI}}$ coming
from the uncertainty of the overall coupling $f_{N}=0.30\pm 0.03$; see Ref.
[68].

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 $m_{t}^{\text{MC}}=173.1\pm 0.6\,\text{GeV}$ [76].
However, the relation between $m_{t}^{\text{MC}}$ 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 $m_{t}=173.5\pm 1.1\,\text{GeV}$
[76]. Hereafter we take conservatively two ranges:

$\displaystyle 171\,\text{GeV}<m_{t}$ $\displaystyle<176\,\text{GeV},$ (11)
$\displaystyle 169\,\text{GeV}<m_{t}$ $\displaystyle<178\,\text{GeV},$ (12)

which roughly corresponds to $2\sigma$ and $4\sigma$ ranges of the above
combination, respectively.

Figure 3: Upper: Allowed regions for $\lambda_{S}=0$. For each $m_{t}$, the
region above the line with corresponding color is allowed. Each vertical line
denotes the lower bound on $m_{\text{DM}}$ from the positivity of potential:
$V_{\varphi\leq\Lambda}>0$. The envelope of the rainbow-colored lines,
indicated by the black line, gives the lower bound on $r$ for each
$m_{\text{DM}}$ when one varies $m_{t}$. Lower: Excluded regions for
$\lambda_{S}=0$ (lower left) and $\lambda_{S}=0.6$ (lower right). The blue and
red regions are excluded by the upper bound on the tensor-to-scalar ratio
$r<0.09$ [78] and by lower bound on the DM mass
$m_{\text{DM}}>0.7\,\text{TeV}$ from PandaX-II [70], respectively.

##  4 Analysis without heavy right-handed neutrinos

As shown in the previous section, we need to compute $V_{\varphi\leq\Lambda}$
in order to put the lower bound on $r$. We may find the excluded region in the
$r$-$m_{\text{DM}}$ plane by obtaining $V_{\varphi\leq\Lambda}^{\text{max}}$
as a function of $\kappa$ for each fixed set of $(\lambda_{S},m_{t})$ and
converting $\kappa$ to $m_{\text{DM}}$ via Eq. (8). We first present our
method of analysis in Sec. 4.1. Then we show our results in Sec. 4.2.

###  4.1 Method of analysis

We employ the two-loop RGEs, which are summarized in Appendix A. The input
parameters for them are $\lambda_{S}$, $\kappa$, and the pole mass of top
quark $m_{t}$.777 The couplings are given at the scale $\mu=m_{t}$ unless
otherwise stated.  From the obtained running couplings, we determine
$V_{\varphi\leq\Lambda}$. Then we exclude the parameter region in which $V$
becomes negative in $\varphi\leq\Lambda$ or the perturbativity of couplings is
violated. For the perturbativity, we demand that all the couplings are smaller
than $\sqrt{4\pi}\simeq 3.5$ in all the region $\varphi\leq\Lambda$. This
condition chooses the region $\kappa\leq 0.5$ ($m_{\text{DM}}\lesssim
1.6\,\text{TeV}$) for $\lambda_{S}=0$.888 See Fig. 1 in Ref. [69] for the
allowed region in the $\lambda_{S}$–$\kappa$ plane.  In this paper, we
restrict to the case $\lambda_{S}=0$ except for the right of Fig. 3 in which
we instead take $\lambda_{S}=0.6$ for comparison.999 We will see in the right
of Fig. 3 that the large $\lambda_{S}$ tends to narrower the allowed region.
Therefore, it is more conservative to set $\lambda_{S}=0$.

For numerical computation, we first neglected the wave-function
renormalization $\Gamma\\!\left(\varphi\right)$ given in Eq. (20). We have
also estimated the largest possible deviation due to
$\Gamma\\!\left(\varphi\right)$ by setting $\varphi=\Lambda$; see Appendix A
for details.

###  4.2 Results

We plot the allowed region in $r$-$m_{\text{DM}}$ plane for $\lambda_{S}=0$ in
the upper panel of Fig. 3. A solid and dashed lines denote the results with
and without the effects of $\Gamma$, 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
$\lambda_{S}=0.6$ on the envelope of the left for a comparison with larger
value of $\lambda_{S}$. To understand the form of envelope, we need to know
how the Higgs potential changes its shape with parameters, see Appendix
B.101010 The same explanation can be adapted for the other envelopes appearing
hereafter.

Fig. 3 shows that the Planck constraint $r<0.09$ [78] leads to bounds on
$m_{t}$ and $m_{\text{DM}}$:

$\displaystyle 171\,\text{GeV}<m_{t}$ $\displaystyle<175\,\text{GeV},$ (13)
$\displaystyle m_{\text{DM}}$ $\displaystyle\lesssim 1.1\,\text{TeV}.$ (14)

This bound on $m_{\text{DM}}$ is stricter than the above-mentioned
perturbativity bound $m_{\text{DM}}\lesssim 1.6\,\text{TeV}$.

When we take the lower bound by the PandaX-II experiment [70],
$m_{\text{DM}}\gtrsim 0.7\,\text{TeV}$ ($\kappa\gtrsim 0.2$), we obtain the
lower bound on tensor-to-scalar ratio

$\displaystyle r\gtrsim 2\times 10^{-3}.$ (15)

We can explore this possibility in near-future experiments such as the
POLARBEAR-2 [79], LiteBIRD [80] and CORE [81].

##  5 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.

###  5.1 Method of analysis

|  $m_{1}$ [eV] |  $m_{2}$ [eV] |  $m_{3}$ [eV] | Pattern
---|---|---|---|---
1\. Normal Hierarchy | 0 (set) | $8.6\times 10^{-3}$ | $5.1\times 10^{-2}$ | $m_{1}\ll m_{2}<m_{3}$
2\. Inverted Hierarchy | $5.0\times 10^{-2}$ | $5.0\times 10^{-2}$ | 0 (set) | $m_{1}\simeq m_{2}\gg m_{3}$
3\. Degenerate (NO) | 0.1 (set) | $1.0\times 10^{-1}$ | $1.1\times 10^{-1}$ | $m_{1}\simeq m_{2}\simeq m_{3}$
3\. Degenerate (IO) | $1.1\times 10^{-1}$ | $1.1\times 10^{-1}$ | 0.1 (set) | $m_{1}\simeq m_{2}\simeq m_{3}$
Table 1: Neutrino masses obtained from the absolute values of mass-squared differences in the notation of Ref. [82]. | Number of effective $\nu$ | Common mass $m_{\nu}$ [eV]
---|---|---
1\. Normal Hierarchy | $n_{\nu}=1$ | $5.1\times 10^{-2}$
2\. Inverted Hierarchy | $n_{\nu}=2$ | $5.0\times 10^{-2}$
3\. Degenerate | $n_{\nu}=3$ | $1.1\times 10^{-1}$
Table 2: Common neutrino mass that we use as an input.

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:

  1. 1.

Normal Hierarchy (NH, $m_{1}$ the lightest),

  2. 2.

Inverted Hierarchy (IH, $m_{3}$ the lightest),

  3. 3.

Degenerate (all masses comparable).

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
$m_{i}\lesssim 0.1\,\text{eV}$.111111 The left-handed neutrino mass
$0.1\,\text{eV}$ for the three degenerate neutrinos corresponds to
$\sum_{i}m_{i}=0.3\,\text{eV}$. The $2\sigma$ upper bound from the TT-only
analysis is $\sum_{i}m_{i}<0.715\,\text{eV}$, while that from the
TT+lensing+ext gives $\sum_{i}m_{i}<0.234\,\text{eV}$ [78]. See also Refs.
[83, 84] for more recent analyses that give a tighter bound
$\sum_{i}m_{i}<0.12\,\text{eV}$.  In Table 1, we show the 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 $n_{\nu}$ neutrinos as having a
common mass $m_{\nu}$ and the remaining $3-n_{\nu}$ ones as being massless as
shown in Table 2.121212 If we want to consider a different $m_{\nu}$, we may
simply rescale the right-handed neutrino mass $M_{\text{R}}$ in our results,
since $m_{\nu}\propto M_{\text{R}}^{-1}$ by the seesaw mechanism.

Under the existence of heavy right-handed neutrino, the remaining input
parameters to determine $V_{\varphi\leq\Lambda}$ are $\lambda_{S}$, $\kappa$,
$m_{t}$, and the right-handed neutrino mass $M_{\text{R},i}$. For simplicity,
we assume that $M_{\text{R},i}$ ($i=1,2,3$) are identical:
$M_{\text{R},i}=M_{\text{R}}$. The Yukawa coupling of neutrino is given by the
seesaw mechanism: $y_{\nu}=\sqrt{2m_{\nu}M_{\text{R}}}/v$, with $v\simeq
246\,\text{GeV}$. We show the $\beta$-functions in this case in Appendix A.

###  5.2 Results for Normal Hierarchy

Figure 4: Allowed region for Normal Hierarchy with $\lambda_{S}=0$. The right-handed neutrino mass $M_{\text{R}}$ is increased in the order of the upper-left, upper-right, middle-left, …, lower-right panels. The bold line in each panel is the envelope of the $m_{t}$-fixed rainbow-colored lines, and gives the lower bound on $r$ for the fixed $M_{\text{R}}$. See the caption of Fig. 3 for the shaded region. | $m_{t}$ | $M_{\text{R}}$
---|---|---
$m_{\text{DM}}=1\,\text{TeV}$, $r=0.01$ | $m_{t}<174\,\text{GeV}$ | $10^{14}\,\text{GeV}\lesssim M_{\text{R}}\lesssim 10^{14.6}\,\text{GeV}$
$m_{\text{DM}}=1\,\text{TeV}$, $r=0.001$ | $173\,\text{GeV}<m_{t}<174\,\text{GeV}$ | $10^{14.1}\,\text{GeV}\lesssim M_{\text{R}}\lesssim 10^{14.3}\,\text{GeV}$
$m_{\text{DM}}=1.5\,\text{TeV}$, $r=0.01$ | $170\,\text{GeV}<m_{t}<178\,\text{GeV}$ | $10^{14.6}\,\text{GeV}\lesssim M_{\text{R}}\lesssim 10^{14.8}\,\text{GeV}$
$m_{\text{DM}}=1.5\,\text{TeV}$, $r=0.001$ | $m_{t}\simeq 177.8\,\text{GeV}$ | $M_{\text{R}}\simeq 10^{14.6}\,\text{GeV}$
Table 3: Constraints that will be obtained from future observations of
$m_{\text{DM}}$ and $r$ for Normal Hierarchy.

We show the results for Normal Hierarchy, $n_{\nu}=1$, in Fig. 4. The right-
handed neutrino mass $M_{\text{R}}$ is fixed in each panel: $10^{13}$,
$10^{14}$, $10^{14.4}$ ($\simeq 2.5\times 10^{14}$), $10^{14.5}$ ($\simeq
3.2\times 10^{14}$), $10^{14.6}$ ($\simeq 4.0\times 10^{14}$), and $10^{14.7}$
($\simeq 5.0\times 10^{14}$) 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 $M_{\text{R}}$ will
be given. Note that the thick line is obtained by tuning one parameter $m_{t}$
for fixed $M_{\text{R}}$, and its minimum corresponds to the two parameter
tuning of $m_{t}$ and $m_{\text{DM}}$.

With the right-handed neutrino, we have one more theoretical parameter
$M_{\text{R}}$ in addition to $m_{t}$ to determine from the observational
constraints of $r$ and $m_{\text{DM}}$. From Fig. 4, we see that the larger
the $M_{\text{R}}$ 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.131313 In this case, the vacuum stability is violated before the
perturbativity.  If we e.g. set $m_{t}>169\,\text{GeV}$, we have the
constraint $M_{\text{R}}\lesssim 10^{14.8}\,\text{GeV}$ ($\simeq 6.3\times
10^{14}\,\text{GeV}$); see also Fig. 8 in Sec. 6.

As we increase $M_{\text{R}}$ and switch panels in Fig. 4, we see that the
value of $m_{\text{DM}}$ 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 $M_{\text{R}}=10^{14.7}\,\text{GeV}$.
Therefore, if the right-handed neutrino mass is larger than that, we have a
stringent lower bound: $r\gtrsim 10^{-2}$. On the other hand, the plot with
$M_{\text{R}}\lesssim 10^{13}\,\text{GeV}$ 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 $r$:141414 Here,
we fit $M_{\text{R}}$ and $m_{t}$ from the future observation of
$m_{\text{DM}}$ and $r$. Instead, one might narrow down the error on the pole
mass $m_{t}$ e.g. at the High Luminosity LHC (HL-LHC) [85]. Then one may use
$m_{t}$ and $m_{\text{DM}}$ as input parameters to predict $M_{\text{R}}$ and
$r$.

  * •

Suppose that $m_{\text{DM}}=1\,\text{TeV}$ ($\kappa\simeq 0.31$) and $r=0.01$
are found. Then the right-handed neutrino mass is predicted to be in the
narrow range $10^{14}\,\text{GeV}\lesssim M_{\text{R}}\lesssim
10^{14.6}\,\text{GeV}$ and the top-quark mass is constrained from above:
$m_{t}<174\,\text{GeV}$.

  * •

If we discover $m_{\text{DM}}=1.5\,\text{TeV}$ ($\kappa\simeq 0.47$) and
$r=0.01$, we obtain the theoretical lower bound $M_{\text{R}}\gtrsim
10^{14.6}\,\text{GeV}$, while the top quark mass is less constrained:
$171\,\text{GeV}<m_{t}<178\,\text{GeV}$. However, $M_{\text{R}}$ 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 $m_{\text{{DM}}}$ and $r$. Generically the
heavy DM mass tends to predict the heavy top-quark mass and $M_{\text{R}}$.
The smaller the $r$ is, the tighter the range of $m_{t}$. Especially, if we
discover $m_{\text{DM}}=1.5\,\text{TeV}$ and $r=0.001$, $m_{t}$ and
$M_{\text{R}}$ are accurately predicted.

We can predict $r_{\text{bound}}$ or $m_{\text{DM}}$ to some extent by
considering typical input parameters. When we choose $m_{t}=173\,\text{GeV}$
and $M_{\text{R}}=10^{14}\,\text{GeV}$, we obtain the bound $m_{\text{DM}}\sim
860\,\text{GeV}$–$970\,\text{GeV}$ for $r<0.09$.

###  5.3 Results for Inverted Hierarchy

We show the results for the case of Inverted Hierarchy ($n_{\nu}=2$) in Fig.
5. In this case, the right-handed neutrinos lighter than $\sim
10^{13}\,\text{GeV}$ do not affect the analysis, similarly as in the case of
Normal Hierarchy. However, the upper bound on $M_{\text{R}}$ is slightly
different: $M_{\text{R}}\lesssim 10^{14.7}\,\text{GeV}$, see also Fig. 7. Let
us summarize implications of future discoveries of the DM and $r$ again:

  * •

If we discover $m_{\text{DM}}=1\,\text{TeV}$ and $r=0.01$, we obtain
$10^{13.9}\,\text{GeV}\lesssim M_{\text{R}}\lesssim 10^{14.4}\,\text{GeV}$ and
$m_{t}<174\,\text{GeV}$.

  * •

If we discover $m_{\text{DM}}=1.5\,\text{TeV}$ and $r=0.01$, $M_{\text{R}}$
must be larger than $\sim 10^{14.6}\,\text{GeV}$ and
$170\,\text{GeV}<m_{t}<178\,\text{GeV}$. Although we cannot obtain the global
narrow bounds on $M_{\text{R}}$ and $m_{t}$, they are highly correlated as in
the case of Normal Hierarchy.

See Table 4 for other pairs of $m_{\text{{DM}}}$ and $r$.

Figure 5: Allowed region for Inverted Hierarchy with $\lambda_{S}=0$. See caption of Fig. 4 for the explanation.  | $m_{t}$ | $M_{\text{R}}$
---|---|---
$m_{\text{DM}}=1\,\text{TeV}$, $r=0.01$ | $m_{t}<174\,\text{GeV}$ | $10^{13.9}\,\text{GeV}\lesssim M_{\text{R}}<10^{14.5}\,\text{GeV}$
$m_{\text{DM}}=1\,\text{TeV}$, $r=0.001$ | $173\,\text{GeV}<m_{t}<174\,\text{GeV}$ | $10^{13.9}\,\text{GeV}<M_{\text{R}}<10^{14.2}\,\text{GeV}$
$m_{\text{DM}}=1.5\,\text{TeV}$, $r=0.01$ | $m_{t}<178\,\text{GeV}$ | $10^{14.4}\,\text{GeV}<M_{\text{R}}\lesssim 10^{14.7}\,\text{GeV}$
$m_{\text{DM}}=1.5\,\text{TeV}$, $r=0.001$ | $177\,\text{GeV}<m_{t}<178\,\text{GeV}$ | $10^{14.4}\,\text{GeV}<M_{\text{R}}<10^{14.5}\,\text{GeV}$
Table 4: Constraints obtained for Inverted Hierarchy.

###  5.4 Results for Degenerate case

We show the results for Degenerated case ($n_{\nu}=3$) in Fig. 6. The right-
handed neutrinos lighter than $\sim 10^{13}\,\text{GeV}$ do not affect the
analysis, similarly as other cases. The upper bound on $M_{\text{R}}$ is
smaller than in other cases: $M_{\text{R}}\lesssim 10^{14.2}\,\text{GeV}\simeq
1.6\times 10^{14}\,\text{GeV}$.

We summarize implications of future discoveries $m_{\text{DM}}$ and $r$ in
Table 5. The right-handed neutrino mass tend to be lighter than hierarchical
cases due to the heavy $m_{\nu}$. However, the prediction of $m_{t}$ is
similar to the other cases.

Figure 6: Allowed region for Degenerate case with $\lambda_{S}=0$. See caption
of Fig. 4 for the explanation.

| $m_{t}$ | $M_{\text{R}}$
---|---|---
$m_{\text{DM}}=1\,\text{TeV}$, $r=0.01$ | $m_{t}<174\,\text{GeV}$ | $10^{13.5}\,\text{GeV}\lesssim M_{\text{R}}\lesssim 10^{14}\,\text{GeV}$
$m_{\text{DM}}=1\,\text{TeV}$, $r=0.001$ | $173\,\text{GeV}\lesssim m_{t}<174\,\text{GeV}$ | $10^{13.5}\,\text{GeV}<M_{\text{R}}<10^{13.8}\,\text{GeV}$
$m_{\text{DM}}=1.5\,\text{TeV}$, $r=0.01$ | $m_{t}<178\,\text{GeV}$ | $10^{14}\,\text{GeV}\lesssim M_{\text{R}}\lesssim 10^{14.2}\,\text{GeV}$
$m_{\text{DM}}=1.5\,\text{TeV}$, $r=0.001$ | $176\,\text{GeV}<m_{t}<178\,\text{GeV}$ | $M_{\text{R}}\simeq 10^{14}\,\text{GeV}$
Table 5: Constraints obtained for degenerate case.

##  6 Allowed region for all parameter space

In the previous section, we have plotted the bounds for various $m_{t}$ in
each panel of fixed $M_{\text{R}}$. 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 $M_{\text{R}}$.

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 $M_{\text{R}}$. In Sec.
6.2, we see the same absolute lower bound in a different way, by changing the
order of fixing $m_{t}$ and $M_{\text{R}}$.

###  6.1 Lower bound on $r$ for each $M_{\text{R}}$

Figure 7: The lower bound on $r$ for each fixed $M_{\text{R}}$ (colored) and
the envelope (black) with $\lambda_{S}=0$ and
$171\,\text{GeV}<m_{t}<176\,\text{GeV}$. The orange line is for
$M_{\text{R}}=10^{13}$ GeV. The vertical colored line comes from the lower
end, $m_{t}>171\,\text{GeV}$. See the caption of Fig. 3 for the shaded region.

In Figs. 7 and 8, we plot our theoretical lower bounds on $r$ for various
$M_{\text{R}}$ when we allow the top-quark pole mass within roughly $2\sigma$
and $4\sigma$ 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 $M_{\text{R}}$. 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.151515 We have plotted the envelope, denoted by the black line, as
follows: 1) Each $M_{\text{R}}$-fixed line has a minimum. Make an
interpolating function which linearly join all these minimum points. 2) Each
$m_{t}$-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 $m_{\text{DM}}$. 4) In large
$m_{\text{DM}}$ region, we replace the interpolated bound with the lower bound
determined by the maximal $m_{t}$; see the caption of Fig. 9 to see how
$m_{t}$ gives the bound. Note that these interpolating functions are evaluated
only for $600\,\text{GeV}<m_{\text{DM}}<1600\,\text{GeV}$, and hence they are
untrustworthy in the extrapolated regions $m_{\text{DM}}<600\,\text{GeV}$ and
$m_{\text{DM}}>1600\,\text{GeV}$. This does not do any harm because these
regions are already excluded by the direct DM search and by the
perturbativity, respectively.

We explain the envelope denoted by the black line in Fig. 7:

  * •

We see that the allowed region is enlarged to

$\displaystyle r\gtrsim 10^{-5}$ (16)

from the $n_{\nu}=0$ case in Eq. (15), which is read from the black line in
Fig. 3 (being close the orange $M_{\text{R}}=10^{13}\,\text{GeV}$ line in Fig.
7). This is because the loop corrections of heavy right-handed neutrinos
reduce $V_{\varphi\leq\Lambda}^{\text{max}}$.

  * •

The lower bound on $r$ increases rapidly in the region $m_{\text{DM}}\gtrsim
1.3\,\text{TeV}$ due to the upper end of the parameter
$m_{t}<176\,\text{GeV}$.

  * •

In the region near the envelope denoted by the black line, the two input
parameters $m_{t}$ and $M_{\text{R}}$ are simultaneously tuned to minimize the
potential height $V_{\varphi\leq\Lambda}^{\text{max}}$.

  * •

If one allows to adjust the three parameters, $m_{t}$, $M_{\text{R}}$, and
$m_{\text{DM}}$ simultaneously, then the lowest point of the black line,
$r\sim 10^{-5}$, 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.

Figure 8: The same plot as in Fig. 7 for
$169\,\text{GeV}<m_{t}<178\,\text{GeV}$.

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
$m_{\text{DM}}\gtrsim 1.3\,\text{TeV}$ in the cases of $n_{\nu}=1$ and 2
because of the difference of potential shapes explained in Appendix B, while
its rise in the region $m_{\text{DM}}\gtrsim 1.5\,\text{TeV}$ is due to the
upper end of the parameter $m_{t}<178\,\text{GeV}$.

###  6.2 Lower bound on $r$ for each $m_{t}$

Figure 9: The lower bound on $r$ for each fixed $m_{t}$ (colored) with
$\lambda_{S}=0$. The black line is their envelope, which is identical to the
ones in Figs. 7 and 8 except for their right-most boundary where they follow
the $m_{t}=176\,\text{GeV}$ (blue) and 178 GeV (purple) lines in this figure,
respectively. See the left of Fig. 3 for the corresponding plot without right-
handed neutrinos and for the explanation of the shaded region.

In Fig. 9, we show the lower bound on $r$ for each fixed $m_{t}$ with
$M_{\text{R}}$ 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 $m_{\text{DM}}$ for each $m_{t}$ does not depend
on the number of neutrinos $n_{\nu}$. Therefore, for any given lower bound on
$m_{t}$, we obtain the corresponding lower bound of $m_{\text{DM}}$ 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 $m_{\text{DM}}$ regardless
of $n_{\nu}$ if $r<10^{-3}$.

##  7 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,
$m_{\text{DM}}$, under the simple assumption that the extrapolation of the
Higgs-field direction plays the role of inflaton at $\varphi>\Lambda$. 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 $m_{\text{DM}}\lesssim 1.1\,\text{TeV}$,
(ii) the tensor-to-scalar ratio $r\gtrsim 2\times 10^{-3}$, and (iii) the pole
mass of the top quark $171\,\text{GeV}<m_{t}<175\,\text{GeV}$ from the current
observational constraints $r<0.09$ and $m_{\text{DM}}\gtrsim 0.7\,\text{TeV}$.
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 $r$-$m_{\text{DM}}$ plane, $r\gtrsim 10^{-5}$ and $m_{\text{DM}}\lesssim
1.6\,\text{TeV}$, if we allow a three-parameter tuning. Altough the region
$r\lesssim 10^{-3}$ 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 $m_{t}$,
$m_{\text{DM}}$ and the right-handed neutrino mass $M_{\text{R}}$.

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.161616
The right-handed neutrinos are regarded massless at
$\mu\sim\varphi_{\text{cl}}\gtrsim M_{\text{R}}$.  In general, the non-minimal
couplings $\xi\varphi^{2}R$ and $\xi_{S}S^{2}R$ cause the mass terms of the
order of $\xi H^{2}\varphi^{2}$ and $\xi_{S}H^{2}S^{2}$, respectively. Under a
classical field value $\varphi_{\text{cl}}$, we get

$\displaystyle\xi H^{2}\varphi^{2}$
$\displaystyle\sim\xi\lambda{\varphi_{\text{cl}}^{4}\over
M_{\text{P}}^{2}}\varphi^{2},$ $\displaystyle\xi_{S}H^{2}S^{2}$
$\displaystyle\sim\xi_{S}\lambda{\varphi_{\text{cl}}^{4}\over
M_{\text{P}}^{2}}S^{2}.$ (17)

Such effective masses
$\sqrt{\xi\lambda}{\varphi_{\text{cl}}^{2}/M_{\text{P}}}$ and
$\sqrt{\xi_{S}\lambda}{\varphi_{\text{cl}}^{2}/M_{\text{P}}}$ are smaller
enough than $\mu\sim\varphi_{\text{cl}}$ since we consider the non-minimal
couplings $\xi,\xi_{S}\lesssim 10^{2}$ and the small quartic coupling
$\lambda\ll 10^{-1}$ in the region $\varphi_{\text{cl}}\lesssim
10^{17}\,\text{GeV}$.

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.171717 In this case, the dynamics of the Higgs
field becomes chaotic in the sense that it is sensitive to the initial
condition at the end of the main inflation.  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.

## Acknowledgement

We thank Tomohiro Abe, Xiangdong Ji, Kiyoharu Kawana, Jinsu Kim, Tae Geun Kim,
Seong Chan Park, Stanislav Rusak, and Minoru Tanaka for useful comments. The
work of Y.H., H.K., and K.O. are supported in part by JSPS KAKENHI Grant Nos.
16J06151 (YH), 16K05322 (HK), and 15K05053 (KO), respectively.

##  Appendix A Renormalization group equations

| Value | Reference
---|---|---
Planck mass $M_{\text{P}}$ |  $2.4353\times 10^{18}$ GeV | [76]
Higgs mass | 125.09 GeV | [76]
$Z$ bozon mass $M_{Z}$ | 91.1876 GeV | [76]
$\alpha_{3}(M_{Z})$ | 0.1184 | [8]
The expectation value of the Higgs field $v$ | 246 GeV | [76]
Table 6: Boundary condition for the RGEs.

We have calculated the lower bound on $r$ as follows:

  1. 1.

Solve the RGEs (21)–(28) (shown in the end of this section) for given
parameters.181818 In this section, we omit the argument of the couplings but
it should be understood that they are not the values at $\mu=m_{t}$; see
footnote 7.  The effects from right-handed neutrino is introduced only at high
energy scale $\varphi\geq M_{\text{R}}$: We set $n_{\nu}=0$ and
$M_{\text{R}}=0$ in $\varphi<M_{\text{R}}$. As the boundary condition to solve
the RGEs, we have used Eqs. (11)–(15) in [69] and the values in Table 6.

  2. 2.

Calculate the one-loop effective Higgs potential

$\displaystyle V_{\varphi\leq\Lambda}={\lambda_{\text{eff}}\over
4}\varphi^{4}$ (18)

where

$\displaystyle\lambda_{\text{eff}}$
$\displaystyle:=e^{4\Gamma}\Bigg{[}\lambda+{1\over
16\pi^{2}}\Bigg{\\{}-3y_{t}^{4}\bigg{(}\ln{y_{t}^{2}\over 2}-{3\over
2}+2\Gamma\bigg{)}+{3g_{2}^{2}\over 8}\bigg{(}\ln{g_{2}^{2}\over 4}-{5\over
6}+2\Gamma\bigg{)}$
$\displaystyle\quad\phantom{e^{4\Gamma}\Bigg{[}}+{3(g_{Y}^{2}+g_{2}^{2})^{2}\over
16}\bigg{(}\ln{g_{Y}^{2}+g_{2}^{2}\over 4}-{5\over
6}+2\Gamma\bigg{)}-n_{\nu}y_{\nu}^{4}\ln\\!\left(M_{\text{R}}+\sqrt{M_{\text{R}}^{2}+4y_{\nu}^{2}\varphi^{2}}\over
2\sqrt{M_{\text{R}}^{2}+\varphi^{2}}\right)\Bigg{\\}}\Bigg{]}$ (19)

is the effective Higgs-self coupling.191919 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.  We set $\mu=\varphi$ when we calculate $\lambda_{\text{eff}}$; see
[52, 86] for the details about the renormalization scale $\mu$ of
$\lambda_{\text{eff}}$. The one-loop wave-function renormalization

$\displaystyle\Gamma\\!\left(\varphi\right)$
$\displaystyle=\int^{\varphi}_{m_{t}}{1\over 16\pi^{2}}\left({9\over
4}g_{2}^{2}+{3\over
4}g_{Y}^{2}-3y_{t}^{2}-n_{\nu}y_{\nu}^{2}\right)\text{d}\ln\mu$ (20)

is neglected due to the calculation cost; its error is estimated as stated
below.

  3. 3.

Change the parameters using the false position method until the value of the
potential minimum becomes sufficiently close to zero.

  4. 4.

Calculate the maximum value of potential and obtain $r_{\text{bound}}$ via Eq.
(5).

  5. 5.

Obtain the value of $\Gamma\\!\left(\varphi\right)$ at $\varphi=\Lambda$ and
use it to estimate the deviation at the potential maximum. This is most
conservative because $\Gamma\\!\left(\varphi\right)$ becomes largest at
$\varphi=\Lambda$. This estimation corresponding to the dashed lines in the
figures.

We have obtained the RGEs for arbitrary $n_{\nu}$ combining the $n_{\nu}=1$
RGEs [87] and the $n_{\nu}=3$ ones [88]:

$\displaystyle\frac{\text{d}g_{Y}}{\text{d}\ln\mu}$ $\displaystyle={1\over
16\pi^{2}}\frac{41}{6}g_{Y}^{3}+\frac{g_{Y}^{3}}{(16\pi^{2})^{2}}\left({199\over
18}g_{Y}^{2}+{9\over 2}g_{2}^{2}+{44\over 3}g_{3}^{2}-{17\over
6}y_{t}^{2}-{n_{\nu}\over 2}y_{\nu}^{2}\right),$ (21)
$\displaystyle\frac{\text{d}g_{2}}{\text{d}\ln\mu}$ $\displaystyle=-{1\over
16\pi^{2}}\frac{19}{6}g_{2}^{3}+\frac{g_{2}^{3}}{(16\pi^{2})^{2}}\left({3\over
2}g_{Y}^{2}+{35\over 6}g_{2}^{2}+12g_{3}^{2}-{3\over 2}y_{t}^{2}-{n_{\nu}\over
2}y_{\nu}^{2}\right),$ (22)
$\displaystyle\frac{\text{d}g_{3}}{\text{d}\ln\mu}$
$\displaystyle=-\frac{7}{16\pi^{2}}g_{3}^{3}+\frac{g_{3}^{3}}{(16\pi^{2})^{2}}\left({11\over
6}g_{Y}^{2}+{9\over 2}g_{2}^{2}-26g_{3}^{2}-2y_{t}^{2}\right),$ (23)
$\displaystyle\frac{\text{d}y_{t}}{\text{d}\ln\mu}$
$\displaystyle=\frac{y_{t}}{16\pi^{2}}\bigg{(}\frac{9}{2}y_{t}^{2}+n_{\nu}y_{\nu}^{2}-\frac{17}{12}g_{Y}^{2}-\frac{9}{4}g_{2}^{2}-8g_{3}^{2}\bigg{)}+\frac{y_{t}}{(16\pi^{2})^{2}}\bigg{\\{}-12y_{t}^{2}-{9n_{\nu}\over
4}y_{\nu}^{4}-{9n_{\nu}\over 4}y_{t}^{2}y_{\nu}^{2}$
$\displaystyle\quad+6\lambda^{2}+{\frac{1}{4}\kappa^{2}}-12\lambda
y_{t}^{2}+g_{Y}^{2}\left(\frac{131}{16}y_{t}^{2}+{5n_{\nu}\over
8}y_{\nu}^{2}\right)+g_{2}^{2}\left(\frac{225}{16}y_{t}^{2}+{5n_{\nu}\over
8}y_{\nu}^{2}\right)+36g_{3}^{2}y_{t}^{2}$
$\displaystyle\quad+\frac{1187}{216}g_{Y}^{4}-\frac{23}{4}g_{2}^{4}-108g_{3}^{4}-\frac{3}{4}g_{Y}^{2}g_{2}^{2}+9g_{2}^{2}g_{3}^{2}+\frac{19}{9}g_{3}^{2}g_{Y}^{2}\bigg{\\}},$
(24) $\displaystyle\frac{\text{d}y_{\nu}}{\text{d}\ln\mu}$
$\displaystyle=\frac{y_{\nu}}{16\pi^{2}}\bigg{\\{}\left(n_{\nu}+{3\over
2}\right)y_{\nu}^{2}+3y_{t}^{2}-{3\over 4}g_{Y}^{2}-{9\over
4}g_{2}^{2}\bigg{\\}}+\frac{y_{\nu}}{(16\pi^{2})^{2}}\bigg{\\{}-\left({9n_{\nu}\over
2}-{3\over 2}\right)y_{\nu}^{4}-{27\over 4}y_{t}^{4}$
$\displaystyle\quad-{27\over 4}y_{t}^{2}y_{\nu}^{2}+6\lambda^{2}+{1\over
4}\kappa^{2}-12\lambda y_{\nu}^{2}+g_{Y}^{2}\left(\left({5n_{\nu}\over
8}+{93\over 16}\right)y_{\nu}^{2}+{85\over 24}y_{t}^{2}\right)$
$\displaystyle\quad+g_{2}^{2}\left(\left({15n_{\nu}\over 8}+{135\over
16}\right)y_{\nu}^{2}+{45\over
8}y_{t}^{2}\right)+20g_{3}^{2}y_{t}^{2}+{35\over 24}g_{Y}^{4}-{23\over
4}g_{2}^{4}-{9\over 4}g_{Y}^{2}g_{2}^{2}\bigg{\\}},$ (25)
$\displaystyle\frac{\text{d}\lambda}{\text{d}\ln\mu}$
$\displaystyle=\frac{1}{16\pi^{2}}\bigg{(}\frac{1}{2}\kappa^{2}+24\lambda^{2}-3g_{Y}^{2}\lambda-9g_{2}^{2}\lambda+4n_{\nu}\lambda
y_{\nu}^{2}+\frac{3}{8}g_{Y}^{4}+\frac{3}{4}g_{Y}^{2}g_{2}^{2}+\frac{9}{8}g_{2}^{4}+12\lambda
y_{t}^{2}-6y_{t}^{4}-2n_{\nu}y_{\nu}^{4}\bigg{)}$
$\displaystyle\quad+\frac{1}{(16\pi^{2})^{2}}\bigg{\\{}-2\kappa^{3}-5\kappa^{2}\lambda-312\lambda^{3}+36\lambda^{2}(g_{Y}^{2}+3g_{2}^{2})-\lambda\left({629\over
24}g_{Y}^{4}-{39\over 4}g_{Y}^{2}g_{2}^{2}+{73\over 8}g_{2}^{4}\right)$
$\displaystyle\phantom{\quad+\frac{1}{(16\pi^{2})^{2}}\bigg{\\{}}+\frac{305}{16}g_{2}^{6}-\frac{289}{48}g_{Y}^{2}g_{2}^{4}-\frac{559}{48}g_{Y}^{4}g_{2}^{2}-\frac{379}{48}g_{Y}^{6}-32g_{3}^{2}y_{t}^{4}-\frac{8}{3}g_{Y}^{2}y_{t}^{4}-\frac{9}{4}g_{2}^{4}y_{t}^{2}-{3n_{\nu}\over
4}g_{2}^{4}y_{\nu}^{2}$
$\displaystyle\phantom{\quad+\frac{1}{(16\pi^{2})^{2}}\bigg{\\{}}+\lambda
y_{t}^{2}\bigg{(}\frac{85}{6}g_{Y}^{2}+\frac{45}{2}g_{2}^{2}+80g_{3}^{2}\bigg{)}+\lambda
y_{\nu}^{2}\left({5n_{\nu}\over 2}g_{Y}^{2}+{15n_{\nu}\over
2}g_{2}^{2}\right)+g_{Y}^{2}y_{t}^{2}\bigg{(}-\frac{19}{4}g_{Y}^{2}+\frac{21}{2}g_{2}^{2}\bigg{)}$
$\displaystyle\phantom{\quad+\frac{1}{(16\pi^{2})^{2}}\bigg{\\{}}-g_{Y}^{2}y_{\nu}^{2}\left({n_{\nu}\over
4}g_{Y}^{2}+{n_{\nu}\over
2}g_{2}^{2}\right)-144\lambda^{2}y_{t}^{2}-48n_{\nu}y_{\nu}^{2}-3\lambda
y_{t}^{4}-n_{\nu}\lambda
y_{\nu}^{4}+30y_{t}^{6}+10n_{\nu}y_{\nu}^{6}\bigg{\\}},$ (26)
$\displaystyle\frac{\text{d}\kappa}{\text{d}\ln\mu}$
$\displaystyle=\frac{\kappa}{16\pi^{2}}\left(12\lambda+\lambda_{S}+4\kappa+6y_{t}^{2}+2n_{\nu}y_{\nu}^{2}-\frac{3}{2}g_{Y}^{2}-\frac{9}{2}g_{2}^{2}\right)+\frac{\kappa}{(16\pi^{2})^{2}}\bigg{\\{}-\frac{21}{2}\kappa^{2}$
$\displaystyle\quad-72\kappa\lambda-60\lambda^{2}-6\kappa\lambda_{S}-\frac{5}{6}\lambda_{S}^{2}-y_{t}^{2}(12\kappa+72\lambda)-4n_{\nu}\kappa
y_{\nu}^{2}-24n_{\nu}\lambda y_{\nu}^{2}$
$\displaystyle\quad-\frac{27}{2}y_{t}^{4}-{9n_{\nu}\over
2}y_{\nu}^{4}+g_{Y}^{2}(\kappa+24\lambda)+g_{2}^{2}(3\kappa+72\lambda)+y_{t}^{2}\left(\frac{85}{12}g_{Y}^{2}+\frac{45}{4}g_{2}^{2}+40g_{3}^{2}\right)$
$\displaystyle\quad+y_{\nu}^{2}\left({5n_{\nu}\over
4}g_{Y}^{2}+{15n_{\nu}\over 4}g_{2}^{2}\right)+{557\over
48}g_{Y}^{4}-{145\over 16}g_{2}^{4}+\frac{15}{8}g_{Y}^{2}g_{2}^{2}\bigg{\\}},$
(27) $\displaystyle\frac{\text{d}\lambda_{S}}{\text{d}\ln\mu}$
$\displaystyle=\frac{1}{16\pi^{2}}\left(3\lambda_{S}^{2}+12\kappa^{2}\right)+\frac{1}{(16\pi^{2})^{2}}\bigg{\\{}-\frac{17}{3}\lambda_{S}^{3}$
$\displaystyle\quad-20\kappa^{2}\lambda_{S}-48\kappa^{3}-72\kappa^{2}y_{t}^{2}-24n_{\nu}\kappa^{2}y_{\nu}^{2}+24\kappa^{2}g_{Y}^{2}+72\kappa^{2}g_{2}^{2}\bigg{\\}}.$
(28)

##  Appendix 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.

Figure 10: The shape of Higgs potential with the values of $m_{t}$ and
$m_{\text{DM}}$ ($\kappa$) that corresponds to a point near the minimum of the
envelope in the left of Fig. 3.

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
$\varphi\simeq 5\times 10^{16}\,\text{GeV}$ in the case of Fig. 10) becomes
identical to the height at $\varphi=\Lambda$. If the local potential minimum
(at $\varphi=9\times 10^{16}\,\text{GeV}$ in the case of Fig. 10) is moved
left, then the height at $\varphi=\Lambda$ 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 $\varphi=\Lambda$.

Figure 11: Left: Typical potential shape when its minimum is at
$\varphi>\Lambda$. Center: Typical potential shape when the minimum at larger
$\varphi$ gives lower height. Right: Typical potential shape when the minimum
at smaller $\varphi$ gives lower height.

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 $n_{\nu}=1$ panel, at
$\left(m_{\text{DM}},\log_{10}r\right)\sim\left(1050\,\text{GeV},-1.5\right)$
for $M_{\text{R}}=10^{14}\,\text{GeV}$, $\left(1100\,\text{GeV},-1.7\right)$
for $M_{\text{R}}=10^{14.1}\,\text{GeV}$, $\left(1150\,\text{GeV},-2\right)$
for $M_{\text{R}}=10^{14.2}\,\text{GeV}$, and
$\left(1220\,\text{GeV},-2.6\right)$ for $M_{\text{R}}=10^{14.3}\,\text{GeV}$.
This is because the Higgs potential has two local minima in general: The one
at higher (lower) $\varphi$ is due to the neutrino (top quark)
contribution.202020 Here we let the word “minimum” also stands for mere a
concavity, namely, even when it is not really a minimum.  There are the
following three kinds of potential shapes:

  1. (i)

On the left side of the cusp of each envelope, the potential minimum at higher
$\varphi$ is located at $\varphi>\Lambda$; see the left of Fig. 11.

  2. (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 $\varphi$ is smaller; see
the center of Fig. 11.

  3. (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 $\varphi$ 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.

Figure 12:  The same plot as in Fig. 9. The black $n_{\nu}=0$ 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 $m_{t}\gtrsim 175\,\text{GeV}$.

We can see this non-smooth point from another point of view in Fig. 12. The
left end of each rainbow-colored line for $m_{t}\lesssim 174\,\text{GeV}$ is
the $M_{\text{R}}\rightarrow 0$ limit, and is the same as the black line in
Fig. 3 of $n_{\nu}=0$. However, they do not touch the black $n_{\nu}=0$ line
for $m_{t}\gtrsim 175\,\text{GeV}$. This is because there arises the two local
minima of the potential as in the case (iii): The minimum at lower $\varphi$
can be tuned to be zero by the top-quark contribution, and the neutrino
contribution may produce the minimum at higher $\varphi$, 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
$m_{t}\gtrsim 175\,\text{GeV}$, 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 $m_{\text{DM}}$ region. At the point where the dotted magenta line
touches the envelope (black, lower), the potential height of the two
degenerate minima becomes zero.212121 At this point, the EW and these two
vacua are all degenerate to (nearly) zero, and the vacuum with highest
$\varphi$ is located at $\varphi=\Lambda$. This point may be interesting in
term of the multiple-point principle [15, 16, 17].  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.

## References

  * [1] M. Holthausen, K. S. Lim, and M. Lindner, _Planck scale Boundary Conditions and the Higgs Mass_ , JHEP 1202 (2012), 037, 1112.2415.
  * [2] F. Bezrukov, M. Y. Kalmykov, B. A. Kniehl, and M. Shaposhnikov, _Higgs Boson Mass and New Physics_ , JHEP 1210 (2012), 140, 1205.2893.
  * [3] G. Degrassi, S. Di Vita, J. Elias-Miro, J. R. Espinosa, G. F. Giudice, G. Isidori, and A. Strumia, _Higgs mass and vacuum stability in the Standard Model at NNLO_ , JHEP 08 (2012), 098, 1205.6497.
  * [4] S. Alekhin, A. Djouadi, and S. Moch, _The top quark and Higgs boson masses and the stability of the electroweak vacuum_ , Phys.Lett. B716 (2012), 214–219, 1207.0980.
  * [5] I. Masina, _The Higgs boson and Top quark masses as tests of Electroweak Vacuum Stability_ , Phys.Rev. D87 (2013), 053001, 1209.0393.
  * [6] Y. Hamada, H. Kawai, and K.-y. Oda, _Bare Higgs mass at Planck scale_ , Phys. Rev. D87 (2013), no. 5, 053009, 1210.2538, [Erratum: Phys. Rev.D89,no.5,059901(2014)].
  * [7] F. Jegerlehner, _The Standard model as a low-energy effective theory: what is triggering the Higgs mechanism?_ , (2013), 1304.7813.
  * [8] D. Buttazzo, G. Degrassi, P. P. Giardino, G. F. Giudice, F. Sala, A. Salvio, and A. Strumia, _Investigating the near-criticality of the Higgs boson_ , JHEP 12 (2013), 089, 1307.3536.
  * [9] V. Branchina and E. Messina, _Stability, Higgs Boson Mass and New Physics_ , Phys.Rev.Lett. 111 (2013), 241801, 1307.5193.
  * [10] A. Kobakhidze and A. Spencer-Smith, _The Higgs vacuum is unstable_ , (2014), 1404.4709.
  * [11] A. Spencer-Smith, _Higgs Vacuum Stability in a Mass-Dependent Renormalisation Scheme_ , (2014), 1405.1975.
  * [12] V. Branchina, E. Messina, and A. Platania, _Top mass determination, Higgs inflation, and vacuum stability_ , (2014), 1407.4112.
  * [13] V. Branchina, E. Messina, and M. Sher, _The lifetime of the electroweak vacuum and sensitivity to Planck scale physics_ , (2014), 1408.5302.
  * [14] A. V. Bednyakov, B. A. Kniehl, A. F. Pikelner, and O. L. Veretin, _Stability of the Electroweak Vacuum: Gauge Independence and Advanced Precision_ , Phys. Rev. Lett. 115 (2015), no. 20, 201802, 1507.08833.
  * [15] C. D. Froggatt and H. B. Nielsen, _Standard model criticality prediction: Top mass 173 +- 5-GeV and Higgs mass 135 +- 9-GeV_ , Phys. Lett. B368 (1996), 96–102, hep-ph/9511371.
  * [16] C. Froggatt, H. B. Nielsen, and Y. Takanishi, _Standard model Higgs boson mass from borderline metastability of the vacuum_ , Phys.Rev. D64 (2001), 113014, hep-ph/0104161.
  * [17] H. B. Nielsen, _PREdicted the Higgs Mass_ , (2012), 94–126, 1212.5716.
  * [18] N. Haba, H. Ishida, N. Okada, and Y. Yamaguchi, _Multiple-point principle with a scalar singlet extension of the Standard Model_ , PTEP 2017 (2017), no. 1, 013B03, 1608.00087.
  * [19] N. Haba and T. Yamada, _Multiple-point principle realized with strong dynamics_ , Phys. Rev. D95 (2017), no. 11, 115015, 1703.04235.
  * [20] K. A. Meissner and H. Nicolai, _Effective action, conformal anomaly and the issue of quadratic divergences_ , Phys.Lett. B660 (2008), 260–266, 0710.2840.
  * [21] R. Foot, A. Kobakhidze, K. L. McDonald, and R. R. Volkas, _A Solution to the hierarchy problem from an almost decoupled hidden sector within a classically scale invariant theory_ , Phys.Rev. D77 (2008), 035006, 0709.2750.
  * [22] S. Iso, N. Okada, and Y. Orikasa, _Classically conformal $B-L$ extended Standard Model_, Phys.Lett. B676 (2009), 81–87, 0902.4050.
  * [23] S. Iso, N. Okada, and Y. Orikasa, _The minimal $B-L$ model naturally realized at TeV scale_, Phys.Rev. D80 (2009), 115007, 0909.0128.
  * [24] T. Hur and P. Ko, _Scale invariant extension of the standard model with strongly interacting hidden sector_ , Phys.Rev.Lett. 106 (2011), 141802, 1103.2571.
  * [25] S. Iso and Y. Orikasa, _TeV Scale B-L model with a flat Higgs potential at the Planck scale - in view of the hierarchy problem -_ , PTEP 2013 (2013), 023B08, 1210.2848.
  * [26] M. Hashimoto, S. Iso, and Y. Orikasa, _Radiative symmetry breaking from flat potential in various U(1)’ models_ , Phys. Rev. D89 (2014), no. 5, 056010, 1401.5944.
  * [27] P. H. Chankowski, A. Lewandowski, K. A. Meissner, and H. Nicolai, _Softly broken conformal symmetry and the stability of the electroweak scale_ , (2014), 1404.0548.
  * [28] A. Kobakhidze and K. L. McDonald, _Comments on the Hierarchy Problem in Effective Theories_ , (2014), 1404.5823.
  * [29] A. Gorsky, A. Mironov, A. Morozov, and T. Tomaras, _Is the Standard Model saved asymptotically by conformal symmetry?_ , Zh.Eksp.Teor.Fiz. 147 (2015), 399–409, 1409.0492.
  * [30] J. Kubo, K. S. Lim, and M. Lindner, _Electroweak Symmetry Breaking via QCD_ , Phys.Rev.Lett. 113 (2014), 091604, 1403.4262.
  * [31] R. Foot, A. Kobakhidze, and A. Spencer-Smith, _Criticality in the scale invariant standard model (squared)_ , (2014), 1409.4915.
  * [32] K. Kawana, _Criticality and Inflation of the Gauged B-L Model_ , (2015), 1501.04482.
  * [33] A. Latosinski, A. Lewandowski, K. A. Meissner, and H. Nicolai, _Conformal Standard Model with an extended scalar sector_ , JHEP 10 (2015), 170, 1507.01755.
  * [34] N. Haba, H. Ishida, R. Takahashi, and Y. Yamaguchi, _Gauge coupling unification in a classically scale invariant model_ , JHEP 02 (2016), 058, 1511.02107.
  * [35] N. Haba, H. Ishida, N. Kitazawa, and Y. Yamaguchi, _A new dynamics of electroweak symmetry breaking with classically scale invariance_ , Phys. Lett. B755 (2016), 439–443, 1512.05061.
  * [36] J. Kubo and M. Yamada, _Scale genesis and gravitational wave in a classically scale invariant extension of the standard model_ , JCAP 1612 (2016), no. 12, 001, 1610.02241.
  * [37] N. Haba and T. Yamada, _Strong dynamics in a classically scale invariant extension of the standard model with a flat potential_ , Phys. Rev. D95 (2017), no. 11, 115016, 1701.02146.
  * [38] S. Iso, P. D. Serpico, and K. Shimada, _QCD-Electroweak First-Order Phase Transition in a Supercooled Universe_ , Phys. Rev. Lett. 119 (2017), no. 14, 141301, 1704.04955.
  * [39] A. Lewandowski, K. A. Meissner, and H. Nicolai, _Conformal Standard Model, Leptogenesis and Dark Matter_ , (2017), 1710.06149.
  * [40] M. Shaposhnikov and C. Wetterich, _Asymptotic safety of gravity and the Higgs boson mass_ , Phys.Lett. B683 (2010), 196–200, 0912.0208.
  * [41] C. Wetterich and M. Yamada, _Gauge hierarchy problem in asymptotically safe gravity–the resurgence mechanism_ , Phys. Lett. B770 (2017), 268–271, 1612.03069.
  * [42] A. Eichhorn, Y. Hamada, J. Lumma, and M. Yamada, _Quantum gravity fluctuations flatten the Planck-scale Higgs-potential_ , (2017), 1712.00319.
  * [43] Y. Kawamura, _Naturalness, Conformal Symmetry and Duality_ , PTEP 2013 (2013), no. 11, 113B04, 1308.5069.
  * [44] Y. Kawamura, _Gauge hierarchy problem, supersymmetry and fermionic symmetry_ , (2013), 1311.2365.
  * [45] H. Kawai and T. Okada, _Solving the Naturalness Problem by Baby Universes in the Lorentzian Multiverse_ , Prog.Theor.Phys. 127 (2012), 689–721, 1110.2303.
  * [46] H. Kawai, _Low energy effective action of quantum gravity and the naturalness problem_ , Int.J.Mod.Phys. A28 (2013), 1340001.
  * [47] Y. Hamada, H. Kawai, and K. Kawana, _Evidence of the Big Fix_ , Int.J.Mod.Phys. A29 (2014), no. 17, 1450099, 1405.1310.
  * [48] K. Kawana, _Reconsideration of the Coleman’s Baby Universe_ , (2014), 1405.2743.
  * [49] Y. Hamada, H. Kawai, and K. Kawana, _Weak Scale From the Maximum Entropy Principle_ , PTEP 2015 (2014), no. 3, 033B06, 1409.6508.
  * [50] Y. Hamada, H. Kawai, K.-y. Oda, and S. C. Park, _Higgs inflation still alive_ , Phys.Rev.Lett. 112 (2014), 241301, 1403.5043.
  * [51] Bezrukov, Fedor and Shaposhnikov, Mikhail, _Higgs inflation at the critical point_ , (2014), 1403.6078.
  * [52] Y. Hamada, H. Kawai, K.-y. Oda, and S. C. Park, _Higgs inflation from Standard Model criticality_ , Phys. Rev. D91 (2015), 053008, 1408.4864.
  * [53] J. L. Cook, L. M. Krauss, A. J. Long, and S. Sabharwal, _Is Higgs inflation ruled out?_ , Phys. Rev. D89 (2014), no. 10, 103525, 1403.4971.
  * [54] D. S. Salopek, J. R. Bond, and J. M. Bardeen, _Designing Density Fluctuation Spectra in Inflation_ , Phys. Rev. D40 (1989), 1753.
  * [55] F. Bezrukov and M. Shaposhnikov, _The Standard Model Higgs boson as the inflaton_ , Phys.Lett. B659 (2008), 703–706, 0710.3755.
  * [56] Y. Hamada, H. Kawai, and K.-y. Oda, _Minimal Higgs inflation_ , PTEP 2014 (2014), 023B02, 1308.6651.
  * [57] Y. Hamada, H. Kawai, and K.-y. Oda, _Eternal Higgs inflation and the cosmological constant problem_ , Phys. Rev. D92 (2015), 045009, 1501.04455.
  * [58] V. Silveira and A. Zee, _SCALAR PHANTOMS_ , Phys.Lett. B161 (1985), 136.
  * [59] J. McDonald, _Gauge singlet scalars as cold dark matter_ , Phys.Rev. D50 (1994), 3637–3649, hep-ph/0702143.
  * [60] C. Burgess, M. Pospelov, and T. ter Veldhuis, _The Minimal model of nonbaryonic dark matter: A Singlet scalar_ , Nucl.Phys. B619 (2001), 709–728, hep-ph/0011335.
  * [61] H. Davoudiasl, R. Kitano, T. Li, and H. Murayama, _The New minimal standard model_ , Phys.Lett. B609 (2005), 117–123, hep-ph/0405097.
  * [62] B. Patt and F. Wilczek, _Higgs-field portal into hidden sectors_ , (2006), hep-ph/0605188.
  * [63] B. Grzadkowski and J. Wudka, _Pragmatic approach to the little hierarchy problem: the case for Dark Matter and neutrino physics_ , Phys.Rev.Lett. 103 (2009), 091802, 0902.0628.
  * [64] A. Drozd, B. Grzadkowski, and J. Wudka, _Multi-Scalar-Singlet Extension of the Standard Model - the Case for Dark Matter and an Invisible Higgs Boson_ , JHEP 1204 (2012), 006, 1112.2582.
  * [65] N. Haba, K. Kaneta, and R. Takahashi, _Planck scale boundary conditions in the standard model with singlet scalar dark matter_ , JHEP 1404 (2014), 029, 1312.2089.
  * [66] S. Baek, P. Ko, and W.-I. Park, _Invisible Higgs Decay Width vs. Dark Matter Direct Detection Cross Section in Higgs Portal Dark Matter Models_ , Phys. Rev. D90 (2014), no. 5, 055014, 1405.3530.
  * [67] J. Kim, P. Ko, and W.-I. Park, _Higgs-portal assisted Higgs inflation with a sizeable tensor-to-scalar ratio_ , JCAP 1702 (2017), no. 02, 003, 1405.1635.
  * [68] J. M. Cline, K. Kainulainen, P. Scott, and C. Weniger, _Update on scalar singlet dark matter_ , Phys. Rev. D88 (2013), 055025, 1306.4710, [Erratum: Phys. Rev.D92,no.3,039906(2015)].
  * [69] Y. Hamada, H. Kawai, and K.-y. Oda, _Predictions on mass of Higgs portal scalar dark matter from Higgs inflation and flat potential_ , JHEP 07 (2014), 026, 1404.6141.
  * [70] PandaX-II, X. Cui et al., _Dark Matter Results From 54-Ton-Day Exposure of PandaX-II Experiment_ , Phys. Rev. Lett. 119 (2017), no. 18, 181302, 1708.06917.
  * [71] P. Minkowski, _$\mu\to e\gamma$ at a Rate of One Out of $10^{9}$ Muon Decays?_, Phys. Lett. 67B (1977), 421–428.
  * [72] T. Yanagida, in Proceedings of the Workshop on Unified Theory and Baryon Number of the Universe, edited by O. Sawada and A. Sugamoto (KEK, Tokyo, 1979), p. 95; M. Gell-Mann, P. Ramond, and R. Slanski, in Supergravity, edited by P. van Nieuwenhuizen and D. Freedman (North-Holland, Amsterdam, 1979) arXiv:1306.4669; S. L. Glashow, in Proceedings of the Cargése Summer Institute on Quarks and Leptons, Cargése, July 9–29, 1979, eds. M. Lévy et al. (Plenum, 1980, New York), p. 707; R. N. Mohapatra and G. Senjanovic, Neutrino Mass and Spontaneous Parity Violation, Phys. Rev. Lett. 44 (1980) 912.
  * [73] Planck, P. A. R. Ade et al., _Planck 2015 results. XX. Constraints on inflation_ , Astron. Astrophys. 594 (2016), A20, 1502.02114.
  * [74] LUX, D. S. Akerib et al., _Results from a search for dark matter in the complete LUX exposure_ , Phys. Rev. Lett. 118 (2017), no. 2, 021303, 1608.07648.
  * [75] XENON, E. Aprile et al., _First Dark Matter Search Results from the XENON1T Experiment_ , (2017), 1705.06655.
  * [76] Particle Data Group, C. Patrignani et al., _Review of Particle Physics_ , Chin. Phys. C40 (2016), no. 10, 100001.
  * [77] G. Cortiana, _Top-quark mass measurements: review and perspectives_ , Rev. Phys. 1 (2016), 60–76, 1510.04483.
  * [78] Planck, P. A. R. Ade et al., _Planck 2015 results. XIII. Cosmological parameters_ , Astron. Astrophys. 594 (2016), A13, 1502.01589.
  * [79] POLARBEAR, Y. Inoue et al., _POLARBEAR-2: an instrument for CMB polarization measurements_ , Proc. SPIE Int. Soc. Opt. Eng. 9914 (2016), 99141I, 1608.03025.
  * [80] T. Matsumura et al., _Mission design of LiteBIRD_ , (2013), 1311.2847, [J. Low. Temp. Phys.176,733(2014)].
  * [81] CORE, J. Delabrouille et al., _Exploring Cosmic Origins with CORE: Survey requirements and mission design_ , (2017), 1706.04516.
  * [82] F. Capozzi, E. Di Valentino, E. Lisi, A. Marrone, A. Melchiorri, and A. Palazzo, _Global constraints on absolute neutrino masses and their ordering_ , (2017), 1703.04471.
  * [83] E. Giusarma, M. Gerbino, O. Mena, S. Vagnozzi, S. Ho, and K. Freese, _Improvement of cosmological neutrino mass bounds_ , Phys. Rev. D94 (2016), no. 8, 083522, 1605.04320.
  * [84] S. Vagnozzi, E. Giusarma, O. Mena, K. Freese, M. Gerbino, S. Ho, and M. Lattanzi, _Unveiling $\nu$ secrets with cosmological data: neutrino masses and mass hierarchy_, Phys. Rev. D96 (2017), no. 12, 123503, 1701.08172.
  * [85] CMS Collaboration, _Updates on Projections of Physics Reach with the Upgraded CMS Detector for High Luminosity LHC_ , (2016), https://cds.cern.ch/record/2221747.
  * [86] Y. Hamada, H. Kawai, Y. Nakanishi, and K.-y. Oda, _Meaning of the field dependence of the renormalization scale in Higgs inflation_ , Phys. Rev. D95 (2017), no. 10, 103524, 1610.05885.
  * [87] N. Haba, H. Ishida, and R. Takahashi, _Higgs inflation and Higgs portal dark matter with right-handed neutrinos_ , PTEP 2015 (2015), no. 5, 053B01, 1405.5738.
  * [88] K. Kawana, _Multiple Point Principle of the Standard Model with Scalar Singlet Dark Matter and Right Handed Neutrinos_ , PTEP 2015 (2015), 023B04, 1411.2097.