# The hierarchy problem and the vacuum stability in two-scalar dark matter
model

Zohre Habibolahi∗ ∗Physics Department, Faculty of Sciences, Arak University,
Arak 38156-8-8349, Iran Karim Ghorbani∗ and Parsa Ghorbani† ∗Physics
Department, Faculty of Sciences, Arak University, Arak 38156-8-8349, Iran

###### Abstract

We consider an extension to the Standard Model (SM) with two extra real
singlet scalars which both interact with the Higgs particle in the SM and one
of the scalars is taken as the dark matter (DM) candidate. We show that the
model successfully explains the relic abundance of the DM in the universe and
evades the strong bounds from direct detection experiments while respecting
the perturbativity and the vacuum stability conditions. In addition, we study
the hierarchy problem within the Veltman approach by solving the
renormalization group equations at one-loop. It is shown that the addition of
the real singlet scalars contributes to the Veltman parameters which in turn
results in satisfying the Veltman conditions much lower than the Planck scale
$\Lambda_{\text{Pl}}$. Therefore, the presence of the extra scalars not only
reduce the fine-tuning problem of the Higgs mass to a given scale
$\Lambda\ll\Lambda_{\text{Pl}}$, but also the singlet scalars do not receive
large corrections up to the scale $\Lambda$. For the case of two-scalar DM
model we find $\Lambda=10^{8}$ GeV.

###### Contents

  1. 1 Introduction
  2. 2 Two Real Scalars
  3. 3 Invisible Higgs decay, relic abundance and direct detection bounds
  4. 4 Veltman parameters
  5. 5 Renormalization group equations and Veltman conditions
  6. 6 Conclusions
  7. 7 Appendix: Positivity Condition
    1. 7.1 Biquadratic potential
    2. 7.2 Positivity in bi-quadratic potential with N scalars

##  1 Introduction

The unnatural Higgs mass of $\sim 125$ GeV [1, 2], yet consistent with the
electroweak precision test [3], has been one of the driving forces behind
developing models beyond the Standard Model (SM) of particle physics. In fact,
the SM Higgs mass determines the scale of the electroweak symmetry breaking,
and on the other hand, the amount of the Higgs mass strongly depends on the
quadratic divergences from the loop corrections. In order to get a negligible
one loop correction within the SM (known as Vetman condition), Veltman found
that the Higgs mass should be $\sim 314$ GeV, which is of course not
consistent with the observed value. Let us assume that the SM is an effective
field theory valid up to a scale $\Lambda$, with $\Lambda$ being the scale of
new physics beyond the Standard Model. Then the Higgs mass radiative
corrections of ${\cal O}(\Lambda^{2})$ for the leading contributions, receives
a huge fine-tuning mechanism to keep the renormalized mass as small as the
measured Higgs mass or the electroweak scale [4]. Early attempts to get around
the hierarchy problem was to expect new physics at the electroweak scale [5,
6, 7, 8, 9, 10]. However, no new physics signal is detected so far at the
electroweak scale, presumably demanding a reinterpretation of our notion for
naturalness. Before the discovery of the SM Higgs, the possibility that the
Veltman condition can be satisfied at high scale while violated at low energy
is studied in [11, 12, 13]. There are also investigations indicating that the
Veltman condition can be fulfilled close to the Planck scale [14, 15, 16, 17].

A prominent and ubiquitous mechanism to protect the Higgs mass correction from
large amounts is through underlying symmetries, e.g. supersymmetric scenarios.
This type of mechanism works if its dynamical scale is not large. Another well
known solution to the fine-tuning problem is the scale invariant models [18,
19, 20]. The hierarchy problem is addressed in [21] in which the SM and its
minimal extensions are merged with a high-scale supersymmetric theory at a
scale where the Veltman condition is satisfied. Fine-tuning problem is
discussed in Two-Higgs doublet model and left-right symmetric model in [22,
23, 24, 25]. The Veltman condition is adapted to extended Higgs sectors in
[26, 27] to lower the fine-tuning of the SM Higgs mass. In the same context
there are studies with extended Higgs sector and various scenarios [28, 29,
30, 31, 32, 33, 34, 35, 36, 37].

There are strong evidences that the SM cannot be an ultimate theory of
particle physics and new physics might be at work at scales smaller than the
Planck mass. The size of the Higgs mass corrections may be controlled by
adding new degrees of freedom in order to cause the cancellation of the
radiative correction at the quadratic level at the scale
$\Lambda\ll\Lambda_{\text{planck}}$. An intriguing question is whether another
weak scale paradigm, the WIMP (Weakly Interacting Massive Particle) scenario,
would be interconnected to the naturalness problem intrinsically. In the
present work we exploit a dark matter model with two extra real singlet
scalars both coupled to the SM Higgs. We consider a set-up in which one of the
singlet scalars is the dark matter (DM) candidate and the other singlet scalar
is unstable. By running the couplings at high scale we study the hierarchy
problem and the effect of extra scalars on the Veltman conditions while
respecting the stability of the extended Higgs potential. There can be found
various scenarios with the inclusion of two real singlet scalars emphasising
on the dark matter phenomenology [38, 39, 40, 41, 42, 43, 44, 45, 46]. As
experimental bounds, we consider strong constraints from observed relic
density granted by Planck and WMAP observations, bounds from direct detection
experiments XENON1t and LUX and the invisible Higgs decay width. On the other
hand, the theoretical constraints entail bounds from the perturbativity and
the bounded from below condition (BFB condition) which we also call it the
“positivity condition” in this article.

The setup of the paper is the following. In the next section we describe the
two scalar model and impose bounds one the couplings from bounded from below
(BFB) condition. Section 3 is devoted to the phenomenology of the dark matter
including bounds from the observed relic density, direct detection experiments
and the invisible Higgs decay. The relevant Veltman conditions for the SM
Higgs and the two singlet scalars are obtained in section 4. The
renormalization group equations and the numerical results for running of the
couplings are given in section 5. We conclude our results in section 6. In the
Appendix we provide bounded from below conditions for a generic potential with
two singlet scalars and N scalars added to the SM Higgs potential.

##  2 Two Real Scalars

The two real singlet scalar extension of the SM has the potential,

$\begin{split}{\cal
V_{\text{DM}}}=&\frac{1}{2}m^{2}_{1}\phi_{1}^{2}+\frac{1}{2}m^{2}_{2}\phi_{2}^{2}+(\lambda_{H1}\phi_{1}^{2}+\lambda_{H2}\phi_{2}^{2})H^{\dagger}H+\lambda_{H}(H^{\dagger}H)^{2}\\\
&+\frac{1}{4}\lambda_{1}\phi_{1}^{4}+\frac{1}{4}\lambda_{2}\phi_{2}^{4}+\frac{1}{2}\lambda_{12}\phi_{1}^{2}\phi_{2}^{2}\,,\end{split}$
(1)

keeping only terms with dimensionless couplings. This potential is symmetric
under $\mathbb{Z}_{2}\times\mathbb{Z}_{2}$. In general we can include
dimension-4 terms like $\phi_{1}\phi_{2}^{3}$ and $\phi_{1}^{3}\phi_{2}$.
However, without considering these interactions and thus rendering our
computations less complicated, we are able to reach our goal defined in this
work. The new singlet scalars interact with the SM Higgs beside having self-
interactions. For the SM Higgs doublet we expand it around the minimum in the
unitary gauge, $H^{\dagger}=(0~{}~{}v+h^{\prime})/\sqrt{2}$, and for the
singlet scalars we assume that only the scalar $\phi_{1}$ gets nonzero vacuum
expectation value (VEV) which breaks the $Z_{2}$ symmetry spontaneously.
Therefore, $\phi_{1}=s^{\prime}_{1}+w$ and $\phi_{2}=s_{2}$ with
$\braket{\phi_{1}}=w$ and $\langle\phi_{2}\rangle=0$. The dark matter
candidate is then the scalar $s_{2}$. Moreover, the coupling $\lambda_{H2}$ is
taken to be negligible at tree level, so we set $\lambda_{H2}=0$. The running
of the coupling $\lambda_{H2}$ will be studied by solving the renormalization
group equations (RGE) in the next sections. In order to find the physical mass
eigenstates, $h$ and $s$, the following rotation is necessary in the space of
the singlet scalars, $h^{\prime}$ and $s^{\prime}_{1}$,

$h=h^{\prime}\cos\theta-s^{\prime}_{1}\sin\theta\,,~{}~{}s_{1}=s^{\prime}_{1}\cos\theta+h^{\prime}\sin\theta\,,$
(2)

where $\theta$ is the mixing angle. The physical masses corresponding to the
scalar fields, $h,s_{1}$ and $s_{2}$ are respectively $m_{h},m_{s_{1}}$ and
$m_{s_{2}}$. In the following we adapt $m_{h}=125$ GeV and
$m_{s_{2}}=m_{\text{DM}}$ being the mass of the Higgs and the DM mass
respectively. The quartic couplings $\lambda_{H},\lambda_{1}$ and
$\lambda_{H1}$ are obtained in terms of the mixing angle $\theta$ and the
physical masses $m_{h}$ and $m_{s_{1}}$,

$\begin{split}\lambda_{H}=&\frac{1}{2v^{2}}(m_{h}^{2}\cos^{2}\theta+m_{s_{1}}^{2}\sin^{2}\theta)\\\
\lambda_{1}=&\frac{1}{2w^{2}}(m_{h}^{2}\sin^{2}\theta+m_{s_{1}}^{2}\cos^{2}\theta)\\\
\lambda_{H1}=&\frac{\sin 2\theta}{4vw}(m_{s_{1}}^{2}-m_{h}^{2})\,.\end{split}$
(3)

The potential of the model must satisfy BFB condition (or the positivity
condition) as a requirement for the vacuum stability. For the potential in Eq.
1 the BFB condition becomes
$\lambda_{1}>0\wedge\lambda_{2}>0\wedge\lambda_{\text{H}}>0$, and

if
$-\sqrt{\lambda_{1}\lambda_{2}}<\lambda_{12}<\sqrt{\lambda_{1}\lambda_{2}}$,
then

$\begin{split}&\Big{(}-\sqrt{\lambda_{\text{H}}\lambda_{1}}<\lambda_{\text{H1}}<-\lambda_{12}\sqrt{\lambda_{\text{H}}/\lambda_{2}}\\\
&~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\wedge\lambda_{\text{H2}}>\frac{\lambda_{12}\lambda_{\text{H1}}}{\lambda_{1}}-\frac{\sqrt{\lambda_{1}^{2}\lambda_{2}\lambda_{\text{H}}+\lambda_{12}^{2}\lambda_{\text{H1}}^{2}-\lambda_{1}\lambda_{12}^{2}\lambda_{\text{H}}-\lambda_{1}\lambda_{2}\lambda_{\text{H1}}^{2}}}{\lambda_{1}}\Big{)}\\\
&\vee\left(\lambda_{\text{H1}}>-\lambda_{12}\sqrt{\lambda_{\text{H}}/\lambda_{2}}\wedge\lambda_{\text{H2}}>-\sqrt{\lambda_{2}\lambda_{\text{H}}}\right)\,,\end{split}$
(4)

and if $\lambda_{12}>\sqrt{\lambda_{1}\lambda_{2}}$, then

$\lambda_{\text{H1}}>-\sqrt{\lambda_{1}\lambda_{\text{H}}}\wedge\lambda_{\text{H2}}>-\sqrt{\lambda_{2}\lambda_{\text{H}}}\,.$
(5)

Setting $\lambda_{\text{H2}}=0$ in Eq. 1 the BFB condition becomes
$\lambda_{1}>0\wedge\lambda_{2}>0\wedge\lambda_{\text{H}}>0$, and

if $-\sqrt{\lambda_{1}\lambda_{2}}<\lambda_{12}<0$, then

$\lambda_{\text{H1}}>-\sqrt{\frac{(\lambda_{1}\lambda_{2}-\lambda^{2}_{12})\lambda_{\text{H}}}{\lambda_{2}}}\,,$
(6)

and if $\lambda_{12}>0$, then

$\lambda_{\text{H1}}>-\sqrt{\lambda_{1}\lambda_{2}}\,.$ (7)

When scanning the parameter space of the model the positivity conditions are
imposed. The general conditions of the positivity for a potential with two
scalars and N scalars are presented in the Appendix.

##  3 Invisible Higgs decay, relic abundance and direct detection bounds

We are interested in finding the viable regions in the coupling space for
further use in the next sections. To this end, beside theoretical constraints,
we impose additionally the constraints from WMAP [47] and Planck [48]
observations on the DM relic density with $\Omega h^{2}\sim 0.12$ and the
constraints from the latest direct detection (DD) experimental bounds by XENON
[49] and LUX [50].

Figure 1: The relevant Feynman diagrams for dark matter annihilation at the
freeze-out epoch.

We exploit the package MicrOMEGAs [51] to compute the relic density and the
DM-nucleus elastic scattering cross sections. The relevant Feynman diagrams
for computation of the DM annihilation cross section is depicted in Fig. 1.
The DM particles annihilate into the SM particles ($h$,$W^{\pm}$,$Z$,$f$)
through $s$-channel, and annihilate through $t$\- and $u$-channel into two SM
Higgs ($hh$) and into $hs_{1}$. Since it is assumed that the DM candidates are
thermal relic, its number density, $n_{2}$, is governed by the Boltzmann
equation,

$\frac{dn_{2}}{dt}+3Hn_{2}=-\langle\sigma_{\text{ann}}v_{\text{rel}}\rangle[n^{2}_{2}-(n^{\text{EQ}}_{2})^{2}]\,,$
(8)

where $H$ in the second term is the Hubble parameter. The thermal average of
the annihilation cross section times the velocity at temperature $T$ is
obtained by integration over the center of mass energy,

$\langle\sigma_{\text{ann}}v_{\text{rel}}\rangle=\frac{1}{8m_{s_{2}}^{4}TK^{2}_{2}(\frac{m_{s_{2}}}{T})}\int^{\infty}_{4m^{2}_{s_{2}}}ds~{}(s-4m^{2}_{s_{2}})\sqrt{s}~{}K_{1}(\frac{\sqrt{s}}{T})~{}\sigma_{\text{ann}}(s)\,,$
(9)

in which $K_{1}$ and $K_{2}$ are modified Bessel functions.

Figure 2: The Feynman diagram for the DM direct detection scattering.

Moreover, DM candidates interact directly with normal matter through
$t$-channel by exchanging a SM Higgs or the singlet scalar $s_{1}$, as shown
in Fig. 2. The DM elastic scattering cross section is spin-independent (SI)
and is given by

$\sigma_{\text{SI}}=\frac{1}{4\pi}\frac{m_{N}^{4}f_{N}^{2}}{(m_{N}+m_{s_{1}})^{2}}\frac{w^{2}\lambda_{12}^{2}\sin
2\theta}{v^{2}}(\frac{1}{m_{h}^{2}}-\frac{1}{m_{s_{1}}^{2}})^{2}\,,$ (10)

where $m_{N}=0.938$ GeV is the nucleon mass and the scalar form-factor is
$f_{N}=0.3$. When DM mass is small such that $m_{s_{2}}<m_{h}/2$, then there
are limits from invisible Higgs decay. The ATLAS experiment provides upper
limits for the invisible branching ratio as
$\text{Br}^{\text{exp}}_{\text{inv}}<0.26$ at $95\%$ C.L.[52], and the CMS
experiment gives a stronger limit as
$\text{Br}^{\text{exp}}_{\text{inv}}<0.19$ at $95\%$ C.L [53]. In this work we
take the stronger limit from the CMS. The decay width of the SM Higgs is
$\Gamma_{\text{SM}}=4.07$ GeV [54]. When Higgs decay to the singlet scalar
$s_{1}$ is kinematically allowed then the total decay width modifies as
$\Gamma_{\text{SM}}\cos\theta+\Gamma_{\text{inv}}$. Given the result from the
CMS, the theoretical invisible decay width should satisfy the relation
$\Gamma_{\text{inv}}<0.95\cos^{2}\theta$ GeV. The invisible Higgs decay
through the process $H\to s_{1}s_{1}$ is given by

$\Gamma_{\text{inv}}=\frac{\lambda_{12}w^{2}\sin^{2}\theta}{8\pi
m_{h}}\sqrt{1-4m_{s_{1}}^{2}/m_{h}^{2}}\,.$ (11)

Figure 3: The viable region restricted by the WMAP/Planck observed relic
density and direct detection bounds from LUX/XENON1t for the scalar dark
matter mass in the range $10~{}\text{GeV}$-$2000~{}\text{GeV}$. The BFB
conditions are imposed on the couplings. Here we took $\lambda_{\text{H2}}=0$

As free parameters of the model, we choose, $m_{\text{DM}}$, $\delta$,
$\theta$, $\lambda_{12}$, $\lambda_{2}$, $\lambda_{\text{H2}}$ and $w$, where
$\delta=m_{s_{1}}-m_{\text{DM}}$. For the computation in this section we fix
$w=250$ GeV. First, we set $\lambda_{\text{H2}}=0$ and scan over the parameter
space within these ranges, $0.969<\cos\theta<1$, $0<\lambda_{2}<2$,
$-\sqrt{\lambda_{1}\lambda_{2}}<\lambda_{12}<2$
,$10~{}\text{GeV}<m_{\text{DM}}<2~{}$TeV and $0<\delta<100$ GeV. The couplings
$\lambda_{H}$, $\lambda_{1}$ and $\lambda_{\text{H1}}$ are given in terms of
the free parameters in Eq. 3. In Fig. 3 the viable scalar couplings are shown
after placing constraints from the BFB conditions, the observed relic density
and the DD upper bounds. Next, we relax the condition of taking
$\lambda_{\text{H2}}$ negligible and in our scan we choose
$-1<\lambda_{\text{H2}}<1$. Respecting the ranges discussed earlier for the
parameters, we present the limits on the couplings in Fig. 4 after imposing
the BFB condition, the observed relic density and the DD bounds.

The results found here are beyond the expectation of the model being simply a
duplication of the singlet scalar model. In the singlet scalar model, the DM
annihilation amplitude and the elastic scattering amplitude in the DM-nucleon
collision, share the same interaction coupling for the entire DM mass range.
This property makes it hard to get small direct detection cross section and at
the same time large annihilation cross section needed for the correct DM relic
density. The situation is changed significantly in the two real singlet model.
This point is discussed in [40, 42] for various setups. Beside having strong
constraints on the couplings from the BFB conditions of the potential there
exist a large parameter space respecting both the observed DM density and the
DD bounds.

Figure 4: The viable region restricted by the WMAP/Planck observed relic
density and direct detection bounds from LUX/XENON1t for the scalar dark
matter mass in the range $10~{}\text{GeV}$-$2000~{}\text{GeV}$. Positivity
conditions are imposed on the couplings. Here $\lambda_{H2}\neq 0$.

##  4 Veltman parameters

To obtain the mass correction of the scalars we rewrite the Lagrangian in
terms of the fields in the mass eigenstates. The Higgs mass correction at one
loop keeping only quadratic correction in the mass scale $\Lambda$,
incorporates the new scalar couplings $\lambda_{H1}$, $\lambda_{H2}$,
$\lambda_{1}$ and $\lambda_{12}$ as,

$\begin{split}\delta
m_{h}^{2}=&\frac{\Lambda^{2}}{16\pi}{\Large[}6\lambda_{H}\cos^{2}\theta+6\lambda_{1}\sin^{2}\theta+12\lambda_{H1}\sin^{2}\theta\cos^{2}\theta+(\frac{3}{4}g_{1}^{2}+\frac{9}{4}g_{2}^{2}-4\lambda_{t}^{2})\cos^{2}\theta\\\
&+\lambda_{H1}+\lambda_{H2}-6\lambda_{H1}\sin^{2}\theta\cos^{2}\theta+3\lambda_{1}\sin^{2}\theta\cos^{2}\theta+3\lambda_{H}\sin^{2}\theta\cos^{2}\theta+\lambda_{12}\sin^{2}\theta{\large]}\\\
\equiv&\frac{\Lambda^{2}}{16\pi}V_{H}\,.\end{split}$ (12)

The measurement of the Higgs boson signal strengths [54] constrains the mixing
angle to small values, $|\cos\theta|>0.969$. Therefore, we will approximate
the Veltman parameter $V_{H}$, at small mixing angles,

$V_{H}=6\lambda_{H}+\lambda_{H1}+\lambda_{H2}+\frac{3}{4}g_{1}^{2}+\frac{9}{4}g_{2}^{2}-4\lambda_{t}^{2}\,.$
(13)

In the same way the new singlet scalar masses get quadratic corrections as ,

$\begin{split}\delta
m^{2}_{s_{1}}=&\frac{\Lambda^{2}}{16\pi}{\large[}\lambda_{H1}+\lambda_{H2}+3\lambda_{1}+\lambda_{12}{\large]}\equiv\frac{\Lambda^{2}}{16\pi}V_{s_{1}}\\\
\delta
m^{2}_{s_{2}}=&\frac{\Lambda^{2}}{16\pi}{\large[}3\lambda_{2}+\lambda_{12}{\large]}\equiv\frac{\Lambda^{2}}{16\pi}V_{s_{2}}\,,\end{split}$
(14)

The corresponding Veltman conditions at some high energy scale, $\Lambda$, are
$V_{h}(\Lambda)\sim 0$, $V_{s_{1}}(\Lambda)\sim 0$ and $V_{s_{2}}(\Lambda)\sim
0$. In case that $V_{s_{1}}(\Lambda),V_{s_{2}}(\Lambda)>0$ at all scales, then
we set the criteria such that $V_{s_{1}}$ and $V_{s_{2}}$ should experience a
very slow variation with increasing the scale $\Lambda$.

##  5 Renormalization group equations and Veltman conditions

The running of the couplings are controlled by the $\beta$ functions in the
renormalization group equations. The $\beta$ function for the coupling
$\lambda$ is defined as

$\beta_{\lambda}=\frac{d\lambda}{d\ln(\mu/\mu_{0})}\,,$ (15)

where $\mu$ is the renormalization scale with the initial value $\mu_{0}$. In
this work the $\beta$ functions are computed at one loop order. The gauge
couplings, $g_{1},g_{2}$ and $g_{3}$, remain intact and are similar to those
in the SM,

$16\pi^{2}\beta_{g_{i}}=b_{i}g_{i}^{3}\,,$ (16)

where $b_{i}=\frac{41}{6},-\frac{19}{6},-7$. There are exact formulas for the
running of the gauge couplings,

$g_{i}(\mu)=\frac{g_{i}(\mu_{0})}{\Large(1-2b_{i}g_{i}^{2}\ln(\mu/\mu_{0})\Large)^{1/2}}\,.$
(17)

The $\beta$ function for the top quark Yukawa coupling at one loop is also the
same as the SM one,

$16\pi^{2}\beta_{\lambda_{t}}=\lambda_{t}(\frac{9}{2}\lambda_{t}^{2}-\frac{17}{12}g_{1}^{2}-\frac{9}{4}g_{2}^{2}-8g_{3}^{2})\,.$
(18)

The $\beta$ functions for the new dimensionless couplings in the potential and
also for the quartic Higgs self-coupling at one loop are as follows111We have
checked our results applying the package SARAH [55].,

$\begin{split}16\pi^{2}\beta_{\lambda_{H}}=&\frac{3}{8}g_{1}^{4}+\frac{3}{4}g_{1}^{2}g_{2}^{2}+\frac{9}{8}g_{2}^{4}-3g_{1}^{2}\lambda_{H}-9g_{2}^{2}\lambda_{H}\\\
&+24\lambda_{H}^{2}+2\lambda_{H1}^{2}+2\lambda_{H2}^{2}+12\lambda_{H}\lambda_{t}^{2}-6\lambda_{t}^{4}\,,\\\
16\pi^{2}\beta_{\lambda_{H1}}=&-\frac{3}{2}g_{1}^{2}\lambda_{H1}-\frac{9}{2}g_{2}^{2}\lambda_{H1}+2\lambda_{12}\lambda_{H2}+6\lambda_{1}\lambda_{H1}\\\
&+12\lambda_{H}\lambda_{H1}+8\lambda_{H1}^{2}+6\lambda_{t}^{2}\lambda_{H1}\,,\\\
16\pi^{2}\beta_{\lambda_{H2}}=&-\frac{3}{2}g_{1}^{2}\lambda_{H2}-\frac{9}{2}g_{2}^{2}\lambda_{H2}+2\lambda_{12}\lambda_{H1}+6\lambda_{2}\lambda_{H2}\\\
&+12\lambda_{H}\lambda_{H2}+8\lambda_{H2}^{2}+6\lambda_{t}^{2}\lambda_{H2}\,,\\\
16\pi^{2}\beta_{\lambda_{1}}=&18\lambda_{1}^{2}+2\lambda_{12}^{2}+8\lambda_{H1}^{2}\,,\\\
16\pi^{2}\beta_{\lambda_{2}}=&18\lambda_{2}^{2}+2\lambda_{12}^{2}+8\lambda_{H2}^{2}\,,\\\
16\pi^{2}\beta_{\lambda_{12}}=&8\lambda_{12}^{2}+6\lambda_{1}\lambda_{12}+6\lambda_{2}\lambda_{12}+8\lambda_{H1}\lambda_{H2}\,,\end{split}$
(19)

Figure 5: The running of all the SM couplings and Veltman parameter are shown.

where we have set all the SM Yukawa couplings equal to zero and only retained
the top quark Yukawa coupling, since the light quark and lepton Yukawa
couplings are expected to have quite small effects in the runnings. We first
show the running of the SM couplings in the set
$\\{g_{1},g_{2},g_{3},\lambda_{t},\lambda_{H}\\}$ in Fig. 5. As it is evident,
the coupling $\lambda_{H}$ tends to zero at $\Lambda\sim 10^{5}$ GeV and
therefore violates the bounded from below condition of the potential. In
addition we take a look at the running of the Veltman parameter, $V_{H}$, in
Fig. 5 and notice that only at $\Lambda\sim 10^{20}$ GeV, the Veltman
condition is fulfilled. This scale is larger than the Planck scale. Next, we
consider two representative points in the viable parameter space in the two-
scalar model. We take these two points as Benchmark-I and Benchmark-II as the
following.

Figure 6: In the left panel the running of all the scalar couplings, and in
the right panel the running of the Veltman parameters are shown for
Benchmark-I in the parameter space which fulfill the constraints from observed
relic abundance and direct detection bounds.

Benchmark-I:

$m_{\text{DM}}=362$ GeV, $m_{s_{1}}=402$ GeV, $\lambda_{H}=0.166$,
$\lambda_{H1}=0.041$,

$\lambda_{H2}=0.038$, $\lambda_{1}=0.052$, $\lambda_{2}=0.026$,
$\lambda_{12}=0.042$ , $\sin\theta=0.17$ .

The running of the scalar couplings and the Veltman conditions of the
benchmark-I are presented in Fig. 6. Here we note that the coupling
$\lambda_{H}$ is positive at all energies up to the Planck scale and beyond,
and the Veltman parameter $V_{H}$ approaches zero at a high energy scale
$\Lambda\sim 10^{17}$ GeV. The singlet scalar Veltman parameters $V_{s_{1}}$
and $V_{s_{1}}$ are positive at all scales and their variations are quite
smooth leaving the scalar masses natural even at very high energies.

Benchmark-II:

$m_{\text{DM}}=406$ GeV, $m_{s_{1}}=482$ GeV, $\lambda_{H}=0.229$,
$\lambda_{H1}=0.064$,

$\lambda_{H2}=0.027$, $\lambda_{1}=0.044$, $\lambda_{2}=0.037$,
$\lambda_{12}=0.43$ , $\sin\theta=0.21$ .

Figure 7: In the left panel the running of all the scalar couplings, and in
the right panel the running of the Veltman parameters are shown for Benchmark-
II in the parameter space which fulfills the constraints from observed relic
abundance and direct detection bounds.

Moreover, as shown in Fig 7, we can find in our scan over the viable parameter
space, a point called Benchmark-II, where the Higgs Veltman condition is
satisfied at a lower scale $\Lambda\sim 10^{8}$ GeV, while the Veltman
parameters for the two singlet scalars vary very slowly with energy. For this
point, the quadratic Higgs coupling remains positive at all energies.

##  6 Conclusions

In this work we considered a dark matter model with two real singlet scalars.
Within the present model one can find regions in the parameter space which
respect the observed relic density and the upper bounds from direct detection
experiments. Furthermore, we aimed at considering the vacuum stability and
hierarchy problem simultaneously. In contrast to the singlet scalar dark
matter model, in the two-scalar model there exist DM candidates which evade
the latest bounds from LUX and XENON1t direct detection experiments. After
imposing the relic density, the DD bounds and BFB conditions, the viable
scalar couplings are obtained at the electroweak scale.

Upon running of the couplings we restrict ourself to points which respect the
vacuum stability and perturbativity conditions. Another theoretical
restriction is to find points which render the Veltman conditions satisfied at
energies much lower than the Planck scale. We studied the running of the
scalar couplings and the Veltman conditions for two points in the parameter
space, Benchmark-I and Benchmark-II. We conclude that it is possible in this
model to find a point in which the Higgs Veltman condition is respected at an
intermediate energy scale e.g. $\Lambda\sim 10^{8}$ GeV, while at the same
time satisfying all the DM constraints, the BFB and perturbativity conditions
for the extended potential.

##  7 Appendix: Positivity Condition

In this Appendix we follow [56]. Let us write the generic two-scalar quartic
potential $V_{4}$ as,

$\mathcal{V}_{4}=\lambda_{\text{H}}|H|^{4}+M(\phi_{1},\phi_{2})|H|^{2}+V(\phi_{1},\phi_{2}),$
(20)

where,

$M(\phi_{1},\phi_{2})=\lambda_{\text{H1}}\phi_{1}^{2}+\lambda_{\text{H2}}\phi_{2}^{2}+2\lambda_{\text{H12}}\phi_{1}\phi_{2},$
(21)

and

$V(\phi_{1},\phi_{2})=\frac{1}{4}\lambda_{1}\phi_{1}^{4}+\frac{1}{4}\lambda_{2}\phi_{2}^{4}+\frac{1}{2}\lambda_{12}\phi_{1}^{2}\phi_{2}^{2}+\lambda_{13}\phi_{1}\phi_{2}^{3}+\lambda_{31}\phi_{1}^{3}\phi_{2}$
(22)

The BFB condition (positivity condition) $\mathcal{V}_{4}>0$ immediately holds
if,

$\lambda_{\text{H}}>0~{}~{}\wedge~{}~{}M(\phi_{1},\phi_{2})>0~{}~{}\wedge~{}V(\phi_{1},\phi_{2})>0.$
(23)

Alternatively we can write
$\mathcal{V}_{4}=(2\lambda_{\text{H}}|H|^{2}+M^{2})^{2}/4\lambda_{\text{H}}+V_{\text{min}}$
where $\mathcal{V}_{\text{min}}=V-M^{2}/4\lambda_{\text{H}}$ is the value of
$\mathcal{V}_{4}$ at the minimum. Therefore the positivity condition becomes,

$\lambda_{\text{H}}>0~{}~{}\wedge~{}~{}\mathcal{V}_{\text{min}}\equiv
V(\phi_{1},\phi_{2})-\frac{M^{2}(\phi_{1},\phi_{2})}{4\lambda_{\text{H}}}>0.$
(24)

Therefore the positivity condition $\mathcal{V}_{4}>0$ is given either by Eq.
(23) or Eq. (24).

We still need to restrict the parameters with conditions $M>0$, $V>0$ in Eq.
(23) and $V_{\text{min}}>0$ in Eq. (24). Making a rotation in
$(\phi_{1},\phi_{2})$ space, the polynomial $M$ can be diagonalized:
$M(\phi_{1}^{\prime},\phi_{2}^{\prime})=\lambda_{\text{H1}}^{\prime}\phi_{1}^{\prime
2}+\lambda_{\text{H2}}^{\prime}\phi_{2}^{\prime 2}$ where
$\lambda_{\text{H1}}^{\prime}$ and $\lambda_{\text{H2}}^{\prime}$ are
eigenvalues of the Hessian matrix for $M(\phi_{1},\phi_{2})$. Therefore,
$M(\phi_{1},\phi_{2})>0$ is equivalent to,

$\displaystyle\lambda_{\text{H1}}^{\prime}\equiv\lambda_{\text{H1}}+\lambda_{\text{H12}}-\sqrt{(\lambda_{\text{H1}}-\lambda_{\text{H12}})^{2}+4\lambda_{\text{H2}}\lambda_{\text{H12}}}>0\,,$
(25a)
$\displaystyle\lambda_{\text{H1}}^{\prime}\equiv\lambda_{\text{H1}}+\lambda_{\text{H12}}+\sqrt{(\lambda_{\text{H1}}-\lambda_{\text{H12}})^{2}+4\lambda_{\text{H2}}\lambda_{\text{H12}}}>0\,.$
(25b)

The condition $V>0$ turns into the question of positivity of a quartic
polynomial if we divide $V$ by $\phi_{1}^{4}$ or $\phi_{2}^{4}$, e.g.,
$V/\phi_{2}^{4}\equiv f(x)=ax^{4}+bx^{3}+cx^{2}+dx+e>0$ with
$x\equiv\phi_{1}^{4}/\phi_{2}^{4}$, and

$a=\frac{\lambda_{1}}{4}~{}~{}~{}b=\lambda_{31}~{}~{}~{}c=\frac{\lambda_{12}}{2}~{}~{}~{}d=\lambda_{13}~{}~{}~{}e=\frac{\lambda_{2}}{4}\,.$
(26)

At $x=0$ and $x\to\infty$ limits, the positivity condition requires
$\lambda_{1}>0$ and $\lambda_{2}>0$. Then $f(x)$ remains positive if it never
intersects the real axis. In other words, the positivity holds when the roots
are always complex. The complexity of the roots is guaranteed by the
conditions,

$\Delta>0\wedge(P>0\vee D>0)$ (27)

where $\Delta$ is the discriminant,

$\begin{split}\Delta&=256a^{3}e^{3}-192a^{2}bde^{2}-128a^{2}c^{2}e^{2}+144a^{2}cd^{2}e-27a^{2}d^{4}+144ab^{2}ce^{2}\\\
&-6ab^{2}d^{2}e-80abc^{2}de+18abcd^{3}+16ac^{4}e-4ac^{3}d^{2}-27b^{4}e^{2}+18b^{3}cde\\\
&-4b^{3}d^{3}-4b^{2}c^{3}e+b^{2}c^{2}d^{2}\,,\end{split}$ (28)

and

$P=8ac-3b^{2},~{}~{}~{}D=64a^{3}e-16a^{2}c^{2}+16ab^{2}c-16a^{2}bd-3b^{4}\,.$
(29)

The same approach can be used to find regions with $V_{\text{min}}>0$. From
(24) $V_{\text{min}}=ax^{4}+bx^{3}+cx^{2}+dx+e$ with

$\begin{split}a=&\frac{\lambda_{1}}{4}-\frac{\lambda_{\text{H1}}^{2}}{4\lambda_{\text{H}}},~{}~{}~{}b=\lambda_{31}-\frac{\lambda_{\text{H1}}\lambda_{\text{H12}}}{\lambda_{\text{H}}},~{}~{}~{}c=\frac{\lambda_{12}}{2}-\frac{2\lambda_{\text{H12}}^{2}+\lambda_{\text{H1}}\lambda_{\text{H2}}}{2\lambda_{\text{H}}},\\\
&d=\lambda_{13}-\frac{\lambda_{\text{H2}}\lambda_{\text{H12}}}{\lambda_{\text{H}}},~{}~{}~{}e=\frac{\lambda_{2}}{4}-\frac{\lambda_{\text{H2}}^{2}}{4\lambda_{\text{H}}}\,.\end{split}$
(30)

These parameters again must satisfy the conditions in Eq. (27) with $\Delta$,
$P$ and $D$ evaluated using Eq. (30).

###  7.1 Biquadratic potential

For bi-quadratic potential or in the absence of the odd terms in Eq. (1), i.e.
by setting $\lambda_{H12}=\lambda_{13}=\lambda_{31}=0$, the positivity
conditions will be determined as before, with $b$ and $d$ being vanishing in
the parameters $\Delta$, $P$ and $D$ given in Eqs. (28) and (29). Therefore,
in bi-quadratic potentials we have,

$\Delta=256a^{3}e^{3}-128a^{2}c^{2}e^{2}+16ac^{4}e,~{}~{}~{}P=8ac,~{}~{}~{}D=64a^{3}e-16a^{2}c^{2}$
(31)

The condition $V>0$ is given by Eq. (27) with,

$a=\frac{\lambda_{1}}{4},~{}~{}~{}c=\frac{\lambda_{12}}{2},~{}~{}~{}e=\frac{\lambda_{2}}{4}\,,$
(32)

and $V_{\text{min}}>0$ with,

$a=\frac{\lambda_{1}}{4}-\frac{\lambda_{\text{H1}}^{2}}{4\lambda_{\text{H}}},~{}~{}~{}c=\frac{\lambda_{12}}{2}-\frac{\lambda_{\text{H1}}\lambda_{\text{H2}}}{2\lambda_{\text{H}}},~{}~{}~{}e=\frac{\lambda_{2}}{4}-\frac{\lambda_{\text{H2}}^{2}}{4\lambda_{\text{H}}}\,.$
(33)

###  7.2 Positivity in bi-quadratic potential with N scalars

Let us consider a generic potential as
$\mathcal{V}_{4}=1/2\lambda_{ij}\phi_{i}^{2}\phi_{j}^{2}$ where $i,j=1,...,N$
with the convention that $\lambda_{ii}\equiv\lambda_{i}/2$. When only
$\phi_{1}\to\infty$ while other fields are finite, we need to impose the
condition $\lambda_{11}\equiv\lambda_{1}>0$. When two scalars are large, say
$\phi_{1}\to\infty,\phi_{2}\to\infty$ and the rest of the scalars are finite,
the positivity imposes,

$\lambda_{1}>0~{}~{}~{}~{}~{}\wedge~{}~{}~{}~{}~{}\lambda_{1}\lambda_{2}-\lambda_{12}^{2}>0$
(34)

This trend should continue until N scalars. The Hessian matrix
$\mathcal{H}_{ij}\equiv\partial^{2}V_{4}/\partial\phi^{2}_{i}\partial\phi^{2}_{j}$
in general can be written as

$\mathcal{H}=\begin{pmatrix}\lambda_{11}&\lambda_{12}&\dots&\lambda_{1N}\\\
\lambda_{21}&\lambda_{22}&\dots&\lambda_{2N}\\\ \vdots&\vdots&\ddots&\vdots\\\
\lambda_{N1}&\lambda_{N2}&\dots&\lambda_{NN}\\\ \end{pmatrix}$ (35)

that is a symmetric matrix, i.e. $\lambda_{ij}=\lambda_{ji}$. The positivity
condition of $N\times N$ Hermitian matrix (symmetric matrix in our case) à la
Sylvester is the positivity of determinants for all principal minors of the
matrix $\mathcal{H}$,

$\lambda_{1}>0~{}~{}~{}~{}~{}\wedge~{}~{}~{}~{}~{}\begin{vmatrix}\lambda_{1}&\lambda_{12}\\\
\lambda_{21}&\lambda_{2}\\\
\end{vmatrix}>0~{}~{}~{}~{}\wedge~{}~{}~{}~{}\dots~{}~{}~{}~{}\wedge~{}~{}~{}~{}\begin{vmatrix}\lambda_{1}&\lambda_{12}&\dots&\lambda_{1N}\\\
\lambda_{12}&\lambda_{2}&\dots&\lambda_{2N}\\\ \vdots&\vdots&\ddots&\vdots\\\
\lambda_{1N}&\lambda_{2N}&\dots&\lambda_{N}\\\ \end{vmatrix}>0\,.$ (36)

## References

  * [1] CMS Collaboration, A. M. Sirunyan et al., “A measurement of the Higgs boson mass in the diphoton decay channel,” Phys. Lett. B 805 (2020) 135425, arXiv:2002.06398 [hep-ex].
  * [2] ATLAS Collaboration, M. Aaboud et al., “Measurement of the Higgs boson mass in the $H\rightarrow ZZ^{*}\rightarrow 4\ell$ and $H\rightarrow\gamma\gamma$ channels with $\sqrt{s}=13$ TeV $pp$ collisions using the ATLAS detector,” Phys. Lett. B 784 (2018) 345–366, arXiv:1806.00242 [hep-ex].
  * [3] M. Ciuchini, E. Franco, S. Mishima, and L. Silvestrini, “Electroweak Precision Observables, New Physics and the Nature of a 126 GeV Higgs Boson,” JHEP 08 (2013) 106, arXiv:1306.4644 [hep-ph].
  * [4] C. F. Kolda and H. Murayama, “The Higgs mass and new physics scales in the minimal standard model,” JHEP 07 (2000) 035, arXiv:hep-ph/0003170.
  * [5] E. Gildener, “Gauge-symmetry hierarchies,” Phys. Rev. D 14 (Sep, 1976) 1667–1672. https://link.aps.org/doi/10.1103/PhysRevD.14.1667.
  * [6] M. J. G. Veltman, “The Infrared - Ultraviolet Connection,” Acta Phys. Polon. B 12 (1981) 437.
  * [7] E. Witten, “Dynamical Breaking of Supersymmetry,” Nucl. Phys. B 188 (1981) 513.
  * [8] E. Witten, “Dynamical breaking of supersymmetry,” Nuclear Physics B 188 no. 3, (1981) 513–554. https://www.sciencedirect.com/science/article/pii/0550321381900067.
  * [9] M. Sher, “Electroweak higgs potential and vacuum stability,” Physics Reports 179 no. 5, (1989) 273–418. https://www.sciencedirect.com/science/article/pii/0370157389900616.
  * [10] M. B. Einhorn and D. R. T. Jones, “The Effective potential and quadratic divergences,” Phys. Rev. D 46 (1992) 5206–5208.
  * [11] I. Jack and D. R. T. Jones, “NATURALNESS WITHOUT SUPERSYMMETRY?,” Phys. Lett. B 234 (1990) 321–323.
  * [12] M. S. Al-sarhi, I. Jack, and D. R. T. Jones, “Quadratic divergences in gauge theories,” Z. Phys. C 55 (1992) 283–288.
  * [13] M. Chaichian, R. Gonzalez Felipe, and K. Huitu, “On quadratic divergences and the Higgs mass,” Phys. Lett. B 363 (1995) 101–105, arXiv:hep-ph/9509223.
  * [14] M. Holthausen, K. S. Lim, and M. Lindner, “Planck scale Boundary Conditions and the Higgs Mass,” JHEP 02 (2012) 037, arXiv:1112.2415 [hep-ph].
  * [15] Y. Hamada, H. Kawai, and K.-y. Oda, “Bare Higgs mass at Planck scale,” Phys. Rev. D 87 no. 5, (2013) 053009, arXiv:1210.2538 [hep-ph]. [Erratum: Phys.Rev.D 89, 059901 (2014)].
  * [16] F. Jegerlehner, “The Standard model as a low-energy effective theory: what is triggering the Higgs mechanism?,” Acta Phys. Polon. B 45 no. 6, (2014) 1167, arXiv:1304.7813 [hep-ph].
  * [17] F. Jegerlehner, “The hierarchy problem of the electroweak Standard Model revisited,” arXiv:1305.6652 [hep-ph].
  * [18] A. D. Plascencia, “Classical scale invariance in the inert doublet model,” JHEP 09 (2015) 026, arXiv:1507.04996 [hep-ph].
  * [19] A. Kobakhidze and K. L. McDonald, “Comments on the Hierarchy Problem in Effective Theories,” JHEP 07 (2014) 155, arXiv:1404.5823 [hep-ph].
  * [20] O. Antipin, M. Mojaza, and F. Sannino, “Conformal Extensions of the Standard Model with Veltman Conditions,” Phys. Rev. D 89 no. 8, (2014) 085015, arXiv:1310.0957 [hep-ph].
  * [21] I. Masina and M. Quiros, “On the Veltman Condition, the Hierarchy Problem and High-Scale Supersymmetry,” Phys. Rev. D 88 (2013) 093003, arXiv:1308.1242 [hep-ph].
  * [22] D. Boer, R. Peeters, and S. Zeinstra, “Analysis of fine-tuning measures in models with extended Higgs sectors,” Nucl. Phys. B 946 (2019) 114695, arXiv:1902.01258 [hep-ph].
  * [23] A. Biswas and A. Lahiri, “Masses of physical scalars in two Higgs doublet models,” Phys. Rev. D 91 no. 11, (2015) 115012, arXiv:1412.6187 [hep-ph].
  * [24] I. Chakraborty and A. Kundu, “Two-Higgs doublet models confront the naturalness problem,” Phys. Rev. D 90 no. 11, (2014) 115017, arXiv:1404.3038 [hep-ph].
  * [25] R. Jora, S. Nasri, and J. Schechter, “Naturalness in a simple two Higgs doublet model,” Int. J. Mod. Phys. A 28 (2013) 1350036, arXiv:1302.6344 [hep-ph].
  * [26] S. El Hedri, A. E. Nelson, and D. G. E. Walker, “Reducing the quadratic divergence in the Higgs boson mass squared without top partners,” Phys. Rev. D 98 no. 3, (2018) 035029, arXiv:1805.04523 [hep-ph].
  * [27] E. Katz, A. E. Nelson, and D. G. E. Walker, “The Intermediate Higgs,” JHEP 08 (2005) 074, arXiv:hep-ph/0504252.
  * [28] Y. G. Kim, K. Y. Lee, and S.-H. Nam, “Singlet fermionic dark matter with Veltman conditions,” Phys. Lett. B 782 (2018) 316–323, arXiv:1801.04074 [hep-ph].
  * [29] I. Chakraborty and A. Kundu, “Controlling the fine-tuning problem with singlet scalar dark matter,” Phys. Rev. D 87 no. 5, (2013) 055015, arXiv:1212.0394 [hep-ph].
  * [30] J. O. aali, B. Manaut, L. Rahili, and S. Semlali, “Naturalness implications within the two-real-scalar-singlet beyond the SM,” Eur. Phys. J. C 81 no. 11, (2021) 1045, arXiv:2010.10899 [hep-ph].
  * [31] L. Bian, “Renormalization group equation, the naturalness problem, and the understanding of the Higgs mass term,” Phys. Rev. D 88 no. 5, (2013) 056022, arXiv:1308.2783 [hep-ph].
  * [32] C. N. Karahan and B. Korutlu, “Effects of a Real Singlet Scalar on Veltman Condition,” Phys. Lett. B 732 (2014) 320–324, arXiv:1404.0175 [hep-ph].
  * [33] N. Chakrabarty and I. Chakraborty, “On the Higgs mass fine-tuning problem with multi-Higgs doublet models,” Int. J. Mod. Phys. A 34 no. 05, (2019) 1950025, arXiv:1801.05272 [hep-ph].
  * [34] N. Darvishi and M. Krawczyk, “Implication of Quadratic Divergences Cancellation in the Two Higgs Doublet Model,” Nucl. Phys. B 926 (2018) 167–178, arXiv:1709.07219 [hep-ph].
  * [35] I. Chakraborty and A. Kundu, “Triplet-extended scalar sector and the naturalness problem,” Phys. Rev. D 89 no. 9, (2014) 095032, arXiv:1404.1723 [hep-ph].
  * [36] F. Bazzocchi and M. Fabbrichesi, “A simple inert model solves the little hierarchy problem and provides a dark matter candidate,” Eur. Phys. J. C 73 no. 2, (2013) 2303, arXiv:1207.0951 [hep-ph].
  * [37] E. Ma, “Vanishing Higgs one-loop quadratic divergence in the scotogenic model and beyond,” Phys. Lett. B 732 (2014) 167–168, arXiv:1401.3284 [hep-ph].
  * [38] A. Abada, D. Ghaffor, and S. Nasri, “A Two-Singlet Model for Light Cold Dark Matter,” Phys. Rev. D 83 (2011) 095021, arXiv:1101.0365 [hep-ph].
  * [39] A. Ahriche, A. Arhrib, and S. Nasri, “Higgs Phenomenology in the Two-Singlet Model,” JHEP 02 (2014) 042, arXiv:1309.5615 [hep-ph].
  * [40] K. Ghorbani and H. Ghorbani, “Scalar split WIMPs in future direct detection experiments,” Phys. Rev. D93 no. 5, (2016) 055012, arXiv:1501.00206 [hep-ph].
  * [41] B. Díaz Sáez, K. Möhling, and D. Stöckinger, “Two real scalar WIMP model in the assisted freeze-out scenario,” JCAP 10 (2021) 027, arXiv:2103.17064 [hep-ph].
  * [42] J. Claude and S. Godfrey, “Exploring Direct Detection Suppressed Regions in a Simple 2-Scalar Mediator Model of Scalar Dark Matter,” Eur. Phys. J. C 81 no. 5, (2021) 405, arXiv:2104.01096 [hep-ph].
  * [43] T. Basak, B. Coleppa, and K. Loho, “An update on the two singlet dark matter model,” JHEP 06 (2021) 104, arXiv:2105.09044 [hep-ph].
  * [44] J. McDowall and D. J. Miller, “The Multiple Point Principle and Extended Higgs Sectors,” Front. in Phys. 7 (2019) 135, arXiv:1909.10459 [hep-ph].
  * [45] T. Robens, T. Stefaniak, and J. Wittbrodt, “Two-real-scalar-singlet extension of the SM: LHC phenomenology and benchmark scenarios,” Eur. Phys. J. C 80 no. 2, (2020) 151, arXiv:1908.08554 [hep-ph].
  * [46] K. Ghorbani and P. H. Ghorbani, “A Simultaneous Study of Dark Matter and Phase Transition: Two-Scalar Scenario,” JHEP 12 (2019) 077, arXiv:1906.01823 [hep-ph].
  * [47] WMAP Collaboration, G. Hinshaw et al., “Nine-year wilkinson microwave anisotropy probe (wmap) observations: Cosmological parameter results,” Astrophys.J.Suppl. 208 (2013) 19, arXiv:1212.5226 [astro-ph].
  * [48] Planck Collaboration, P. A. R. Ade et al., “Planck 2013 results. XVI. Cosmological parameters,” Astron. Astrophys. 571 (2014) A16, arXiv:1303.5076 [astro-ph.CO].
  * [49] XENON Collaboration, E. Aprile et al., “First Dark Matter Search Results from the XENON1T Experiment,” Phys. Rev. Lett. 119 no. 18, (2017) 181301, arXiv:1705.06655 [astro-ph.CO].
  * [50] LUX Collaboration, D. S. Akerib et al., “Results from a search for dark matter in the complete LUX exposure,” Phys. Rev. Lett. 118 no. 2, (2017) 021303, arXiv:1608.07648 [astro-ph.CO].
  * [51] D. Barducci, G. Belanger, J. Bernon, F. Boudjema, J. Da Silva, S. Kraml, U. Laa, and A. Pukhov, “Collider limits on new physics within micrOMEGAs4.3” Comput. Phys. Commun. 222 (2018) 327–338, arXiv:1606.03834 [hep-ph].
  * [52] ATLAS Collaboration, M. Aaboud et al., “Combination of searches for invisible Higgs boson decays with the ATLAS experiment,” Phys. Rev. Lett. 122 no. 23, (2019) 231801, arXiv:1904.05105 [hep-ex].
  * [53] CMS Collaboration, A. M. Sirunyan et al., “Search for invisible decays of a Higgs boson produced through vector boson fusion in proton-proton collisions at $\sqrt{s}=$ 13 TeV,” Phys. Lett. B 793 (2019) 520–551, arXiv:1809.05937 [hep-ex].
  * [54] Particle Data Group Collaboration, P. A. Zyla et al., “Review of Particle Physics,” PTEP 2020 no. 8, (2020) 083C01.
  * [55] F. Staub, “SARAH 3.2: Dirac Gauginos, UFO output, and more,” Comput. Phys. Commun. 184 (2013) 1792–1809, arXiv:1207.0906 [hep-ph].
  * [56] K. Kannike, “Vacuum Stability of a General Scalar Potential of a Few Fields,” Eur. Phys. J. C 76 no. 6, (2016) 324, arXiv:1603.02680 [hep-ph]. [Erratum: Eur.Phys.J.C 78, 355 (2018)].