#  The simplest scoto-seesaw model:
WIMP dark matter phenomenology and Higgs vacuum stability

Sanjoy Mandal<EMAIL_ADDRESS>AHEP Group, Institut de Física Corpuscular –
CSIC/Universitat de València, Parc Científic de Paterna.
C/ Catedrático José Beltrán, 2 E-46980 Paterna (Valencia) - SPAIN  Rahul
Srivastava<EMAIL_ADDRESS>Department of Physics, Indian Institute of
Science Education and Research - Bhopal,
Bhopal Bypass Road, Bhauri, Bhopal 462066, India  José W. F. Valle
<EMAIL_ADDRESS>AHEP Group, Institut de Física Corpuscular – CSIC/Universitat
de València, Parc Científic de Paterna.
C/ Catedrático José Beltrán, 2 E-46980 Paterna (Valencia) - SPAIN

###### Abstract

We analyze the consistency of electroweak breaking, neutrino and dark matter
phenomenology within the simplest scoto-seesaw model. By adding the minimal
dark sector to the simplest “missing partner” type-I seesaw one has a physical
picture for the neutrino oscillation lengths: the “atmospheric” mass scale
arises from the tree-level seesaw, while the “solar” scale is induced
radiatively, mediated by the dark sector. We identify parameter regions
consistent with theoretical constraints, as well as dark matter relic
abundance and direct detection searches. Using two-loop renormalization group
equations we explore the stability of the vacuum and the consistency of the
underlying dark parity symmetry. One also has a lower bound for the
neutrinoless double beta decay amplitude.

##  1 Introduction

The discovery of neutrino oscillations [1, 2] implies that at least two
neutrinos are massive. There has by now been strong evidence, at different
scales, for the existence of cosmological dark matter, the basic understanding
and interpretation of which we also lack [3]. The main current neutrino mass
generation paradigms are the seesaw and the scotogenic mechanism, which also
accounts for dark matter as the mediator of neutrino mass, as a result of an
assumed $\mathbb{Z}_{2}$ symmetry. Both mechanisms give mass “democratically”
to all neutrino states, according to the structure of the relevant Yukawa
couplings.

The simplest “scoto-seesaw” extension of the Standard Model [4] combines these
two main paradigms within its minimal $\mathrm{SU(3)_{c}\otimes
SU(2)_{L}\otimes U(1)_{Y}}$ framework. In such hybrid scenario the atmospheric
scale comes from the tree level seesaw, while the solar scale is mediated by
the radiative exchange of dark states, i.e.

$\displaystyle\Delta
m_{\text{ATM}}^{2}=\left(\frac{v^{2}}{2M_{N}}\mathbb{Y}_{N}^{2}\right)^{2},\,\,\,\Delta
m_{\text{SOL}}^{2}\approx\left(\frac{1}{32\pi^{2}}\right)^{2}\left(\frac{\lambda_{5}v^{2}}{M_{f}^{2}-m_{\eta}^{2}}M_{f}\mathbb{Y}_{f}^{2}\right)^{2},$
(1)

where $M_{N}$ is the “right-handed” neutrino mass, $M_{f}$ and $m_{\eta}$ are
“dark-sector” masses and $\mathbb{Y}_{N}^{2}$, $\mathbb{Y}_{f}^{2}$ are
corresponding Yukawa coupling strengths.

One sees that the solar splitting will be non-zero as long as $\lambda_{5}\neq
0$. Moreover, one accounts naturally for the hierarchy between the solar and
atmospheric scales observed in the experimental data [5]. The corresponding
ratio of squared solar-to-atmospheric mass splittings for normal and inverted
mass hierarchy are found to be [6, 5]:

$\displaystyle\textbf{NO:}\,\frac{\Delta m_{\text{SOL}}^{2}}{\Delta
m_{\text{ATM}}^{2}}=0.0294^{+0.0027}_{-0.0023},\,\,\,\,\textbf{IO:}\,\frac{\Delta
m_{\text{SOL}}^{2}}{\Delta m_{\text{ATM}}^{2}}=0.0306^{+0.0028}_{-0.0025}.$
(2)

Altogether, the interplay of the “seesaw” and “dark-sectors” provide an
interesting way to describe lepton number violation and neutrino mass
generation. Indeed, the scoto-seesaw model has a viable weakly interacting
massive particle (WIMP) dark matter candidate and accounts for the observed
neutrino masses, including the solar-to-atmospheric hierarchy. The aim of this
work is to explore the scoto-seesaw model in more detail.

The paper is organized as follows. In section 2 we briefly describe the model,
giving the details of the new fields and their interactions. In section 3, we
describe the tree level and radiative neutrino mass generation. In section 4
we study the parameter space for the case of a scalar dark matter candidate.
In section 6 we look at vacuum stability in the scoto-seesaw model and show
that, over large parameter regions, the vacuum is stable all the way up to the
Planck scale. In section 7 we examine the robustness of the dark parity
symmetry under renormalization group (RG) evolution of the parameters. We
finally conclude in section 8.

##  2 Minimal scoto-seesaw model

The minimal combination of the seesaw mechanism and the scotogenic model was
proposed in Ref. [4] 111One can have scoto-seesaw realizations based on (3,2)
seesaw extensions [7]. While they have new interesting features, one looses
the interesting prediction in Eq. (1).. It clones the simplest “missing
partner” (3,1) version of the Standard Model seesaw mechanism suggested in [8,
9] with the minimal scotogenic model proposed in [10]. We now describe in
detail both the fermionic and the scalar sectors of the model.

###  2.1 The Yukawa Sector

We now briefly recall the basic features of the minimal scoto-seesaw model
[4]. The new particles and their charges are given in Table 1, where the
family index $a$ runs from 1 to 3.

| Standard Model | New Fermions | New Scalar
---|---|---|---
|  $L_{a}$ |  $e_{a}$ |  $H$ |  $N$ |  $f$ |  $\eta$
$SU(2)_{L}$ | 2 | 1 | 2 | 1 | 1 | 2
$U(1)_{Y}$ | -1/2 | -1 | 1/2 | 0 | 0 | 1/2
$\mathbb{Z}_{2}$ | $+$ | $+$ | $+$ | $+$ | $-$ | $-$
Table 1: Matter content and charge assignment of the minimal scoto-seesaw
model.

In Table. 1 the additional $\mathbb{Z}_{2}$ symmetry is the “dark parity”
responsible for the stablity of the dark matter candidate. All the Standard
Model particles and $N$ are even under this dark $\mathbb{Z}_{2}$ parity,
while the dark sector, consisting of one fermion $f$ and one scalar $\eta$, is
odd under $\mathbb{Z}_{2}$.

The full Yukawa sector can be split as

$\displaystyle\mathcal{L}=\mathcal{L}_{\text{SM}}+\mathcal{L}_{\text{ATM}}+\mathcal{L}_{\text{DM,SOL}}$
(3)

where $\mathcal{L}_{\text{SM}}$ is the Standard Model Lagrangian, while

$\displaystyle\mathcal{L}_{\text{ATM}}=-Y_{N}^{a}\bar{L}^{a}\tilde{H}N+\frac{1}{2}M_{N}\overline{N^{c}}N+h.c,$
(4)

induces the type-I seesaw neutrino mass (atmospheric neutrino mass scale)
after the electroweak symmetry breaking. Note also that throughtout this work
repeated indices imply summation, with $\tilde{H}=i\sigma_{2}H^{*}$,
$\sigma_{2}$ being the second Pauli matrix.

The Lagrangian responsible for the solar and dark sector is given by

$\displaystyle\mathcal{L}_{\text{DM,SOL}}=Y_{f}^{a}\bar{L}^{a}\tilde{\eta}~{}f+\frac{1}{2}M_{f}\overline{f^{c}}f+h.c.$
(5)

It induces the solar neutrino mass scale as discussed in Sec. 3.

###  2.2 The Scalar Sector

Apart from the Standard Model (SM) Higgs doublet $H$ we have a scalar doublet
$\eta$ carrying the same quantum numbers, but with $\mathbb{Z}_{2}$-odd
parity. The $\mathrm{SU(3)_{c}\otimes SU(2)_{L}\otimes U(1)_{Y}}$ gauge
invariant scalar potential is given by

$\displaystyle V$
$\displaystyle=-\mu_{H}^{2}H^{\dagger}H+m_{\eta}^{2}\eta^{\dagger}\eta+\lambda(H^{\dagger}H)^{2}+\lambda_{\eta}(\eta^{\dagger}\eta)^{2}+\lambda_{3}(H^{\dagger}H)(\eta^{\dagger}\eta)+\lambda_{4}(H^{\dagger}\eta)(\eta^{\dagger}H)$
$\displaystyle+\frac{\lambda_{5}}{2}\left((H^{\dagger}\eta)^{2}+h.c.\right)$
(6)

We now turn to the consistency conditions of the potential. The following
restrictions must hold so as to ensure that the scalar potential is bounded
from below and has a stable vacuum at any given energy scale $\mu$:

$\displaystyle\lambda(\mu)>0,\,\,\lambda_{\eta}(\mu)>0,$ (7)
$\displaystyle\lambda_{A}\equiv\lambda_{3}(\mu)+\sqrt{4\lambda(\mu)\lambda_{\eta}(\mu)}>0,$
(8)
$\displaystyle\lambda_{B}\equiv\lambda_{3}(\mu)+\lambda_{4}(\mu)+\sqrt{4\lambda(\mu)\lambda_{\eta}(\mu)}-|\lambda_{5}(\mu)|>0.$
(9)

where $\lambda_{i}(\mu)$ are the values of the quartic couplings at the
running scale $\mu$. In order to have an absolutely stable vacuum, one must
satisfy the conditions given in Eqs. (7), (8) and (9) at each and every energy
scale. To ensure perturbativity, we take a conservative approach of simply
requiring that the scalar quartic couplings in Eq. (6) obey $\leq 4\pi$.

###  2.3 Mass Spectrum

In order to ensure dark matter stability the $\mathbb{Z}_{2}$ symmetry should
remain unbroken. This means that the $\mathbb{Z}_{2}$ odd scalar $\eta$ should
not acquire a nonzero vacuum expectation value (VEV). As a result, electroweak
symmetry breaking is driven simply by the VEV of $H$. The fields $\eta$ and
$H$ can be expanded as follows

$\displaystyle H=\begin{pmatrix}H^{+}\\\
(v+h+i\phi^{0})/\sqrt{2}\end{pmatrix},\,\,\eta=\begin{pmatrix}\eta^{+}\\\
(\eta^{R}+i\eta^{I})/\sqrt{2}\end{pmatrix}$ (10)

Exact conservation of the $\mathbb{Z}_{2}$ symmetry forbids the mixing between
the Higgs and the dark doublet $\eta$. The components of $\eta$ have the
following masses

$\displaystyle m_{\eta^{R}}^{2}$
$\displaystyle=m_{\eta}^{2}+\frac{1}{2}\left(\lambda_{3}+\lambda_{4}+\lambda_{5}\right)v^{2}$
(11) $\displaystyle m_{\eta^{I}}^{2}$
$\displaystyle=m_{\eta}^{2}+\frac{1}{2}\left(\lambda_{3}+\lambda_{4}-\lambda_{5}\right)v^{2}$
(12) $\displaystyle m_{\eta^{+}}^{2}$
$\displaystyle=m_{\eta}^{2}+\frac{1}{2}\lambda_{3}v^{2}.$ (13)

The difference $m_{\eta^{R}}^{2}-m_{\eta^{I}}^{2}$ depends only on the
parameter $\lambda_{5}$ which, we will show later, is also responsible for
smallness of solar neutrino mass scale. The conservation of the
$\mathbb{Z}_{2}$ symmetry also makes the lightest of the two eigenstates
$\eta^{R}$ and $\eta^{I}$ a viable scalar dark matter candidate222Throughout
this work we assume that the dark fermion $f$ is heavier than the dark scalars
$\eta^{R}$ and $\eta^{I}$., as we explore in what follows.

##  3 Neutrino Masses

At tree-level this model gives rise to the following neutrino mass matrix
$\mathcal{M}_{\nu}^{ab}$,

$\displaystyle\mathcal{M}_{\nu}^{ab}=\begin{pmatrix}0&0&0&\frac{Y_{N}^{1}v}{\sqrt{2}}\\\
0&0&0&\frac{Y_{N}^{2}v}{\sqrt{2}}\\\ 0&0&0&\frac{Y_{N}^{3}v}{\sqrt{2}}\\\
\frac{Y_{N}^{1}v}{\sqrt{2}}&\frac{Y_{N}^{2}v}{\sqrt{2}}&\frac{Y_{N}^{3}v}{\sqrt{2}}&M_{N}\\\
\end{pmatrix}$ (14)

in the basis $(L^{a},N)^{T}$. Notice the (3,1) structure of the seesaw [8, 9],
as a result of which one sees that the $N$ pairs off with one combination of
the doublets in $L_{a}$ through their Dirac-like couplings.

This clearly leads to a projective structure for the tree-level neutrino mass
matrix

$\displaystyle\mathcal{M}_{\nu\text{TREE}}=-\frac{v^{2}}{2M_{N}}Y_{N}^{a}Y_{N}^{b},$
(15)

where $a,b=1,2,3$ are family indices of the lepton doublets. One sees from Eq.
(15) that for “sizeable” Yukawa couplings, $Y_{N}\sim\mathcal{O}(1)$, in order
to reproduce the required value of the atmospheric scale, heavy neutrinos must
lie at mass scale $M_{N}\sim\mathcal{O}(10^{14}\,\text{GeV})$. Smaller values
of the Yukawa coupling $Y_{N}$ would require correspondingly lower seesaw
scale $M_{N}$.

The solar mass scale arises from Fig. 1 involving the exchange of the scalar
and fermionic dark mediators $\eta$ and $f$.

Figure 1: Solar neutrino mass scale induced from radiative dark-sector
exchange.

Although these corrections also have a projective structure, they break the
“missing-partner” nature of the (3,1) type-I seesaw mechanism333The situation
is analogous to neutrino mass generation in bilinear broken R-parity
supersymmetry [11, 12, 13].. The total neutrino mass has the form

$\mathcal{M}_{\nu\text{TOT}}^{ab}=-\frac{v^{2}}{2M_{N}}Y_{N}^{a}Y_{N}^{b}+\mathcal{F}(m_{\eta^{R}},m_{\eta^{I}},M_{f})M_{f}Y_{f}^{a}Y_{f}^{b},$
(16)

where the first term is the tree-level seesaw part, and the loop function
$\mathcal{F}$ characterizes the quantum correction arising from Fig. 1. This
is responsible for inducing the solar mass scale.

The loop function $\mathcal{F}$ is expressed as the difference of two
$B_{0}$-Veltman functions, namely,

$\displaystyle\mathcal{F}(m_{\eta^{R}},m_{\eta^{I}},M_{f})=\frac{1}{32\pi^{2}}\left(\frac{m_{\eta^{R}}^{2}\,\text{log}\left(M_{f}^{2}/m_{\eta^{R}}^{2}\right)}{M_{f}^{2}-m_{\eta^{R}}^{2}}-\frac{m_{\eta^{I}}^{2}\,\text{log}\left(M_{f}^{2}/m_{\eta^{I}}^{2}\right)}{M_{f}^{2}-m_{\eta^{I}}^{2}}\right)$
(17)

Since both terms in Eq. (16) have a projective nature, one out of the three
neutrinos remains massless.

From the eigenvalues of the neutrino mass matrix $\mathcal{M}_{\nu}^{ab}$ one
can estimate the atmospheric and solar square mass differences as,

$\displaystyle\Delta
m_{\text{ATM}}^{2}=\left(\frac{v^{2}}{2M_{N}}\mathbb{Y}_{N}^{2}\right)^{2},\,\,\,\Delta
m_{\text{SOL}}^{2}\approx\left(\frac{1}{32\pi^{2}}\right)^{2}\left(\frac{\lambda_{5}v^{2}}{M_{f}^{2}-m_{\eta^{R}}^{2}}M_{f}\mathbb{Y}_{f}^{2}\right)^{2}.$
(18)

where we take $M_{f}^{2}$, $m_{\eta^{R}}^{2}$,
$M_{f}^{2}-m_{\eta^{R}}^{2}\gg\lambda_{5}v^{2}$ and
$\mathbb{Y}_{\ell}^{2}=(Y_{\ell}^{e})^{2}+(Y_{\ell}^{\mu})^{2}+(Y_{\ell}^{\tau})^{2}$
for $\ell=N,\,f$. It follows that the ratio between the solar and atmospheric
square mass differences can be written as:

$\displaystyle\frac{\Delta m_{\text{SOL}}^{2}}{\Delta
m_{\text{ATM}}^{2}}\approx\left(\frac{1}{16\pi^{2}}\right)^{2}\left(\lambda_{5}\frac{M_{N}M_{f}}{M_{f}^{2}-m_{\eta^{R}}^{2}}\right)^{2}\left(\frac{\mathbb{Y}_{f}^{2}}{\mathbb{Y}_{N}^{2}}\right)^{2}$
(19)

From Eq. (18) it is clear that one can fit the observed atmospheric and solar
mass square differences in many ways as long as one takes an adequately small
value for $\lambda_{5}$. Moreover, Eq. (19) nicely reproduces Eq. (2). In the
following we list some choices which can satisfy both the solar and
atmospheric scales, as well as have $\eta^{R}$ as the scalar WIMP dark matter:

  * •

$M_{N}\sim 10^{14}$ GeV, $M_{f}\sim 10^{12}$ GeV, $m_{\eta^{R}}\sim 10^{3}$
GeV, $\mathbb{Y}_{N}\sim 0.4$, $\mathbb{Y}_{f}\sim 0.4$,

  * •

$M_{N}\sim 10^{12}$ GeV, $M_{f}\sim 10^{4}$ GeV, $m_{\eta^{R}}\sim 10^{3}$
GeV, $\mathbb{Y}_{N}\sim 0.1$, $\mathbb{Y}_{f}\sim 10^{-4}$,

  * •

$M_{N}\sim 10^{14}$ GeV, $M_{f}\sim 10^{5}$ GeV, $m_{\eta^{R}}\sim 10^{3}$
GeV, $\mathbb{Y}_{N}\sim 0.4$, $\mathbb{Y}_{f}\sim 10^{-4}$,

  * •

$M_{N}\sim 10^{6}$ GeV, $M_{f}\sim 10^{6}$ GeV, $m_{\eta^{R}}\sim 10^{3}$ GeV,
$\mathbb{Y}_{N}\sim 10^{-5}$, $\mathbb{Y}_{f}\sim 10^{-4}$.

The upshot of this discussion is that one can easily fit the solar and
atmospheric scales for reasonable parameter choices. For example, for
sufficiently small $\lambda_{5}$ values, one can choose a reasonable Yukawa
coupling $\mathbb{Y}_{f}$ and $M_{f}\raise
1.29167pt\hbox{$\;>$\kern-7.5pt\raise-4.73611pt\hbox{$\sim\;$}}\mathcal{O}$(TeV).
In section 4 we show that either $\eta^{I}$ or $\eta^{R}$ can, indeed, be
taken as a consistent WIMP dark matter candidate.

##  4 Phenomenology of Scalar WIMP Dark Matter

In this section we collect the results of our analysis of dark matter
phenomenology. In addition to ensuring radiative generation of neutrino
masses, the $\mathbb{Z}_{2}$ symmetry in the dark sector ensures the stability
of “lightest dark particle” (LDP). Such LDP is in principle a viable dark
matter candidate. There are three LDP options. The first is the dark fermion
$f$. The others are the real and imaginary parts of the $\eta$ scalar,
$\eta^{R}$ and $\eta^{I}$. In our analysis, we assume scalar dark matter, with
the condition $\lambda_{5}<0$ on the quartic coupling $\lambda_{5}$. As a
result $\eta^{R}$ will be our dark matter candidate (the opposite scenario
with $\lambda_{5}>0$ would have $\eta^{I}$ as the dark matter particle).

In order to calculate all the vertices, mass matrices, tadpole equations etc
the model is implemented in the SARAH package [14]. On the other hand, the
thermal component of the dark matter relic abundance, as well as the dark
matter-nucleon scattering cross section, are determined using micrOMEGAS-5.0.8
[15].

###  4.1 Relic density

As shown in Fig. 9 (Appendix A), there are several dark matter annihilation
and coannihilation diagrams present in the scoto-seesaw model. They involve
annilation to quarks and leptons, SM gauge bosons and the Higgs boson.
Altogether, they determine the relic abundance of our assumed LDP, $\eta^{R}$.
Our numerical scan is performed varying the input parameters as given in Table
2.

Parameters | Range
---|---
$m_{\eta}^{2}$ | $[100^{2},5000^{2}]\,(\text{GeV}^{2})$
$\lambda_{3}$ | $[10^{-5},1]$
$\lambda_{4}$ | $[10^{-5},1]$
$|\lambda_{5}|$ | $[10^{-5},10^{-3}]$
Table 2: Ranges of variation of the input parameters used in our numerical
scan.

In Fig. 2 we show the relic density as a function of the mass of the scalar
dark matter candidate $\eta^{R}$.

Figure 2:  Relic abundance as a function of the dark matter mass
$m_{\eta^{R}}$. Cyan points inside the black lines fall within the measured
$3\sigma$ cold dark matter relic density range given by Planck satellite data,
Eq. (20). Gray (blue) points outside the narrow band give under (over)
abundance of dark matter, respectively.

The narrow horizontal band is the $3\sigma$ range for cold dark matter derived
from the Planck satellite data [16]:

$\displaystyle 0.1126\leq\Omega_{\eta^{R}}h^{2}\leq 0.1246.$ (20)

Only for solutions falling exactly within this band the totality of the dark
matter can be explained by $\eta^{R}$. The relic density for the cyan points
in Fig. 2 lies within the above $3\sigma$ range, whereas the relic density for
blue and gray points is above and below the $3\sigma$ range. One sees from
Fig. 2 that the correct relic density can be obtained in three mass ranges:
$m_{\eta^{R}}<50$ GeV, $70\,\text{GeV}<m_{\eta^{R}}<100$ GeV and
$m_{\eta^{R}}>550$ GeV. The reasons for these mass gaps can be understood by
looking in detail into the $\eta^{R}$ annihilation channels (see Appendix A).
The first dip occurs at $m_{\eta^{R}}\sim M_{Z}/2$ and corresponds to
annihilation via s-channel $Z$ exchange. The second depletion of the relic
density happens around $m_{\eta^{R}}\sim m_{h}/2$ and corresponds to
annihilations via s-channel Higgs boson exchange. This becomes very efficient
when the SM-like Higgs $h$ is on-shell, precluding us from obtaining a relic
density matching Planck observations. Notice that the second dip is more
efficient than the first one, as the Z-mediated dip is momentum suppressed.
For heavier $\eta^{R}$ masses, quartic interactions with gauge bosons become
effective. For $m_{\eta^{R}}\raise
1.29167pt\hbox{$\;>$\kern-7.5pt\raise-4.73611pt\hbox{$\sim\;$}}80$ GeV,
annihilations of $\eta^{R}$ into $W^{+}W^{-}$ and $ZZ$ via quartic couplings
are particularly important, thus explaining the third drop in the relic
abundance. In the mass range $m_{\eta^{R}}\geq 120$ GeV, $\eta^{R}$ can
annihilate also into two Higgs bosons, $hh$. When $m_{\eta^{R}}\geq m_{t}$, a
new channel $\eta^{R}\eta^{R}\to t\bar{t}$ opens up. All these annihilation
channels make dark matter annihilation very efficient, and it is difficult to
obtain the correct relic density. For very heavy $m_{\eta^{R}}$ the relic
density increases due to the suppressed annihilation cross section, which
drops as $\sim\frac{1}{m_{\eta^{R}}^{2}}$. Notice also that the coannihilation
channels with $\eta^{I}$ and $\eta^{\pm}$ may occur in all regions of the
parameter space, with the effect of lowering the relic dark matter density.

###  4.2 Direct detection

Let us now study the direct detection prospects of our dark matter $\eta^{R}$.
In our model, the tree-level spin-independent $\eta^{R}$-nucleon cross section
is mediated by the Higgs and the $Z$ portals, see Fig. 3.

Figure 3: Higgs and $Z$-mediated tree-level Feynman diagrams contributing to
the elastic scattering of $\eta^{R}$ off nuclei.

Notice that, as the $\eta$ doublet has non-zero hypercharge, the
$\eta^{R}$-nucleon spin-independent (SI) cross-section is mediated by the
Z-boson. Generally this exceeds the current limit from direct detection
experiments like XENON1T [17]. However this can be easily avoided by taking
non-zero $\lambda_{5}$. In this case there is a small mass splitting between
$\eta^{I}$ and $\eta^{R}$, so that the interaction through the Z-boson is
kinematically forbiden or leads to inelastic scattering. As a result, for
nonzero $\lambda_{5}$, the $\eta^{R}$-nucleon interaction via the Higgs will
be the dominant one. The coupling between $\eta^{R}$ and the Higgs boson
depends on $\lambda_{345}=\lambda_{3}+\lambda_{4}+\lambda_{5}$ and the
$\eta^{R}$-nucleon cross section is given by

$\displaystyle\sigma^{\text{SI}}=\frac{\lambda_{345}^{2}}{4\pi
m_{h}^{4}}\frac{m_{N}^{4}f_{N}^{2}}{(m_{\eta^{R}}+m_{N})^{2}}$ (21)

where $m_{h}$ is the mass of SM Higgs boson and $m_{N}$ is the nucleon mass,
i.e. the average of the proton and neutron masses. Here $f_{N}$ is the form
factor, which depends on hadronic matrix elements. In Fig. 4 we show the spin-
independent $\eta^{R}$-nucleon cross section as a function of the $\eta^{R}$
mass, for the range of parameters covered by our scan given in Table 2.

Figure 4: Spin-independent WIMP-nucleon elastic scattering cross section
versus the dark matter mass $m_{\eta^{R}}$, with the same colour code as in
Fig. 2. The solid red line denotes the recent upper bound from the XENON1T
experiment [17]. A broad region of experimentally viable scoto-seesaw model
points in cyan fall within future projected sensitivities for the PandaX-4t
[18] (green), LZ [19] (dashed purple), XENONnT with 20 ton-yr exposure [20]
(blue), DarkSide-20k [21] (magenta), DARWIN [22] (black) and ARGO [23] (brown)
proposals. The dot-dashed orange line corresponds to the “neutrino floor”
coming from coherent elastic neutrino scattering [24]. The upper triangular
white region violates perturbativity.

The color code in Fig. 4 is the same as in Fig. 2. The red line denotes the
latest upper bound from the XENON1T collaboration. There are constraints from
other experiments as well, such as LUX [25] and PandaX-II [26], but weaker
when compared to the XENON1T limit. We also show the projected sensitivities
for the PandaX-4t [18], LUX-ZEPLIN(LZ) [19], XENONnT [20], DarkSide-20k [21],
DARWIN [22] and ARGO [23] experiments. The lower limit corresponding to the
“neutrino floor” from coherent elastic neutrino scattering is also indicated.
We see from Fig. 4 that there are low-mass solutions with the correct dark
matter relic density. However, most of these are ruled out by the XENON1T
direct detection cross section upper limits. Moreover, there are also tight
constraints on low mass dark matter from collider searches, as we will discuss
in the next section.

##  5 Collider constraints

In this section we confront our scalar dark matter candidate $\eta^{R}$ with
the latest data from particle colliders, in particular the LHC. First we note
that, if $\eta^{R}/\eta^{I}$ are light enough, there are two additional decay
channels for the SM-like Higgs boson,

$\displaystyle\Gamma(h\to\eta^{R}\eta^{R})$
$\displaystyle=\frac{v^{2}\lambda_{345}^{2}}{32\pi
m_{h}}\sqrt{1-\frac{4m_{\eta^{R}}^{2}}{m_{h}^{2}}}$ (22)
$\displaystyle\Gamma(h\to\eta^{I}\eta^{I})$
$\displaystyle=\frac{(m_{\eta^{I}}^{2}-m_{\eta^{R}}^{2}+\frac{\lambda_{345}}{2}v^{2})^{2}}{8\pi
v^{2}m_{h}}\sqrt{1-\frac{4m_{\eta^{I}}^{2}}{m_{h}^{2}}}$ (23)

Note that due to LEP limit $m_{\eta^{\pm}}>m_{W}$ [27], there is no phase
space for the two body decay $h\to\eta^{\pm}\eta^{\pm}$. The decay mode
$\Gamma(h\to\eta^{R}\eta^{R})$ contributes to the invisible Higgs decay width,
constrained by the LHC experiments, e.g. the CMS experiment [28],

$\text{BR}(h\to\text{Inv})\leq 0.19.$

The SM-like Higgs boson $h$ also couples to the charged Higgs $\eta^{\pm}$,
contributing to the diphoton decay channel $h\to\gamma\gamma$ 444Note that
this invisible Higgs decay and charged scalar contributions to
$h\to\gamma\gamma$ are generic features of inert doublet schemes [29, 30] as
well as scotogenic models [31, 32].. To quantify the deviation from the
Standard Model prediction, we define the following parameter

$\displaystyle
R_{\gamma\gamma}=\frac{\text{BR}(h\to\gamma\gamma)}{\text{BR}(h\to\gamma\gamma)^{\text{SM}}}.$
(24)

The value we use for the Standard Model is
$\text{BR}(h\to\gamma\gamma)^{\text{SM}}\approx 2.27\times 10^{-3}$.

Figure 5: Invisible Higgs branching ratio (left panel) and $R_{\gamma\gamma}$
(right panel) as a function of the dark matter mass $m_{\eta^{R}}$. The color
code is same as in Fig. 2. The shaded red region in the left panel is excluded
from the LHC constraint on the invisible Higgs decay [28], while the shaded
green band in the right is allowed by ATLAS measurements of $R_{\gamma\gamma}$
[33].

The ATLAS and CMS collaborations have studied this decay mode and their
combined analysis with 8 TeV data gives
$R_{\gamma\gamma}^{\text{exp}}=1.16_{-0.18}^{+0.20}$ [34]. For the 13 TeV
Run-2, there is no combined final data so far, and the available data is
separated by production processes [35]. In our analysis we have used 13 TeV
ATLAS result which gives the global signal strength measurement of
$R_{\gamma\gamma}^{\text{exp}}=0.99_{-0.14}^{+0.15}$ [33]. In the left panel
of Fig. 5 we show the invisible Higgs branching ratio
$\text{BR}(h\to\eta^{R}\eta^{R})$ as a function of the dark matter candidate
mass $m_{\eta^{R}}$. In the right panel we give the expected
$R_{\gamma\gamma}$ values for the same random scan of parameters. One sees
that for low dark matter masses $m_{\eta^{R}}<60$ GeV the invisible decay mode
$h\to\eta^{R}\eta^{R}$ is open and violates the LHC limit
$\text{BR}(h\to\text{Inv})\leq 0.19$ [28].

Likewise, the $R_{\gamma\gamma}$ measurement rules out the lower dark matter
mass region. For intermediate dark matter masses in the range
$70\,\text{GeV}\leq m_{\eta^{R}}\leq 100\,\text{GeV}$, there are acceptable
solutions with acceptable $R_{\gamma\gamma}\approx 1$. In the large mass
region $m_{\eta^{R}}>550$ GeV, the charged Higgs $\eta^{\pm}$ contribution to
the diphoton decay mode $h\to\gamma\gamma$ is negligible, so that
$R_{\gamma\gamma}$ is close to unity. From the above discussion, one can say
that low mass dark matter with $m_{\eta^{R}}<60$ GeV is ruled out by LHC
constraints. However, they can not completely rule out the intermediate mass
region $70\,\text{GeV}\leq\,m_{\eta^{R}}\leq\,100\,\text{GeV}$. Moreover, the
heavy dark matter mass region $m_{\eta^{R}}>550$ GeV is completely allowed by
LHC constraints.

Before concluding this section we should note that there are also constraints
from LEP-I and LEP-II experiments. The precise LEP-I measurements rule out SM-
gauge bosons decays to dark sector particles [30, 36]. This requires that

$\displaystyle
m_{\eta^{R}}+m_{\eta^{I}},2m_{\eta^{\pm}}>m_{Z},\,\,\text{and}\,\,m_{\eta^{R}/\eta^{I}}+m_{\eta^{\pm}}>m_{W}$
(25)

Although there is no dedicated analysis of LEP-II data in the context of
scotogenic dark matter models, Ref. [37] has discussed LEP II limits for the
case of the Inert Doublet Model, leading to the limits $m_{\eta^{R}}\raise
1.29167pt\hbox{$\;>$\kern-7.5pt\raise-4.73611pt\hbox{$\sim\;$}}80$ GeV and
$m_{\eta^{I}}\raise
1.29167pt\hbox{$\;>$\kern-7.5pt\raise-4.73611pt\hbox{$\sim\;$}}100$ GeV and a
small $\eta_{R}-\eta_{I}$ mass splitting. Altogether, in view of the above,
one can say that intermediate dark matter masses in the range
$80\,\text{GeV}\leq m_{\eta^{R}}\leq 100\,\text{GeV}$ are not inconsistent
with collider constraints. Dark matter heavier than 550 GeV is perfectly
allowed. Dark matter masses in between 100 and 550 GeV could also be possible
in the presence of another dark matter component.

##  6 Electroweak Vacuum Stability

The detailed analysis of the Higgs vacuum within the Standard Model has been
carried out in [38, 39, 40, 41, 42, 43]. Taking into account the updated input
top and Higgs boson mass values one finds that the Standard Model Higgs
quartic coupling $\lambda_{\text{SM}}$ becomes negative at $\mu\simeq 10^{10}$
GeV. This would imply that the Higgs potential is unbounded from below and the
Higgs vacuum is unstable. A dedicated analysis shows that, actually, the
Standard Model Higgs vacuum is metastable with very long lifetime [43].

Here our aim is to determine the parameter region consistent both with dark
matter observations as well as vacuum stability. We first fix the parameters
(quartic couplings and mass of the dark matter candidate) which are consistent
with the present day relic density and the direct dark matter detection
constraints.

In Fig. 6 we have shown the quartic coupling
$\lambda_{345}=\lambda_{3}+\lambda_{4}+\lambda_{5}$ as a function of dark
matter mass $m_{\eta^{R}}$.

Figure 6: Coupling $\lambda_{345}=\lambda_{3}+\lambda_{4}+\lambda_{5}$ as a
function of the dark mass $m_{\eta^{R}}$. The color code is same as in Fig. 2.

The color code is same as in Fig. 2. The lower mass region is already excluded
by experiment. One sees that in the large mass regime above $m_{\eta^{R}}>550$
GeV, the allowed $\lambda_{345}$ values that successfully explain the relic
density while obeying the direct detection limit cover a wide range. For
relatively large couplings, the evolution of quartic couplings can make them
exceed the perturbativity limit even before the Planck scale. We therefore
choose relatively small $\lambda_{3,4,5}$ values. This way $\lambda_{345}$ is
small enough to match the required relic density and to satisfy the direct
detection cross section bound for $m_{\eta^{R}}>550$ GeV.

We now examine the effect of the new particles present in the scoto-seesaw
model upon the stability of the electroweak vacuum. As a “missing partner”
(3,1) type-I seesaw cloned with the simplest scotogenic sector, the scoto-
seesaw contains the a “right-handed” neutrino $N$, together with the dark
particles $f$ and $\eta$. Using SARAH [14] we have computed the two-loop RGEs
of the full theory for all the quartic scalar couplings, as well as Yukawa
couplings, as given in Appendix B.

We now summarise our results. To begin with, in the effective theory where the
heavy singlet fermions $N$ and $f$ are integrated out, we have two natural
threshold scales $\Lambda_{N}\approx M_{N}$ and $\Lambda_{f}\approx M_{f}$.
These masses are obtained from Eq. (4) and (5). Hence, in the RGEs of the full
theory, we can simply take the $Y_{N}$ and $Y_{f}$ Yukawa parameters to be
given as $\theta\left(\mu-M_{N}\right)Y_{N}$, $\theta\left(\mu-
M_{f}\right)Y_{f}$. Clearly they do not run in the effective theory.

Our aim is to study the effect of large as well as small Yukawa couplings on
vacuum stability. As discussed before, we take an adequately small but nonzero
value for $\lambda_{5}$, as required for having a reasonable direct detection
cross section. For example, with Yukawa couplings $Y_{N}\sim
Y_{f}\sim\mathcal{O}(1)$ and very large $M_{N}$, $M_{f}$ values, one sees from
Eq. (18) that one can easily reproduce the solar and atmospheric scale with
$m_{\eta}^{R}\sim\mathcal{O}(1\,\text{TeV})$.

We now illustrate in more detail the relevant parameter space of the scoto-
seesaw model which is consistent with vacuum stability as well as neutrino and
dark matter phenomenology. In order to do so we have chosen three sets of
benchmarks, given as:

  * •

BP1: $M_{N}\sim 10^{14}$ GeV, $M_{f}\sim 10^{12}$ GeV, $m_{\eta^{R}}\sim
10^{3}$ GeV, $\mathbb{Y}_{N}\sim 0.45$, $\mathbb{Y}_{f}\sim 0.45$,

  * •

BP2: $M_{N}\sim 10^{14}$ GeV, $M_{f}\sim 10^{5}$ GeV, $m_{\eta^{R}}\sim
10^{3}$ GeV, $\mathbb{Y}_{N}\sim 0.45$, $\mathbb{Y}_{f}\sim 10^{-4}$,

  * •

BP3: $M_{N}\sim 10^{6}$ GeV, $M_{f}\sim 10^{6}$ GeV, $m_{\eta^{R}}\sim 10^{3}$
GeV, $\mathbb{Y}_{N}\sim 10^{-5}$, $\mathbb{Y}_{f}\sim 10^{-4}$.

In the upper left, upper right and bottom panel of Fig. 7, we have shown the
results for benchmark points BP1, BP2 and BP3, respectively. We have taken
$\lambda_{\eta}=0.1$ and $\lambda=\frac{m_{h}^{2}}{2v^{2}}$ at the electroweak
scale. Recall that in order to have an absolutely stable vacuum, one needs to
satisfy the conditions given in Eqs. (7), (8) and (9) at all energy scales.

Assuming small $\lambda_{5}$ we show in Fig. 7 the evolution of the remaining
four quartic couplings $\lambda$, $\lambda_{\eta}$, $\lambda_{A}$ and
$\lambda_{B}$.

Figure 7: The RGE-evolution of quartic couplings $\lambda$, $\lambda_{\eta}$,
$\lambda_{A}$, $\lambda_{B}$ defined in Eqs. (7),(8),(9). The Yukawas $Y_{N}$,
$Y_{f}$ are also shown. The values given in the boxes are the initial values
at the respective threshold scales. All couplings remain perturbative up to
the Planck Scale. Note that we have fixed $m_{\eta^{R}}=1$ TeV and taken small
$\lambda_{5}=10^{-3}$. See text for more details.

One sees that, with reasonable initial choices, all of the quartic couplings
can remain positive and perturbative all the way up to the Planck scale. Since
the Yukawa couplings have a negative effect in the RGE evolution of $\lambda$,
the required value of the quartic couplings $\lambda_{3}$ and $\lambda_{4}$ is
correspondingly larger, as seen when going from the upper left to the right
panel and finally to the bottom panel in Fig. 7.

To sum up, one sees that the minimal scoto-seesaw model has improved stability
properties compared to the type-I seesaw scenario, due to the new scalars
needed to realize the scotogenic “completion”. The model can explain both
solar and atmospheric neutrino mass scales as well as dark matter, upgrading
the (3, 1) type-I seesaw mechanism, which can only generate the atmospheric
neutrino mass scale. Moreover, the minimal scoto-seesaw model leads to a
stable and perturbative electroweak vacuum all the way up to the Planck scale.
It can therefore be considered as a full consistent theory for neutrino masses
and dark matter.

##  7 High energy behavior of the dark parity

The conservation of the dark parity is a key feature of the scoto-seesaw
model, ensuring dark matter stability as well as the radiative origin of the
solar mass scale. Without this $\mathbb{Z}_{2}$ symmetry conservation, the LDP
would no longer be stable, and also the solar neutrino splitting would not be
“calculable” from Eq. (18). It was first pointed out in Ref. [44] that
renormalization group evolution can alter the scalar potential at high
energies, leading to $\mathbb{Z}_{2}$ breaking. It is easy to understand the
source of $\mathbb{Z}_{2}$ symmetry breaking from the one-loop $\beta$
function of the $m_{\eta}^{2}$ parameter. This is given in Appendix B [45,
46]. One needs to focus on the terms which contribute negatively to the
evolution of $m_{\eta}^{2}$, given in Eq. (42). We see that with relatively
large Yukawa coupling $Y_{f}$ (i.e. $\lambda_{5}\ll 1$) and
$M_{f}^{2}>m_{\eta}^{2}$, the term $-|M_{f}|^{2}|Y_{f}|^{2}$ dominates the
running of $m_{\eta}^{2}$. This can quickly drive $m_{\eta}^{2}$ towards
negative values and induce a minimum of the scalar potential with
$\braket{\eta}\neq 0$. Notice, however, that there are terms in Eq. (42) which
can counter this negative effect. For example, terms proportional to the
quartic scalar couplings $\lambda_{3}$ and $\lambda_{4}$ may do so if their
signs are properly chosen. The contribution to the $m_{\eta}^{2}$ evolution
will be positive for $\lambda_{\eta}>0$ and $\lambda_{3,4}<0$.

Figure 8: RGE evolution of $m_{\eta}$ as a function of the energy scale $\mu$.
For both panels we start running at $m_{\eta}=250\,\text{GeV}$, taking
$Y_{N}=0$, $\lambda_{3}=\lambda_{4}=0.1$ and $\lambda_{5}=10^{-3}$. Different
lines correspond to various Yukawa coupling values, as indicated. Note that
for $M_{f}=10^{6}$GeV (right panel) the Yukawas are much smaller. Notice also
that all curves terminate when $m_{\eta}^{2}$ becomes negative ($m_{\eta}$
imaginary) indicating $\mathbb{Z}_{2}$ breakdown.

Fig. 8 shows the evolution of the scalar mass $m_{\eta}$ as a function of
energy scale $\mu$. The results have been obtained for two values of $M_{f}$,
$M_{f}=10^{3}$ GeV (left panel) and $10^{6}$ GeV (right panel). We have fixed
$m_{\eta}^{2}=250^{2}\,\text{GeV}^{2}$, $\lambda_{3}=\lambda_{4}=0.1$,
$\lambda_{5}\approx 0$ and $Y_{N}=0$, for simplicity. The red-solid, blue-dot-
dashed, orange-dashed and green-dotted lines in the left (right) panel
correspond to four values of the Yukawa coupling $Y_{f}$, as indicated. As
expected, the $\mathbb{Z}_{2}$ breaking scale decreases for larger $M_{f}$ due
to the effect of the term $-|M_{f}|^{2}|Y_{f}|^{2}$. In other words, the
larger the scale $M_{f}$, the smaller the allowed value of the Yukawa coupling
$Y_{f}$ in order to have the $\mathbb{Z}_{2}$ symmetry preserved all the way
up to the Planck scale. From Fig. 8, one sees that the allowed value of this
Yukawa coupling is $Y_{f}\leq 0.2$ for $M_{f}=10^{3}$ GeV, whereas for
$M_{f}=10^{6}$ GeV it is $Y_{f}\leq 10^{-4}$. Although the different quartic
couplings such as $\lambda_{3}$ and $\lambda_{4}$ may alter the details, this
generic behavior remains. To sum up, we found that in the scoto-seesaw model
the dark parity can be preserved up to the Planck scale over large portions of
the parameter space.

##  8 Summary and discussion

We have examined the minimal combination of the the seesaw and scotogenic
paradigms for neutrino mass generation and dark matter able to “explain” the
solar and atmospheric oscillation wavelengths, in Eqs. 1 and 2. The model
provides a simple picture where the “atmospheric” mass scale arises from the
tree-level “missing partner” seesaw, while the “solar” scale is induced
radiatively by the dark sector, see Fig. 1. We have derived the full two-loop
RGEs for the relevant parameters, such as the quartic Higgs self-coupling
$\lambda$ of the Standard Model. The new scalars present in the scoto-seesaw
mechanism improve the stability properties of the electroweak vacuum, as seen
in Fig. 7. We have also explored the consistency of the underlying dark
symmetry, as seen in Fig. 8.

Concerning phenomenology we have discussed scalar dark matter including the
experimental restrictions that follow from colliders, Fig. 5. By taking into
account the relevant annihiliation channels in Fig. 9 we identified viable
parameter regions consistent with the required dark matter relic abundance,
Fig. 2. Direct dark matter detection by nucleon recoil proceeds through the
diagrams in Fig.3. The expected rates are given in Fig. 4 and offer promising
results for upcoming dark matter experiments. We found that the low dark
matter mass region $m_{\eta^{R}}\raise
1.29167pt\hbox{$\;<$\kern-7.5pt\raise-4.73611pt\hbox{$\sim\;$}}60$ GeV is
ruled out by LHC data, but the intermediate region $80\,\text{GeV}\,\raise
1.29167pt\hbox{$\;<$\kern-7.5pt\raise-4.73611pt\hbox{$\sim\;$}}m_{\eta^{R}}\raise
1.29167pt\hbox{$\;<$\kern-7.5pt\raise-4.73611pt\hbox{$\sim\;$}}100\,\text{GeV}$
is still allowed both by LHC and LEP data. The heavier mass region
$m_{\eta^{R}}\raise
1.29167pt\hbox{$\;>$\kern-7.5pt\raise-4.73611pt\hbox{$\sim\;$}}550$ GeV is
free from collider constraints.

Our construction is very attractive from the point of view of neutrino
physics. In contrast to the original scotogenic model where 2 (or 3) species
of either dark fermion or dark scalars are needed to generate masses for 2 (or
3) neutrinos, our dark sector is truly minimal, with only one dark fermion and
one dark scalar. Therefore, the allowed parameter space differs from the
original scotogenic model, though these differences do not translate into a
phenomenological smoking-gun signature which can easily distinguish it from
canonical scotogenic model in Ref. [10].

Before closing we also note an important phenomenological implication of the
minimal scoto-seesaw model that can make it testable. Namely, it implies that
one of the neutrinos is massless, as can be readily seen from Eq. (16). This
leads to a lower bound on the neutrinoless double beta decay rates even for a
normal-ordered neutrino mass spectrum [47, 48, 49, 32]. As shown in Fig.2 of
Ref. [32], for an inverted mass spectrum the $0\nu\beta\beta$ lower bound lies
substantially higher than in the generic case for that ordering. As a result
it falls within the sensitivity of future experiments such as nEXO [50]. In
the scoto-seesaw one has that a positive $0\nu\beta\beta$ decay discovery
would open bright prospects for the underpinning of the value of the relevant
elusive Majorana phase. As discussed in [4], the expected rates for lepton
flavour violation processes can also lie within reach of experiments,
providing additional signatures. In summary, the scoto-seesaw model is a
theoretically consistent and phenomenologicaly interesting “dark matter
completion” of the type-I seesaw mechanism.

###### Acknowledgements.

This work is supported by the Spanish grant FPA2017-85216-P (AEI/FEDER, UE),
PROMETEO/2018/165 (Generalitat Valenciana). R.S. is supported by SERB,
Government of India grant SRG/2020/002303.

##  Appendix A Scalar dark matter annihilation mechanisms

In the minimal scoto-seesaw model the relic abundance of the lightest dark
particle $\eta^{R}$ is determined by the following annihilation and
coannihilation diagrams.

Figure 9: Annihilation and coannihilation diagrams contributing to the relic
abundance of $\eta^{R}$.

##  Appendix B Renormalization group equations

The evolution of a given parameter c in the theory is described by the
appropriate $\beta$ function, given by,

$\displaystyle\frac{dc}{dt}\equiv\beta_{c}=\frac{1}{16\pi^{2}}\beta_{c}^{(1)}+\frac{1}{(16\pi^{2})^{2}}\beta_{c}^{(2)}\,.$

where $\beta_{c}^{(1)}$ are the one-loop renormalization group (RG)
coefficients, while $\beta_{c}^{(2)}$ correspond to the two-loop RG
corrections.

###  2.1 Higgs quartic scalar self coupling

The scalar potential of the scoto-seesaw model is given in Eq. (6). The model
contains five quartic couplings
$\lambda,\lambda_{\eta},\lambda_{3},\lambda_{4},\lambda_{5}$. The one-loop and
two-loop RG equations of the Higgs quartic self-coupling $\lambda$ are given
by

$\displaystyle\beta_{\lambda}^{(1)}$
$\displaystyle=+\frac{27}{200}g_{1}^{4}+\frac{9}{20}g_{1}^{2}g_{2}^{2}+\frac{9}{8}g_{2}^{4}+2\lambda_{3}^{2}+2\lambda_{3}\lambda_{4}+\lambda_{4}^{2}+\lambda_{5}^{2}-\frac{9}{5}g_{1}^{2}\lambda-9g_{2}^{2}\lambda+24\lambda^{2}$
$\displaystyle+12\lambda
y_{t}^{2}+4\lambda\mbox{Tr}\Big{(}{Y_{N}Y_{N}^{\dagger}}\Big{)}-6y_{t}^{4}-2\mbox{Tr}\Big{(}{Y_{N}Y_{N}^{\dagger}Y_{N}Y_{N}^{\dagger}}\Big{)}\,,$
(26) $\displaystyle\beta_{\lambda}^{(2)}$
$\displaystyle=-\frac{3537}{2000}g_{1}^{6}-\frac{1719}{400}g_{1}^{4}g_{2}^{2}-\frac{303}{80}g_{1}^{2}g_{2}^{4}+\frac{291}{16}g_{2}^{6}+\frac{9}{10}g_{1}^{4}\lambda_{3}+\frac{15}{2}g_{2}^{4}\lambda_{3}+\frac{12}{5}g_{1}^{2}\lambda_{3}^{2}+12g_{2}^{2}\lambda_{3}^{2}$
$\displaystyle-8\lambda_{3}^{3}+\frac{9}{20}g_{1}^{4}\lambda_{4}+\frac{3}{2}g_{1}^{2}g_{2}^{2}\lambda_{4}+\frac{15}{4}g_{2}^{4}\lambda_{4}+\frac{12}{5}g_{1}^{2}\lambda_{3}\lambda_{4}+12g_{2}^{2}\lambda_{3}\lambda_{4}-12\lambda_{3}^{2}\lambda_{4}+\frac{6}{5}g_{1}^{2}\lambda_{4}^{2}$
$\displaystyle+3g_{2}^{2}\lambda_{4}^{2}-16\lambda_{3}\lambda_{4}^{2}-6\lambda_{4}^{3}-\frac{3}{5}g_{1}^{2}\lambda_{5}^{2}-20\lambda_{3}\lambda_{5}^{2}-22\lambda_{4}\lambda_{5}^{2}+\frac{1953}{200}g_{1}^{4}\lambda+\frac{117}{20}g_{1}^{2}g_{2}^{2}\lambda$
$\displaystyle-\frac{51}{8}g_{2}^{4}\lambda-20\lambda_{3}^{2}\lambda-20\lambda_{3}\lambda_{4}\lambda-12\lambda_{4}^{2}\lambda-14\lambda_{5}^{2}\lambda+\frac{108}{5}g_{1}^{2}\lambda^{2}+108g_{2}^{2}\lambda^{2}-312\lambda^{3}$
$\displaystyle-4\lambda_{3}^{2}\mbox{Tr}\Big{(}{Y_{f}Y_{f}^{\dagger}}\Big{)}-4\lambda_{3}\lambda_{4}\mbox{Tr}\Big{(}{Y_{f}Y_{f}^{\dagger}}\Big{)}-2\lambda_{4}^{2}\mbox{Tr}\Big{(}{Y_{f}Y_{f}^{\dagger}}\Big{)}-2\lambda_{5}^{2}\mbox{Tr}\Big{(}{Y_{f}Y_{f}^{\dagger}}\Big{)}-\frac{171}{100}g_{1}^{4}y_{t}^{2}$
$\displaystyle+\frac{63}{10}g_{1}^{2}g_{2}^{2}y_{t}^{2}-\frac{9}{4}g_{2}^{4}y_{t}^{2}+\frac{17}{2}g_{1}^{2}\lambda
y_{t}^{2}+\frac{45}{2}g_{2}^{2}\lambda y_{t}^{2}+80g_{3}^{2}\lambda
y_{t}^{2}-144\lambda^{2}y_{t}^{2}-\frac{9}{100}g_{1}^{4}\mbox{Tr}\Big{(}{Y_{N}Y_{N}^{\dagger}}\Big{)}$
$\displaystyle-\frac{3}{10}g_{1}^{2}g_{2}^{2}\mbox{Tr}\Big{(}{Y_{N}Y_{N}^{\dagger}}\Big{)}-\frac{3}{4}g_{2}^{4}\mbox{Tr}\Big{(}{Y_{N}Y_{N}^{\dagger}}\Big{)}+\frac{3}{2}g_{1}^{2}\lambda\mbox{Tr}\Big{(}{Y_{N}Y_{N}^{\dagger}}\Big{)}+\frac{15}{2}g_{2}^{2}\lambda\mbox{Tr}\Big{(}{Y_{N}Y_{N}^{\dagger}}\Big{)}$
$\displaystyle-48\lambda^{2}\mbox{Tr}\Big{(}{Y_{N}Y_{N}^{\dagger}}\Big{)}-3\lambda\mbox{Tr}\Big{(}{Y_{f}Y_{f}^{\dagger}Y_{N}Y_{N}^{\dagger}}\Big{)}-\frac{8}{5}g_{1}^{2}y_{t}^{4}-32g_{3}^{2}y_{t}^{4}-3\lambda
y_{t}^{4}-\lambda\mbox{Tr}\Big{(}{Y_{N}Y_{N}^{\dagger}Y_{N}Y_{N}^{\dagger}}\Big{)}$
$\displaystyle+30y_{t}^{6}+2\mbox{Tr}\Big{(}{Y_{f}Y_{f}^{\dagger}Y_{N}Y_{N}^{\dagger}Y_{N}Y_{N}^{\dagger}}\Big{)}+10\mbox{Tr}\Big{(}{Y_{N}Y_{N}^{\dagger}Y_{N}Y_{N}^{\dagger}Y_{N}Y_{N}^{\dagger}}\Big{)}\,.$
(27) $\displaystyle\beta_{\lambda_{\eta}}^{(1)}$
$\displaystyle=+\frac{27}{200}g_{1}^{4}+\frac{9}{8}g_{2}^{4}+2\lambda_{3}^{2}+2\lambda_{3}\lambda_{4}+\lambda_{4}^{2}+\lambda_{5}^{2}+\frac{9}{20}g_{1}^{2}\Big{(}-4\lambda_{\eta}+g_{2}^{2}\Big{)}-9g_{2}^{2}\lambda_{\eta}+24\lambda_{\eta}^{2}$
$\displaystyle+4\lambda_{\eta}\text{Tr}({Y_{f}Y_{f}^{\dagger}})-2\text{Tr}(\Big{(}{Y_{f}Y_{f}^{\dagger}Y_{f}Y_{f}^{\dagger}}\Big{)}\Big{)}$
(28)

$\displaystyle\beta_{\lambda_{\eta}}^{(2)}$
$\displaystyle=-\frac{3537}{2000}g_{1}^{6}-\frac{1719}{400}g_{1}^{4}g_{2}^{2}-\frac{303}{80}g_{1}^{2}g_{2}^{4}+\frac{291}{16}g_{2}^{6}+\frac{9}{10}g_{1}^{4}\lambda_{3}+\frac{15}{2}g_{2}^{4}\lambda_{3}+\frac{12}{5}g_{1}^{2}\lambda_{3}^{2}+12g_{2}^{2}\lambda_{3}^{2}$
$\displaystyle-8\lambda_{3}^{3}+\frac{9}{20}g_{1}^{4}\lambda_{4}+\frac{3}{2}g_{1}^{2}g_{2}^{2}\lambda_{4}+\frac{15}{4}g_{2}^{4}\lambda_{4}+\frac{12}{5}g_{1}^{2}\lambda_{3}\lambda_{4}+12g_{2}^{2}\lambda_{3}\lambda_{4}-12\lambda_{3}^{2}\lambda_{4}+\frac{6}{5}g_{1}^{2}\lambda_{4}^{2}$
$\displaystyle+3g_{2}^{2}\lambda_{4}^{2}-16\lambda_{3}\lambda_{4}^{2}-6\lambda_{4}^{3}-\frac{3}{5}g_{1}^{2}\lambda_{5}^{2}-20\lambda_{3}\lambda_{5}^{2}-22\lambda_{4}\lambda_{5}^{2}+\frac{1953}{200}g_{1}^{4}\lambda_{\eta}+\frac{117}{20}g_{1}^{2}g_{2}^{2}\lambda_{\eta}$
$\displaystyle-\frac{51}{8}g_{2}^{4}\lambda_{\eta}-20\lambda_{3}^{2}\lambda_{\eta}-20\lambda_{3}\lambda_{4}\lambda_{\eta}-12\lambda_{4}^{2}\lambda_{\eta}-14\lambda_{5}^{2}\lambda_{\eta}+\frac{108}{5}g_{1}^{2}\lambda_{\eta}^{2}+108g_{2}^{2}\lambda_{\eta}^{2}-312\lambda_{\eta}^{3}$
$\displaystyle-\lambda_{\eta}\text{Tr}\Big{(}{Y_{f}Y_{f}^{\dagger}Y_{f}Y_{f}^{\dagger}}\Big{)}+10\text{Tr}(\Big{(}{Y_{f}Y_{f}^{\dagger}}Y_{f}Y_{f}^{\dagger}Y_{f}Y_{f}^{\dagger}\Big{)}-3\lambda_{\eta}\text{Tr}\Big{(}{Y_{f}Y_{N}^{\dagger}}{Y_{N}Y_{f}^{\dagger}}\Big{)}-4\lambda_{3}^{2}\text{Tr}\Big{(}{Y_{N}Y_{N}^{\dagger}}\Big{)}$
$\displaystyle-4\lambda_{3}\lambda_{4}\text{Tr}\Big{(}{Y_{N}Y_{N}^{\dagger}}\Big{)}-2\lambda_{4}^{2}\text{Tr}\Big{(}{Y_{N}Y_{N}^{\dagger}}\Big{)}-2\lambda_{5}^{2}\text{Tr}\Big{(}{Y_{N}Y_{N}^{\dagger}}\Big{)}+\text{Tr}\Big{(}{Y_{f}Y_{f}^{\dagger}}\Big{)}\Big{(}2\text{Tr}\Big{(}{Y_{f}Y_{N}^{\dagger}}{Y_{N}Y_{f}^{\dagger}}\Big{)}$
$\displaystyle-\frac{3}{100}\Big{(}10g_{1}^{2}\Big{(}-5\lambda_{\eta}+g_{2}^{2}\Big{)}+25\Big{(}-10g_{2}^{2}\lambda_{\eta}+64\lambda_{\eta}^{2}+g_{2}^{4}\Big{)}+3g_{1}^{4}\Big{)}\Big{)}-12\lambda_{3}^{2}y_{t}^{2}$
$\displaystyle-12\lambda_{3}\lambda_{4}y_{t}^{2}-6\lambda_{4}^{2}y_{t}^{2}-6\lambda_{5}^{2}y_{t}^{2}$
(29) $\displaystyle\beta_{\lambda_{4}}^{(1)}$
$\displaystyle=+\frac{9}{5}g_{1}^{2}g_{2}^{2}-\frac{9}{5}g_{1}^{2}\lambda_{4}-9g_{2}^{2}\lambda_{4}+8\lambda_{3}\lambda_{4}+4\lambda_{4}^{2}+8\lambda_{5}^{2}+4\lambda_{4}\lambda_{\eta}+4\lambda_{4}\lambda+2\lambda_{4}\text{Tr}\Big{(}{Y_{f}Y_{f}^{\dagger}}\Big{)}$
$\displaystyle-4\text{Tr}\Big{(}{Y_{f}Y_{N}^{\dagger}}{Y_{N}Y_{f}^{\dagger}}\Big{)}+2\lambda_{4}\text{Tr}\Big{(}{Y_{N}Y_{N}^{\dagger}}\Big{)}+6\lambda_{4}y_{t}^{2}$
(30) $\displaystyle\beta_{\lambda_{4}}^{(2)}$
$\displaystyle=-\frac{657}{50}g_{1}^{4}g_{2}^{2}-\frac{42}{5}g_{1}^{2}g_{2}^{4}+\frac{6}{5}g_{1}^{2}g_{2}^{2}\lambda_{3}+\frac{1413}{200}g_{1}^{4}\lambda_{4}+\frac{153}{20}g_{1}^{2}g_{2}^{2}\lambda_{4}-\frac{231}{8}g_{2}^{4}\lambda_{4}+\frac{12}{5}g_{1}^{2}\lambda_{3}\lambda_{4}$
$\displaystyle+36g_{2}^{2}\lambda_{3}\lambda_{4}-28\lambda_{3}^{2}\lambda_{4}+\frac{24}{5}g_{1}^{2}\lambda_{4}^{2}+18g_{2}^{2}\lambda_{4}^{2}-28\lambda_{3}\lambda_{4}^{2}+\frac{48}{5}g_{1}^{2}\lambda_{5}^{2}+54g_{2}^{2}\lambda_{5}^{2}-48\lambda_{3}\lambda_{5}^{2}$
$\displaystyle-26\lambda_{4}\lambda_{5}^{2}+6g_{1}^{2}g_{2}^{2}\lambda_{\eta}+\frac{24}{5}g_{1}^{2}\lambda_{4}\lambda_{\eta}-80\lambda_{3}\lambda_{4}\lambda_{\eta}-40\lambda_{4}^{2}\lambda_{\eta}-48\lambda_{5}^{2}\lambda_{\eta}-28\lambda_{4}\lambda_{\eta}^{2}+6g_{1}^{2}g_{2}^{2}\lambda$
$\displaystyle+\frac{24}{5}g_{1}^{2}\lambda_{4}\lambda-80\lambda_{3}\lambda_{4}\lambda-40\lambda_{4}^{2}\lambda-48\lambda_{5}^{2}\lambda-28\lambda_{4}\lambda^{2}-\frac{9}{2}\lambda_{4}\text{Tr}\Big{(}{Y_{f}Y_{f}^{\dagger}Y_{f}Y_{f}^{\dagger}}\Big{)}+\text{Tr}\Big{(}Y_{N}Y_{N}^{\dagger}\Big{)}$
$\displaystyle\Big{(}-\frac{3}{5}g_{1}^{2}g_{2}^{2}+\frac{3}{4}g_{1}^{2}\lambda_{4}+\frac{15}{4}g_{2}^{2}\lambda_{4}-8\lambda_{3}\lambda_{4}-4\lambda_{4}^{2}-8\lambda_{5}^{2}-8\lambda_{4}\lambda\Big{)}-\frac{9}{2}\lambda_{4}\text{Tr}\Big{(}{Y_{N}Y_{N}^{\dagger}Y_{N}Y_{N}^{\dagger}}\Big{)}$
$\displaystyle+\text{Tr}\Big{(}{Y_{f}Y_{N}^{\dagger}}{Y_{N}Y_{f}^{\dagger}}\Big{)}\Big{(}-3\lambda_{4}+8\lambda_{3}+8\text{Tr}\Big{(}{Y_{N}Y_{N}^{\dagger}}\Big{)}\Big{)}+\text{Tr}\Big{(}{Y_{f}Y_{f}^{\dagger}}\Big{)}\Big{(}-\frac{3}{5}g_{1}^{2}g_{2}^{2}+\frac{3}{4}g_{1}^{2}\lambda_{4}$
$\displaystyle+\frac{15}{4}g_{2}^{2}\lambda_{4}-8\lambda_{3}\lambda_{4}-4\lambda_{4}^{2}-8\lambda_{5}^{2}-8\lambda_{4}\lambda_{\eta}+8\text{Tr}\Big{(}{Y_{f}Y_{N}^{\dagger}}{Y_{N}Y_{f}^{\dagger}}\Big{)}\Big{)}+y_{t}^{2}\Big{(}\frac{63}{5}g_{1}^{2}g_{2}^{2}+\frac{17}{4}g_{1}^{2}\lambda_{4}$
$\displaystyle+\frac{45}{4}g_{2}^{2}\lambda_{4}+40g_{3}^{2}\lambda_{4}-24\lambda_{3}\lambda_{4}-12\lambda_{4}^{2}-24\lambda_{5}^{2}-24\lambda_{4}\lambda-\frac{27}{2}\lambda_{4}y_{t}^{2}\Big{)}$
(31) $\displaystyle\beta_{\lambda_{3}}^{(1)}$
$\displaystyle=+\frac{27}{100}g_{1}^{4}-\frac{9}{10}g_{1}^{2}g_{2}^{2}+\frac{9}{4}g_{2}^{4}-\frac{9}{5}g_{1}^{2}\lambda_{3}-9g_{2}^{2}\lambda_{3}+4\lambda_{3}^{2}+2\lambda_{4}^{2}+2\lambda_{5}^{2}+12\lambda_{3}\lambda_{\eta}$
$\displaystyle+4\lambda_{4}\lambda_{\eta}+12\lambda_{3}\lambda+4\lambda_{4}\lambda+2\lambda_{3}\text{Tr}\Big{(}{Y_{f}Y_{f}^{\dagger}}\Big{)}+2\lambda_{3}\text{Tr}\Big{(}{Y_{N}Y_{N}^{\dagger}}\Big{)}+6\lambda_{3}y_{t}^{2}$
(32)

$\displaystyle\beta_{\lambda_{3}}^{(2)}$
$\displaystyle=-\frac{3537}{1000}g_{1}^{6}+\frac{909}{200}g_{1}^{4}g_{2}^{2}+\frac{33}{40}g_{1}^{2}g_{2}^{4}+\frac{291}{8}g_{2}^{6}+\frac{1773}{200}g_{1}^{4}\lambda_{3}+\frac{33}{20}g_{1}^{2}g_{2}^{2}\lambda_{3}-\frac{111}{8}g_{2}^{4}\lambda_{3}+\frac{6}{5}g_{1}^{2}\lambda_{3}^{2}$
$\displaystyle+6g_{2}^{2}\lambda_{3}^{2}-12\lambda_{3}^{3}+\frac{9}{10}g_{1}^{4}\lambda_{4}-\frac{9}{5}g_{1}^{2}g_{2}^{2}\lambda_{4}+\frac{15}{2}g_{2}^{4}\lambda_{4}-12g_{2}^{2}\lambda_{3}\lambda_{4}-4\lambda_{3}^{2}\lambda_{4}-\frac{6}{5}g_{1}^{2}\lambda_{4}^{2}$
$\displaystyle+6g_{2}^{2}\lambda_{4}^{2}-16\lambda_{3}\lambda_{4}^{2}-12\lambda_{4}^{3}+\frac{12}{5}g_{1}^{2}\lambda_{5}^{2}-18\lambda_{3}\lambda_{5}^{2}-44\lambda_{4}\lambda_{5}^{2}+\frac{27}{10}g_{1}^{4}\lambda_{\eta}-3g_{1}^{2}g_{2}^{2}\lambda_{\eta}$
$\displaystyle+\frac{45}{2}g_{2}^{4}\lambda_{\eta}+\frac{72}{5}g_{1}^{2}\lambda_{3}\lambda_{\eta}+72g_{2}^{2}\lambda_{3}\lambda_{\eta}-72\lambda_{3}^{2}\lambda_{\eta}+\frac{24}{5}g_{1}^{2}\lambda_{4}\lambda_{\eta}+36g_{2}^{2}\lambda_{4}\lambda_{\eta}-32\lambda_{3}\lambda_{4}\lambda_{\eta}$
$\displaystyle-28\lambda_{4}^{2}\lambda_{\eta}-36\lambda_{5}^{2}\lambda_{\eta}-60\lambda_{3}\lambda_{\eta}^{2}-16\lambda_{4}\lambda_{\eta}^{2}+\frac{27}{10}g_{1}^{4}\lambda-3g_{1}^{2}g_{2}^{2}\lambda+\frac{45}{2}g_{2}^{4}\lambda+\frac{72}{5}g_{1}^{2}\lambda_{3}\lambda$
$\displaystyle+72g_{2}^{2}\lambda_{3}\lambda-72\lambda_{3}^{2}\lambda+\frac{24}{5}g_{1}^{2}\lambda_{4}\lambda+36g_{2}^{2}\lambda_{4}\lambda-32\lambda_{3}\lambda_{4}\lambda-28\lambda_{4}^{2}\lambda-36\lambda_{5}^{2}\lambda-60\lambda_{3}\lambda^{2}$
$\displaystyle-16\lambda_{4}\lambda^{2}-\frac{9}{2}\lambda_{3}\text{Tr}\Big{(}{Y_{f}Y_{f}^{\dagger}}Y_{f}Y_{f}^{\dagger}\Big{)}-\frac{9}{100}g_{1}^{4}\text{Tr}\Big{(}{Y_{N}Y_{N}^{\dagger}}\Big{)}+\frac{3}{10}g_{1}^{2}g_{2}^{2}\text{Tr}\Big{(}{Y_{N}Y_{N}^{\dagger}}\Big{)}-\frac{3}{4}g_{2}^{4}\text{Tr}\Big{(}{Y_{N}Y_{N}^{\dagger}}\Big{)}$
$\displaystyle+\frac{3}{4}g_{1}^{2}\lambda_{3}\text{Tr}\Big{(}{Y_{N}Y_{N}^{\dagger}}\Big{)}+\frac{15}{4}g_{2}^{2}\lambda_{3}\text{Tr}\Big{(}{Y_{N}Y_{N}^{\dagger}}\Big{)}-4\lambda_{3}^{2}\text{Tr}\Big{(}{Y_{N}Y_{N}^{\dagger}}\Big{)}-2\lambda_{4}^{2}\text{Tr}\Big{(}{Y_{N}Y_{N}^{\dagger}}\Big{)}-2\lambda_{5}^{2}\text{Tr}\Big{(}{Y_{N}Y_{N}^{\dagger}}\Big{)}$
$\displaystyle-24\lambda_{3}\lambda\text{Tr}\Big{(}{Y_{N}Y_{N}^{\dagger}}\Big{)}-8\lambda_{4}\lambda\text{Tr}\Big{(}{Y_{N}Y_{N}^{\dagger}}\Big{)}-\frac{9}{2}\lambda_{3}\text{Tr}\Big{(}{Y_{N}Y_{N}^{\dagger}}Y_{N}Y_{N}^{\dagger}\Big{)}+\text{Tr}\Big{(}{Y_{f}Y_{f}^{\dagger}}\Big{)}\Big{(}-\frac{9}{100}g_{1}^{4}$
$\displaystyle+\frac{3}{10}g_{1}^{2}g_{2}^{2}-\frac{3}{4}g_{2}^{4}+\frac{3}{4}g_{1}^{2}\lambda_{3}+\frac{15}{4}g_{2}^{2}\lambda_{3}-4\lambda_{3}^{2}-2\lambda_{4}^{2}-2\lambda_{5}^{2}-24\lambda_{3}\lambda_{\eta}-8\lambda_{4}\lambda_{\eta}+4\text{Tr}\Big{(}{Y_{f}Y_{N}^{\dagger}}{Y_{N}Y_{f}^{\dagger}}\Big{)}\Big{)}$
$\displaystyle+\text{Tr}\Big{(}{Y_{f}Y_{N}^{\dagger}}\Big{)}\Big{(}\text{Tr}\Big{(}{Y_{N}Y_{f}^{\dagger}}\Big{)}\Big{(}-3\lambda_{3}+4\text{Tr}\Big{(}{Y_{N}Y_{N}^{\dagger}}\Big{)}+8\lambda_{4}\Big{)}\Big{)}+y_{t}^{2}\Big{(}-\frac{171}{100}g_{1}^{4}-\frac{63}{10}g_{1}^{2}g_{2}^{2}-\frac{9}{4}g_{2}^{4}$
$\displaystyle+\frac{17}{4}g_{1}^{2}\lambda_{3}+\frac{45}{4}g_{2}^{2}\lambda_{3}+40g_{3}^{2}\lambda_{3}-12\lambda_{3}^{2}-6\lambda_{4}^{2}-6\lambda_{5}^{2}-72\lambda_{3}\lambda-24\lambda_{4}\lambda-\frac{27}{2}\lambda_{3}y_{t}^{2}\Big{)}$
(33) $\displaystyle\beta_{\lambda_{5}}^{(1)}$
$\displaystyle=-\frac{9}{5}g_{1}^{2}\lambda_{5}-9g_{2}^{2}\lambda_{5}+8\lambda_{3}\lambda_{5}+12\lambda_{4}\lambda_{5}+4\lambda_{5}\lambda_{\eta}+4\lambda_{5}\lambda+2\lambda_{5}\text{Tr}\Big{(}{Y_{f}Y_{f}^{\dagger}}\Big{)}+2\lambda_{5}\text{Tr}\Big{(}{Y_{N}Y_{N}^{\dagger}}\Big{)}$
$\displaystyle+6\lambda_{5}y_{t}^{2}$ (34)

$\displaystyle\beta_{\lambda_{5}}^{(2)}$
$\displaystyle=+\frac{1413}{200}g_{1}^{4}\lambda_{5}+\frac{57}{20}g_{1}^{2}g_{2}^{2}\lambda_{5}-\frac{231}{8}g_{2}^{4}\lambda_{5}+\frac{48}{5}g_{1}^{2}\lambda_{3}\lambda_{5}+36g_{2}^{2}\lambda_{3}\lambda_{5}-28\lambda_{3}^{2}\lambda_{5}+\frac{72}{5}g_{1}^{2}\lambda_{4}\lambda_{5}$
$\displaystyle+72g_{2}^{2}\lambda_{4}\lambda_{5}-76\lambda_{3}\lambda_{4}\lambda_{5}-32\lambda_{4}^{2}\lambda_{5}+6\lambda_{5}^{3}-\frac{12}{5}g_{1}^{2}\lambda_{5}\lambda_{\eta}-80\lambda_{3}\lambda_{5}\lambda_{\eta}-88\lambda_{4}\lambda_{5}\lambda_{\eta}-28\lambda_{5}\lambda_{\eta}^{2}$
$\displaystyle-\frac{12}{5}g_{1}^{2}\lambda_{5}\lambda-80\lambda_{3}\lambda_{5}\lambda-88\lambda_{4}\lambda_{5}\lambda-28\lambda_{5}\lambda^{2}+\frac{1}{4}\lambda_{5}\Big{(}15g_{2}^{2}-16\Big{(}2\lambda_{3}+2\lambda_{\eta}+3\lambda_{4}\Big{)}+3g_{1}^{2}\Big{)}\text{Tr}\Big{(}{Y_{f}Y_{f}^{\dagger}}\Big{)}$
$\displaystyle-\frac{1}{2}\lambda_{5}\text{Tr}\Big{(}{Y_{f}Y_{f}^{\dagger}Y_{f}Y_{f}^{\dagger}}\Big{)}-3\lambda_{5}\text{Tr}\Big{(}{Y_{f}Y_{N}^{\dagger}}{Y_{N}Y_{f}^{\dagger}}\Big{)}+\frac{3}{4}g_{1}^{2}\lambda_{5}\text{Tr}\Big{(}{Y_{N}Y_{N}^{\dagger}}\Big{)}+\frac{15}{4}g_{2}^{2}\lambda_{5}\text{Tr}\Big{(}{Y_{N}Y_{N}^{\dagger}}\Big{)}$
$\displaystyle-8\lambda_{3}\lambda_{5}\text{Tr}\Big{(}{Y_{N}Y_{N}^{\dagger}}\Big{)}-12\lambda_{4}\lambda_{5}\text{Tr}\Big{(}{Y_{N}Y_{N}^{\dagger}}\Big{)}-8\lambda_{5}\lambda\text{Tr}\Big{(}{Y_{N}Y_{N}^{\dagger}}\Big{)}-\frac{1}{2}\lambda_{5}\text{Tr}\Big{(}{Y_{N}Y_{N}^{\dagger}}Y_{N}Y_{N}^{\dagger}\Big{)}$
$\displaystyle+y_{t}^{2}\Big{(}\frac{17}{4}g_{1}^{2}\lambda_{5}+\frac{45}{4}g_{2}^{2}\lambda_{5}+40g_{3}^{2}\lambda_{5}-24\lambda_{3}\lambda_{5}-36\lambda_{4}\lambda_{5}-24\lambda_{5}\lambda-\frac{3}{2}\lambda_{5}y_{t}^{2}\Big{)}$
(35)

###  2.2 Yukawa Couplings

The one-loop and two-loop RG equations for the Yukawa couplings $Y_{f}$,
$Y_{N}$ and $y_{t}$ are given by

$\displaystyle\beta_{Y_{f}}^{(1)}$
$\displaystyle=\frac{1}{20}\Big{(}10\left(3{Y_{f}Y_{f}^{\dagger}Y_{f}}+{Y_{N}Y_{N}^{\dagger}Y_{f}}\right)+Y_{f}\Big{(}20\mbox{Tr}\Big{(}{Y_{f}Y_{f}^{\dagger}}\Big{)}-9\Big{(}5g_{2}^{2}+g_{1}^{2}\Big{)}\Big{)}\Big{)}\,,$
(36) $\displaystyle\beta_{Y_{f}}^{(2)}$
$\displaystyle=+\frac{1}{80}\Big{(}33g_{1}^{2}{Y_{N}Y_{N}^{\dagger}Y_{f}}+165g_{2}^{2}{Y_{N}Y_{N}^{\dagger}Y_{f}}-160\lambda_{3}{Y_{N}Y_{N}^{\dagger}Y_{f}}-320\lambda_{4}{Y_{N}Y_{N}^{\dagger}Y_{f}}$
$\displaystyle+120{Y_{f}Y_{f}^{\dagger}Y_{f}Y_{f}^{\dagger}Y_{f}}-20{Y_{f}Y_{f}^{\dagger}Y_{N}Y_{N}^{\dagger}Y_{f}}-20{Y_{N}Y_{N}^{\dagger}Y_{N}Y_{N}^{\dagger}Y_{f}}-180{Y_{N}Y_{N}^{\dagger}Y_{f}}y_{t}^{2}$
$\displaystyle+3{Y_{f}Y_{f}^{\dagger}Y_{f}}\Big{(}225g_{2}^{2}-320\lambda_{\eta}-60\mbox{Tr}\Big{(}{Y_{f}Y_{f}^{\dagger}}\Big{)}+93g_{1}^{2}\Big{)}+Y_{f}\Big{(}\frac{117}{200}g_{1}^{4}-\frac{27}{20}g_{1}^{2}g_{2}^{2}$
$\displaystyle-\frac{21}{4}g_{2}^{4}+\lambda_{3}^{2}+\lambda_{3}\lambda_{4}+\lambda_{4}^{2}+\frac{3}{2}\lambda_{5}^{2}+6\lambda_{\eta}^{2}+\frac{3}{8}\Big{(}5g_{2}^{2}+g_{1}^{2}\Big{)}\mbox{Tr}\Big{(}{Y_{f}Y_{f}^{\dagger}}\Big{)}$
$\displaystyle-\frac{9}{4}\mbox{Tr}\Big{(}{Y_{f}Y_{f}^{\dagger}Y_{f}Y_{f}^{\dagger}}\Big{)}-\frac{3}{4}\mbox{Tr}\Big{(}{Y_{f}Y_{f}^{\dagger}Y_{N}Y_{N}^{\dagger}}\Big{)}\Big{)}\,,$
(37) $\displaystyle\beta_{Y_{N}}^{(1)}$
$\displaystyle=+\frac{1}{2}\Big{(}3{Y_{N}Y_{N}^{\dagger}Y_{N}}+{Y_{f}Y_{f}^{\dagger}Y_{N}}\Big{)}+Y_{N}\Big{(}3y_{t}^{2}-\frac{9}{20}g_{1}^{2}-\frac{9}{4}g_{2}^{2}+\mbox{Tr}\Big{(}{Y_{N}Y_{N}^{\dagger}}\Big{)}\Big{)}\,,$
(38) $\displaystyle\beta_{Y_{N}}^{(2)}$
$\displaystyle=+\frac{1}{80}\Big{(}279g_{1}^{2}{Y_{N}Y_{N}^{\dagger}Y_{N}}+675g_{2}^{2}{Y_{N}Y_{N}^{\dagger}Y_{N}}-960\lambda{Y_{N}Y_{N}^{\dagger}Y_{N}}-20{Y_{f}Y_{f}^{\dagger}Y_{f}Y_{f}^{\dagger}Y_{N}}$
$\displaystyle-20{Y_{N}Y_{N}^{\dagger}Y_{f}Y_{f}^{\dagger}Y_{N}}+120{Y_{N}Y_{N}^{\dagger}Y_{N}Y_{N}^{\dagger}Y_{N}}+{Y_{f}Y_{f}^{\dagger}Y_{N}}\Big{(}-160\lambda_{3}+165g_{2}^{2}-320\lambda_{4}$
$\displaystyle+33g_{1}^{2}-60\mbox{Tr}\Big{(}{Y_{f}Y_{f}^{\dagger}}\Big{)}\Big{)}-540{Y_{N}Y_{N}^{\dagger}Y_{N}}y_{t}^{2}-180{Y_{N}Y_{N}^{\dagger}Y_{N}}\mbox{Tr}\Big{(}{Y_{N}Y_{N}^{\dagger}}\Big{)}$
$\displaystyle+Y_{N}\Big{(}\frac{117}{200}g_{1}^{4}-\frac{27}{20}g_{1}^{2}g_{2}^{2}-\frac{21}{4}g_{2}^{4}+\lambda_{3}^{2}+\lambda_{3}\lambda_{4}+\lambda_{4}^{2}+\frac{3}{2}\lambda_{5}^{2}+6\lambda^{2}+\frac{17}{8}g_{1}^{2}y_{t}^{2}$
$\displaystyle+\frac{45}{8}g_{2}^{2}y_{t}^{2}+20g_{3}^{2}y_{t}^{2}+\frac{3}{8}g_{1}^{2}\mbox{Tr}\Big{(}{Y_{N}Y_{N}^{\dagger}}\Big{)}+\frac{15}{8}g_{2}^{2}\mbox{Tr}\Big{(}{Y_{N}Y_{N}^{\dagger}}\Big{)}-\frac{3}{4}\mbox{Tr}\Big{(}{Y_{f}Y_{f}^{\dagger}Y_{N}Y_{N}^{\dagger}}\Big{)}$
$\displaystyle-\frac{27}{4}y_{t}^{4}-\frac{9}{4}\mbox{Tr}\Big{(}{Y_{N}Y_{N}^{\dagger}Y_{N}Y_{N}^{\dagger}}\Big{)}\Big{)}\,,$
(39) $\displaystyle\beta_{y_{t}}^{(1)}$
$\displaystyle=\frac{3}{2}y_{t}^{3}+y_{t}\Big{(}3y_{t}^{2}-8g_{3}^{2}-\frac{17}{20}g_{1}^{2}-\frac{9}{4}g_{2}^{2}+\mbox{Tr}\Big{(}{Y_{N}Y_{N}^{\dagger}}\Big{)}\Big{)}\,,$
(40) $\displaystyle\beta_{y_{t}}^{(2)}$
$\displaystyle=+\frac{1}{80}\Big{(}120y_{t}^{5}+y_{t}^{3}\Big{(}1280g_{3}^{2}-180\mbox{Tr}\Big{(}{Y_{N}Y_{N}^{\dagger}}\Big{)}+223g_{1}^{2}-540y_{t}^{2}+675g_{2}^{2}-960\lambda\Big{)}\Big{)}$
$\displaystyle+y_{t}\Big{(}\frac{1267}{600}g_{1}^{4}-\frac{9}{20}g_{1}^{2}g_{2}^{2}-\frac{21}{4}g_{2}^{4}+\frac{19}{15}g_{1}^{2}g_{3}^{2}+9g_{2}^{2}g_{3}^{2}-108g_{3}^{4}+\lambda_{3}^{2}+\lambda_{3}\lambda_{4}+\lambda_{4}^{2}+\frac{3}{2}\lambda_{5}^{2}$
$\displaystyle+6\lambda^{2}+\frac{17}{8}g_{1}^{2}y_{t}^{2}+\frac{45}{8}g_{2}^{2}y_{t}^{2}+20g_{3}^{2}y_{t}^{2}+\frac{3}{8}g_{1}^{2}\mbox{Tr}\Big{(}{Y_{N}Y_{N}^{\dagger}}\Big{)}+\frac{15}{8}g_{2}^{2}\mbox{Tr}\Big{(}{Y_{N}Y_{N}^{\dagger}}\Big{)}$
$\displaystyle-\frac{3}{4}\mbox{Tr}\Big{(}{Y_{f}Y_{f}^{\dagger}Y_{N}Y_{N}^{\dagger}}\Big{)}-\frac{27}{4}y_{t}^{4}-\frac{9}{4}\mbox{Tr}\Big{(}{Y_{N}Y_{N}^{\dagger}Y_{N}Y_{N}^{\dagger}}\Big{)}\Big{)}$
(41)

###  2.3 Scalar Mass term

The evolution of the scalar mass-squared term ${m_{\eta}^{2}}$ is dictated by
the $\beta$ function

$\displaystyle\beta_{m_{\eta}^{2}}^{(1)}$
$\displaystyle=12\lambda_{\eta}m_{\eta}^{2}+2\left(-2|M_{f}|^{2}+m_{\eta}^{2}\right)\text{Tr}\left({Y_{f}^{\dagger}Y_{f}}\right)-2(\lambda_{4}+2\lambda_{3})\mu_{H}^{2}$
$\displaystyle-\left(\frac{9}{10}g_{1}^{2}+\frac{9}{2}g_{2}^{2}\right)m_{\eta}^{2}$
(42)

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