#  Electroweak symmetry breaking in the inverse seesaw mechanism

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, 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 investigate the stability of Higgs potential in inverse seesaw models. We
derive the full two-loop RGEs of the relevant parameters, such as the quartic
Higgs self-coupling, taking thresholds into account. We find that for
relatively large Yukawa couplings the Higgs quartic self-coupling goes
negative well below the Standard Model instability scale $\sim 10^{10}$ GeV.
We show, however, that the “dynamical” inverse seesaw with spontaneous lepton
number violation can lead to a completely consistent and stable Higgs vacuum
up to the Planck scale.

##  1 Introduction

The historical discovery of the Higgs boson [1, 2] and the subsequent precise
measurements of its properties [3] can be used to shed light on the
electroweak symmetry breaking mechanism. In particular, we can now not only
determine the value of the quartic coupling of the Standard Model scalar
potential at the electroweak scale, but also use it to shed light on possible
new physics all the way up to Planck scale. Given the present measured top
quark and Higgs boson masses, one can calculate the corresponding Yukawa
$y_{t}$ and Higgs quartic $\lambda_{\text{SM}}$ couplings within the Standard
Model. These, along with the $\mathrm{SU(3)_{c}\otimes SU(2)_{L}\otimes
U(1)_{Y}}$ gauge couplings $g_{1},g_{2},g_{3}$ respectively, are the most
important input parameters characterizing the Standard Model renormalization
group equations (RGEs). Given the values of these input parameters111The
numbers given in Table 1 are the central values. We use them as the input
parameters for our RGEs. The importance of errors has been studied in Ref.
[4], to which we refer the reader for more details., as shown in Table 1, the
Higgs quartic coupling tends to run negative between the electroweak and
Planck scales, as seen in Fig. 1.

| $g_{1}$ | $g_{2}$ | $g_{3}$ | $y_{t}$ | $\lambda_{\text{SM}}$
---|---|---|---|---|---
$\mu(m_{t})$ | 0.462607 | 0.647737 | 1.16541 | 0.93519 | 0.126115
Table 1:  $\overline{\text{MS}}$ values of the input parameters at the top
quark mass scale, $\mu(m_{t})=173\pm 0.4$ GeV [3].  Figure 1: The
renormalization group evolution of the Standard Model gauge couplings $g_{1}$,
$g_{2}$, $g_{3}$, the top quark Yukawa coupling $y_{t}$ and the quartic Higgs
boson self-coupling $\lambda_{\text{SM}}$. Here we adopt the
$\overline{\text{MS}}$ scheme, taking the parameter values at low scale as
input, see [5] for details.

One sees that the Standard Model Higgs quartic coupling $\lambda_{\text{SM}}$
becomes negative at an energy scale $\sim 10^{10}$ GeV. This would imply that
the Standard Model Higgs potential is unbounded from below. Hence, the
Standard Model vacuum is not absolutely stable [6, 7, 4]. Instead, these next-
to-next-to-leading order analyses of the Standard Model Higgs potential
suggest that the vacuum is actually metastable.

Moreover, despite its many successes, the Standard Model cannot be the final
theory of nature. One of its main shortcomings is its inability to account for
neutrino mass generation, needed to describe neutrino oscillations [8]. The
Higgs vacuum stability problem in neutrino mass models can become worse than
in the Standard Model [9, 10, 11, 12, 13, 14, 15, 16]. Here we follow Ref. [5]
and confine ourselves to the Standard-Model-based seesaw mechanism using the
simplest $\mathrm{SU(3)_{c}\otimes SU(2)_{L}\otimes U(1)_{Y}}$ gauge group.

The latter can be realized in “high-scale” schemes with explicit [17] or
spontaneous violation of lepton number [18, 19]. These typicaly involve
messenger masses much larger than the electroweak scale. Alternatively,
neutrino mass may result from “low-scale” physics [20]. For example, the
type-I seesaw mechanism can be mediated by “low-scale” messengers. This
happens in the inverse seesaw mechanism. Lepton number is broken by
introducing extra $\mathrm{SU(3)_{c}\otimes SU(2)_{L}\otimes U(1)_{Y}}$
singlet fermions with small Majorana mass terms, in addition to the
conventional “right-handed” neutrinos. Again, one can have either explicit
[21] or spontaneous lepton number violation [22].

Figure 2: Destabilizing effect of Weinberg’s effective operator on the Higgs
quartic interaction.

Any theory with massive neutrinos has an intrinsic effect, illustrated in Fig.
2, that may potentially destabilize the electroweak vacuum 222In the presence
of very specific symmetries this model-independent argument might be
circumvented.. This vacuum stability problem becomes severe in low-scale-
seesaw schemes [5]. Indeed, if the heavy mediator neutrino lies in the TeV
scale, its Yukawa coupling will run for much longer than in the high-scale
type-I seesaw. As a consequence, the quartic coupling $\lambda$ tends to
become negative sooner, much before the Standard Model instability sets in.

Here we examine the consistency of the electroweak symmetry breaking vacuum
within the inverse seesaw mechanism. Apart from the destabilizing effect
illustrated in Fig. 2 there will in general be other, model-dependent, and
possibly leading contributions that can reverse this trend. We note that the
spontaneous violation of lepton number, implying the existence of a physical
Nambu-Goldstone boson, dubbed majoron [18, 19], can substantially improve the
electroweak vacuum stability properties. Indeed, the extended scalar sector of
low-scale-majoron-seesaw schemes plays a key role in improving their vacuum
stability. This sharpens the results presented in Ref. [11]. Indeed, we find
that renormalization group (RG) evolution can cure the vacuum stability
problem in inverse seesaw models also in the presence of threshold effects.
These can be associated both with the scalar as well as the fermion sector of
the theory 333Notice that, while Ref. [5] included threshold effects, in the
high-scale seesaw framework such effects appear only at high energies, and do
not affect low-scale physics..

The paper is organized as follows. In Section 2, we describe neutrino mass
generation in the inverse-seesaw model. In Section 3 we show that the vacuum
stability problem becomes worse within the simplest inverse-seesaw extensions
with explicitly broken lepton number. In Section 4, we then focus on the
majoron completion of the inverse seesaw. We then show in Section 5 how the
majoron helps stabilize the Higgs vacuum, all the way up to Planck scale. In
Section 6, we compare the vacuum stability properties of the various missing-
partner-inverse-seesaw variants with those of the sequential case. Finally, we
conclude and summarize our main results in Section 7.

##  2 The Inverse Seesaw mechanism

The issue of vacuum stability must be studied on a model-by-model basis. In
this work we examine it in the context of inverse-seesaw extensions of the
Standard Model. The inverse seesaw mechanism is realized by adding two sets of
electroweak singlet “left-handed” fermions $\nu^{c}_{i}$ and $S_{i}$ [21, 22].
The relevant part of the Lagrangian is given by

$\displaystyle-\mathcal{L}=\sum_{ij}Y_{\nu}^{ij}L_{i}\tilde{\Phi}\nu^{c}_{j}+M^{ij}\nu^{c}_{i}S_{j}+\frac{1}{2}\mu^{ij}_{S}S_{i}S_{j}+\text{H.c.}$
(1)

where $L_{i}=\begin{pmatrix}\nu&\ell\end{pmatrix}^{T}$;$i=1,2,3$ are the
lepton doublets, $\Phi$ is the Standard Model Higgs doublet, $M$ is the Dirac
mass term. The two sets of fields $\nu^{c}$ and $S$ transform under the lepton
number symmetry $U(1)_{L}$ as $\nu^{c}\sim-1$ and $S\sim+1$, respectively. The
$M$ and $\mu_{S}$ terms are both gauge invariant mass matrices, but only $M$
is invariant under lepton number symmetry, since $\mu_{S}$ violates lepton
number by two units. Light neutrino masses are generated through the tiny
lepton number violation. Indeed, after electroweak symmetry breaking, the
effective light neutrino mass matrix has the following form

$\displaystyle\mathcal{M}_{\nu}=\begin{pmatrix}0&m_{D}&0\\\ m_{D}^{T}&0&M\\\
0&M^{T}&\mu_{S}\\\ \end{pmatrix},$ (2)

with $m_{D}=\frac{v}{\sqrt{2}}Y_{\nu}$. Neutrino masses arise by block-
diagonalizing Eq. 2 as,

$\displaystyle\mathcal{U}^{T}.\mathcal{M}_{\nu}.\mathcal{U}=\mathcal{M}_{D}$
(3)

through the unitary transformation matrix $\mathcal{U}$, where
$\mathcal{M}_{D}$ has a block-diagonal form. Since the lepton number is
retored as $\mu_{S}\to 0$, the symmetry breaking entries of $\mu_{S}$ can be
made naturally small in the sense of t’Hooft. Apart from symmetry protection,
the smallness of $\mu_{S}$ may also result from having a radiative origin
associated to new physics such as supersymmetry, left-right symmetry or dark
matter physics [23, 24, 25]. In contrast, being gauge and lepton-number
invariant, the elements of $M$ are expected to be naturally large. Thus we
obtain the hierarchy $M\gg m_{D}\gg\mu_{S}$. Under this hierarchy assumption
we perform the standard seesaw diagonalization procedure [19], to obtain the
effective light neutrino mass matrix $m_{\nu}$ as

$\displaystyle m_{\nu}\approx
m_{D}M^{-1}\mu_{S}(M^{T})^{-1}m_{D}^{T}=\frac{v^{2}}{2}Y_{\nu}M^{-1}\mu_{S}(M^{T})^{-1}Y_{\nu}^{T}$
(4)

Furthermore, in contrast to conventional type-I seesaw, the scale of lepton
number violating parameter $\mu_{S}$ is much smaller than the characteristic
mediators scale $M$. As a result, the heavy singlet neutrinos become quasi-
Dirac-type fermions 444The concept of quasi-Dirac fermions was first suggested
for the light neutrinos in [26]. It constitutes a common feature of all low-
scale seesaw models.. Note that, the small lepton number violating Majorana
mass parameters in $\mu_{S}$ control the smallness of light neutrino masses.
As $\mu_{S}\to 0$, the global lepton number symmetry is restored, and as a
result, all the three light neutrinos are strictly massless. Small neutrino
masses are “symmetry-protected” by the tiny value of $\mu_{S}\neq 0$. The
smallness of $\mu_{S}$ allows the Yukawa couplings $Y_{\nu}$ to be sizeable,
even when the messenger mass scale $M$ lies in the TeV scale, without
conflicting with the observed smallness of neutrino masses.

In contrast to the high-scale type-I seesaw, in inverse-seesaw schemes one can
have a very rich phenomenology that makes them potentially testable in current
or upcoming experiments. For example, the mediators would be accessible to
high-energy collider experiments [27, 28, 29, 30], with stringent bounds, e.g.
from the Delphi and L3 collaborations [28, 29]. Moreover, they would induce
lepton flavour and leptonic CP violating processes with potentially large
rates, unsuppressed by the small neutrino masses [31, 32, 33, 34, 35].
Finally, since the mediators would not take part in low-energy weak processes,
the light-neutrino mixing matrix describing oscillations would be effectively
non-unitary [36, 37, 38, 39, 40]. In short, in contrast to the conventional
high-scale seesaw, the inverse seesaw mechanism could harbor a rich plethora
of accessible new physics processes, that could be just around the corner.

As $\nu^{c}$ and $S$’s are Standard Model gauge singlets, carrying no
anomalies, there is no theoretical limit on their multiplicity. Many
possibilities can arise depending on the number of $\nu^{c}$ and $S$ in a
given model. In the sequential inverse seesaw model the number of $\nu^{c}$
matches that of $S$, and there are three “heavy” quasi-Dirac leptons in
addition to the three light neutrinos. For the case of different number of
$\nu^{c}$ and $S$, in addition to the light and heavy neutrinos, the spectrum
will also contain intermediate states with mass proportional to $\mu_{S}$.
These could be warm dark matter candidate if their mass lies in KeV scale
[41].

For the sake of simplicity, here we consider only the case where $\nu^{c}$ and
$S$ come with the same multiplicity. Moroever, since adding more fermion
species will only worsen the Higgs vacuum stability problem, in section 3 we
opt for the minimal (3,1,1) case, namely a single pair of lepton mediators. In
such minimal “missing-partner” seesaw [17] two of the light neutrinos will be
left massless. In Section 6 we examine the quantitative differences between
the different multiplicity choices concerning the issue of vacuum stability.
Moroever, we briefly discuss the phenomenological viability of the various
options.

##  3 Higgs vacuum stability in inverse seesaw

In the above preliminary considerations we have briefly summarized the main
features of the inverse seesaw model. We now examine the effect of the new
fermions $\nu^{c}$ and $S$ upon the stability of the electroweak Higgs vacuum.
We take into account the effect of the thresholds associated with the extra
fermions $\nu^{c}$ and $S$, as well as the scalars (in Section 4 and 5)
responsible for the spontaneous breaking of lepton number.

###  3.1 Effective Theory

To begin with, in the effective theory where the heavy singlet fermions
$\nu^{c}$ and $S$ are integrated out we have a natural threshold scale
$\Lambda\approx M$ given by their mass, see Eq. (1). As as a result, below
this scale the theory is the Standard Model plus an effective dimension five
Weinberg operator [42], given by

$\displaystyle-\mathcal{L}_{\nu}^{d=5}=\frac{\kappa}{2}\,L\,L\,\Phi\,\Phi\,+\,\text{H.c.}$
(5)

where $\kappa=(Y_{\nu}M^{-1}\mu_{S}(M^{T})^{-1}Y_{\nu}^{T})$ is the $3\times
3$ effective coupling matrix. Unless they are needed, in what follows we will
suppress the generation indices. Note that $\kappa$ has negative mass
dimension. The above Lagrangian leads to a left-handed neutrino Majorana mass
matrix as

$\displaystyle m_{\nu}\equiv\kappa\frac{v^{2}}{2}$ (6)

As a result, below the scale $\Lambda$, only the Standard Model couplings and
$\kappa$ will run. Neglecting lepton and light quark Yukawa couplings, the
one-loop RGEs [43, 44, 45] are given by [5]

$\displaystyle
16\pi^{2}\beta_{\kappa}=6y_{t}^{2}\kappa-3g_{2}^{2}\kappa+\lambda_{\kappa}\kappa$
(7)

Due to the large top Yukawa coupling, $\kappa$ slowly increases with the
threshold scale $\Lambda$. We denote the Higgs quartic coupling in this case
as $\lambda_{\kappa}$ to distinguish it from the pure Standard Model case. The
above Weinberg operator also gives a correction to the Higgs quartic coupling
$\lambda_{\kappa}$ below the scale $\Lambda$. The contribution of the coupling
$\kappa$ to the running of $\lambda_{\kappa}$ is of order $v^{2}\kappa^{2}$
and thus negligible, as shown in [14, 45, 5]. Hence, below the scale
$\Lambda$, the evolution of $\lambda_{\kappa}$ will be almost the same as in
the Standard Model.

###  3.2 Full Theory

We now turn to the region above the threshold scale $\Lambda$. In this regime
we have the full Ultra-Violet (UV) complete theory. Hence one must take into
account the RGEs of all the new couplings present in the model, as they will
affect the evolution of the Higgs quartic coupling. In particular, we will see
that the stability of the electroweak vacuum limits how large the Yukawa
coupling $Y_{\nu}$ can be. The Higgs quartic self-coupling in full UV-complete
theory will be denoted by $\lambda$, to distinguish it from the Standard Model
coupling $\lambda_{\text{SM}}$ and from the effective theory quartic coupling
$\lambda_{\kappa}$ discussed above.

For simplicity we will first study the case of just one species of $\nu^{c}$
and $S$, which we call the $(3,1,1)$ inverse seesaw. As mentioned, this of
course is not – by itself – realistic, as in this case only one of the light
neutrinos obtains mass. However, the missing mass parameter may arise from a
different mechanism [25] associated, say, with dark matter. Moreover, the
$(3,1,1)$ case provides the simplest reference scheme, that brings out all the
relevant features. In Section 6 we will compare with the $(3,2,2)$ and the
$(3,3,3)$ – the sequential inverse seesaw mechanim – with two and three
species of $\nu^{c}$ and $S$, respectively.

Figure 3: The destabilizing effect of right-handed neutrinos on the evolution
of the Higgs quartic coupling.

The running of $Y_{\nu}$ above the threshold scale is governed by the RGEs
given in Appendix. A. Apart from the RG evolution, one must also take into
account the threshold corrections, associated with integrating the heavy
fermions in the effective theory. The tree-level Higgs potential is given by

$\displaystyle
V=-\mu_{\Phi}^{2}(\Phi^{\dagger}\Phi)+\lambda(\Phi^{\dagger}\Phi)^{2}$ (8)

This will get corrections from higher loop diagrams of Standard Model
particles as well as from the extra fermions present in the inverse seesaw
model. It introduces a threshold correction to the Higgs quartic coupling
$\lambda$ at $\Lambda=M$. Here we follow Ref. [5] in estimating this threshold
correction as $\Delta\lambda_{\text{TH}}=-\frac{5}{32\pi^{2}}|Y_{\nu}|^{4}$.
We take into consideration this shift in $\lambda$ at $\Lambda=M$ when solving
the RGEs,

$\displaystyle\lambda(\Lambda)\to\lambda(\Lambda)-\frac{5}{32\pi^{2}}|Y_{\nu}|^{4}.$
(9)

Having set up our basic scheme, let us start by looking at the impact of the
Yukawa coupling $Y_{\nu}$ on the stability of the Higgs vacuum. As already
discussed, in the Standard Model, the running of the Higgs quartic coupling
$\lambda_{\text{SM}}$ is dominated by the top quark Yukawa coupling and
becomes negative around energy scale $\sim 10^{10}$ GeV. However, within the
inverse seesaw, the Yukawa coupling $Y_{\nu}$ in Eq.(1) can dominate the
evolution of $\lambda$ above the threshold scale $\Lambda=M$, as seen in Fig.
4.

Figure 4: Evolution of the Higgs quartic self-coupling $\lambda$ (solid-red)
and Yukawa coupling $Y_{\nu}$ (dotted-green) within the minimal (3,1,1)
inverse seesaw scheme. $\lambda_{\kappa}$ is the quartic coupling in the
effective theory with the Weinberg operator. For comparison, we also plot the
running of $\lambda_{\text{SM}}$, the SM quartic coupling, indicated by the
dashed-red line.

In Fig. 4 we have shown the RG evolution of the relevant coupling parameters
assuming the Yukawa coupling $Y_{\nu}=0.6$ at the threshold scale, taken to be
$\Lambda=M=10^{3}$ GeV (left panel) and $10^{5}$ GeV (right panel). We see
that $\lambda$ becomes negative at around energy scales $3.27\times 10^{7}$
GeV and $3.16\times 10^{8}$ GeV for the threshold scale $\Lambda=10^{3}$ GeV
and $10^{5}$ GeV, respectively. By comparing this with the running of the
Standard Model Higgs quartic coupling $\lambda_{\text{SM}}$ (red dashed), one
sees how the Higgs vacuum stability problem becomes more acute in the inverse
seesaw model. This was expected, since the new fermions tend to destabilize
the Higgs vacuum, as illustrated in Fig. 3. It should also be noted that in
the effective theory regime the evolution of the quartic coupling
$\lambda_{\kappa}$ almost coincides with that of $\lambda_{\text{SM}}$, due to
the negligible effect of the Weinberg operator on its running. Finally, note
that all couplings in Fig. 4 remain within the perturbative region up to
Planck scale.

### Consistency Restrictions

We now turn to the issue of the general self-consistency of the inverse seesaw
mechanism. In order to ensure a perturbative and mathematically consistent
model, the tree-level couplings must satisfy certain conditions, e.g. all of
them should have a perturbative value, and the potential should be bounded
from below. However, once we take into account the quantum corrections, these
conditions also get corrected. In this section we analyze these modified
conditions in more detail.

We start by examining the restrictions coming from perturbativity at tree-
level, which require $|Y_{\nu}|<\sqrt{4\pi}$. The RG evolution of $Y_{\nu}$
increases its value with increasing scale. Fig. 5 shows the evolution of
$Y_{\nu}$ and $\lambda$. From the left panel of Fig. 5 one sees that demanding
that $|Y_{\nu}|<\sqrt{4\pi}$ up to the Planck scale implies that
$|Y_{\nu}|\raise
1.29167pt\hbox{$\;<$\kern-7.5pt\raise-4.73611pt\hbox{$\sim\;$}}0.8$ at the
threshold scale $\Lambda=10^{3}$ GeV. However, as one can see from Fig. 5, the
Higgs quartic coupling $\lambda$ becomes negative much before the Planck
scale. Therefore, demanding pertubativity of $Y_{\nu}$ all the way up to the
Planck scale does not ensure full consistency of the scalar potential. If one
demands perturbativity only till, say, 100 TeV, as shown in right panel of
Fig. 5, one finds that the pertubativity limit on $Y_{\nu}$ is relaxed to
$|Y_{\nu}|\raise
1.29167pt\hbox{$\;<$\kern-7.5pt\raise-4.73611pt\hbox{$\sim\;$}}2$ at the
threshold scale $\Lambda=10^{3}$ GeV. Such large $Y_{\nu}$ values lead to
large threshold corrections for $\lambda$ – the negative jump shown in the
right panel – making it negative even before turning on its RG evolution.

Figure 5:  Perturbativity limits on the Yukawa coupling $Y_{\nu}$. The left
panel requires $Y_{\nu}<\sqrt{4\pi}$ up to the Planck scale, so that only RG
evolution is relevant. The right panel demands $Y_{\nu}<\sqrt{4\pi}$ only up
to 100 TeV. In this case $Y_{\nu}$ is large enough that threshold effects make
$\lambda$ negative even before running. In both cases the vacuum is unstable,
i.e. $\lambda<0$, before $Y_{\nu}$ reaches the perturbative limit, see text
for details.

This highlights the importance of taking into account the threshold
corrections for $\lambda$. From Fig. 5 one sees that a large $Y_{\nu}$ value
can lead to an unbounded potential already at the threshold scale, even before
RG evolution. Taking the Yukawa coupling $Y_{\nu}(\Lambda)=1.58$ at
$\Lambda=10^{3}$ GeV makes $\lambda(\Lambda)=0$ due to threshold corrections.
RG running will further decrease $\lambda$ above the threshold scale, making
the vacuum unstable. It is clear that threshold corrections are crucial when
considering large Yukawa couplings and that a true limit on $Y_{\nu}$ requires
one to take into account both RG evolution as well as the threshold
corrections it induces on the quartic coupling $\lambda$.

Figure 6: Limiting $Y_{\nu}$ by demanding $Y_{\nu}$ to remain perturbative and
$\lambda$ to remain positive up to 100 TeV. Left (right) panel correspond to
threshold scales $\Lambda=1$ TeV ($\Lambda=10$ TeV). See text for details.

As an example, in Fig. 6 we show the result of demanding that neither
$Y_{\nu}$ goes non-perturbative, nor $\lambda$ goes negative up to 100 TeV. To
quantify the implications of this demand, we have taken two threshold scales,
$\Lambda=10^{3}$GeV (left panel), and $\Lambda=10^{4}$GeV (right panel),
respectively. With this combined requirement we obtain the limit
$Y_{\nu}\raise
1.29167pt\hbox{$\;<$\kern-7.5pt\raise-4.73611pt\hbox{$\sim\;$}}0.87$ (left
panel) and $Y_{\nu}\raise
1.29167pt\hbox{$\;<$\kern-7.5pt\raise-4.73611pt\hbox{$\sim\;$}}1.02$ (right
panel). This illustrates that the limit on $Y_{\nu}$ also depends on the
choices of threshold scale, for higher threshold scales the limit on $Y_{\nu}$
gets relaxed.

##  4 The Majoron Completion of the inverse Seesaw

In the previous section we saw that the addition of new fermions to the
Standard Model in order to mediate neutrino mass generation via the inverse
seesaw mechanism [21] has a destabilizing effect on the Higgs vacuum. This
problem can be potentially cured if there are other particles in the theory
providing a “positive” contribution to the RGEs governing the evolution of the
Higgs quartic coupling. A well-motivated way to do this is to assume the
dynamical version of the inverse seesaw mechanism [22].

Building up on the work of Ref. [11] here we focus on low-scale generation of
neutrino mass through the inverse seesaw mechanism with spontaneous lepton
number violation. Lepton number is promoted to a spontaneously broken symmetry
within the minimal $SU(3)_{c}\otimes SU(2)_{L}\otimes U(1)_{Y}$ gauge
framework. To achieve this, in addition to the Standard Model singlets
$\nu^{c}$ and $S$, we now add a complex scalar singlet $\sigma$ carrying two
units of lepton number. Lepton number symmetry is then spontaneously broken by
the vacuum expectation value of $\sigma$. The relevant Lagrangian is given by

$\displaystyle-\mathcal{L}=\sum_{i,j}^{3}Y_{\nu}^{ij}L_{i}\tilde{\Phi}\nu^{c}_{j}+M^{ij}\nu^{c}_{i}S_{j}+Y_{S}^{ij}\sigma
S_{i}S_{j}+\text{H.c.}$ (10)

After the electroweak and lepton number symmetry breaking the neutrino mass
matrix has the following form

$\displaystyle\mathcal{M}_{\nu}=\begin{pmatrix}0&m_{D}&0\\\ m_{D}^{T}&0&M\\\
0&M^{T}&\mu_{S}\\\ \end{pmatrix}$ (11)

where $m_{D}=\frac{Y_{\nu}v_{\Phi}}{\sqrt{2}}$,
$\mu_{S}=2\frac{Y_{S}v_{\sigma}}{\sqrt{2}}$ with
$\left\langle\Phi\right\rangle=\frac{v_{\Phi}}{\sqrt{2}}$ and
$\left\langle\sigma\right\rangle=\frac{v_{\sigma}}{\sqrt{2}}$ being the vacuum
expectation values (vevs) of the $\Phi$ and $\sigma$ fields respectively.
Again, within the standard seesaw approximation, the effective neutrino mass
is obtained as

$\displaystyle
m_{\nu}\simeq\frac{v_{\Phi}^{2}}{\sqrt{2}}Y_{\nu}M^{-1}Y_{S}v_{\sigma}(M^{T})^{-1}Y_{\nu}^{T}$
(12)

Light neutrino masses of $\mathcal{O}(0.1)$ eV, are generated for reasonable
choices of $v_{\sigma}$ and $M$, small Yukawa couplings $Y_{S}$, and sizeable
$Y_{\nu}\sim\mathcal{O}(1)$.

Turning to the scalar sector, in the presence of the complex scalar singlet
$\sigma$ and doublet $\Phi$, the most general potential driving electroweak
and lepton number symmetry breaking is given by

$\displaystyle
V=-\mu_{\Phi}^{2}\Phi^{\dagger}\Phi-\mu_{\sigma}^{2}\sigma^{\dagger}\sigma+\lambda_{\Phi}(\Phi^{\dagger}\Phi)^{2}+\lambda_{\sigma}(\sigma^{\dagger}\sigma)^{2}+\lambda_{\Phi\sigma}(\Phi^{\dagger}\Phi)(\sigma^{\dagger}\sigma).$
(13)

As already noted, in addition to the $SU(3)_{c}\otimes SU(2)_{L}\otimes
U(1)_{Y}$ gauge invariance, $V(\Phi,\sigma)$ also has a global $U(1)$ lepton
number symmetry.

This potential is bounded from below if $\lambda_{\sigma}$, $\lambda_{\Phi}$
and $\lambda_{\Phi\sigma}+2\sqrt{\lambda_{\sigma}\lambda_{\Phi}}$ are all
positive, and has a minimum for non-zero vacuum expectation values of both
$\Phi$ and $\sigma$ provided $\lambda_{\Phi}$, $\lambda_{\sigma}$ and
$4\lambda_{\Phi}\lambda_{\sigma}-\lambda_{\Phi\sigma}^{2}$ are all positive.
After the breaking of electroweak and lepton number symmetries, we end up with
a physical Goldstone boson, the Majoron $J$ [18, 19], which is a pure gauge
singlet. After symmetry breaking one has, in the unitary gauge,

$\displaystyle\Phi\to\frac{1}{\sqrt{2}}\begin{pmatrix}0\\\
v_{\Phi}+h^{\prime}\\\ \end{pmatrix},\hskip
28.45274pt\sigma\to\frac{v_{\sigma}+\sigma^{\prime}+iJ}{\sqrt{2}}.$ (14)

The CP even fields $h^{\prime}$ and $\sigma^{\prime}$ will mix, so the mass
matrix for neutral scalar $M_{ns}$ is given by

$\displaystyle
M_{ns}^{2}=\begin{pmatrix}2\lambda_{\Phi}v_{\Phi}^{2}&\lambda_{\Phi\sigma}v_{\Phi}v_{\sigma}\\\
\lambda_{\Phi\sigma}v_{\Phi}v_{\sigma}&2\lambda_{\sigma}v_{\sigma}^{2}\\\
\end{pmatrix}$ (15)

We can diagonalise the above mass matrix to obtain the mass eigenstates
$(h\,\,H)^{T}$ through the rotation matrix $O_{R}$ as

$\displaystyle\begin{pmatrix}h\\\ H\\\
\end{pmatrix}=O_{R}\begin{pmatrix}h^{\prime}\\\ \sigma^{\prime}\\\
\end{pmatrix}\equiv\begin{pmatrix}\text{cos}~{}\alpha&\text{sin}~{}\alpha\\\
-\text{sin}~{}\alpha&\text{cos}~{}\alpha\\\
\end{pmatrix}\begin{pmatrix}h^{\prime}\\\ \sigma^{\prime}\\\ \end{pmatrix},$
(16)

Here $\alpha$ is the CP-even scalar mixing angle, and its range of allowed
values is constrained by LHC data [46, 47]. The rotation matrix satisfies

$\displaystyle O_{R}M_{ns}^{2}O_{R}^{T}=\text{diag}(m_{h}^{2},m_{H}^{2})$ (17)

where the masses $m_{h},m_{H}$ of the scalars $h,H$ respectively, are given by

$\displaystyle
m_{h}^{2}=\lambda_{\Phi}v_{\Phi}^{2}+\lambda_{\sigma}v_{\sigma}^{2}-\sqrt{(\lambda_{\Phi}v_{\Phi}^{2}-\lambda_{\sigma}v_{\sigma}^{2})^{2}+(\lambda_{\Phi\sigma}vv_{\sigma})^{2}}$
(18) $\displaystyle
m_{H}^{2}=\lambda_{\Phi}v_{\Phi}^{2}+\lambda_{\sigma}v_{\sigma}^{2}+\sqrt{(\lambda_{\Phi}v_{\Phi}^{2}-\lambda_{\sigma}v_{\sigma}^{2})^{2}+(\lambda_{\Phi\sigma}vv_{\sigma})^{2}}$
(19)

The lighter of these two mass eigenstates $h$ is identified with the $125$ GeV
scalar discovered at the LHC [1, 2].

We can use Eqs. (18) and (19) along with (15)-(16) to solve for the parameters
$\lambda_{\Phi}$, $\lambda_{\sigma}$ and $\lambda_{\Phi\sigma}$ in terms of
physical quantitites i.e. masses $m_{h}^{2}$, $m_{H}^{2}$ and the mixing angle
$\alpha$ as

$\displaystyle\lambda_{\Phi}$
$\displaystyle=\frac{m_{h}^{2}\cos^{2}\alpha+m_{H}^{2}\sin^{2}\alpha}{2v_{\Phi}^{2}},$
(20) $\displaystyle\lambda_{\sigma}$
$\displaystyle=\frac{m_{h}^{2}\sin^{2}\alpha+m_{H}^{2}\cos^{2}\alpha}{2v_{\sigma}^{2}},$
(21) $\displaystyle\lambda_{\Phi\sigma}$
$\displaystyle=\frac{(m_{h}^{2}-m_{H}^{2})\sin\alpha\cos\alpha}{v_{\Phi}v_{\sigma}}.$
(22)

##  5 Vacuum stability in inverse seesaw with majoron

In this section we will explore the consequences of spontaneous breaking of
the lepton number symmetry on the stability of the electroweak vacuum. Due to
the presence of the scalar $\sigma$, the RGE of the $\Phi$ quartic coupling
receives a new 1-loop contribution through the diagram shown in Fig. 7. This
“positive” contribution plays a crucial role in counteracting the “negative”
contribution coming from the extra fermions of the inverse seesaw model, see
Fig. 3.

Figure 7: One-loop correction to the $\Phi$ quartic coupling due to its
interaction with the singlet $\sigma$ that drives spontaneous lepton number
violation in inverse seesaw models. This diagram leads to a “positive” term in
the RGE of the $\Phi$ quartic coupling, that can overcome the destabilizing
effect of the fermions in Fig. 3.

Vacuum stability in this model can be studied in two different regimes namely
i) $v_{\sigma}\gg v_{\Phi}$ and ii) $v_{\sigma}\approx\mathcal{O}(v_{\Phi})$.
We start with the first possibility. As before, we focus on the missing
partner (3, 1, 1) inverse seesaw, other possibilities will be taken up in
Section 6.

###  5.1 Case I: $v_{\sigma}\gg v_{\Phi}$

In the limit $v_{\sigma}\gg v_{\Phi}$ the heavy CP-even Higgs boson $H$ almost
decouples, with its mass $m_{H}$ given as $m_{H}\equiv
M_{H}\approx\sqrt{2\lambda_{\sigma}v_{\sigma}}$. Moreover, in this limit small
neutrino masses require small $Y_{S}$, so the two heavy singlet fermions
$\nu^{c}$ and $S$ form a quasi-Dirac pair with nearly degenerate mass $M$. We
assume, for simplicity of the analysis, that $M_{H}$ and $M$, have a common
value, so that we deal with just one threshold scale $\Lambda=M=M_{H}$. Below
this scale we have an effective theory with the Standard Model structure,
suplemented by the Weinberg operator for neutrino mass generation555Note that
the majoron $J$ will also be present in this effective theory. Even though
massless or fairly light, it will pratically decouple from the Higgs boson,
and will not affect vacuum stability.. Thus, below the threshold scale, we
need to integrate out $\sqrt{2}\text{Re}(\sigma)$ at tree-level [48]. As a
result, at the scale $\Lambda$, there is a tree-level threshold correction
which induces a shift in the Higgs quartic coupling,
$\delta\lambda=\frac{\lambda_{\Phi\sigma}^{2}}{4\lambda_{\sigma}}$. This will
lead to the following effective Higgs potential below the threshold scale
$\Lambda$

$\displaystyle
V_{\text{eff}}=\lambda_{\Phi}^{\prime}\Big{(}\Phi^{\dagger}\Phi-\frac{v^{2}}{2}\Big{)}^{2},$
(23)

where the effective Higgs quartic coupling $\lambda_{\Phi}^{\prime}$ below the
threshold scale is defined as

$\displaystyle\lambda_{\Phi}^{\prime}\equiv\lambda_{\kappa}=\lambda_{\Phi}-\frac{\lambda_{\Phi\sigma}^{2}}{4\lambda_{\sigma}}.$
(24)

Here $\lambda_{\kappa}$ is the effective quartic coupling for the case of
explicit lepton number breaking, see Section 3. The evolution of the Higgs
quartic coupling $\lambda^{\prime}_{\Phi}$ in the effective theory is shown in
Fig. 8. One can appreciate the jump in the value of the Higgs quartic coupling
due to threshold corrections. Since only the dimension-five Weinberg operator
runs below the scale $\Lambda$, the RG evolution of $\lambda_{\text{SM}}$ is
essentially the same as that of $\lambda^{\prime}_{\Phi}$. Both are very close
to the RG running of $\lambda_{\kappa}$ of the effective theory with explicit
lepton number breaking. Moreover, at tree-level the numerical value of
$\lambda_{\kappa}(M_{Z})$ and $\lambda_{\text{SM}}(M_{Z})$ is the same, since
in both cases one must reproduce the 125 GeV Higgs mass.

Moving on to the full theory at the threshold scale $\Lambda=M$, the first
thing to note is the impact of threshold corrections, Eq. 24. They lead to a
positive shift in value of the Higgs quartic coupling above the threshold
scale $\Lambda=M$, enhancing the chances of keeping $\lambda_{\Phi}$ positive
[5]. Furthermore, to understand the evolution of $\lambda_{\Phi}$ in the full
theory above the scale $\Lambda=M$ one must perform the RG evolution of all
parameters. Above the scale $\Lambda$ one needs to include
$\beta_{\lambda_{\Phi\sigma}}$, $\beta_{\lambda_{\sigma}}$ and evolve the
quartic coupling $\lambda_{\Phi}$ using the full RGEs with the matching
condition Eq. 24 at $\Lambda$. In Appendix. B, we give the two-loop RGEs of
the full theory.

Figure 8:  The RG evolution of the quartic couplings and right-handed neutrino
Yukawa couplings within the Majoron extension of (3,1,1) inverse seesaw
scheme. For comparison, we also show the evolution of $\lambda_{\text{SM}}$
(red-dashed). Here $\lambda_{\Phi}^{\prime}\equiv\lambda_{\kappa}$ is the
effective Higgs quartic coupling below threshold, see Eq. 24.

In Fig. 8 we show the evolution of various couplings in the majoron inverse
seesaw model for given benchmark points. We have taken the threshold scale as
$M=M_{H}=10$ TeV and $M=M_{H}=100$ TeV for the left and right panels,
respectively. For the sake of comparison, the initial values of other
parameters have been kept the same in both panels. The Yukawa coupling has
been fixed at $Y_{\nu}=0.45$. We have taken
$\lambda_{\sigma},\lambda_{\Phi\sigma}=0.1$ at the scale $\Lambda$. The
positive shift in the evolution of $\lambda$ at the threshold scale is coming
from the matching condition given in Eq. 24. Notice that below threshold the
running of $\lambda_{\Phi}^{\prime}$ and $\lambda_{\text{SM}}$ almost coincide
with each other, due to the tiny effective Weinberg operator. Finally, since
$Y_{S}$ has been taken to be very small, it has no direct impact on vacuum
stability.

In summary, it is clear from Fig. 8 that the dynamical variant of the inverse
seesaw mechanism can be free from the Higgs vaccum instability problem. This
is possible thanks to the positive contribution of the scalar $\sigma$ both to
the threshold corrections, as well as to the RG evolution of the Higgs quartic
coupling. These effects are enough to counteract the negative contribution of
the new fermions present in inverse seesaw model, even for sizeable Yukawa
couplings $Y_{\nu}\sim\mathcal{O}(1)$. These could lead to a plethora of new
phenomena [27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40]. Thus, in
contrast to the case of inverse seesaw with explicitly broken lepton number,
the dynamical variants can have a completely stable Higgs vacuum.

###  5.2 Case II: $v_{\sigma}=\mathcal{O}(v_{\Phi})$

In this case, the mass of the heavy scalar $m_{H}$ is of the order of the
electroweak scale. Hence we can neglect the small range between $M_{Z}$ and
$m_{H}$, starting instead with Eq. 22, which already includes the threshold
effect of Eq. 24. Thus in this case only the fermions are integrated out at
the threshold scale $\Lambda=M$, while all the scalars remain in the resulting
theory below threshold. Thus the scalar couplings evolve over a larger range,
and have better chance of curing the Higgs vacuum instability problem.
Needless to say that, as before, the Higgs vaccum instability can be avoided
if the mixed quartic $\lambda_{\Phi\sigma}$ is sufficiently large,
$\mathcal{O}(0.1)$. This in turn implies a sizeable mixing
$\alpha\sim\mathcal{O}(0.1)$ between the two CP-even Higgs bosons.

Figure 9:  Evolution of the quartic couplings and right-handed neutrino
Yukawas within the Majoron extension of the missing partner (3,1,1) inverse
seesaw scheme. For comparison, the evolution of $\lambda_{\text{SM}}$ is shown
in the red dashed curve. Here only the fermion singlets are integrated out at
the threshold scale $\Lambda=M$, all scalars are part of the effective theory
below threshold, taken as the weak scale.

The evolution of the couplings in this case is shown in Fig. 9. In these
plots, we have fixed the singlet neutrino scale $\Lambda=10$ TeV in the left
panel, and 100 TeV in the right panel. In contrast to the scalar couplings,
the Yukawa coupling $Y_{\nu}$ starts running only above threshold. Notice that
for relatively large mediator scale, the allowed value of $Y_{\nu}$ will also
be large as there is not enough range, in terms of RGEs evolution, to sizeably
alter the $Y_{\nu}$. We found that for large Yukawa couplings, $Y_{\nu}\geq
0.7$ (0.8) for threshold scale $\Lambda=10$ TeV (100 TeV), respectively, we
get either unstable vacuum or non-perturbative dynamics.

Moreover, as shown in Fig. 9, we can have positive $\lambda_{\Phi}$,
$\lambda_{\sigma}$ and $\lambda_{\Phi\sigma}$ all the way up to the Planck
scale, even for sizeable Yukawa couplings. We found that for small $m_{H}$ the
required mixing angle is relatively large, in contrast to the large $m_{H}$
case. For small $\alpha$ or $m_{H}$ the potential becomes unbounded from below
at high energies. In other words, experimental limits on $\alpha$, e.g. coming
from the LHC [46, 47], can be used to place a lower limit on the mass $m_{H}$.
The interplay between the experimental limits, and the restrictions from
vacuum stability deserve a dedicated study, outside the scope of this paper.
Finally, we also note that the lepton number breaking scale $v_{\sigma}$
should not be too small to prevent the existence of a Landau pole in the
running parameters.

##  6 Comparing sequential and missing partner inverse seesaw

For simplicity we have so far only analyzed the explicit and dynamical lepton
number breaking within the simplest (3,1,1) missing partner inverse seesaw
mechanism. We now compare the stability properties of this minimal
construction with those of (3,2,2) and (3,3,3) inverse seesaw mechanisms.

###  6.1 Sequential versus missing partner seesaw: electroweak vacuum
stability

As already mentioned, the problem of Higgs vacuum stability only gets worse
with the addition of extra fermions. This fact is clearly illustrated in Fig.
10 where we compare the RG evolution of the Higgs quartic coupling $\lambda$
within the Standard Model (dashed, red) with the $(3,n,n)$ inverse seesaw
completions, with $n=1$ (solid, blue), $n=2$ (dot-dash, magenta) and $n=3$
(dot, green).

In Fig. 10 we have taken the initial Yukawa coupling values in such a way as
to facilitate a proper comparison of the different cases. To do this for
(3,1,1) case, we have fixed the Yukawa coupling $|Y_{\nu}|=0.4$. For (3,2,2)
and (3,3,3) case, we have taken the diagonal entries of the $Y_{\nu}$ matrix
to be $Y_{\nu}^{ii}=0.4$, while all off-diagonal ones, $Y_{\nu}^{ij}$ for
$i\neq j$, were neglected in the RGEs. Clearly one sees how $(3,n,n)$ inverse
seesaw scenarios with $n>1$ have worse Higgs vacuum stability properties than
the $n=1$ case.

Figure 10:  Comparing the evolution of the quartic Higgs self-coupling
$\lambda$ in the Standard Model (dashed, red) with various inverse-seesaw
extensions with explicit lepton number violation: (3,1,1) denoted in solid
(blue), (3,2,2) dot-dashed (magenta) and (3,3,3) dotted (green), see text for
details.

Figure 11:  Comparing the evolution of the quartic Higgs self-coupling
$\lambda$ in the Standard Model (red dashed) with the majoron inverse seesaw
mechanism: the minimal (3,1,1) is denoted in solid (blue), (3,2,2) is dot-
dashed (magenta) and (3,3,3) is dotted (green). See text for details.

In Fig. 11, we display our vacuum stability results for the majoron inverse
seesaw models. One can compare the Standard Model case (dashed, red) with the
(3,1,1) (solid, blue), (3,2,2) (dot-dash, magenta) and (3,3,3) (dot, green)
majoron inverse seesaw schemes. As before, to ensure a consistent comparison,
we have taken the Yukawa coupling $|Y_{\nu}|=0.4$ for (3,1,1) case, while for
the (3,2,2) and (3,3,3) cases, we have taken $Y_{\nu}^{ii}=0.4$ and neglected
off-diagonal $Y_{\nu}^{ij}$. In the left panel we have taken the case of
$\Lambda=M=m_{H}=10$ TeV. Below threshold we have integrated out the fields
$\sqrt{2}\text{Re}(\sigma)$, $\nu^{c}$ and $S$ and included the threshold
effects. This leads to the jump in the quartic coupling seen in the figure. In
contrast, for the right panel, we have fixed $v_{\sigma}=1$ TeV and
$m_{H}=500$ GeV. In this case the scalars are not integrated out and the
quartic coupling runs smoothly from electroweak scale till Planck scale.

Fig. 11 clearly illustrates that even for $n\geq 2$, we can have a stable
electroweak vacuum for adequate choices of $\alpha$ and $m_{H}$. Indeed, even
in the higher (3,2,2) and (3,3,3) majoron inverse seesaw, the positive
contribution from the new scalar is enough to overcome the negative
contribution from the new fermions of the inverse seesaw. In short, the Higgs
vacuum can be kept stable all the way up to the Planck scale even for
appreciable Yukawa coupling $Y_{\nu}$.

###  6.2 Sequential versus missing partner seesaw: brief phenomenological
discussion

Here we note that neither the explicit nor the dynamical variant of the
minimal (3,1,1) inverse seesaw mechanism is phenomenologically realistic. The
reason is that (3,1,1) leads to only one massive neutrino (lying say, at the
atmospheric scale), hence inconsistent with oscillation data [8]. This minimal
scheme is simply the inverse seesaw embedding of the minimum “missing partner”
(3,1) seesaw mechanism of Sec.III in Ref. [17]. This lack of the solar
neutrino mass splitting can be avoided by the presence of a complementary
radiative mechanism. To implement such “radiative completion” of the minimal
scheme one would need to invoke new physics. The latter could be associated,
say, to the presence of a dark matter sector [49]. This would provide an
elegant theory with a tree-level atmospheric scale, and a radiatively-induced
solar neutrino mass scale, very much analogous to the case of the bilinear
breaking of R-parity in supersymmetry [50, 51, 52].

Alternatively, one can generate non-zero tree-level masses for two neutrinos
by going directly to the (3,2,2) “missing partner” seesaw scheme. Again, this
would be the inverse-seesaw-analogue of the (3,2) seesaw mechanism in Ref.
[17]. Finally, the sequential (3,3,3) inverse seesaw mechanism will generate
tree-level masses for all three light neutrinos. Any of these would be totally
consistent with neutrino oscillations 666Modulo, of course, explaining the
detailed pattern of mixing angles indicated by the oscillation data [8]. Such
a challenging task would require a family symmetry, whose detailed nature is
not yet fully understood..

Concerning neutrinoless double beta decay, here lies an important
phenomenological difference between the “missing partner” and the “sequential”
seesaw mechanism. In the missing partner seesaw there can be no cancellation
amongst the individual light-neutrino amplitudes leading to the decay [53]
777This feature may also be implemented in some radiative schemes of
scotogenic type, see e.g. [54, 55, 56].. As a result, there is a lower bound
on the neutrinoless double beta decay rates that could be testable in the
upcoming generation of searches.

There are other implications of low-scale seesaw schemes, such as our inverse-
seesaw, that could be potentially testable in current or upcoming experiments.
For example, the associated heavy neutrino mediators could be accessible at
high energy experiments such as $e^{+}e^{-}$ collider [27, 28, 29, 30], with
stringent bounds, e.g. from the Delphi and L3 collaborations [28, 29].
Likewise, they could produce interesting signatures at the LHC [57, 58].
Moreover, these mediators would also induce lepton flavour and leptonic CP
violation effects with potentially detectable rates, unsuppressed by the small
neutrino masses [31, 32, 33, 34, 35]. Finally, since the heavy singlet
neutrinos would not take part in oscillations, these could reveal new features
associated to unitarity violation in the lepton mixing matrix [36, 37, 38, 39,
40]. A dedicated study would be required to scrutinize whether these
signatures could be used to distinguish missing partner from sequential
seesaw.

##  7 Conclusions

We have examined the consistency of electroweak symmetry breaking within the
inverse seesaw mechanism. We have derived the full two-loop RGEs of the
relevant parameters within inverse seesaw schemes, examining both the simplest
inverse seesaw with explicit violation of lepton number, as well as the
majoron extension of inverse seesaw. The addition of fermion singlets
($\nu^{c}$ and $S$) has a destabilizing effect on the running of the Higgs
quartic coupling $\lambda$. We found that for inverse seesaw with sizeable
Yukawa coupling $Y_{\nu}$ the quartic coupling $\lambda$ becomes negative much
before the Standard Model instability scale $\sim 10^{10}$ GeV. We have taken
as our simplest benchmark neutrino model the “incomplete” (3,1,1) inverse
seesaw scheme, as it has the “best” stability properties within the class of
inverse seesaw schemes. We compared this reference case, in which only one
oscillation scale is generated at tree-level, with the “higher” inverse seesaw
constructions $(3,n,n)$ with $n=2,3$, in which the other mass scale also
arises from the tree-level seesaw mechanism. We showed how, in contrast to
simplest inverse seesaw with explicit lepton number violation, the stability
properties improve when this violation is spontaneous, and there is a physical
Nambu-Goldstone boson, the majoron.

###### Acknowledgements.

This work is supported by the Spanish grant FPA2017-85216-P (AEI/FEDER, UE),
PROMETEO/2018/165 (Generalitat Valenciana), Fundação para a Ciência e a
Tecnologia (FCT, Portugal) under project CERN/FIS-PAR/0004/2019, and the
Spanish Red Consolider MultiDark FPA2017-90566-REDC.

##  Appendix A RGEs: Inverse seesaw

In our work we have used the package SARAH [59] to perform the RG analysis.
The $\beta$ function of a given parameter $c$ is given by,

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

where $\mu$ is the running scale and $\beta_{c}^{(1)}$, $\beta_{c}^{(2)}$ are
the one-loop and two-loop RG corrections.

###  1.1 Higgs quartic scalar self coupling

The one-loop and two-loop RG corrections to the Higgs quartic self-coupling
are given by

$\displaystyle\beta_{\lambda}^{(1)}=+\frac{27}{200}g_{1}^{4}+\frac{9}{20}g_{1}^{2}g_{2}^{2}+\frac{9}{8}g_{2}^{4}-\frac{9}{5}g_{1}^{2}\lambda-9g_{2}^{2}\lambda+24\lambda^{2}+12\lambda
y_{t}^{2}+4\lambda\mbox{Tr}\Big{(}{Y_{\nu}Y_{\nu}^{\dagger}}\Big{)}$
$\displaystyle-6y_{t}^{4}-2\mbox{Tr}\Big{(}{Y_{\nu}Y_{\nu}^{\dagger}Y_{\nu}Y_{\nu}^{\dagger}}\Big{)}$
(25)
$\displaystyle\beta_{\lambda}^{(2)}=-\frac{3411}{2000}g_{1}^{6}-\frac{1677}{400}g_{1}^{4}g_{2}^{2}-\frac{289}{80}g_{1}^{2}g_{2}^{4}+\frac{305}{16}g_{2}^{6}+\frac{1887}{200}g_{1}^{4}\lambda+\frac{117}{20}g_{1}^{2}g_{2}^{2}\lambda-\frac{73}{8}g_{2}^{4}\lambda$
$\displaystyle+\frac{108}{5}g_{1}^{2}\lambda^{2}+108g_{2}^{2}\lambda^{2}-312\lambda^{3}-\frac{171}{100}g_{1}^{4}y_{t}^{2}+\frac{63}{10}g_{1}^{2}g_{2}^{2}y_{t}^{2}-\frac{9}{4}g_{2}^{4}y_{t}^{2}+\frac{17}{2}g_{1}^{2}\lambda
y_{t}^{2}+\frac{45}{2}g_{2}^{2}\lambda y_{t}^{2}$
$\displaystyle+80g_{3}^{2}\lambda
y_{t}^{2}-144\lambda^{2}y_{t}^{2}-\frac{9}{100}g_{1}^{4}\mbox{Tr}\Big{(}{Y_{\nu}Y_{\nu}^{\dagger}}\Big{)}-\frac{3}{10}g_{1}^{2}g_{2}^{2}\mbox{Tr}\Big{(}{Y_{\nu}Y_{\nu}^{\dagger}}\Big{)}-\frac{3}{4}g_{2}^{4}\mbox{Tr}\Big{(}{Y_{\nu}Y_{\nu}^{\dagger}}\Big{)}$
(26)
$\displaystyle+\frac{3}{2}g_{1}^{2}\lambda\mbox{Tr}\Big{(}{Y_{\nu}Y_{\nu}^{\dagger}}\Big{)}+\frac{15}{2}g_{2}^{2}\lambda\mbox{Tr}\Big{(}{Y_{\nu}Y_{\nu}^{\dagger}}\Big{)}-48\lambda^{2}\mbox{Tr}\Big{(}{Y_{\nu}Y_{\nu}^{\dagger}}\Big{)}-\frac{8}{5}g_{1}^{2}y_{t}^{4}-32g_{3}^{2}y_{t}^{4}-3\lambda
y_{t}^{4}$
$\displaystyle-\lambda\mbox{Tr}\Big{(}{Y_{\nu}Y_{\nu}^{\dagger}Y_{\nu}Y_{\nu}^{\dagger}}\Big{)}+30y_{t}^{6}+10\mbox{Tr}\Big{(}{Y_{\nu}Y_{\nu}^{\dagger}Y_{\nu}Y_{\nu}^{\dagger}Y_{\nu}Y_{\nu}^{\dagger}}\Big{)}$

###  1.2 Yukawa Couplings

The one-loop and two-loop RG corrections for the most relevant Yukawa
couplings in the simplest inverse seesaw model are given by

$\displaystyle\beta_{Y_{\nu}}^{(1)}$
$\displaystyle=\frac{3}{2}{Y_{\nu}Y_{\nu}^{\dagger}Y_{\nu}}+Y_{\nu}\Big{(}3y_{t}^{2}-\frac{9}{20}g_{1}^{2}-\frac{9}{4}g_{2}^{2}+\mbox{Tr}\Big{(}{Y_{\nu}Y_{\nu}^{\dagger}}\Big{)}\Big{)}$
(27) $\displaystyle\beta_{Y_{\nu}}^{(2)}$
$\displaystyle=\frac{1}{80}\Big{(}279g_{1}^{2}{Y_{\nu}Y_{\nu}^{\dagger}Y_{\nu}}+675g_{2}^{2}{Y_{\nu}Y_{\nu}^{\dagger}Y_{\nu}}-960\lambda{Y_{\nu}Y_{\nu}^{\dagger}Y_{\nu}}+120{Y_{\nu}Y_{\nu}^{\dagger}Y_{\nu}Y_{\nu}^{\dagger}Y_{\nu}}-540{Y_{\nu}Y_{\nu}^{\dagger}Y_{\nu}}y_{t}^{2}$
(28)
$\displaystyle-180{Y_{\nu}Y_{\nu}^{\dagger}Y_{\nu}}\mbox{Tr}\Big{(}{Y_{\nu}Y_{\nu}^{\dagger}}\Big{)}+2Y_{\nu}\Big{(}21g_{1}^{4}-54g_{1}^{2}g_{2}^{2}-230g_{2}^{4}+240\lambda^{2}+85g_{1}^{2}y_{t}^{2}+225g_{2}^{2}y_{t}^{2}$
$\displaystyle+800g_{3}^{2}y_{t}^{2}+15g_{1}^{2}\mbox{Tr}\Big{(}{Y_{\nu}Y_{\nu}^{\dagger}}\Big{)}+75g_{2}^{2}\mbox{Tr}\Big{(}{Y_{\nu}Y_{\nu}^{\dagger}}\Big{)}-270y_{t}^{4}-90\mbox{Tr}\Big{(}{Y_{\nu}Y_{\nu}^{\dagger}Y_{\nu}Y_{\nu}^{\dagger}}\Big{)}\Big{)}\Big{)}$
$\displaystyle\beta_{y_{t}}^{(1)}$
$\displaystyle=\frac{3}{2}y_{t}^{3}+y_{t}\Big{(}3y_{t}^{2}-8g_{3}^{2}-\frac{17}{20}g_{1}^{2}-\frac{9}{4}g_{2}^{2}+\mbox{Tr}\Big{(}{Y_{\nu}Y_{\nu}^{\dagger}}\Big{)}\Big{)}$
(29) $\displaystyle\beta_{y_{t}}^{(2)}$
$\displaystyle=+\frac{1}{80}\Big{(}120y_{t}^{5}+y_{t}^{3}\Big{(}1280g_{3}^{2}-180\mbox{Tr}\Big{(}{Y_{\nu}Y_{\nu}^{\dagger}}\Big{)}+223g_{1}^{2}-540y_{t}^{2}+675g_{2}^{2}-960\lambda\Big{)}$
$\displaystyle+y_{t}\Big{(}\frac{1187}{600}g_{1}^{4}-\frac{9}{20}g_{1}^{2}g_{2}^{2}-\frac{23}{4}g_{2}^{4}+\frac{19}{15}g_{1}^{2}g_{3}^{2}+9g_{2}^{2}g_{3}^{2}-108g_{3}^{4}+6\lambda^{2}+\frac{17}{8}g_{1}^{2}y_{t}^{2}+\frac{45}{8}g_{2}^{2}y_{t}^{2}$
$\displaystyle+20g_{3}^{2}y_{t}^{2}+\frac{3}{8}g_{1}^{2}\mbox{Tr}\Big{(}{Y_{\nu}Y_{\nu}^{\dagger}}\Big{)}+\frac{15}{8}g_{2}^{2}\mbox{Tr}\Big{(}{Y_{\nu}Y_{\nu}^{\dagger}}\Big{)}-\frac{27}{4}y_{t}^{4}-\frac{9}{4}\mbox{Tr}\Big{(}{Y_{\nu}Y_{\nu}^{\dagger}Y_{\nu}Y_{\nu}^{\dagger}}\Big{)}\Big{)}$
(30)

##  Appendix B RGEs: Inverse seesaw with majoron

In the presence of the majoron the one- and two-loop RG corrections for the
quartic scalar couplings in the inverse seesaw model are modified to

###  2.1 Quartic scalar couplings

$\displaystyle\beta_{\lambda_{\Phi}}^{(1)}$
$\displaystyle=+\frac{27}{200}g_{1}^{4}+\frac{9}{20}g_{1}^{2}g_{2}^{2}+\frac{9}{8}g_{2}^{4}+\lambda_{\Phi\sigma}^{2}-\frac{9}{5}g_{1}^{2}\lambda_{\Phi}-9g_{2}^{2}\lambda_{\Phi}+24\lambda_{\Phi}^{2}+12\lambda_{\Phi}y_{t}^{2}+4\lambda_{\Phi}\mbox{Tr}\Big{(}{Y_{\nu}Y_{\nu}^{\dagger}}\Big{)}-6y_{t}^{4}$
$\displaystyle-2\mbox{Tr}\Big{(}{Y_{\nu}Y_{\nu}^{\dagger}Y_{\nu}Y_{\nu}^{\dagger}}\Big{)}$
(31) $\displaystyle\beta_{\lambda_{\Phi}}^{(2)}$
$\displaystyle=-\frac{3411}{2000}g_{1}^{6}-\frac{1677}{400}g_{1}^{4}g_{2}^{2}-\frac{289}{80}g_{1}^{2}g_{2}^{4}+\frac{305}{16}g_{2}^{6}-4\lambda_{\Phi\sigma}^{3}+\frac{1887}{200}g_{1}^{4}\lambda_{\Phi}+\frac{117}{20}g_{1}^{2}g_{2}^{2}\lambda_{\Phi}-\frac{73}{8}g_{2}^{4}\lambda_{\Phi}$
(32)
$\displaystyle-10\lambda_{\Phi\sigma}^{2}\lambda_{\Phi}+\frac{108}{5}g_{1}^{2}\lambda_{\Phi}^{2}+108g_{2}^{2}\lambda_{\Phi}^{2}-312\lambda_{\Phi}^{3}-4\lambda_{\Phi\sigma}^{2}\mbox{Tr}\Big{(}{Y_{S}Y_{S}^{*}}\Big{)}-\frac{171}{100}g_{1}^{4}y_{t}^{2}+\frac{63}{10}g_{1}^{2}g_{2}^{2}y_{t}^{2}$
$\displaystyle-\frac{9}{4}g_{2}^{4}y_{t}^{2}+\frac{17}{2}g_{1}^{2}\lambda_{\Phi}y_{t}^{2}+\frac{45}{2}g_{2}^{2}\lambda_{\Phi}y_{t}^{2}+80g_{3}^{2}\lambda_{\Phi}y_{t}^{2}-144\lambda_{\Phi}^{2}y_{t}^{2}-\frac{9}{100}g_{1}^{4}\mbox{Tr}\Big{(}{Y_{\nu}Y_{\nu}^{\dagger}}\Big{)}$
$\displaystyle-\frac{3}{10}g_{1}^{2}g_{2}^{2}\mbox{Tr}\Big{(}{Y_{\nu}Y_{\nu}^{\dagger}}\Big{)}-\frac{3}{4}g_{2}^{4}\mbox{Tr}\Big{(}{Y_{\nu}Y_{\nu}^{\dagger}}\Big{)}+\frac{3}{2}g_{1}^{2}\lambda_{\Phi}\mbox{Tr}\Big{(}{Y_{\nu}Y_{\nu}^{\dagger}}\Big{)}+\frac{15}{2}g_{2}^{2}\lambda_{\Phi}\mbox{Tr}\Big{(}{Y_{\nu}Y_{\nu}^{\dagger}}\Big{)}-48\lambda_{\Phi}^{2}\mbox{Tr}\Big{(}{Y_{\nu}Y_{\nu}^{\dagger}}\Big{)}$
$\displaystyle-\frac{8}{5}g_{1}^{2}y_{t}^{4}-32g_{3}^{2}y_{t}^{4}-3\lambda_{\Phi}y_{t}^{4}-\lambda_{\Phi}\mbox{Tr}\Big{(}{Y_{\nu}Y_{\nu}^{\dagger}Y_{\nu}Y_{\nu}^{\dagger}}\Big{)}+30y_{t}^{6}+10\mbox{Tr}\Big{(}{Y_{\nu}Y_{\nu}^{\dagger}Y_{\nu}Y_{\nu}^{\dagger}Y_{\nu}Y_{\nu}^{\dagger}}\Big{)}$
$\displaystyle\beta_{\lambda_{\Phi\sigma}}^{(1)}$
$\displaystyle=\frac{1}{10}\lambda_{\Phi\sigma}\Big{(}-9g_{1}^{2}-45g_{2}^{2}+40\lambda_{\Phi\sigma}+80\lambda_{\sigma}+120\lambda_{\Phi}+40\mbox{Tr}\Big{(}{Y_{S}Y_{S}^{*}}\Big{)}+60y_{t}^{2}+20\mbox{Tr}\Big{(}{Y_{\nu}Y_{\nu}^{\dagger}}\Big{)}\Big{)}$
(33) $\displaystyle\beta_{\lambda_{\Phi\sigma}}^{(2)}$
$\displaystyle=+\frac{1671}{400}g_{1}^{4}\lambda_{\Phi\sigma}+\frac{9}{8}g_{1}^{2}g_{2}^{2}\lambda_{\Phi\sigma}-\frac{145}{16}g_{2}^{4}\lambda_{\Phi\sigma}+\frac{3}{5}g_{1}^{2}\lambda_{\Phi\sigma}^{2}+3g_{2}^{2}\lambda_{\Phi\sigma}^{2}-11\lambda_{\Phi\sigma}^{3}-48\lambda_{\Phi\sigma}^{2}\lambda_{\sigma}$
$\displaystyle-40\lambda_{\Phi\sigma}\lambda_{\sigma}^{2}+\frac{72}{5}g_{1}^{2}\lambda_{\Phi\sigma}\lambda_{\Phi}+72g_{2}^{2}\lambda_{\Phi\sigma}\lambda_{\Phi}-72\lambda_{\Phi\sigma}^{2}\lambda-60\lambda_{\Phi\sigma}\lambda_{\Phi}^{2}-8\lambda_{\Phi\sigma}^{2}\mbox{Tr}\Big{(}{Y_{S}Y_{S}^{*}}\Big{)}$
$\displaystyle-32\lambda_{\Phi\sigma}\lambda_{\sigma}\mbox{Tr}\Big{(}{Y_{S}Y_{S}^{*}}\Big{)}+\frac{17}{4}g_{1}^{2}\lambda_{\Phi\sigma}y_{t}^{2}+\frac{45}{4}g_{2}^{2}\lambda_{\Phi\sigma}y_{t}^{2}+40g_{3}^{2}\lambda_{\Phi\sigma}y_{t}^{2}-12\lambda_{\Phi\sigma}^{2}y_{t}^{2}$
$\displaystyle-72\lambda_{\Phi\sigma}\lambda_{\Phi}y_{t}^{2}+\frac{3}{4}g_{1}^{2}\lambda_{\Phi\sigma}\mbox{Tr}\Big{(}{Y_{\nu}Y_{\nu}^{\dagger}}\Big{)}+\frac{15}{4}g_{2}^{2}\lambda_{\Phi\sigma}\mbox{Tr}\Big{(}{Y_{\nu}Y_{\nu}^{\dagger}}\Big{)}-4\lambda_{\Phi\sigma}^{2}\mbox{Tr}\Big{(}{Y_{\nu}Y_{\nu}^{\dagger}}\Big{)}$
$\displaystyle-24\lambda_{\Phi\sigma}\lambda_{\Phi}\mbox{Tr}\Big{(}{Y_{\nu}Y_{\nu}^{\dagger}}\Big{)}-24\lambda_{\Phi\sigma}\mbox{Tr}\Big{(}{Y_{S}Y_{S}^{*}Y_{S}Y_{S}^{*}}\Big{)}-\frac{27}{2}\lambda_{\Phi\sigma}y_{t}^{4}-\frac{9}{2}\lambda_{\Phi\sigma}\mbox{Tr}\Big{(}{Y_{\nu}Y_{\nu}^{\dagger}Y_{\nu}Y_{\nu}^{\dagger}}\Big{)}$
(34) $\displaystyle\beta_{\lambda_{\sigma}}^{(1)}$
$\displaystyle=2\Big{(}10\lambda_{\sigma}^{2}+4\lambda_{\sigma}\mbox{Tr}\Big{(}{Y_{S}Y_{S}^{*}}\Big{)}-8\mbox{Tr}\Big{(}{Y_{S}Y_{S}^{*}Y_{S}Y_{S}^{*}}\Big{)}+\lambda_{\Phi\sigma}^{2}\Big{)}$
(35) $\displaystyle\beta_{\lambda_{\sigma}}^{(2)}$
$\displaystyle=+\frac{12}{5}g_{1}^{2}\lambda_{\Phi\sigma}^{2}+12g_{2}^{2}\lambda_{\Phi\sigma}^{2}-8\lambda_{\Phi\sigma}^{3}-20\lambda_{\Phi\sigma}^{2}\lambda_{\sigma}-240\lambda_{\sigma}^{3}-80\lambda_{\sigma}^{2}\mbox{Tr}\Big{(}{Y_{S}Y_{S}^{*}}\Big{)}-12\lambda_{\Phi\sigma}^{2}y_{t}^{2}$
$\displaystyle-4\lambda_{\Phi\sigma}^{2}\mbox{Tr}\Big{(}{Y_{\nu}Y_{\nu}^{\dagger}}\Big{)}+16\lambda_{\sigma}\mbox{Tr}\Big{(}{Y_{S}Y_{S}^{*}Y_{S}Y_{S}^{*}}\Big{)}+256\mbox{Tr}\Big{(}{Y_{S}Y_{S}^{*}Y_{S}Y_{S}^{*}Y_{S}Y_{S}^{*}}\Big{)}$
(36)

###  2.2 Yukawa Couplings

Likewise, in the presence of the majoron the one- and two-loop RG corrections
for the Yukawas in the inverse seesaw model are modified to

$\displaystyle\beta_{Y_{\nu}}^{(1)}$
$\displaystyle=\frac{3}{2}{Y_{\nu}Y_{\nu}^{\dagger}Y_{\nu}}+Y_{\nu}\Big{(}3y_{t}^{2}-\frac{9}{20}g_{1}^{2}-\frac{9}{4}g_{2}^{2}+\mbox{Tr}\Big{(}{Y_{\nu}Y_{\nu}^{\dagger}}\Big{)}\Big{)}$
(37) $\displaystyle\beta_{Y_{\nu}}^{(2)}$
$\displaystyle=\frac{1}{80}\Big{(}279g_{1}^{2}{Y_{\nu}Y_{\nu}^{\dagger}Y_{\nu}}+675g_{2}^{2}{Y_{\nu}Y_{\nu}^{\dagger}Y_{\nu}}-960\lambda_{\Phi}{Y_{\nu}Y_{\nu}^{\dagger}Y_{\nu}}+120{Y_{\nu}Y_{\nu}^{\dagger}Y_{\nu}Y_{\nu}^{\dagger}Y_{\nu}}-540{Y_{\nu}Y_{\nu}^{\dagger}Y_{\nu}}y_{t}^{2}$
$\displaystyle-180{Y_{\nu}Y_{\nu}^{\dagger}Y_{\nu}}\mbox{Tr}\Big{(}{Y_{\nu}Y_{\nu}^{\dagger}}\Big{)}+2Y_{\nu}\Big{(}21g_{1}^{4}-54g_{1}^{2}g_{2}^{2}-230g_{2}^{4}+20\lambda_{\Phi\sigma}^{2}+240\lambda_{\Phi}^{2}+85g_{1}^{2}y_{t}^{2}$
$\displaystyle+225g_{2}^{2}y_{t}^{2}+800g_{3}^{2}y_{t}^{2}+15g_{1}^{2}\mbox{Tr}\Big{(}{Y_{\nu}Y_{\nu}^{\dagger}}\Big{)}+75g_{2}^{2}\mbox{Tr}\Big{(}{Y_{\nu}Y_{\nu}^{\dagger}}\Big{)}-270y_{t}^{4}-90\mbox{Tr}\Big{(}{Y_{\nu}Y_{\nu}^{\dagger}Y_{\nu}Y_{\nu}^{\dagger}}\Big{)}\Big{)}\Big{)}$
(38) $\displaystyle\beta_{y_{t}}^{(1)}$
$\displaystyle=\frac{3}{2}y_{t}^{3}+y_{t}\Big{(}3y_{t}^{2}-8g_{3}^{2}-\frac{17}{20}g_{1}^{2}-\frac{9}{4}g_{2}^{2}+\mbox{Tr}\Big{(}{Y_{\nu}Y_{\nu}^{\dagger}}\Big{)}\Big{)}$
(39) $\displaystyle\beta_{y_{t}}^{(2)}$
$\displaystyle=+\frac{1}{80}\Big{(}120y_{t}^{5}+y_{t}^{3}\Big{(}1280g_{3}^{2}-180\mbox{Tr}\Big{(}{Y_{\nu}Y_{\nu}^{\dagger}}\Big{)}+223g_{1}^{2}-540y_{t}^{2}+675g_{2}^{2}-960\lambda_{\Phi}\Big{)}$
$\displaystyle+y_{t}\Big{(}\frac{1187}{600}g_{1}^{4}-\frac{9}{20}g_{1}^{2}g_{2}^{2}-\frac{23}{4}g_{2}^{4}+\frac{19}{15}g_{1}^{2}g_{3}^{2}+9g_{2}^{2}g_{3}^{2}-108g_{3}^{4}+\frac{1}{2}\lambda_{\Phi\sigma}^{2}+6\lambda_{\Phi}^{2}+\frac{17}{8}g_{1}^{2}y_{t}^{2}$
(40)
$\displaystyle+\frac{45}{8}g_{2}^{2}y_{t}^{2}+20g_{3}^{2}y_{t}^{2}+\frac{3}{8}g_{1}^{2}\mbox{Tr}\Big{(}{Y_{\nu}Y_{\nu}^{\dagger}}\Big{)}+\frac{15}{8}g_{2}^{2}\mbox{Tr}\Big{(}{Y_{\nu}Y_{\nu}^{\dagger}}\Big{)}-\frac{27}{4}y_{t}^{4}-\frac{9}{4}\mbox{Tr}\Big{(}{Y_{\nu}Y_{\nu}^{\dagger}Y_{\nu}Y_{\nu}^{\dagger}}\Big{)}\Big{)}$
$\displaystyle\beta_{Y_{S}}^{(1)}$
$\displaystyle=2Y_{S}\mbox{Tr}\Big{(}{Y_{S}Y_{S}^{*}}\Big{)}+4{Y_{S}Y_{S}^{*}Y_{S}}$
(41) $\displaystyle\beta_{Y_{S}}^{(2)}$
$\displaystyle=28{Y_{S}Y_{S}^{*}Y_{S}Y_{S}^{*}Y_{S}}-4{Y_{S}Y_{S}^{*}Y_{S}}\Big{(}3\mbox{Tr}\Big{(}{Y_{S}Y_{S}^{*}}\Big{)}+8\lambda_{\sigma}\Big{)}+Y_{S}\Big{(}-12\mbox{Tr}\Big{(}{Y_{S}Y_{S}^{*}Y_{S}Y_{S}^{*}}\Big{)}$
$\displaystyle+4\lambda_{\sigma}^{2}+\lambda_{\Phi\sigma}^{2}\Big{)}$ (42)

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