monthyeardate[],

Study on the global minimum and $H\to\gamma\gamma$ in the Dirac scotogenic
model

Raghavendra Srikanth Hundi

Department of Physics, Indian Institute of Technology Hyderabad,

Kandi - 502 284, India.

E-mail<EMAIL_ADDRESS>
###### Abstract

We have analyzed the vacuum structure of the Dirac scotogenic model, whose
scalar sector consists of two complex Higgs doublets and a real singlet field.
In this model, the standard model like Higgs doublet acquires non-zero vacuum
expectation value (VEV), whereas, the other two fields acquire zero VEVs. This
pattern of VEVs constitute a minimum, which is the desired vacuum of the
model. After analyzing the scalar potential of this model, we have found that
other vacua are also possible in this model. We have shown that plenty of
parameter space exist where the desired vacuum of this model is the global
minimum. We have studied the implications of scalar sector of this model on
the observable quantity of signal strength of Higgs to diphoton decay. After
evaluating this quantity, we have found that the current experimental values
of this quantity can be fitted in this model. Lastly, we have studied on the
possibility of making any of the additional scalar fields of this model as a
candidate for dark matter.

##  1 Introduction

With the discovery of Higgs boson at the LHC [1, 2], the spontaneous breaking
of electroweak symmetry has been verified. In the standard model (SM), the
breaking of this symmetry is explained by postulating a single scalar Higgs
doublet [3, 4, 5, 6, 7], which acquires non-zero VEV at the minimum of the
scalar potential. A consequence of this breaking mechanism is the existence of
the Higgs boson, which is found in the LHC. As of now, the properties of the
Higgs boson, which are measured at the LHC, agrees with the SM predictions
[8]. However, more work is to be done in order to precisely measure the
couplings of Higgs boson to all the SM particles. On the other hand, several
reasons exist for the extension of SM [9, 10]. As a result of this, it is
worth to explore theories by proposing additional Higgs doublets. A minimal
extension to the SM, in this aspect, is the two Higgs doublet model (2HDM)
[11], where the field content is as same that of SM apart from an extra scalar
Higgs doublet.

In the SM, at the minimum of the scalar potential, the VEV of the scalar
doublet breaks only the electroweak symmetry. In contrast to this, in the
2HDM, both the Higgs doublets can acquire VEVs in such a way that, in addition
to the electroweak symmetry, $CP$ and charge symmetries can also be broken
spontaneously [12, 13]. Different forms of VEVs to the Higgs doublets are
possible in the 2HDM, due to the parameter choice of the model. As a result of
this, in the 2HDM, the possible vacua are categorized as follows [12, 13]: (i)
neutral minimum, (ii) $CP$-violating minimum, (iii) charge-breaking minimum.
Here, neutral minimum breaks only the electroweak symmetry. Whereas, $CP$\-
and charge-breaking minima break the respective symmetries, in addition to the
electroweak symmetry. $CP$-violating minimum is phenomenologically acceptable,
however, charge-breaking minimum should be avoided, since violation of charge
symmetry is not found in experiments. Theoretically it is demonstrated that,
in the 2HDM, a neutral minimum do not coexist with either $CP$\- or charge-
breaking minima [12, 13, 14, 15]. In other words, it is possible to choose a
parameter region of the scalar potential of 2HDM in such a way that the
minimum breaks only the electroweak symmetry spontaneously, and moreover, this
can be the global minimum. This result is appealing and it makes the 2HDM as a
viable candidate for the extension of SM.

The result mentioned above is valid in any model where the scalar sector
contains only two Higgs doublets. Several models are proposed with only the
two Higgs doublets, in order to explain the limitations of SM [9, 10]. One
among these is the scotogenic model [16], whose motivation is to explain the
smallness of neutrino masses and the existence of dark matter. This model
contains an additional and exact discrete symmetry $Z_{2}$, whose purpose is
to generate neutrino masses at 1-loop level and also to accommodate a
candidate for dark matter. In order to achieve the motivation of this model,
one of the two Higgs doublets of this model should acquire non-zero VEV and
the other one should acquire zero VEV. We can consider this pattern of VEVs to
the Higgs doublets as the desired vacuum of the scotogenic model. However, by
minimizing the scalar potential of this model, it is possible for both the
Higgs doublets to acquire different patterns of VEVs, and thus generate
different possible vacua. Topics on this subject are discussed in [17, 18] and
it is shown that it is possible to make the desired vacuum of the scotogenic
model as the global minimum by restricting the parameter space of the model.
See [19, 20], for alternative proposals on scotogenic mechanism in composite
Higgs models.

In the scotogenic model [16], neutrinos are Majorana particles. Since there is
no indication from experiments on the Majorana nature of neutrinos, a priori,
it is worth to construct models based on Dirac nature of neutrinos. It is for
this reason, the scotogenic model has been modified into Dirac scotogenic
model [21], where the neutrinos are purely Dirac particles. In this later
model, three copies of Weyl singlet fermions $\nu^{c}_{\alpha}$, $N_{k}$,
$N^{c}_{k}$ are introduced. Here, $\alpha=e,\mu,\tau$, which is the generation
index of lepton family and $k=1,2,3$. $N_{k}$ and $N^{c}_{k}$ combine to give
massive Dirac fermions $N^{D}_{k}$. Whereas, $\nu^{c}_{\alpha}$ combine with
left-handed neutrinos of the lepton doublets to form Dirac neutrinos
$\nu^{D}_{\alpha}$. To forbid Majorana masses for the above Weyl fermions and
to conserve lepton number, an additional and exact symmetry $U(1)_{B-L}$ is
proposed. In the scalar sector of this model, there exist two complex Higgs
doublets ($\Phi$, $\eta$) and a real scalar singlet ($\chi$). Here, $\Phi$ is
the SM-like Higgs doublet. This model has an additional symmetry
$Z_{2}^{(A)}\times Z_{2}^{(B)}$, which prevents masses to Dirac neutrinos at
tree level and generate them at 1-loop level [21]. The construction of the
model is such that the $Z_{2}^{(A)}$ symmetry is softly broken but
$Z_{2}^{(B)}$ is exact symmetry. Hence, the lightest charged particle under
$Z_{2}^{(B)}$ can be a viable candidate for dark matter.

Like in the case of scotogenic model, in the Dirac scotogenic model as well,
the scalar fields should acquire VEVs in a specific pattern in order to
generate masses for neutrinos at 1-loop level in a consistent way. This
pattern is such that, only $\Phi$ acquires non-zero VEV, whereas, $\eta$ and
$\chi$ acquire zero VEVs [21]. We expect this pattern of VEVs to constitute a
minimum of the model in some parameter region of it. On the other hand, with
the description we have given for the cases of 2HDM and scotogenic model, one
can expect other possible minima for Dirac scotogenic model, apart from the
desired minimum which is mentioned above. A noteworthy point is that the
scalar content of Dirac scotogenic model is different from that of 2HDM.
Hence, the results we described above for the case of 2HDM need not be
applicable to the Dirac scotogenic model. More specifically, we may expect
some charge-breaking minima to coexist with the desired minimum of this model.
As a result of this, we need to know if the desired minimum of this model can
be made as the global minimum.

In this work, after analyzing the scalar potential, we describe all possible
inequivalent vacua of the Dirac scotogenic model. We have found that,
including the desired minimum of this model, there can exist eleven different
vacua, which includes three charge-breaking minima. As part of our
investigation on global minimum, we have studied if the desired minimum of
this model can coexist with other possible vacua of the model. In our
numerical analysis, we have found that the desired minimum of this model do
not coexist with charge-breaking minima in the viable parameter space of this
model. We have justified this result by giving an analytical proof to it. On
the other hand, the desired minimum of this model is found to coexist with
certain other minima of the model. In the case that the desired minimum of
this model coexist with other minima, we have given the conditions that need
to be satisfied in order to make the desired minimum of this model as the
global minimum. We shown that there exist plenty of parameter space where the
desired minimum of this model is the global minimum.

The study on global minimum of the Dirac scotogenic model will have
implications on the scalar sector of this model, since the analysis is mainly
concerned with the parameters of the scalar potential. One of the
phenomenological implications of the scalar sector of this model is on the
signal strength of the Higgs to diphoton decay $H\to\gamma\gamma$. The signal
strength of $H\to\gamma\gamma$, $R_{\gamma\gamma}$, is measured in the LHC
experiment and its value is around one [8]. The additional contribution to the
decay $H\to\gamma\gamma$ in the Dirac scotogenic model [21] is due to the
trilinear coupling of the Higgs with the charged component of $\eta$ field. As
a result of this, the contribution to $R_{\gamma\gamma}$ is determined by the
above trilinear coupling and also by the masses of components of $\eta$ field.
Since the couplings and masses of scalar fields are affected by the above
described analysis of global minimum, we have studied its implications on
$R_{\gamma\gamma}$. In our analysis, we have found that the experimental value
of $R_{\gamma\gamma}$ can be fitted in this model, irrespective of the fact
that the desired minimum of this model coexist with other minima or not. The
fitted value to $R_{\gamma\gamma}$ in this model, is found to be either less
or greater than one, depending on the parameter choice.

Another implication of the scalar sector of the Dirac scotogenic model is on
the dark matter phenomenology [8]. As described above, in this model,
$Z_{2}^{(B)}$ is an exact symmetry. Hence, the lightest particle charged under
the $Z_{2}^{(B)}$ can be a candidate for dark matter. We have studied on the
possibility of making any of the additional scalar fields of this model as a
candidate for thermal cold dark matter.

The paper is organized as follows. In the next section, we give a brief
description on the Dirac scotogenic model. In Sec. 3, we describe all
different possible minima of this model. In Secs. 4 and 5, we discuss on
making the desired minimum of this model as the global minimum. In Sec. 6, we
present our study on the signal strength of Higgs to diphoton decay. In Sec.
7, we discuss on the possibility of a scalar dark matter candidate in this
model. In Sec. 8, we have compared the phenomenology of the Dirac scotogenic
model with that of scotogenic model. We present the conclusions of our work in
Sec. 9. In appendix A, we describe analytical arguments in order to justify
some of the numerical results of Sec. 5.

##  2 Dirac scotogenic model

We have given a brief introduction to the Dirac scotogenic model [21] in Sec.
1. In this work, we follow the original model of this, which is proposed in
[21]. Apart from this, other models are also proposed which have the idea of
scotogenic masses to Dirac neutrinos [22, 23, 24, 25, 26, 27, 28]. In the
Dirac scotogenic model [21], additional scalars and fermionic fields are
introduced along with the additional symmetry $U(1)_{B-L}\times
Z_{2}^{(A)}\times Z_{2}^{(B)}$. The field content of this model, which is
relevant to lepton sector, and their charge assignments are given in Tab. 1.

Field | $SU(2)_{L}$ | $U(1)_{Y}$ | $U(1)_{B-L}$ | $Z_{2}^{(A)}$ | $Z_{2}^{(B)}$
---|---|---|---|---|---
$L_{\alpha}=(\nu_{\alpha},\ell_{\alpha})$ | $2$ | $-1/2$ | $-1$ | $+$ | $+$
$\ell^{c}_{\alpha}$ | $1$ | $1$ | $1$ | $+$ | $+$
$\nu^{c}_{\alpha}$ | $1$ | $0$ | $1$ | $-$ | $+$
$\Phi^{T}=(\phi^{+},\phi^{0})$ | $2$ | $1/2$ | $0$ | $+$ | $+$
$\eta^{T}=(\eta^{+},\eta^{0})$ | $2$ | $1/2$ | $0$ | $+$ | $-$
$\chi$ | $1$ | $0$ | $0$ | $-$ | $-$
$N_{k}$ | $1$ | $0$ | $-1$ | $+$ | $-$
$N_{k}^{c}$ | $1$ | $0$ | $1$ | $+$ | $-$
Table 1: Fields in the lepton sector of Dirac scotogenic model [21] along with
their charge assignments. As described in Sec. 1, $\Phi,\eta$ and $\chi$ are
scalars. Rest of the fields are fermionic.

Here, the symmetry $U(1)_{B-L}$ can be either global or gauged [21]. In this
work, we have taken this to be global and it is an exact symmetry. With the
charge assignments of Tab. 1, the allowed interaction terms in the Lagrangian
are

$-{\cal
L}=y_{\alpha\beta}(\nu_{\alpha}\phi^{+^{*}}+\ell_{\alpha}\phi^{0^{*}})\ell^{c}_{\beta}+f_{\alpha
k}(\nu_{\alpha}\eta^{0}-\ell_{\alpha}\eta^{+})N^{c}_{k}+h_{k\alpha}N_{k}\nu^{c}_{\alpha}\chi+m_{N_{k}}N_{k}N^{c}_{k}+h.c.$
(1)

Here, $m_{N_{k}}$ is the Dirac mass for $N^{D}_{k}=(N_{k},N^{c}_{k})$. The
couplings $y_{\alpha\beta}$ in Eq. (1) generate masses to charged leptons
after the electroweak symmetry breaking, where $\Phi$ acquires non-zero VEV.
On the other hand, the terms of Eq. (1) do not generate masses to light
neutrinos at tree level, since $\eta$ and $\chi$ acquire zero VEVs, which is
due to the fact that both these fields are charged under the exact symmetry
$Z_{2}^{(B)}$. However, if there exist a trilinear term among $\Phi$, $\eta$
and $\chi$, neutrinos acquire masses at 1-loop level in this model. Shortly
below, we explain how such a trilinear term can be present in this model. We
notice from Tab. 1 that all the additional fields of this model are charged
under the $Z_{2}^{(B)}$ symmetry. Since this symmetry is exact, the lightest
among these additional fields can be a candidate for dark matter. Later in
this work, we discuss on the possibility of a scalar dark matter in this
model.

As discussed in the previous section, the Dirac scotogenic model is motivated
from the scotogenic model [16], where neutrinos are Majorana particles. We
notice that there is an analogy between these two models in terms of field
content. Instead of $(N_{k},N^{c}_{k})$, there exist $N^{M}_{k}$ in the
scotogenic model, which is a Majorana field. The scalar sector of the
scotogenic model is same as that of Dirac scotogenic model, except for the
singlet field $\chi$. As a result of this, the third term of Eq. (1) does not
exist in the Lagrangian of scotogenic model. Moreover, by replacing
$(N_{k},N^{c}_{k})$ with a Majorana $N^{M}_{k}$ field in Eq. (1), we get
corresponding terms in the scotogenic model [16]. Later in this work, we
compare the above two models in terms of neutrino masses and other physically
observable quantities.

The scalar potential of the Dirac scotogenic model is [21]

$\displaystyle V$ $\displaystyle=$
$\displaystyle\mu_{1}^{2}\Phi^{\dagger}\Phi+\mu_{2}^{2}\eta^{\dagger}\eta+\frac{1}{2}\mu_{3}^{2}\chi^{2}+\frac{1}{2}\lambda_{1}(\Phi^{\dagger}\Phi)^{2}+\frac{1}{2}\lambda_{2}(\eta^{\dagger}\eta)^{2}+\lambda_{3}(\Phi^{\dagger}\Phi)(\eta^{\dagger}\eta)$
(2)
$\displaystyle+\lambda_{4}(\Phi^{\dagger}\eta)(\eta^{\dagger}\Phi)+\frac{1}{2}[\lambda_{5}(\Phi^{\dagger}\eta)^{2}+h.c.]+\frac{1}{4}\lambda_{6}\chi^{4}+\frac{1}{2}\lambda_{7}(\Phi^{\dagger}\Phi)\chi^{2}+\frac{1}{2}\lambda_{8}(\eta^{\dagger}\eta)\chi^{2}$
$\displaystyle+A\chi[\Phi^{\dagger}\eta+h.c.]$

Here, we have chosen the parameter $A$ to be real by fixing the phases in
$\Phi$ and $\eta$. For this particular choice, the parameter $\lambda_{5}$ can
in general be complex. However, in the analysis of [21], $\lambda_{5}$ is
taken to be real for the sake of simplicity. We discuss about $\lambda_{5}$ in
the context our work, later in the next section. The terms of Eq. (2) generate
masses to physical fields of this model, after the electroweak symmetry
breaking, where $\Phi$ acquires non-zero VEV and the $\eta,\chi$ acquire zero
VEVs. The non-zero VEV for $\Phi$ can be taken as
$\langle\phi^{0}\rangle=v_{EW}=174$ GeV, which is the electroweak symmetry
breaking scale. We see that, after this symmetry breaking, $\phi^{+}$ and
imaginary part of $\phi^{0}$ become Goldstone bosons. The real part of
$\phi^{0}$ is physical and we identify $Re(\phi^{0})=H$ as the Higgs boson.
The charged component of $\eta$ is physical. On the other hand, the neutral
component of $\eta$ has mixing with $\chi$ through the last term of Eq. (2).
As a result of this, we write
$\eta^{0}=(\eta_{R}^{0}+i\eta_{I}^{0})/\sqrt{2}$. Now, the mass spectrum of
the physical scalar fields in the Dirac scotogenic model is [21]

$\displaystyle m_{H}^{2}=2\lambda_{1}v_{EW}^{2},$ $\displaystyle
m_{\eta^{+}}^{2}=\mu_{2}^{2}+\lambda_{3}v_{EW}^{2},$ $\displaystyle
m_{\eta_{I}^{0}}^{2}\equiv
m_{\zeta_{3}}^{2}=\mu_{2}^{2}+(\lambda_{3}+\lambda_{4}-\lambda_{5})v_{EW}^{2},$
$\displaystyle
M_{\eta_{R}^{0},\chi}^{2}=\left(\begin{array}[]{cc}\mu_{2}^{2}+(\lambda_{3}+\lambda_{4}+\lambda_{5})v_{EW}^{2}&\sqrt{2}Av_{EW}\\\
\sqrt{2}Av_{EW}&\mu_{3}^{2}+\lambda_{7}v_{EW}^{2}\end{array}\right)$ (5)

Here, $M_{\eta_{R}^{0},\chi}^{2}$ gives the mixing masses between
$\eta_{R}^{0}$ and $\chi$, whose mass eigenstates are denoted by
$\zeta_{1,2}$. Also here, for the sake of notational simplicity, we have
written $\eta_{I}^{0}=\zeta_{3}$.

The last term of Eq. (2) breaks the $Z_{2}^{(A)}$ symmetry softly. This term
is necessary in order to generate masses to neutrinos at 1-loop level [21],
and moreover, this is the trilinear term that we have discussed above. Now,
with the $A$-term of Eq. (2) and with the interaction terms of Eq. (1),
neutrinos acquire masses at 1-loop, whose expressions are given below [21].

$\displaystyle({\cal M}_{\nu})_{\alpha\beta}$ $\displaystyle=$
$\displaystyle(f\Lambda h)_{\alpha\beta}=\sum_{K=1}^{3}f_{\alpha
k}\Lambda_{k}h_{k\beta},$ $\displaystyle\Lambda_{k}$ $\displaystyle=$
$\displaystyle\frac{\sin(2\theta)}{32\pi^{2}\sqrt{2}}m_{N_{k}}\left[\frac{m_{\zeta_{1}}^{2}}{m_{\zeta_{1}}^{2}-m_{N_{k}}^{2}}\ln\frac{m_{\zeta_{1}}^{2}}{m_{N_{k}}^{2}}-\frac{m_{\zeta_{2}}^{2}}{m_{\zeta_{2}}^{2}-m_{N_{k}}^{2}}\ln\frac{m_{\zeta_{2}}^{2}}{m_{N_{k}}^{2}}\right]$
(6)

Here, $\theta$ is the mixing angle between $\eta^{0}_{R}$ and $\chi$. As a
result of this mixing, the diagonal masses for these fields are denoted by
$m_{\zeta_{1}}$ and $m_{\zeta_{2}}$. As already described before, this mixing
is arising due to the $A$-term of Eq. (2). As a result of this, $\theta$ is
proportional to the $A$ parameter. Since the $A$-term breaks $Z_{2}^{(A)}$
symmetry softly, the parameter $A$, and hence, the $\theta$ can be small. The
small mass for neutrinos in Eq. (6) can be explained either through small
mixing angle $\theta$, or by large value for $m_{N_{k}}$, or by taking
degenerate masses for $\zeta_{1,2}$, apart from the loop suppression factor.
It is to be noted that the masses of $\zeta_{1,2}$ depend on various
parameters, which are given in Eq. (5). It is possible to fine tune these
parameters in such way that $m_{\zeta_{1}}$ and $m_{\zeta_{2}}$ are nearly
degenerate, which gives an additional suppression for neutrino masses in Eq.
(6).

Apart from the smallness of neutrino masses, we need to explain the observed
neutrino mixing angles [8] in the Dirac scotogenic model. In order to explain
this, we use the Casas-Ibarra parametrization [29], and thereby, the Yukawa
couplings in Eq. (6) can be expressed as

$\displaystyle f$ $\displaystyle=$ $\displaystyle
U\sqrt{m_{\nu}}R\sqrt{\Lambda}^{-1},\quad
h=\sqrt{\Lambda}^{-1}S^{\dagger}\sqrt{m_{\nu}}V^{\dagger},$ $\displaystyle
m_{\nu}$ $\displaystyle=$ $\displaystyle{\rm
diag}(m_{\nu_{1}},m_{\nu_{2}},m_{\nu_{3}})$ (7)

Here, $m_{\nu_{i}}$, where $i=1,2,3$, are the neutrino mass eigenvalues. $R$
and $S$ are in general complex matrices, which satisfy $RS^{\dagger}=I$. After
using the above parametrizations for $f$ and $h$ in Eq. (6), we get
$U^{\dagger}{\cal M}_{\nu}V=m_{\nu}$, which is the desired relation for
diagonalizing the ${\cal M}_{\nu}$. Here, $U=U_{PMNS}$ is identified as the
Pontecorvo-Maki-Nakagawa-Sakata matrix, which is parametrized in terms of the
three neutrino mixing angles and a $CP$ violating Dirac phase [8]. $V$ is a
unitary matrix which rotate the right-handed neutrino fields from flavor to
mass eigenstates. As a result of the above given description, the neutrino
mixing angles in the Dirac scotogenic model can be explained by parametrizing
the Yukawa couplings as in Eq. (7). In the parametrizations of $f$ and $h$,
the neutrino mass eigenvalues can be chosen either in normal or inverted
ordering, in order to fit the solar and atmospheric mass-square differences of
neutrinos [8].

The expression for neutrino masses in the Dirac scotogenic model, which is
given in Eq. (6) is similar to the corresponding expression of the scotogenic
model [16]. In the context of neutrino mass generation, the difference between
the above two models is described below. The neutrino masses in the Dirac
scotogenic model are driven by two different Yukawa couplings, whereas, in the
scotogenic model these masses are driven by one kind of Yukawa couplings. The
parametrizations of Yukawa couplings in the Dirac scotogenic model, which are
given in Eq. (7), are similar to that in the scotogenic model, which can be
seen from [30]. In [30], we have worked on the lepton flavor violating (LFV)
decays of $Z$ and Higgs boson in the scotogenic model, where we have also done
numerical analysis on fitting the neutrino masses and mixing angles in this
model. This numerical analysis can be analogously worked in the Dirac
scotogenic model.

To test the Dirac scotogenic model in collider experiments, such as the LHC,
we should take the masses of all additional fields to be around few hundred
GeV. Now, using the discussion given before, for $m_{N_{K}}\sim$ 1 TeV, the
small masses for neutrinos can be explained if either of the following
quantities are taken to be small: $\theta$ or $m_{\zeta_{1}}-m_{\zeta_{2}}$.
In the limit that these quantities are small, we see that $\Lambda_{k}$ of Eq.
(6) becomes small. Hence, from Eq. (7), the Yukawa couplings $f$ and $h$ can
become ${\cal O}(1)$ in some region of parameter space. The couplings
$f_{\alpha k}$ drive LFV processes in this model which can have significant
branching ratios, since $f_{\alpha k}$ are not suppressed and $m_{N_{K}}\sim$
1 TeV. Later in this work, we describe about LFV processes of this model.

As described before, in the Dirac scotogenic model, $\Phi$ acquires non-zero
VEV and $\eta,\chi$ should acquire zero VEVs. This pattern of VEVs can be
achieved by minimizing the potential of Eq. (2) for $\mu_{1}^{2}<0$. Here we
notice that in some parameter region of the scalar potential the above pattern
of VEVs constitute a minimum for the model. By choosing different parameter
regions of the scalar potential, it is possible to find other minima of this
model. In the next section, we argue that the above minimum required for the
Dirac scotogenic model is only one possible minima of this model.

##  3 Possible minima of the Dirac scotogenic model

In this section, we describe different possible minima of the Dirac scotogenic
model, after analyzing the scalar potential of it. The scalar potential of
this model is described in Eq. (2), which consists of the fields $\Phi$,
$\eta$ and $\chi$. At the minimum of the scalar potential, either of these
scalar fields can acquire non-zero VEVs. First, let us consider the case where
$\langle\Phi\rangle\neq 0$ and $\langle\eta\rangle=0$. In this case, using the
$SU(2)$ transformation, the VEV of $\Phi$ can always be brought into a form
where the neutral component of it acquires non-zero and real VEV. As a result
of this, there can exist two different minima, depending on whether $\chi$
acquires a VEV or not. These possible minima are

$\displaystyle N1:\quad\langle\Phi\rangle=\left(\begin{array}[]{c}0\\\
v_{EW}\end{array}\right),\quad\langle\eta\rangle=\left(\begin{array}[]{c}0\\\
0\end{array}\right),\quad\langle\chi\rangle=0.$ (12) $\displaystyle
N2:\quad\langle\Phi\rangle=\left(\begin{array}[]{c}0\\\
v_{\phi(2)}\end{array}\right)\quad\langle\eta\rangle=\left(\begin{array}[]{c}0\\\
0\end{array}\right),\quad\langle\chi\rangle=v_{\chi(2)},$ (17)

where $v_{\phi(2)}$ and $v_{\chi(2)}$ are some non-zero real variables. Here,
$N1,N2$ are neutral minima which break only the electroweak symmetry. $N1$ is
the desired minimum of the Dirac scotogenic model, which we have considered to
be the true model of our universe. Hence, we have equated the VEV of $\Phi$
field to $v_{EW}$. On the other hand, $N2$ is one possible minima of the
scalar potential of the Dirac scotogenic model. But otherwise, this minimum is
not consistent with the model framework. Hence, $N2$ does not represent the
observable world of our universe. As a result of this, the VEVs of $\Phi$ and
$\chi$ in this minimum are some variables and need not represent the
electroweak symmetry breaking scale. Now, let us consider the case where
$\langle\Phi\rangle=0$ and $\langle\eta\rangle\neq 0$. In this case, in
analogy to the description given above, the following two minima are possible:

$\displaystyle N3:\quad\langle\Phi\rangle=\left(\begin{array}[]{c}0\\\
0\end{array}\right),\quad\langle\eta\rangle=\left(\begin{array}[]{c}0\\\
v_{\eta(3)}\end{array}\right),\quad\langle\chi\rangle=0.$ (22) $\displaystyle
N4:\quad\langle\Phi\rangle=\left(\begin{array}[]{c}0\\\
0\end{array}\right)\quad\langle\eta\rangle=\left(\begin{array}[]{c}0\\\
v_{\eta(4)}\end{array}\right),\quad\langle\chi\rangle=v_{\chi(4)}.$ (27)

The non-zero entries in the above equations are real and arbitrary. The minima
$N3$ and $N4$ are unphysical, since they do not generate masses to SM fermions
of this model.

In the case where $\langle\Phi\rangle=0=\langle\eta\rangle$, there can exist
one non-trivial minimum, which is given below.

$N5:\quad\langle\Phi\rangle=\left(\begin{array}[]{c}0\\\
0\end{array}\right)\quad\langle\eta\rangle=\left(\begin{array}[]{c}0\\\
0\end{array}\right),\quad\langle\chi\rangle=v_{\chi(5)}.$ (28)

Here, the VEV of $\chi$ is real and non-zero variable. The minimum $N5$ is
clearly unphysical, since it doesn’t break the electroweak symmetry. Finally,
in the case where $\langle\Phi\rangle\neq 0\neq\langle\eta\rangle$, the
following six minima are possible:

$\displaystyle N6:\quad\langle\Phi\rangle=\left(\begin{array}[]{c}0\\\
v_{\phi(6)}\end{array}\right),\quad\langle\eta\rangle=\left(\begin{array}[]{c}0\\\
v_{\eta(6)}\end{array}\right),\quad\langle\chi\rangle=0.$ (33) $\displaystyle
N7:\quad\langle\Phi\rangle=\left(\begin{array}[]{c}0\\\
v_{\phi(7)}\end{array}\right),\quad\langle\eta\rangle=\left(\begin{array}[]{c}0\\\
iv_{\eta(7)}\end{array}\right),\quad\langle\chi\rangle=0.$ (38) $\displaystyle
N8:\quad\langle\Phi\rangle=\left(\begin{array}[]{c}0\\\
v_{\phi(8)}\end{array}\right)\quad\langle\eta\rangle=\left(\begin{array}[]{c}0\\\
v_{\eta(8)}\end{array}\right),\quad\langle\chi\rangle=v_{\chi(8)}.$ (43)
$\displaystyle C9:\quad\langle\Phi\rangle=\left(\begin{array}[]{c}0\\\
v_{\phi(9)}\end{array}\right),\quad\langle\eta\rangle=\left(\begin{array}[]{c}c_{\eta(9)}\\\
v_{\eta(9)}\end{array}\right),\quad\langle\chi\rangle=0.$ (48) $\displaystyle
C10:\quad\langle\Phi\rangle=\left(\begin{array}[]{c}0\\\
v_{\phi(10)}\end{array}\right)\quad\langle\eta\rangle=\left(\begin{array}[]{c}c_{\eta(10)}\\\
iv_{\eta(10)}\end{array}\right),\quad\langle\chi\rangle=0.$ (53)
$\displaystyle C11:\quad\langle\Phi\rangle=\left(\begin{array}[]{c}0\\\
v_{\phi(11)}\end{array}\right)\quad\langle\eta\rangle=\left(\begin{array}[]{c}c_{\eta(11)}\\\
v_{\eta(11)}\end{array}\right),\quad\langle\chi\rangle=v_{\chi(11)}.$ (58)

Here, $C9$, $C10$ and $C11$ are charge-breaking minima, where the charged
component of $\eta$ acquires non-zero VEV. On the other hand, $N6$, $N7$ and
$N8$ break only the electroweak symmetry. In obtaining the forms of VEVs in
Eqs. (33)$-$(58), we have used the $SU(2)$ transformation [31] on $\Phi$ and
$\eta$. It is to be noticed that the variables of the form
$v_{\phi(k)},v_{\eta(k)},c_{\eta(k)},v_{\chi(k)}$ in Eqs. (33)$-$(58) are real
and non-zero. While obtaining that these variables are real, we have taken the
$\lambda_{5}$ parameter of Eq. (2) to be real. Shortly below, we give a
demonstration about this. We notice that the VEV of $\eta$ in Eqs. (38) and
(53) is complex, which tells something about $CP$ symmetry in the minima of
$N7$ and $C10$. As we have taken $\lambda_{5}$ to be real, the minimum $N7$
respect the $CP$ symmetry, since Eqs. (2) and (38) are invariant under the
following $CP$ transformation:
$\Phi\to\Phi^{*},\eta\to-\eta^{*},\chi\to-\chi$. On the other hand, $C10$
breaks the $CP$ symmetry spontaneously, apart from the charge and electroweak
symmetries.

After using the $SU(2)$ transformation [31], the general structure of the VEVs
of the scalar fields in the Dirac scotogenic model can be written as

$\langle\Phi\rangle=\left(\begin{array}[]{c}0\\\
v_{\phi}\end{array}\right)\quad\langle\eta\rangle=\left(\begin{array}[]{c}c_{\eta}\\\
v_{\eta}\end{array}\right),\quad\langle\chi\rangle=v_{\chi}$ (59)

In the above equation, $v_{\phi},c_{\eta},v_{\chi}$ are real variables and
$v_{\eta}$ is in general a complex variable. Apart from $v_{\eta}$, the other
complex variable in this model is the $\lambda_{5}$ parameter of Eq. (2). As a
result of this, we write the forms for $\lambda_{5}$ and $v_{\eta}$ as

$\lambda_{5}=|\lambda_{5}|e^{i\theta_{5}},\quad
v_{\eta}=|v_{\eta}|e^{i\theta_{\eta}}$ (60)

Here, $\theta_{5}$ and $\theta_{\eta}$ are phases in $\lambda_{5}$ and
$v_{\eta}$, respectively. After plugging Eq. (59) in Eq. (2), the relevant
part of the potential is

$\langle
V\rangle=|\lambda_{5}|v_{\phi}^{2}|v_{\eta}|^{2}\cos(\theta_{5}+2\theta_{\eta})+2Av_{\chi}v_{\phi}|v_{\eta}|\cos\theta_{\eta}$
(61)

In the above equation, we have written only that part of the potential which
contains only the phases $\theta_{5}$ and $\theta_{\eta}$. The extremum for
$\langle V\rangle$ with respect to these phases is found to be

$\theta_{5}=(m-2n)\pi,\quad\theta_{\eta}=n\pi$ (62)

Here, $m,n$ are any integers. Using the above relations, we see that
$\lambda_{5}$ and $v_{\eta}$ are real quantities at any extremum of the
potential. To see if the values of Eq. (62) correspond to the minimum of Eq.
(61), we need to evaluate the second order derivatives of $\langle V\rangle$
with respect to $\theta_{5}$ and $\theta_{\eta}$. As a result of this, we get
the following matrix

$M_{V}=\left.\left(\begin{array}[]{cc}\frac{\partial^{2}\langle
V\rangle}{\partial\theta_{5}^{2}}&\frac{\partial^{2}\langle
V\rangle}{\partial\theta_{5}\partial\theta_{\eta}}\\\
\frac{\partial^{2}\langle
V\rangle}{\partial\theta_{5}\partial\theta_{\eta}}&\frac{\partial^{2}\langle
V\rangle}{\partial\theta_{\eta}^{2}}\end{array}\right)\right|_{\theta_{5}=(m-2n)\pi,\theta_{\eta}=n\pi}$
(63)

After demanding that the eigenvalues of $M_{V}$ are positive, and since
$\lambda_{5}$ and $v_{\eta}$ are real at the minimum of the potential, we get
the following conditions

$\lambda_{5}<0,\quad Av_{\phi}v_{\eta}v_{\chi}<0$ (64)

Using the analysis, which is described in the previous paragraph, we notice
that in the minima of Eqs. (43) and (58), the quantities $v_{\eta(8)}$ and
$v_{\eta(11)}$ should be real. Moreover, in these minima, $\lambda_{5}$ should
be a real parameter and the conditions of Eq. (64) should be satisfied. Now,
in the analysis of previous paragraph, let us consider the case where
$v_{\chi}=0$. In this case, at the minimum of the potential, the individual
phases in $\lambda_{5}$ and $v_{\eta}$ cannot be determined. However, if we
choose $\lambda_{5}$ to be real, then $v_{\eta}$ can be either real or purely
imaginary at the minimum of the potential. As a result of this, the VEV of the
neutral component of $\eta$ is taken to be real in Eqs. (33) and (48),
whereas, this quantity is taken to be purely imaginary in Eqs. (38) and (53).
Let us mention here that $\lambda_{5}$ can in general be complex in the minima
of Eqs. (12)$-$(28). However, to simplify our numerical analysis, we have
taken $\lambda_{5}$ to be real in the rest of this work.

We have described that the VEV structures given in Eqs. (12)$-$(58) as
possible minima of the model. In order to clarify this point, we refer each of
these VEV structures as a stationary point (SP). An SP becomes a minimum if
the following two conditions are satisfied: (1) minimization conditions of the
scalar potential, (2) mass-square eigenvalues of scalar fields, which are not
Goldstone bosons, are positive. The minimization conditions, which should be
satisfied by the SPs of Eqs. (12)$-$(58), are given in Tab. 2.

$N1$ | $\mu_{1}^{2}+\lambda_{1}v_{EW}^{2}=0$
---|---
$N2$ | $A=0,\quad\mu_{1}^{2}+\lambda_{1}v_{\phi(2)}^{2}+\frac{1}{2}\lambda_{7}v_{\chi(2)}^{2}=0,\quad\mu_{3}^{2}+\lambda_{6}v_{\chi(2)}^{2}+\lambda_{7}v_{\phi(2)}^{2}=0$
$N3$ | $\mu_{2}^{2}+\lambda_{2}v_{\eta(3)}^{2}=0$
$N4$ | $A=0,\quad\mu_{2}^{2}+\lambda_{2}v_{\eta(4)}^{2}+\frac{1}{2}\lambda_{8}v_{\chi(4)}^{2}=0,\quad\mu_{3}^{2}+\lambda_{6}v_{\chi(4)}^{2}+\lambda_{8}v_{\eta(4)}^{2}=0$
$N5$ | $\mu_{3}^{2}+\lambda_{6}v_{\chi(5)}^{2}=0$
$N6$ |  $A=0,\quad\mu_{1}^{2}+\lambda_{1}v_{\phi(6)}^{2}+(\lambda_{3}+\lambda_{4}+\lambda_{5})v_{\eta(6)}^{2}=0$,
|
$\mu_{2}^{2}+\lambda_{2}v_{\eta(6)}^{2}+(\lambda_{3}+\lambda_{4}+\lambda_{5})v_{\phi(6)}^{2}=0$
$N7$ | $\mu_{1}^{2}+\lambda_{1}v_{\phi(7)}^{2}+(\lambda_{3}+\lambda_{4}-\lambda_{5})v_{\eta(7)}^{2}=0,\quad\mu_{2}^{2}+\lambda_{2}v_{\eta(7)}^{2}+(\lambda_{3}+\lambda_{4}-\lambda_{5})v_{\phi(7)}^{2}=0$
$N8$ | $\mu_{1}^{2}+\lambda_{1}v_{\phi(8)}^{2}+(\lambda_{3}+\lambda_{4}+\lambda_{5})v_{\eta(8)}^{2}+\frac{1}{2}\lambda_{7}v_{\chi(8)}^{2}+Av_{\chi(8)}v_{\eta(8)}/v_{\phi(8)}=0,$
|
$\mu_{2}^{2}+\lambda_{2}v_{\eta(8)}^{2}+(\lambda_{3}+\lambda_{4}+\lambda_{5})v_{\phi(8)}^{2}+\frac{1}{2}\lambda_{8}v_{\chi(8)}^{2}+Av_{\chi(8)}v_{\phi(8)}/v_{\eta(8)}=0,$
|
$\mu_{3}^{2}+\lambda_{6}v_{\chi(8)}^{2}+\lambda_{7}v_{\phi(8)}^{2}+\lambda_{8}v_{\eta(8)}^{2}+2Av_{\phi(8)}v_{\eta(8)}/v_{\chi(8)}=0$
$C9$ | $A=0,\quad\lambda_{4}+\lambda_{5}=0,\quad\mu_{1}^{2}+\lambda_{1}v_{\phi(9)}^{2}+\lambda_{3}(c_{\eta(9)}^{2}+v_{\eta(9)}^{2})=0,$
|
$\mu_{2}^{2}+\lambda_{2}(c_{\eta(9)}^{2}+v_{\eta(9)}^{2})+\lambda_{3}v_{\phi(9)}^{2}=0$
$C10$ | $\lambda_{4}-\lambda_{5}=0,\quad\mu_{1}^{2}+\lambda_{1}v_{\phi(10)}^{2}+\lambda_{3}(c_{\eta(10)}^{2}+v_{\eta(10)}^{2})=0,$
|
$\mu_{2}^{2}+\lambda_{2}(c_{\eta(10)}^{2}+v_{\eta(10)}^{2})+\lambda_{3}v_{\phi(10)}^{2}=0$
$C11$ | $Av_{\chi(11)}+(\lambda_{4}+\lambda_{5})v_{\eta(11)}v_{\phi(11)}=0,$
|
$\mu_{1}^{2}+\lambda_{1}v_{\phi(11)}^{2}+\lambda_{3}(c_{\eta(10)}^{2}+v_{\eta(11)}^{2})+\frac{1}{2}\lambda_{7}v_{\chi(11)}^{2}=0,$
|
$\mu_{2}^{2}+\lambda_{2}(c_{\eta(11)}^{2}+v_{\eta(11)}^{2})+\lambda_{3}v_{\phi(11)}^{2}+\frac{1}{2}\lambda_{8}v_{\chi(11)}^{2}=0,$
|
$\mu_{3}^{2}+\lambda_{6}v_{\chi(11)}^{2}+\lambda_{7}v_{\phi(11)}^{2}+\lambda_{8}(c_{\eta(11)}^{2}+v_{\eta(11)}^{2})+2Av_{\phi(11)}v_{\eta(11)}/v_{\chi(11)}=0$
Table 2: Minimization conditions which should be satisfied at the SPs of Eqs.
(12)$-$(58).

Shortly below, we describe on how we compute the mass-square eigenvalues of
scalar fields at the SPs of Eqs. (12)$-$(58). It is to be noted that, in the
Dirac scotogenic model, the scalar sector consists of nine real degrees of
freedom. Out of these nine, some of them may become Goldstone bosons, since
the electroweak and charge symmetries are spontaneously broken by some of the
SPs of Eqs. (12)$-$(58). The SP $N5$ of Eq. (28) does not break either of
these symmetries, and hence, in this case all the nine scalar fields become
massive.

As stated before, the general structure of VEVs of scalar fields in the Dirac
scotogenic model is given by Eq. (59). Hence, in order to compute the mass
eigenstates, we parametrize these fields as

$\Phi=\left(\begin{array}[]{c}(\phi_{R}^{1}+i\phi_{I}^{1})/\sqrt{2}\\\
v_{\phi}+(\phi_{R}^{0}+i\phi_{I}^{0})/\sqrt{2}\end{array}\right),\quad\eta=\left(\begin{array}[]{c}c_{\eta}+(\eta_{R}^{1}+i\eta_{I}^{1})/\sqrt{2}\\\
v_{\eta}+(\eta_{R}^{0}+i\eta_{I}^{0})/\sqrt{2}\end{array}\right),\quad\chi=v_{\chi}+\chi_{R}$
(65)

Now, in the case of Eq. (12), which corresponds to the desired minimum of the
Dirac scotogenic model, $\phi_{R}^{1},\phi_{I}^{1},\phi_{I}^{0}$ become
Goldstone bosons. Moreover, for this minimum, we have $v_{\phi}=v_{EW}$ and
$c_{\eta}=v_{\eta}=v_{\chi}=0$. Now, after identify $\phi_{R}^{0}=H$ as the
Higgs boson and $\eta^{+}=(\eta_{R}^{1}+i\eta_{I}^{1})/\sqrt{2}$, we see that
the mass spectrum of physical fields in the case of $N1$ matches with that of
Eq. (5). Similar to what we described above, the mass spectrum of scalar
fields at other SPs of Eqs. (17)$-$(58) are computed accordingly. It should be
noted here that, even if any of these SPs become minima, the scalar fields in
these minima are unphysical, since these minima do not represent the physical
world of ours. In our analysis, which we will present later, we have computed
the scalar masses at the SPs of Eqs. (17)$-$(58) numerically. We have found in
our analysis that each of the SPs given in Eqs. (12)$-$(58) can become a
minimum in some region of parameter space of the scalar potential. Since
multiple minima can exist for the scalar potential of the Dirac scotogenic
model, we need to know if $N1$, which is the desired minimum of this model,
can become the global minimum. The next few sections discuss about this.

##  4 Relative depths in potential

At the end of the last section, we have noted that $N1$ is one possible
minimum among other minima of the Dirac scotogenic model. In order to address
if $N1$ is the global minimum of this model, we need to know if $N1$ coexist
with other minima. If it coexist with other minima, we demand that the
potential depth at $N1$ is lower as compared to that at other minima, so that
$N1$ is the global minimum of this model. In this section, we assume that the
minimum $N1$ to coexist with other minima in some region of parameter space.
We then calculate the differences in the depth of potential at $N1$ and at
other minima. Using these quantities, we predict on the possibility of making
$N1$ as the global minimum of this model. The results obtained in this section
are helpful for the next section, where we study on the coexistence of $N1$
with other minima.

In order to compute the difference between the value of potential at $N1$ and
at any other SP, we follow the work of [12, 13, 32], which is based on a
formalism of bilinears. For some works using the bilinear formalism, see [33,
34, 35, 36, 37]. Using this formalism, we notice that, except for the last
term of Eq. (2), other terms of the scalar potential are either quadratic or
quartic. As a result of this, we define the following bilinears:

$x_{1}=\Phi^{\dagger}\Phi,\quad x_{2}=\eta^{\dagger}\eta,\quad
x_{3}=\chi^{2},\quad x_{4}=Re(\Phi^{\dagger}\eta),\quad
x_{5}=Im(\Phi^{\dagger}\eta)$ (66)

Apart from these, we also define the following matrices:

$X=\left(\begin{array}[]{c}x_{1}\\\ x_{2}\\\ x_{3}\\\ x_{4}\\\
x_{5}\end{array}\right),\quad M_{2}=\left(\begin{array}[]{c}\mu_{1}^{2}\\\
\mu_{2}^{2}\\\ \frac{1}{2}\mu_{3}^{2}\\\ 0\\\ 0\end{array}\right),\quad
M_{4}=\left(\begin{array}[]{ccccc}\lambda_{1}&\lambda_{3}&\frac{1}{2}\lambda_{7}&0&0\\\
\lambda_{3}&\lambda_{2}&\frac{1}{2}\lambda_{8}&0&0\\\
\frac{1}{2}\lambda_{7}&\frac{1}{2}\lambda_{8}&\frac{1}{2}\lambda_{6}&0&0\\\
0&0&0&2(\lambda_{4}+\lambda_{5})&0\\\ 0&0&0&0&2(\lambda_{4}-\lambda_{5})\\\
\end{array}\right)$ (67)

Now, using Eqs. (66) and (67), the scalar potential of Eq. (2) can be
expressed as

$\displaystyle V=V_{2}+V_{3}+V_{4},$ $\displaystyle V_{2}=M_{2}^{T}X,\quad
V_{3}=2A\chi x_{4},\quad V_{4}=\frac{1}{2}X^{T}M_{4}X$ (68)

At any SP of Eqs. (12)$-$(58), the scalar potential has to satisfy the
minimization conditions. Hence, we get the following relation [32]

$\sum_{i}\varphi_{i}\left.\frac{\partial
V}{\partial\varphi_{i}}\right|_{SP}=0\implies
2(V_{2})_{SP}+3(V_{3})_{SP}+4(V_{4})_{SP}=0$ (69)

In the above equation, $\varphi_{i}$ represent any real scalar degree of
freedom of the Dirac scotogenic model. Here, $(V_{i})_{SP}$, where $i=2,3,4$,
is the value of $V_{i}$ evaluated at an SP. Using the above relation, the
value of scalar potential evaluated at an SP is found to be

$V_{SP}=\frac{1}{2}(V_{2})_{SP}+\frac{1}{4}(V_{3})_{SP}$ (70)

We define $X_{SP}$ as the matrix $X$ evaluated at an SP. We then define the
following quantity at any SP:

$V^{\prime}_{SP}=M_{2}+M_{4}X_{SP}$ (71)

Using the above definitions, for the case of $N1$, we get

$X_{N1}=\left(\begin{array}[]{c}v_{EW}^{2}\\\ 0\\\ 0\\\ 0\\\
0\end{array}\right),\quad V^{\prime}_{N1}=\left(\begin{array}[]{c}0\\\
m_{\eta^{+}}^{2}\\\ \frac{1}{2}(\mu_{3}^{2}+\lambda_{7}v_{EW}^{2})\\\ 0\\\
0\end{array}\right)$ (72)

While obtaining the form of $V^{\prime}_{N1}$, we have used the minimization
condition for $N1$, which is given in Tab. 2. Similarly, for other SPs of Eqs.
(17)$-$(58), we have obtained $X_{SP}$ and $V^{\prime}_{SP}$ accordingly.
Below we describe the expression for difference in the value of potential at
$N1$ and at any other SP. To do this computation, we first consider the
following products: $X_{N1}^{T}V^{\prime}_{SP}$ and
$X_{SP}^{T}V^{\prime}_{N1}$. Now, using the quantities described in the
previous paragraph, these products can be expressed as

$\displaystyle
X_{N1}^{T}V^{\prime}_{SP}=2V_{N1}-\frac{1}{2}(V_{3})_{N1}+X_{N1}^{T}M_{4}X_{SP},$
$\displaystyle
X_{SP}^{T}V^{\prime}_{N1}=2V_{SP}-\frac{1}{2}(V_{3})_{SP}+X_{SP}^{T}M_{4}X_{N1}$
(73)

From the relations in the above equation, we get the following expression,
which gives the relative depth in potential between a SP and $N1$.

$V_{SP}-V_{N1}=\frac{1}{2}(X_{SP}^{T}V^{\prime}_{N1}-X_{N1}^{T}V^{\prime}_{SP})+\frac{1}{4}((V_{3})_{SP}-(V_{3})_{N1})$
(74)

It is to remind here that the above expression is obtained after using the
minimization conditions for the SPs of Eqs. (12)$-$(58), but otherwise, these
SPs need not be minima.

Using the general expression given in Eq. (74), we have computed the
differences in the value of potential at $N1$ and at any other SP of Eqs.
(17)$-$(58). These expressions are given below.

$\displaystyle V_{N2}-V_{N1}$ $\displaystyle=$
$\displaystyle\frac{v_{\chi(2)}^{2}}{4}(\mu_{3}^{2}+\lambda_{7}v_{EW}^{2}),$
(75) $\displaystyle V_{N3}-V_{N1}$ $\displaystyle=$
$\displaystyle\frac{v_{\eta(3)}^{2}}{2}\mu_{2}^{2}-\frac{v_{EW}^{2}}{2}\mu_{1}^{2},$
(76) $\displaystyle V_{N4}-V_{N1}$ $\displaystyle=$
$\displaystyle\frac{v_{\eta(4)}^{2}}{2}\mu_{2}^{2}-\frac{v_{EW}^{2}}{2}\mu_{1}^{2}+\frac{v_{\chi(4)}^{2}}{4}\mu_{3}^{2},$
(77) $\displaystyle V_{N5}-V_{N1}$ $\displaystyle=$
$\displaystyle\frac{v_{\chi(5)}^{2}}{4}\mu_{3}^{2}-\frac{v_{EW}^{2}}{2}\mu_{1}^{2},$
(78) $\displaystyle V_{N6}-V_{N1}$ $\displaystyle=$
$\displaystyle\frac{v_{\eta(6)}^{2}}{2}(\mu_{2}^{2}+(\lambda_{3}+\lambda_{4}+\lambda_{5})v_{EW}^{2}),$
(79) $\displaystyle V_{N7}-V_{N1}$ $\displaystyle=$
$\displaystyle\frac{v_{\eta(7)}^{2}}{2}m_{\eta_{I}^{0}}^{2},$ (80)
$\displaystyle V_{N8}-V_{N1}$ $\displaystyle=$
$\displaystyle\frac{v_{\eta(8)}^{2}}{2}(\mu_{2}^{2}+(\lambda_{3}+\lambda_{4}+\lambda_{5})v_{EW}^{2})+\frac{v_{\chi(8)}^{2}}{4}(\mu_{3}^{2}+\lambda_{7}v_{EW}^{2})$
(81)
$\displaystyle+\frac{Av_{\chi(8)}v_{\eta(8)}}{2v_{\phi(8)}}(v_{EW}^{2}+v_{\phi(8)}^{2}),$
$\displaystyle V_{C9}-V_{N1}$ $\displaystyle=$
$\displaystyle\frac{c_{\eta(9)}^{2}+v_{\eta(9)}^{2}}{2}m_{\eta^{+}}^{2},$ (82)
$\displaystyle V_{C10}-V_{N1}$ $\displaystyle=$
$\displaystyle\frac{c_{\eta(10)}^{2}+v_{\eta(10)}^{2}}{2}m_{\eta^{+}}^{2},$
(83) $\displaystyle V_{C11}-V_{N1}$ $\displaystyle=$
$\displaystyle\frac{c_{\eta(11)}^{2}+v_{\eta(11)}^{2}}{2}m_{\eta^{+}}^{2}+\frac{v_{\chi(11)}^{2}}{4}(\mu_{3}^{2}+\lambda_{7}v_{EW}^{2})+\frac{1}{2}Av_{\chi(11)}v_{\eta(11)}v_{\phi(11)}$
(84)

In the above equations, $m_{\eta^{+}}^{2}$ and $m_{\eta_{I}^{0}}^{2}$ are
mass-square eigenvalues of $\eta^{+}$ and $\eta_{I}^{0}$ for $N1$, whose
expressions are given in Eq. (5). Now, in the region where $N1$ is a minimum,
we should have $m_{\eta^{+}}^{2}>0$ and $m_{\eta_{I}^{0}}^{2}>0$. Hence, in
this region, we get $V_{N7}-V_{N1}>0$, $V_{C9}-V_{N1}>0$ and
$V_{C10}-V_{N1}>0$. This means, if the $N1$ minimum coexist with either of
$N7$, $C9$ and $C10$, the value of potential at $N1$ is always lower than that
at $N7$, $C9$ and $C10$. Now, let us consider a region where the $N1$ minimum
coexist with either $N2$ or $N6$. In this region, we should have $A=0$, since
this is one of the minimization conditions for $N2$ and $N6$, which can be
seen from Tab. 2. Now, using $A=0$ in Eq. (5) and also from the fact that $N1$
is a minimum, we get
$\mu_{2}^{2}+(\lambda_{3}+\lambda_{4}+\lambda_{5})v_{EW}^{2}>0$ and
$\mu_{3}^{2}+\lambda_{7}v_{EW}^{2}>0$. This implies $V_{N2}-V_{N1}>0$ and
$V_{N6}-V_{N1}>0$, which means that the potential depth at $N1$ is deeper than
that at $N2$ and $N6$.

We have argued above that the potential value at $N1$ is always lower than
that at $N2$, $N6$, $N7$, $C9$ and $C10$, in a region where these minima
coexist. However, the situation is different in a region where the $N1$
minimum coexist with either of $N3$, $N4$ and $N5$. In each of Eqs.
(76)$-$(78), there exist both positive and negative terms, and hence, it is
not guaranteed that the potential depth at $N1$ is deeper than that at $N3$,
$N4$ and $N5$. Now, let us look at Eqs. (81) and (84), which give the relative
potential depths for $N8$ and $C11$ with $N1$. As described in Sec. 3, the
minima $N8$ and $C11$ should satisfy the conditions of Eq. (64). As a result
of this, the last term of Eqs. (81) and (84) should give negative
contribution. Hence, it is not guaranteed that the potential depth at $N1$ is
lower than that at $N8$ and $C11$.

In Sec. 3, we have described that eleven different minima can exist in the
Dirac scotogenic model. If we assume that $N1$, which is the desired minimum
of this model, to coexist with other minima of this model, we have shown that
the potential value at $N1$ is not guaranteed to be deeper than that at $N3$,
$N4$, $N5$, $N8$ and $C11$. But otherwise, we have argued that $N1$ minimum is
stable against the other vacua of $N2$, $N6$, $N7$, $C9$ and $C10$. Now, in
order to address if $N1$ can be made the global minimum of this model, we
follow the below described steps. First, we find a parameter region of this
model where $N1$ is a minimum. Now in this parameter region, we check if $N1$
coexist with either of the minima of $N3$, $N4$, $N5$, $N8$ and $C11$. If $N1$
minimum is found to coexist with the above described minima, we demand that
the relative potential depths for these minima against $N1$, which are given
in Eqs. (76), (77), (78), (81) and (84), should be positive. After this
demand, we see that $N1$ becomes the global minimum in the region of
coexistence. In the next section, we present numerical analysis on the above
described steps and give results on the status of $N1$ as the global minimum
of this model.

##  5 Numerical results

As we have described at the end of the last section, we first find a parameter
region where $N1$ is a minimum and then check if $N1$ coexist with the other
minima of $N3$, $N4$, $N5$, $N8$ and $C11$ in this region. This parameter
region depends on the dimensionless $\lambda$ parameters and dimensionful
parameters of the scalar potential, which is given in Eq. (2). While finding
the above mentioned parameter region, we scan over these parameters in such a
way that perturbativity bounds on $\lambda$ parameters and boundedness from
below conditions on the scalar potential are satisfied. As a result of this,
in our scan over parameters, the below perturbativity conditions are satisfied
on $\lambda$ parameters.

$|\lambda_{i}|\leq 4\pi$ (85)

As for the boundedness from below conditions, they are determined by the
quartic part of the scalar potential. In our case, the quartic part of the
scalar potential is same as that considered in [38]. Hence, the allowed region
by the boundedness from below of the scalar potential of Dirac scotogenic
model is given by [38]

$\displaystyle\Omega_{1}\cup\Omega_{2},$ $\displaystyle\Omega_{1}$
$\displaystyle=$
$\displaystyle\left\\{\lambda_{1,2,6}>0;\sqrt{2\lambda_{1}\lambda_{6}}+\lambda_{7}>0;\sqrt{2\lambda_{2}\lambda_{6}}+\lambda_{8}>0;\sqrt{\lambda_{1}\lambda_{2}}+\lambda_{3}+D>0;\right.$
$\displaystyle\left.\lambda_{7}+\sqrt{\frac{\lambda_{1}}{\lambda_{2}}}\lambda_{8}\geq
0\right\\},$ $\displaystyle\Omega_{2}$ $\displaystyle=$
$\displaystyle\left\\{\lambda_{1,2,6}>0;2\lambda_{2}\lambda_{6}\geq\lambda_{8}^{2};\sqrt{2\lambda_{1}\lambda_{6}}>-\lambda_{7}\geq\sqrt{2\lambda_{2}\lambda_{6}}+\lambda_{8};\right.$
$\displaystyle\left.\sqrt{(\lambda_{7}^{2}-2\lambda_{1}\lambda_{6})(\lambda_{8}^{2}-2\lambda_{2}\lambda_{6})}>\lambda_{7}\lambda_{8}-2(D+\lambda_{3})\lambda_{6}\right\\},$
$\displaystyle D$ $\displaystyle=$ $\displaystyle{\rm
min}(\lambda_{4}-|\lambda_{5}|,0)$ (86)

In our numerical analysis, we randomly generate the $\lambda$ parameters in
such a way that the bounds in Eqs. (85) and (86) are satisfied. In addition to
these $\lambda$ parameters, dimensionful parameters are also exist in the
scalar potential of Eq. (2). Among these dimensionful parameters, we fix
$\mu_{1}^{2}=-\lambda_{1}v_{EW}^{2}$ in our numerical analysis. This is due to
the fact that we search for a region where $N1$ is a minimum and the above
relation is the minimization condition for this minimum. Rest of the
dimensionful parameters such as $\mu_{2}^{2},\mu_{3}^{2},A$ are either fixed
to a value or randomly generated. Here, we notice that the $A$-parameter is
necessary in order to explain neutrino masses in the Dirac scotogenic model,
which we have discussed in Sec. 2. It is described that this parameter breaks
$Z_{2}^{(A)}$ symmetry softly, and hence, this parameter should be small. As a
result of this, we have taken this parameter to be small in our numerical
analysis. On the other hand, the parameters $\mu_{2,3}^{2}$, except that for
$N1$, determine the minimization conditions of all minima, which can be seen
from Tab. 2. As a result of this, either we have fixed these parameters in
order to satisfy minimization conditions of a particular minimum, or else, we
have generated them randomly. We discuss details on these parameters later.

In the previous paragraph, we have described our methodology in scanning over
parameters of the scalar potential. We notice that these parameters are
physical, since they can be measured in future experiments. Apart from these
parameters, in our scanning procedure, we have randomly varied the non-zero
VEVs of Eqs. (17)$-$(58). It is to remind here that the minima of Eqs.
(17)$-$(58) are unphysical, since they do not correspond to the vacuum of our
physical world. As a result of this, the non-zero VEVs of Eqs. (17)$-$(58) are
unknown, and a priori, they can take arbitrary values.

As described previously, we scan over parameters of the model and find a
region where $N1$ is a minimum. For the $N1$ minimum, the mass eigenstates of
the scalar fields are physical, since this minimum corresponds to the vacuum
of our physical world. The mass eigenstates of this minimum are described in
Eq. (5). Among these eigenstates, only the Higgs boson is found in the LHC
experiment and rest of the scalar fields are yet to be found. In order to fit
the Higgs boson mass in our analysis, we have taken $m_{H}=$ 125.25 GeV [8].
This value of Higgs boson mass also fixes the $\lambda_{1}$ parameter in our
analysis. Regarding the additional mass eigenstates of the $N1$ minimum, they
should satisfy the phenomenological lower bounds on their masses due to non-
observation of these fields in collider experiments. As a result of this, we
apply the following bounds on these masses in our analysis.

$m_{\eta^{+}}>105~{}{\rm GeV},\quad m_{\zeta_{1,2,3}}>5~{}{\rm GeV}$ (87)

The lower bound of 105 GeV on $m_{\eta^{+}}$ is based on the fact that no non-
SM charged particle is found in the LEP collider.

Below we present our numerical results in the form of percentage of chances on
the coexistence of $N1$ minimum with other minima. To get these results, we
have done multiple scans over the parameter space. In each of these scans, the
percentages are computed after generating a large number of parametric points
(at least 15000), which satisfy the bounds of Eqs. (85) and (86).

In Tab. 3, we have given numerical results on coexistence of the $N1$ and $N3$
minima.

$\frac{A}{v_{EW}}$ | $N1$ |  $N1$ and $N3$ |  $N1$ deeper than $N3$
---|---|---|---
0.1 | 43.3$\%$ | 15.2$\%$ | 11.1$\%$
$10^{-3}$ | 43.6$\%$ | 14.7$\%$ | 10.6$\%$
$10^{-5}$ | 43.7$\%$ | 15.1$\%$ | 11.0$\%$
Table 3: For various values of $A$, percentages of coexistence between the
minima $N1$ and $N3$ are given. In the second column, percentage of existence
of $N1$ minimum is given. In the third column, percentage of coexistence of
$N1$ and $N3$ is given. In the fourth column, percentage of coexistence of
$N1$ and $N3$ with $V_{N1}<V_{N3}$ is given.

In order to search for this coexistence, in our scan over parameters of the
scalar potential, we satisfy the minimization conditions only for $N1$ and
$N3$. As a result of this, from Tab. 2, we notice that the relations for
$\mu_{1}^{2}$ and $\mu_{2}^{2}$ are fixed, but $\mu_{3}^{2}$ can be chosen
arbitrarily. Moreover, the quantity $v_{\eta(3)}$ can also be chosen
arbitrarily. In our scan, we have varied $v_{\eta(3)}$ and $\mu_{3}^{2}$
randomly in the ranges of $(-100,100)$ GeV and $(-10^{5},10^{5})$ GeV2,
respectively. For these ranges of $v_{\eta(3)}$ and $\mu_{3}^{2}$, we have
found that the percentages given in Tab. 3 to be large. In Tab. 3, we have
chosen $A$-parameter to be suppressed as compared to the electroweak scale,
since this parameter should be small, which is described previously. From this
table, we see that nearly 44$\%$ chance is there in finding the $N1$ minimum.
On the other hand, around 15$\%$ chance is there in finding a parametric point
for which both $N1$ and $N3$ are minima. From the last column of Tab. 3, we
see that around 11$\%$ of points correspond to the $N1$ minimum to be deeper
than the $N3$ minimum in terms of potential depth. In this table, the
difference of percentages between the second and third columns, for a
particular value of $A$, is about 29$\%$. This percentage corresponds to the
parametric region in which $N1$ is the only minimum. We see here that, out of
the total area of scan, in a significant fraction of it, $N1$ is the only
minimum. From this table, we see that the numerical results are not sensitive
to the $A$-parameter. Moreover, by changing the sign of $A$, we nearly got the
same percentages that are given in Tab. 3.

After satisfying only the minimization conditions for $N1$ and $N4$ in our
scanning procedure, we have obtained numerical results on the coexistence of
these minima. These results are given in Tab. 4.

$\frac{A}{v_{EW}}$ | $N1$ |  $N1$ and $N4$ |  $N1$ deeper than $N4$
---|---|---|---
0 | 33.5$\%$ | 18.3$\%$ | 12.3$\%$
Table 4: Percentages of coexistence between the minima $N1$ and $N4$ are
given. The columns in this table are analogous to that in Tab. 3.

In a region where $N1$ and $N4$ coexist, the parameters $\mu_{2,3}^{2}$ are
fixed according to the relations given in Tab. 2. Here, the quantities
$v_{\eta(4)}$ and $v_{\chi(4)}$ can be chosen arbitrarily. We have varied
these quantities independently in the range $(-100,100)$ GeV in order to
obtain percentages in Tab. 4. For this particular range of values, the
percentages given in this table are large. We notice that the minima $N1$ and
$N4$ coexist in a region where $A=0$. For this value of $A$, and from the
discussion given below Eq. (6), the neutrino masses in this model become zero.
Hence, in order to explain non-zero masses to neutrinos, we should choose
$A\neq 0$, and thereby, the coexistence between $N1$ and $N4$ can be avoided.

In analogy to the results described for Tabs. 3 and 4, we have searched for
the coexistence of the minima $N1$ and $N5$. The results of this coexistence
are given in Tab. 5.

$\frac{A}{v_{EW}}$ | $N1$ |  $N1$ and $N5$ |  $N1$ deeper than $N5$
---|---|---|---
0.1 | 36.0$\%$ | 9.4$\%$ | 7.8$\%$
$10^{-3}$ | 36.0$\%$ | 9.5$\%$ | 8.1$\%$
$10^{-5}$ | 36.4$\%$ | 9.3$\%$ | 8.0$\%$
Table 5: Percentages of coexistence between the minima $N1$ and $N5$ are
given. The columns in this table are analogous to that in Tab. 3.

In order to get results in this table, we have varied $v_{\chi(5)}$ and
$\mu_{2}^{2}$ in the ranges $(-100,100)$ GeV and $(-10^{5},10^{5})$ GeV2,
respectively. From this table, we see that the percentages are not sensitive
to the value of $A$. Moreover, we have seen that these results are not
sensitive to the sign of $A$. After comparing the results in Tabs. 3 and 5, we
see that the percentage of coexistence between $N1$ and $N5$ is lower as
compared to that between $N1$ and $N3$.

In Tab. 6, we have given the results on the coexistence of the minima $N1$ and
$N8$.

$\frac{|A|}{v_{EW}}$ | $N1$ |  $N1$ and $N8$ |  $N1$ deeper than $N8$
---|---|---|---
0.1 | 26.4$\%$ | 4.8$\%$ | 4.2$\%$
$10^{-3}$ | 21.0$\%$ | 0.57$\%$ | 0.56$\%$
$10^{-5}$ | 21.4$\%$ | 0.14$\%$ | 0.14$\%$
0 | 21.7$\%$ | 0$\%$ | 0$\%$
Table 6: Percentages of coexistence between the minima $N1$ and $N8$ are
given. The columns in this table are analogous to that in Tab. 3.

These results are obtained after satisfying only the minimization conditions
of $N1$ and $N8$. Among these conditions, the relation for $\mu_{1}^{2}$ gives
a constraint relation. This relation is satisfied by solving for unknown
$\lambda$ parameters and later we have checked if these $\lambda$ parameters
obey the conditions of Eqs. (85) and (86). While obtaining the results in Tab.
6, we have varied each of $v_{\phi(8)},v_{\eta(8)},v_{\chi(8)}$ independently
in the range $(-100,100)$ GeV, for which the percentages in this table are
found to be large. After comparing the results in this table with that of
Tabs. 3 and 5, we see that the percentage of coexistence for $N1$ and $N8$ is
lower than that for $N1$ with either $N3$ or $N5$. In Tab. 6, we have given
the value of $|A|$, since the sign of $A$ is determined by the sign of
$v_{\phi(8)}v_{\eta(8)}v_{\chi(8)}$ in our scanning procedure. This is due to
the fact that we have a condition of Eq. (64) for the minimum $N8$. We notice
that by decreasing the value of $|A|$, the percentage for coexistence between
the minima $N1$ and $N8$ is getting decreased. We have found that, for $A=0$,
the minimum $N8$ exist in some region of parameter space. However, $N8$
minimum does not coexist with $N1$ minimum in the region where $A=0$. In the
numerical analysis, we have noticed that $N8$ becomes a saddle point in the
region where $N1$ is a minimum, for $A=0$.

In our scanning procedure, we have searched for the coexistence of minima $N1$
and $C11$. While satisfying the minimization conditions of these minima, the
relation for $\mu_{1}^{2}$ gives a constraint relation. This is solved in an
analogous way of what we have described for Tab. 6. We have varied the
quantities $v_{\phi(11)},c_{\eta(11)},v_{\eta(11)},v_{\chi(11)}$ arbitrarily
in our analysis. Now, we see that the parameter $A$ is determined by the above
quantities through one of the minimization conditions of $C11$, which is given
in Tab. 2. As already described before, the parameter $A$ should be small in
order to be consistent with the model framework. As a result of this, we have
demanded $\frac{|A|}{v_{EW}}\leq 0.1$ in our analysis, and thereafter, we have
not found a region for the coexistence of the minima $N1$ and $C11$. In the
appendix A, we argue that for $A$ to be a negligibly small variable, $C11$
becomes a saddle point in the region where $N1$ is a minimum. The result shown
in this appendix, concurs with our numerical result that the minima $N1$ and
$C11$ do not coexist for $A$ to be a small variable. On the other hand, in our
analysis, for $\frac{|A|}{v_{EW}}>0.1$, we have found a region for the
coexistence of minima $N1$ and $C11$. However, this region is not viable due
to the above mentioned reasons.

In the previous section we have argued that, even if the minimum $N1$ coexist
with either of the minima $N2$, $N6$, $N7$, $C9$ or $C10$, the potential depth
at $N1$ is always deeper than that at the other minima mentioned here. For the
sake of completeness, in our numerical analysis, we have searched if $N1$
coexist with any of the above mentioned minima. This searched is done in an
analogous way of what we have described for the results of Tabs. 3$-$6. In our
analysis, we have not found a region where $N1$ coexist with either of the
minima $N2$, $N6$, $N7$, $C9$ or $C10$. In the appendix A, we argue that some
of these SPs become saddle points in a region where $N1$ is a minimum.

From the numerical results presented so far, we have seen that in the Dirac
scotogenic model, the minimum $N1$ can coexist with either of the minima $N3$,
$N4$, $N5$ or $N8$. Here, it should be noted that the coexistence of the
minima $N1$ and $N4$ happen in a region where $A=0$. This region is not
interesting, since neutrino masses become zero in this region of the model. On
the other hand, the coexistence of $N1$ with the other minima of $N3$, $N5$
and $N8$ can happen in a region where $A\neq 0$, which is an interesting
region to us from the point of neutrino masses. Hence, in this region, we have
searched to see if $N1$ minimum coexist with more than one minima of $N3$,
$N5$ and $N8$. In Tab. 7, we have given results on the coexistence of the
minima among $N1$, $N3$ and $N5$.

$\frac{A}{v_{EW}}$ | $N1$ |  $N1$, $N3$ and $N5$ |  $N1$ deeper than $N3$, $N5$
---|---|---|---
0.1 | 32.9$\%$ | 0.81$\%$ | 0.77$\%$
Table 7: Percentages of coexistence among the minima $N1$, $N3$ and $N5$ are
given. The columns in this table are analogous to that in Tab. 3.

While obtaining results on this coexistence, we have satisfied minimization
conditions only for these minima. Also, for these results, we have varied
$v_{\eta(3)}$ and $v_{\chi(5)}$ independently in the range $(-100,100)$ GeV.
We see that the percentage of coexistence among these minima is far less than
that given in Tabs. 3 and 5. We have noticed that the percentages given in
Tab. 7 are not sensitive to the value of $A$, which is also the case in Tabs.
3 and 5.

We have also searched for the coexistence of minima among $N1$, $N3$ and $N8$
and also among $N1$, $N5$ and $N8$. In our analysis, we have found that the
above mentioned coexistences can happen in a region of
$\frac{|A|}{v_{EW}}>0.1$. However, this region is not viable since $A$ should
be a small parameter. Hence, we have have demanded $\frac{|A|}{v_{EW}}\leq
0.1$ in our analysis, and thereafter, we have not found a region for the above
mentioned coexistences. This result may be understood in the following way. In
Tab. 6, it is shown that in the limit that $A$ is small, the percentage of
chances for $N8$ to coexist with $N1$ is getting decreased. Hence, for small
$A$, it is difficult for $N8$ to coexist with $N1$ and $N3$ or with $N1$ and
$N5$. Finally, we have found that the minima $N1$, $N3$, $N5$ and $N8$ can
coexist in a region of $\frac{|A|}{v_{EW}}>0.1$, but otherwise, we have not
found a region for these minima to coexist.

While describing the results of Tab. 3, we have mentioned that the difference
in the percentages of second and third columns corresponds to the fraction of
total scanned region, where $N1$ is the only minimum. This statement is true
even for Tabs. 4$-$7. In any of these tables, the difference in the
percentages of second and third columns, for a particular value of $A$, is at
least 15$\%$. After comparing percentages in these tables, we notice that in a
significant parameter region of our scanning process, $N1$ is the only
minimum. Moreover, in this region, $N1$ is obviously the global minimum of the
Dirac scotogenic model. On the other hand, there exist a certain parameter
region where $N1$ can coexist with other minima, whose percentage of chances
of finding is given by the third column of Tabs. 3$-$7. In the region where
$N1$ coexist with other minima, we have demanded that the potential depth at
$N1$ is lower than that at the other minima, so that $N1$ can be the global
minimum of this model. The fourth column of Tabs. 3$-$7 gives the percentage
of chances for $N1$ to be the global minimum in the region where it coexist
with other minima.

##  6 Higgs to diphoton decay

In the previous section, we have described the status of the $N1$ minimum as
the global minimum of the Dirac scotogenic model. We have noticed that the
analysis of the previous section is determined by the parameters of the scalar
potential. We now want to study the impact of this analysis on
phenomenological observable quantities. One of the observable quantities is
the decay $H\to\gamma\gamma$ upon which the scalar sector of this model can
have an impact. This decay is in general driven by charged particles through a
loop induced process. In the Dirac scotogenic model, this decay gets
additional contribution due to $\eta^{+}$ field. This decay is the subject of
experimental investigation, since it can distinguish any new physics signal
from that of SM. As part of this investigation, in the LHC experiment, the
signal strength of $H\to\gamma\gamma$ is measured and it is found to be
$1.1\pm 0.07$ [8]. In this section, we compute the signal strength of this
decay in the Dirac scotogenic model and study consequences on this quantity
due to analysis of the previous section.

The signal strength of $H\to\gamma\gamma$ is defined as the ratio of the
observed cross section of $pp\to H\to\gamma\gamma$ against to the same
quantity computed in the SM. The observed cross section of $pp\to
H\to\gamma\gamma$ in the LHC experiment should match with that computed in the
model of our work, which is the Dirac scotogenic model. We notice that the
production cross section for the Higgs boson in the Dirac scotogenic model is
nearly same as that in the SM, since the dominant process for this production
is through the gluon fusion. As a result of this, after using the narrow width
approximation, the signal strength of $H\to\gamma\gamma$ in our work is given
by

$R_{\gamma\gamma}=\frac{Br(H\to\gamma\gamma)_{DSM}}{Br(H\to\gamma\gamma)_{SM}}=\frac{\Gamma(H\to\gamma\gamma)_{DSM}}{\Gamma(H\to\gamma\gamma)_{SM}}\frac{\Gamma_{SM}^{H}}{\Gamma_{DSM}^{H}}$
(88)

Here, the quantities having the suffixes DSM and SM are the ones computed in
the Dirac scotogenic and standard models, respectively. $\Gamma^{H}_{DSM,SM}$
correspond to the total decay widths of the Higgs boson in the above two
models. The decay widths of $H\to\gamma\gamma$, which are required in Eq.
(88), are computed using a general expression for this quantity given in [39].
For the case of Dirac scotogenic model, we have

$\displaystyle\Gamma(H\to\gamma\gamma)_{DSM}$ $\displaystyle=$
$\displaystyle\frac{\alpha^{2}G_{F}m_{H}^{3}}{128\sqrt{2}\pi^{3}}\left|\sum_{f}N_{f}Q_{f}^{2}F_{1/2}(\beta_{f})+F_{1}(\beta_{W})+\frac{\lambda_{3}v_{EW}^{2}}{m^{2}_{\eta^{+}}}F_{0}(\beta_{\eta})\right|^{2},$
(89)
$\displaystyle\quad\beta_{f}=\frac{4m_{f}^{2}}{m_{H}^{2}},\quad\beta_{W}=\frac{4m_{W}^{2}}{m_{H}^{2}},\quad\beta_{\eta}=\frac{4m_{\eta^{+}}^{2}}{m_{H}^{2}},$

where $\alpha$ and $G_{F}$ are fine-structure and Fermi constants,
respectively. Here, $N_{f}$, $Q_{f}$ and $m_{f}$ are color factor, charge and
mass of SM fermion, respectively. After excluding the last term in the modulus
of the above equation, we get the expression for
$\Gamma(H\to\gamma\gamma)_{SM}$. The $F$-functions in Eq. (89) are the form
factors of spin-1/2, -1 and -0 fields, which drive the decay
$H\to\gamma\gamma$. These functions are given below.

$\displaystyle F_{1/2}(\beta)$ $\displaystyle=$
$\displaystyle-2\beta[1+(1-\beta)f(\beta)],$ $\displaystyle F_{1}(\beta)$
$\displaystyle=$ $\displaystyle 2+3\beta+3\beta(2-\beta)f(\beta),$
$\displaystyle F_{0}(\beta)$ $\displaystyle=$ $\displaystyle\beta[1-\beta
f(\beta)],$ $\displaystyle f(\beta)$ $\displaystyle=$
$\displaystyle\left\\{\begin{array}[]{cr}\left(\sin^{-1}\frac{1}{\sqrt{\beta}}\right)^{2},&\beta\geq
1\\\
-\frac{1}{4}\left[\ln\frac{1+\sqrt{1-\beta}}{1-\sqrt{1-\beta}}-i\pi\right]^{2},&\beta<1\end{array}\right.$
(92)

While computing $R_{\gamma\gamma}$ in our analysis, we have taken the total
decay width of Higgs boson in the SM as $\Gamma_{SM}^{H}=4.1\times 10^{-3}$
GeV [40]. Now, the total decay width of Higgs boson in the Dirac scotogenic
model, to a leading order, is given by

$\Gamma_{DSM}^{H}=\Gamma_{SM}^{H}+\Gamma(H\to\zeta_{1}\zeta_{1})+\Gamma(H\to\zeta_{1}\zeta_{2})+\Gamma(H\to\zeta_{2}\zeta_{2})+\Gamma(H\to\zeta_{3}\zeta_{3})+\Gamma(H\to\eta^{+}\eta^{-})$
(93)

In the above equation, the partial decay widths of Higgs boson into scalar
particles of Dirac scotogenic model are computed using tree level couplings of
these processes. These couplings are given below.

$\displaystyle
C_{H\zeta_{1}\zeta_{1}}=-i\sqrt{2}v_{EW}[(\lambda_{3}+\lambda_{4}+\lambda_{5})O_{12}^{2}+\lambda_{7}O_{22}^{2}]-i2AO_{12}O_{22},$
$\displaystyle
C_{H\zeta_{1}\zeta_{2}}=-i\sqrt{2}v_{EW}[(\lambda_{3}+\lambda_{4}+\lambda_{5})O_{11}O_{12}+\lambda_{7}O_{21}O_{22}]-iA(O_{12}O_{21}+O_{11}O_{22}),$
$\displaystyle
C_{H\zeta_{2}\zeta_{2}}=-i\sqrt{2}v_{EW}[(\lambda_{3}+\lambda_{4}+\lambda_{5})O_{11}^{2}+\lambda_{7}O_{21}^{2}]-i2AO_{11}O_{21}$
$\displaystyle
C_{H\zeta_{3}\zeta_{3}}=-i\sqrt{2}(\lambda_{3}+\lambda_{4}-\lambda_{5})v_{EW}$
$\displaystyle C_{H\eta^{+}\eta^{-}}=-i\sqrt{2}\lambda_{3}v_{EW}$ (94)

Here, $O_{ij}$, where $i,j=1,2$, is the element of an orthogonal matrix which
diagonalizes the mixing mass matrix of $\eta_{R}^{0}$ and $\chi$, which is
given in Eq. (5). In our numerical analysis, we follow the below convention
for the diagonalization of this mixing matrix.

$O^{T}M_{\eta_{R}^{0}\chi}^{2}O={\rm diag}(m_{\zeta_{2}},m_{\zeta_{1}})$ (95)

Here we identify $m_{\zeta_{1}}$ to be the lightest.

Using the expressions, which are described above, we have computed
$R_{\gamma\gamma}$ after scanning over parameters of the scalar potential of
Dirac scotogenic model. We notice here that while scanning over these
parameters, it is possible that the $N1$ minimum of this model may coexist
with other minima of the model. In order to see the effect of this
coexistence, we have done the scanning in a way of what we have described in
the previous section. The difference in the scanning of previous section and
the current section is that, in the current analysis, we have taken
$m_{\zeta_{1}}$ as the lightest among the masses of additional particles of
Dirac scotogenic model. It is to remind here that the additional particles of
this model are charged under $Z_{2}^{(B)}$ symmetry, which is exact. Hence,
$\zeta_{1}$ is a possible candidate for the dark matter. As a result of this,
we have applied the following constraints in the current analysis.

$m_{\eta^{+}}>105~{}{\rm GeV},\quad 5~{}{\rm
GeV}<m_{\zeta_{1}}<m_{\zeta_{2,3}},m_{\eta^{+}}$ (96)

The results of our analysis, after doing a generic scan over parameters, are
given in Fig. 1.

Figure 1: The left-hand side plot is for parametric points for which $N1$ is
the only minimum. Right-hand side plot is for parametric points where the $N1$
minimum coexist with other minima of the model, and moreover, for these
points, the potential depth at $N1$ is deeper than that at other minima. In
the right-hand side plot, blue, green, red and yellow are the points where
$N1$ minimum coexist with $N3$, $N5$, $N8$ and $N3+N5$ minima respectively.
The horizontal line in these plots indicates the experimentally allowed lower
3$\sigma$ value of $R_{\gamma\gamma}$. In these plots, we have taken
$\frac{|A|}{v_{EW}}$ = 0.01.

We see that many points exist in the left-hand side plot as compared to that
in the right-hand side plot of Fig. 1. This implies that many parametric
points in our scan correspond to the points where $N1$ is the only minimum.
This result is also described in the numerical analysis of previous section.
From the plots of Fig. 1, we see that for $m_{\zeta_{1}}<m_{H}/2$,
$R_{\gamma\gamma}$ is suppressed. This suppression is due to the factor
$\frac{\Gamma_{SM}^{H}}{\Gamma_{DSM}^{H}}$ in $R_{\gamma\gamma}$. For
$m_{\zeta_{1}}<m_{H}/2$, the decay channel $H\to\zeta_{1}\zeta_{1}$ opens up,
whose decay width is found to be at least about 0.1 GeV, which gives the
necessary suppression in the above mentioned factor. Also, in Fig. 1, we have
found that the points, for which $m_{\zeta_{1}}<m_{H}/2$, do not satisfy the
constraint due to invisible decay of Higgs boson. For $m_{\zeta_{1}}<m_{H}/2$,
the Higgs boson of this model decays invisibly, whose branching ratio is
constrained to be $Br_{H\to inv}<0.145$ [41]. In order to get enhancement of
$R_{\gamma\gamma}$ for $m_{\zeta_{1}}<m_{H}/2$, one has to suppress the
couplings of Higgs to scalar particles, whose expressions are given in Eq.
(94). In the scanning process, after including the above mentioned constraint
on $Br_{H\to inv}$, we have seen that the suppression in couplings is
possible, and thereby, $R_{\gamma\gamma}$ can be enhanced to within the
experimentally allowed region, for $m_{\zeta_{1}}<m_{H}/2$. On the other hand,
for $m_{\zeta_{1}}>m_{H}/2$, there won’t be suppression in
$\frac{\Gamma_{SM}^{H}}{\Gamma_{DSM}^{H}}$. Hence, majority of the points are
within the experimentally allowed region, for $m_{\zeta_{1}}>m_{H}/2$.

In both the plots of Fig. 1, for $m_{\zeta_{1}}>m_{H}/2$, the points are
around the horizontal line, which corresponds to lower 3$\sigma$ allowed value
of $R_{\gamma\gamma}$. Moreover, we see that only few points give the
enhancement of $R_{\gamma\gamma}>1$. In this regard, see [42, 43, 44, 45, 46,
47] where enhancement of $R_{\gamma\gamma}>1$ has been reported in various
models. Below we describe the reasons for not getting much enhancement in
$R_{\gamma\gamma}$ in the plots of Fig. 1. As already explained before, for
$m_{\zeta_{1}}>m_{H}/2$, we get $\frac{\Gamma_{SM}^{H}}{\Gamma_{DSM}^{H}}=1$.
Hence, in the region of $m_{\zeta_{1}}>m_{H}/2$, we get $R_{\gamma\gamma}>1$
only if $\lambda_{3}<0$, since $\beta_{\eta}>1$. The value of $\lambda_{3}$ is
constrained by the bounded from below conditions of Eq. (86). After satisfying
these conditions, we have noticed that $\lambda_{3}>-2$. Since the negative
values of $\lambda_{3}$ are restricted by these conditions, we do not get much
enhancement in $R_{\gamma\gamma}$ in our work. On the other hand, in our
scanning procedure, we have seen that by restricting $m_{\eta^{+}}$ to as low
as 130 GeV, it is possible for $R_{\gamma\gamma}$ to be as high as 1.3.

In the region where $N1$ minimum coexist with $N3$ minimum, we do not get
$R_{\gamma\gamma}>1$. This is due to the fact that, in this region we have
$\mu_{2}^{2}<0$, because of the minimization condition of $N3$. As a result of
this, in order to get $m^{2}_{\eta^{+}}>0$ we should have $\lambda_{3}>0$, and
thereby we get $R_{\gamma\gamma}<1$. Similarly, in the region where $N1$ and
$N8$ minima coexist, we have found $\lambda_{3}>0$ in the scanning analysis,
and hence, we get $R_{\gamma\gamma}<1$. On the other hand, in the region where
$N1$ and $N5$ minima coexist, $\mu_{2}^{2}$ is a free parameter. Hence, in
this region, we can choose $\mu_{2}^{2}>0$, and thereby $\lambda_{3}$ can be
negative, so that $R_{\gamma\gamma}>1$. Now, it should be clear that in a
region where $N1$ is the only minimum, we can get $\mu_{2}^{2}>0$, and
thereby, we get $R_{\gamma\gamma}>1$. From the above described results, we see
that the future determination of $R_{\gamma\gamma}$ by the LHC experiment can
have an impact on the study of global minimum of this model. It is to be noted
that a precise determination of $R_{\gamma\gamma}$ in the LHC experiment can
distinguish the above described vacuum realizations. On the other hand, if the
error bar is large enough so that the allowed value of $R_{\gamma\gamma}$ is
around 1, the above vacuum realizations may not be distinguished. Finally,
from Fig. 1, we notice that the experimentally allowed values of
$R_{\gamma\gamma}$ can be fitted in the Dirac scotogenic model, irrespective
of the fact that the $N1$ minimum either coexist or not with other minima of
this model.

##  7 Possibility of a scalar dark matter

It is described in Sec. 2 that, due to exact symmetry of $Z_{2}^{(B)}$, the
lightest among the additional particles of the Dirac scotogenic model is a
candidate for dark matter. We notice that the lightest among the singlet Dirac
fermions $N^{D}_{k}$ is a possible candidate for dark matter. In the original
model [21], this possibility has been studied and it is shown that $N^{D}_{1}$
can consistently explain the dark matter phenomenology. See [48], for another
work in this direction. In the present work, since our motivation is to study
the scalar sector of the Dirac scotogenic model, we study on the possibility
of scalar dark matter in this model.

In the previous section, it is described that, while obtaining results of Fig.
1, we have taken $\zeta_{1}$ as the lightest particle among the additional
particles of Dirac scotogenic model. From Eq. (5), we notice that $\zeta_{1}$
is an admixture of $\eta_{R}^{0}$ and $\chi$. For sufficiently small $A$,
which is the case in Fig. 1, $\eta_{R}^{0}$ and $\chi$ are nearly equal to the
mass eigenstates of this model. In such a case, $\zeta_{1}$ is dominantly made
of either $\eta_{R}^{0}$ or $\chi$, depending on the parameter choice of the
model. We see that the scalar dark matter in this model is either part of an
$SU(2)_{L}$ doublet or a singlet field. We first consider the case of
$\zeta_{1}$ being dominantly made of $\chi$ and analyze if this case can
consistently explain all the dark matter phenomenology. Later we analyze the
case where $\zeta_{1}$ is dominantly made of $\eta_{R}^{0}$.

In the phenomenology of dark matter, we need to explain the relic abundance of
dark matter in the present universe and also the null results of direct and
indirect searches for dark matter detection. The current relic density of dark
matter is 0.12$\pm$0.0012 [49]. In order to explain this relic density, we
need to compute thermally averaged pair-annihilation cross section of dark
matter times the relative velocity of these particles, which is denoted by
$\langle\sigma v_{rel}\rangle$. To a good approximation, for a cold dark
matter, the above mentioned relic density can be fitted if $\langle\sigma
v_{rel}\rangle\approx 3\times 10^{-26}$ cm3/s = 1 pb [8]. In the Dirac
scotogenic model, for the case of dark matter $\zeta_{1}\approx\chi$, the
possible pair-annihilation processes at tree level are as follows:

$\displaystyle\zeta_{1}\zeta_{1}\to\nu^{D}_{i}\bar{\nu}^{D}_{j},$ (97)
$\displaystyle\zeta_{1}\zeta_{1}\to HH,$ (98)
$\displaystyle\zeta_{1}\zeta_{1}\to H^{*}\to f\bar{f},W^{+}W^{-},ZZ,HH$ (99)

Here, $H^{*}$ is virtual Higgs boson and $f$ is any SM fermion. $\nu^{D}_{i}$
are the mass eigenstates of the neutrino fields of the Dirac scotogenic model.

The reaction in Eq. (97) is driven by the $h_{k\alpha}$ couplings of Eq. (1),
via a $t$-channel process mediated by the singlet fermions $N^{D}_{k}$. We can
take the mass of $N^{D}_{k}$ ($m_{N_{k}}$) to be around 1 TeV and the
couplings $h_{k\alpha}\sim 1$. For these values of $m_{N_{k}}$ and
$h_{k\alpha}$, and from the discussion given below Eq. (6), we see that it is
possible to explain small masses to neutrinos either by suppressing the
$A$-parameter or by taking degenerate masses to $\zeta_{1,2}$. Now the
annihilation cross section for the process in Eq. (97) is given by

$\sigma(\zeta_{1}\zeta_{1}\to\nu^{D}_{i}\bar{\nu}^{D}_{j})v_{rel}=\frac{1}{8\pi}\sum_{k}\frac{|\sum_{\alpha,\beta}h_{k\alpha}V^{*}_{\alpha
i}h^{*}_{k\beta}V_{\beta
j}|^{2}}{(m_{\zeta_{1}}^{2}+m_{N_{k}}^{2})^{2}}(m_{\nu_{i}}^{2}+m_{\nu_{j}}^{2})$
(100)

Here, $V_{\alpha i}$ are the elements of $V$, which diagonalize the neutrino
mass matix, which is discussed in Sec. 2. For neutrino masses around 0.1 eV,
the above pair-annihilation cross section is suppressed by fourteen orders of
magnitude as compared to the required amount. Hence, the process in Eq. (97)
cannot explain the relic density of dark matter.

The processes in Eqs. (98) and (99) are driven due to quartic and trilinear
couplings of $\zeta_{1}$ to the Higgs field and these couplings are
proportional to $\lambda_{7}$, for $\zeta_{1}\approx\chi$. By taking
$m_{\zeta_{1}}=$ 200 GeV and $\lambda_{7}=0.1$, we have found the following
pair-annihilation cross sections for the processes of Eqs. (98) and (99):

$\left.\sigma v_{rel}\right|_{t\bar{t}}=0.17~{}{\rm pb},\quad\left.\sigma
v_{rel}\right|_{WW}=0.94~{}{\rm pb},\quad\left.\sigma
v_{rel}\right|_{ZZ}=0.44~{}{\rm pb},\quad\left.\sigma
v_{rel}\right|_{HH}=0.51~{}{\rm pb}$ (101)

We see that the cross section into $WW$ channel is dominant among all possible
pair-annihilations of $\zeta_{1}$. The above cross section values decrease
with increasing $m_{\zeta_{1}}$. From these cross section values, we see that,
for few hundred GeV of mass to $\zeta_{1}$ and for $\lambda_{7}\sim 0.1$,
using the processes of Eqs. (98) and (99), the relic abundance of dark matter
can be fitted in this model. However, in this model, the trilinear coupling of
$\zeta_{1}$ to the Higgs field also drives the process of dark matter
scattering against a nucleus. This process has be searched in many dark matter
experiments [50, 51, 52], which use xenon as target nucleus. Since no sign of
dark matter is found in these experiments, upper bounds on the cross section
of dark matter against a nucleon have been obtained. In the Dirac scotogenic
model, using [53], we estimate the spin-independent cross section of
$\zeta_{1}\approx\chi$ with xenon nucleus as

$\displaystyle\sigma_{0}$ $\displaystyle=$
$\displaystyle\frac{1}{\pi}\left(\frac{m_{\zeta_{1}}m_{Xe}}{m_{\zeta_{1}}+m_{Xe}}\right)^{2}\left|\frac{54f_{p}+77f_{n}}{131}\right|^{2},$
$\displaystyle\frac{f_{p}}{m_{p}}$ $\displaystyle=$
$\displaystyle\left[0.075+\frac{2}{27}(1-0.075)\right]\frac{\lambda_{7}}{m_{\zeta_{1}}m_{H}^{2}},$
$\displaystyle\frac{f_{n}}{m_{n}}$ $\displaystyle=$
$\displaystyle\left[0.078+\frac{2}{27}(1-0.078)\right]\frac{\lambda_{7}}{m_{\zeta_{1}}m_{H}^{2}}$
(102)

Here, $m_{p}$, $m_{n}$ and $m_{Xe}$ are masses of proton, neutron and xenon
nucleus, respectively. As stated before, upper bounds on $\sigma_{0}$ have
been set due to null results of dark matter detection in various experiments.
Among these, the most stringent limit is $\sigma_{0}<6.5\times 10^{-48}$ cm2
[52]. In order to satisfy this limit, in the Dirac scotogenic model, we get
$\lambda_{7}<4.8\times 10^{-5}$ for $m_{\zeta_{1}}\sim$ 100 GeV. For this
suppressed value of $\lambda_{7}$ and for no fine tuning in the mass of
$m_{\zeta_{1}}$, we see that the cross sections in the processes of Eqs. (98)
and (99) are highly suppressed and we cannot fit the relic abundance of dark
matter in this model. On the other hand, for a fine tuned mass of
$m_{\zeta_{1}}\approx m_{H}/2$, via the process $\zeta_{1}\zeta_{1}\to
H^{*}\to b\bar{b}$, the relic abundance of dark matter can be fitted for
$\lambda_{7}=1.3\times 10^{-5}$, which also satisfies the upper limit on
$\sigma_{0}$. However, from the indirect searches of dark matter, thermal
annihilation cross section of dark matter into $b\bar{b}$ channel has been
ruled out for dark matter mass of up to 300 GeV [54]. As a result of this, we
see that the processes in Eqs. (98) and (99) cannot consistently explain the
dark matter phenomenology, for the case of $\zeta_{1}\approx\chi$.

Now, we consider the case $\zeta_{1}\approx\eta_{R}^{0}$, where the dark
matter is dominantly made up of a component of $SU(2)_{L}$ doublet. In this
case, in order to explain the relic density of dark matter, pair-annihilation
of $\zeta_{1}$ through the processes given in Eqs. (97)$-$(99) can be
analyzed. It should be noted that the process of Eq. (97) happen for the case
$\zeta_{1}\approx\eta_{R}^{0}$ through the $f_{\alpha k}$ couplings of Eq.
(1). As a result of this, pair-annihilation cross section for this process is
analogous to that of Eq. (100), where $h_{k\alpha}V^{*}_{\alpha i}$ should be
replaced by $f^{*}_{\alpha k}U^{*}_{\alpha i}$. Now, we see that, due to small
masses to neutrinos, this pair-annihilation cross section is highly suppressed
as compared to the required amount in order to explain the relic density of
dark matter. The processes in Eqs. (98) and (99) are driven by the quartic and
trilinear couplings of $\zeta_{1}$ to the Higgs field. We see that, for the
case $\zeta_{1}\approx\eta_{R}^{0}$, these couplings are proportional to
$\lambda_{3}+\lambda_{4}+\lambda_{5}$. As a result of this, the cross section
values of Eq. (101) are also applicable to the case
$\zeta_{1}\approx\eta_{R}^{0}$, where we take $m_{\zeta_{1}}=$ 200 GeV and
$\lambda_{3}+\lambda_{4}+\lambda_{5}=0.1$. Hence, the relic density of dark
matter can be fitted for the case $\zeta_{1}\approx\eta_{R}^{0}$. However,
analogous to what we described above, due to non-observation of dark matter in
direct detection experiments, the quantity
$\lambda_{3}+\lambda_{4}+\lambda_{5}$ should be suppressed to around
$10^{-5}$. As a result of this, the processes in Eqs. (98) and (99) cannot
consistently explain the relic density of dark matter, for the case
$\zeta_{1}\approx\eta_{R}^{0}$. Apart from the processes of Eqs. (97)$-$(99),
due to gauge interactions, the following annihilations are also possible for
the case of $\zeta_{1}\approx\eta_{R}^{0}$.

$\zeta_{1}\zeta_{1}\to W^{+}W^{-},\quad\zeta_{1}\zeta_{1}\to
ZZ,\quad\zeta_{1}\zeta_{1}\to W^{+}W^{-}\gamma\gamma$ (103)

In the above equation, the first two annihilations happen due to quartic
interactions and the third annihilation happens due to mediation of $\eta^{+}$
field at tree level. Moreover, for $m_{\zeta_{1}}<m_{W,Z}$, one of the
$V=W^{\pm},Z$ in the first two processes of Eq. (103) can be off-shell, and
thereby, we get 3- and 4-body final states from the annihilation products of
$VV^{*}$ and $V^{*}V^{*}$, respectively. A priori, for
$m_{\zeta_{1}}\mathrel{\raise 2.58334pt\hbox{$>$}\kern-7.7778pt\lower
2.79857pt\hbox{$\sim$}}m_{W}$, the first process of Eq. (103) can give the
required amount of thermal annihilation cross section in order to fit the
relic density of dark matter. Moreover, constraints due to indirect detection
of dark matter from pair-annihilation into gauge bosons are weaker [54].
Nevertheless, the first two processes of Eq. (103) can induce scattering
between $\zeta_{1}$ and a nucleus at 1-loop level. In fact, there exist other
1-loop processes, which are mediated by gauge interactions, for the above
mentioned scattering. Although this scattering happens at 1-loop level, given
the strong constraints on this due to direct detection experiments, we may
expect it is challenge for $\zeta_{1}\approx\eta_{R}^{0}$ to evade these
constraints.

##  8 Comparision with the scotogenic model

In this section, we compare the phenomenology of Dirac scotogenic model with
that of scotogenic model [16], where neutrinos are Majorana particles. It is
described in Sec. 2 that there is an analogy between these two models in terms
of field content and also from the view point of neutrino masses and mixing.
Here, we describe some more analogy between these two models in terms of
phenomenological observables quantities of dark matter, LFV and collider
signals.

In the previous section, we have discussed the status of scalar dark matter in
the Dirac scotogenic model. We have discussed two possibilities, where in one
case $\zeta_{1}\approx\chi$ and in the other case
$\zeta_{1}\approx\eta_{R}^{0}$. Now, below we compare the scalar dark matter
phenomenology of this model with that of scotogenic model [16]. It to remind
here that in the scotogenic model, the singlet field $\chi$ does not exist.
Hence, a possibility for scalar dark matter in the scotogenic model is
$\zeta_{1}=\eta_{R}^{0}$. As a result of this, the phenomenological discussion
we have given in the previous section for the case of
$\zeta_{1}\approx\eta_{R}^{0}$ is applicable to the scotogenic model. A
difference in the scotogenic model is that, in addition to the process of Eq.
(97), the following process $\zeta_{1}\zeta_{1}\to\nu^{M}_{i}\nu^{M}_{j}$ can
also happen, which violates lepton number. Here, $\nu^{M}_{i}$ is a neutrino
field of scotogenic model, which is a Majorana particle. The above process is
mediated in the scotogenic model by the analogous couplings of $f_{\alpha k}$,
which are given in Eq. (1). As already described below Eq. (1), in the
scotogenic model, there exist $N^{M}_{k}$ field, instead of
$(N_{k},N^{c}_{k})$. The field $N^{M}_{k}$ is Majorana in the scotogenic
model. As a result of this, the annihilation cross section for the above
process is found to be

$\sigma(\zeta_{1}\zeta_{1}\to\nu^{M}_{i}\nu^{M}_{j})v_{rel}=\frac{S}{16\pi}\sum_{k}\left|\sum_{\alpha,\beta}f_{\alpha
k}U_{\alpha i}f_{\beta k}U_{\beta
j}\right|^{2}\frac{M_{K}^{2}}{(M_{k}^{2}+m_{\zeta_{1}}^{2})^{2}}$ (104)

Here, $M_{k}$ is the mass of $N^{M}_{k}$ field of the scotogenic model and $S$
is a symmetry factor which is 1(2) for $i\neq j(i=j)$. For $f_{\alpha k}\sim
1$, $m_{\zeta_{1}}\sim$ 100 GeV and $M_{k}=$ 2.78 TeV, the above pair-
annihilation cross section is around 1 pb, which is the required amount in
order to fit the relic abundance of dark matter. Moreover, it looks there
exist no constraints on the above pair-annihilation cross section from
experiments. On the other hand, see [55, 56, 57] for indirect searches of dark
matter, where an upper limit on the pair-annihilation cross section of dark
matter into $\nu\bar{\nu}$ mode is set to around $10^{-24}$ cm3/s. As a result
of the above description, in the scotogenic model, it is possible to fit the
relic density of dark matter and avoid the indirect detection bounds on it.
However, as already described in the previous section, the first two processes
of Eq. (103) can induce scattering of $\zeta_{1}$ with a nucleus at 1-loop
level. The loop diagrams for this scattering are driven by gauge couplings and
are mediated by SM fields. Hence, it appears that the amplitude of these
diagrams have only the loop suppression factor. It may be worth to compute the
cross section for the above scattering in order to see if it satisfies the
direct detection bound [52] on the dark matter.

As described in Sec. 2, there is a region of parameter space where the
couplings $f_{\alpha k}$ can be of order one and these couplings drive LFV
processes [8] in the Dirac scotogenic model. So far none of the LFV processes
are observed in experiments and upper bounds have been set on the branching
ratios of various LFV decays and also on the conversion rate of $\mu$ to $e$
in a nucleus[8]. Among the LFV decays, stringent limits exist on the branching
ratios of $\mu\to e\gamma$ [58] and $\mu\to 3e$ [59]. The above mentioned LFV
processes are driven in the Dirac scotogenic model due to mediation of
$\eta^{+}$ and $N^{D}_{k}$ at 1-loop level. Analyzing LFV processes is out of
the scope of this paper. Nevertheless, in our work, experimental limits on
these processes can be satisfied by taking the masses of $\eta^{+}$ or
$N^{D}_{k}$ to be sufficiently high, for $f_{\alpha k}\sim 1$. In addition to
this, there is also a possibility of suppressing the couplings $f_{\alpha k}$
in order to satisfy the above experimental limits.

Below, we compare the Dirac scotogenic and scotogenic models in terms of LFV
processes. As stated before, Feynman diagrams for LFV processes in the Dirac
scotogenic model are driven by $\eta^{+}$ and $N^{D}_{k}$. Whereas, in the
scotogenic model [16], the corresponding Feynman diagrams are driven by
$\eta^{+}$ and $N^{M}_{k}$, where $N^{M}_{k}$ is an additional Majorana field.
In the scotogenic model, LFV processes have been analyzed in [60]. We see
that, in the Feynman diagrams for LFV processes of scotogenic model, one
should replace the $N^{M}_{k}$-propagator with $N^{D}_{k}$-propagator in order
to get the corresponding Feynman diagrams of Dirac scotogenic model. As a
result of this, by interchanging the mass of $N^{M}_{k}$ with $N^{D}_{k}$, we
get the same branching ratio expressions for $\mu\to e\gamma$ in both these
models. However, the branching ratio expression for $\mu\to 3e$ should be
different in these models, which is explained below. The amplitude for $\mu\to
3e$ arises from the following contributions in both these models:
$\gamma$-penguin, $Z$-penguin, Higgs-penguin and box diagrams. The penguin
diagrams should give same kind of expressions for amplitude in both these
models, due to above mentioned replacement of propagators. On the other hand,
the box diagrams of scotogenic model involve diagrams which are due to Dirac
and Majorana nature of $N^{M}_{k}$-propagator. Whereas, these diagrams are
driven only due to Dirac nature of $N^{D}_{k}$-propagator in the Dirac
scotogenic model. As a result of this, we get additional contribution to box
diagrams in the scotogenic model as compared to that of Dirac scotogenic
model. Now, we see that, the expression for conversion rate of $\mu$ to $e$
should be same in both these models, since this conversion happens due to
penguin diagrams.

The additional fields in the Dirac scotogenic model are $N^{D}_{k}$,
$\eta^{+}$ and $\zeta_{i}$, $i=1,2,3$. The best way to test this model in
collider experiments is by probing the $\eta^{+}$ field. $\eta^{+}$ can be
produced in the LHC experiment via gauge interactions and it decays as
$\eta^{+}\to\ell^{+}N^{D}_{1}$ or $\eta^{+}\to W^{+}\zeta_{1}$. Here,
$N^{D}_{1}$ is the lightest among $N^{D}_{k}$. The above two decays give us
missing energy plus a charged lepton or a di-jet signal, depending on the
decay channel of $W$. Analyzing collider signals of this model is out of the
scope of this work. However, see [61, 62], for some collider analysis on
scalar sector in related models. The above described signal in the Dirac
scotogenic model is also possible in the scotogenic model. The difference
between these two models in terms of field content is the presence of the
singlet scalar field $\chi$ in the Dirac scotogenic model. Since the singlet
field does not experience gauge interactions, it is a challenge to probe the
existence of $\chi$ field in the Dirac scotogenic model. Hence, one needs to
develop some techniques to distinguish the above two models in collider
experiments. It is worth to do a detailed analysis on this aspect in future.

##  9 Conclusions

In this work, we have studied on the vacuum structure of the scalar potential
of Dirac scotogenic model. One of the motivations of this model is to explain
neutrino masses through a radiative mechanism, and also to have Dirac nature
to the neutrinos. The other motivation is to have a stable dark matter
candidate. After analyzing the scalar potential of this model, we have found
that eleven different minima are possible. Out of these eleven, the $N1$
minimum of Eq. (12) is the desired minimum of this model. Only with this
minimum, the motivations of this model can be achieved consistently. As a
result of this, we have worked on to see if the $N1$ minimum can be made as
the global minimum of this model. Through our numerical analysis, we have
shown that plenty of parameter space exist where $N1$ is the global minimum.
In our numerical analysis, we have found that the $N1$ minimum can coexist
with certain other minima in some regions of parameter space. However, in the
viable parameter space of this model, we have not found the coexistence of the
$N1$ minimum with charge-breaking minima. We have justified this statement
with an analytical demonstration to it in the appendix A.

We have studied the Higgs to diphoton decay in this model, since the scalar
sector of this model has an implication on this decay. After doing a generic
scan over parameters of the scalar potential, we have found that the signal
strength of this decay can be within the experimentally allowed region, but
most likely to be around the lower 3$\sigma$ allowed value of this quantity.
With some tuning of the parameters, the signal strength of this decay is found
to be as high as 1.3. In the numerical analysis, we have found that the
experimentally allowed values of this quantity can be explained irrespective
of the fact that the $N1$ minimum coexist with other minima or not.

Finally, we have studied on the possibility of making the lightest among the
additional scalar particles of this model, as a candidate for dark matter. We
have found that the singlet scalar field of this model cannot be a viable
candidate for dark matter. This we have found, due to the fact that, the
current bounds from the direct and indirect detection of dark matter in
experiments rule out the possibility of explaining the relic density of dark
matter. The other possibility for scalar dark matter in this model is the
$\eta_{R}^{0}$ field, which is a component of $SU(2)_{L}$ doublet. In this
case, we have found that, constraints due to direct detection bounds on the
dark matter are difficult to be satisfied.

##  Appendix A Saddle points

From the numerical analysis of Sec. 5, we have noticed that certain minima do
not coexist with the $N1$ minimum. Below we present analytical calculations
through which we justify why some of these minima do not coexist with the $N1$
minimum. Our methodology in these calculations is based on the discussions
given in [12, 38].

In Sec. 5, it is described that, by demanding $\frac{|A|}{v_{EW}}\leq 0.1$ in
our numerical analysis, we have not found a region of coexistence between the
minima $C11$ and $N1$. Here we show that, in the limit that $A$ is negligibly
small parameter, $C11$ becomes a saddle point in the region where $N1$ is a
minimum. From the minimization conditions of $C11$, which are given in Tab. 2,
we notice that $A$ becomes a small variable if either
$\lambda_{4}+\lambda_{5}$ or $v_{\phi(11)}v_{\eta(11)}/v_{\chi(11)}$ is
suppressed. As a result of this, we neglect terms involving the above
mentioned variables in comparison to other terms of the scalar potential. Now,
apart from satisfying the minimization conditions of $C11$, we need to
evaluate the mass-square eigenvalues of the scalar fields in order to check if
$C11$ becomes a minimum or not. For calculating these eigenvalues, we have
described the parametrization of scalar fields of an SP in Eq. (65). We
express these scalar fields in the following basis:
$\varphi=(\phi_{R}^{1},\phi_{I}^{1},\eta_{R}^{1},\eta_{I}^{1},\phi_{R}^{0},\phi_{I}^{0},\eta_{R}^{0},\eta_{I}^{0},\chi_{R})$.
Now, the general form of the mixing mass-square matrix among the scalar fields
of an SP is given by

$[M^{2}]_{ij}=\frac{\partial^{2}V}{\partial\varphi_{i}\partial\varphi_{j}}=\frac{\partial^{2}V}{\partial
x_{l}\partial x_{m}}\frac{\partial x_{l}}{\partial\varphi_{i}}\frac{\partial
x_{m}}{\partial\varphi_{j}}+\frac{\partial V}{\partial
x_{l}}\frac{\partial^{2}x_{l}}{\partial\varphi_{i}\partial\varphi_{j}}$ (105)

Here, $i,j=1,\cdots,9$ and $x_{l}$, where $l=1,\cdots,5$, are defined in Eq.
(66).

The last term of Eq. (105) becomes zero for the case of $C11$, since after
using the minimization conditions, we get $\frac{\partial V}{\partial
x_{l}}=0$. As a result of this, for the SP $C11$, Eq. (105) becomes into
$M^{2}_{C11}=Y^{T}BY$. Here, $Y$ and $B$ are matrices of orders $5\times 9$
and $5\times 5$, respectively. The elements of these matrices are given below.

$[Y]_{li}=\frac{\partial
x_{l}}{\partial\varphi_{i}},\quad[B]_{lm}=\frac{\partial^{2}V}{\partial
x_{l}\partial x_{m}}=[M_{4}]_{lm}+\frac{\partial^{2}V_{3}}{\partial
x_{l}\partial x_{m}}$ (106)

In the limit that $A$ is a small variable, the matrix $B$ takes the following
form.

$B=\left(\begin{array}[]{ccccc}&&&0&0\\\ &B_{1}&&0&0\\\
&&&\frac{A}{v_{\chi(11)}}&0\\\
0&0&\frac{A}{v_{\chi(11)}}&2(\lambda_{4}+\lambda_{5})&0\\\
0&0&0&0&2(\lambda_{4}-\lambda_{5})\end{array}\right),\quad
B_{1}=\left(\begin{array}[]{ccc}\lambda_{1}&\lambda_{3}&\frac{1}{2}\lambda_{7}\\\
\lambda_{3}&\lambda_{2}&\frac{1}{2}\lambda_{8}\\\
\frac{1}{2}\lambda_{7}&\frac{1}{2}\lambda_{8}&\frac{1}{2}\lambda_{6}\end{array}\right)$
(107)

Now, as it is argued in [12], it is possible to express the matrix
$M^{2}_{C11}$ in the following block form.

$M^{2}_{C11}=\left(\begin{array}[]{cc}0&0\\\
0&Y^{\prime^{T}}BY^{\prime}\end{array}\right)$ (108)

Here, $Y^{\prime}$ is a $5\times 5$ matrix which depends only on the VEVs of
$C11$. From the above form of $M^{2}_{C11}$, we see that $C11$ has four
Goldstone bosons. This is expected since the SP $C11$ breaks the electroweak
and charge symmetries, which are continuous. Also, from the above equation, we
see that, if all the eigenvalues of $B$ are positive (negative), then
$M^{2}_{C11}$ is positive (negative) definite, and thus, $C11$ is minimum
(maximum). On the other hand, if $B$ has both positive and negative
eigenvalues, then $C11$ becomes a saddle point. Now, in the limit that $A$ is
a small variable, the first three eigenvalues of $B$ are determined by that of
$B_{1}$. We see that $Tr(B_{1})>0$, since due to the conditions of Eq. (86).
As a result of this, $B$ has at least one positive eigenvalue, in the limit
that $A$ is a small variable.

In Eq. (84), we have given the difference in potential depths at the
stationary points $C11$ and $N1$. In the limit that $A$ is a small variable,
the last term of Eq. (84) can be neglected. Moreover, in this limiting
process, $m_{3}^{2}+\lambda_{7}v_{EW}^{2}$ is nearly equal to one of the
eigenvalues of the scalar fields of the SP $N1$. Hence, in a region where $N1$
is a minimum and in the above limiting process, we get $V_{C11}-V_{N1}>0$. To
get the expression for $V_{C11}-V_{N1}$, we have used the general expression
of Eq. (74). Here, we see that $X_{N1}^{T}V^{\prime}_{C11}=0$. Hence, in the
limit that $A$ is a small variable, we get
$V_{C11}-V_{N1}\approx\frac{1}{2}X_{C11}^{T}V^{\prime}_{N1}$. Now, we define
$\tilde{X}_{C11}$, $\tilde{X}_{N1}$, $\tilde{V}^{\prime}_{N1}$ and
$\tilde{M}_{2}$ as 3-column matrices, whose elements are the first three
elements of $X_{C11}$, $X_{N1}$, $V^{\prime}_{N1}$ and $M_{2}$, respectively.
With these definitions and in the above limiting process, we get the following
relation.

$\left.\frac{\partial V}{\partial
x_{l}}\right|_{C11}=0\implies\tilde{X}_{C11}\approx-B_{1}^{-1}\tilde{M}_{2}$
(109)

After using the above relation, we get

$\displaystyle V_{C11}-V_{N1}\approx\frac{1}{2}X_{C11}^{T}V^{\prime}_{N1}$
$\displaystyle=$
$\displaystyle\frac{1}{2}\tilde{X}_{C11}^{T}\tilde{V}^{\prime}_{N1}$ (110)
$\displaystyle=$
$\displaystyle-\frac{1}{2}\tilde{V}^{\prime^{T}}_{N1}B_{1}^{-1}\tilde{M}_{2}=-\frac{1}{2}\tilde{V}^{\prime^{T}}_{N1}B_{1}^{-1}(\tilde{V}^{\prime}_{N1}-B_{1}\tilde{X}_{N1})$
$\displaystyle=$
$\displaystyle-\frac{1}{2}\tilde{V}^{\prime^{T}}_{N1}B_{1}^{-1}\tilde{V}^{\prime}_{N1}$

Earlier we have argued that $V_{C11}-V_{N1}>0$ in a region where $N1$ is a
minimum. Hence, after using the above relation, we see that $B_{1}$ should not
be a positive definite matrix, and thus, one of the eigenvalues of $B_{1}$ is
negative. As a result of this, in the limit that $A$ is a small variable, one
the eigenvalues of $B$ is negative. Combing this result with the earlier
result that $B$ has at least one positive eigenvalue, we see that $C11$
becomes a saddle point in a region where $N1$ is a minimum and also that $A$
is negligibly small variable.

The above described result on the nature of $C11$ is valid even if $A=0$.
However, for $A=0$, we get $\lambda_{4}+\lambda_{5}=0$, and thus, the inverse
of $B$ does not exist. To circumvent this problem, we have used $B_{1}$, which
is a sub-matrix of $B$, in the above described analysis.

The above described analysis can be applied to other SPs of $N2$, $C9$ and
$C10$, in order to show that these become saddle points in the region where
$N1$ is a minimum. The difference we encounter is that the last term of Eq.
(105) does not vanish for the above mentioned SPs. As a result of this, we
have explicitly computed the mixing masses for the scalar fields of the above
SPs. For instance, in the case of $C9$, we have found that the fields
$\eta_{R}^{1}$, $\phi_{R}^{0}$ and $\eta_{R}^{0}$ mix together, whose masses
are given by

$Z^{T}\tilde{B}Z,\quad\tilde{B}=\left(\begin{array}[]{ccc}\lambda_{2}&\lambda_{3}&\lambda_{2}\\\
\lambda_{3}&\lambda_{1}&\lambda_{3}\\\
\lambda_{2}&\lambda_{3}&\lambda_{2}\end{array}\right),\quad
Z=\left(\begin{array}[]{ccc}c_{\eta(9)}&0&0\\\ 0&v_{\phi(9)}&0\\\
0&0&v_{\eta(9)}\end{array}\right)$ (111)

Now, using an analogous formalism described for the case of $C11$, we can show
that the matrix $\tilde{B}$ has one positive and one negative eigenvalue,
apart from a zero eigenvalue. As a result of this, $C9$ becomes a saddle point
in a region where $N1$ is minimum.

In Sec. 5, we have mentioned that the minima $N6$ and $N7$ do not coexist with
the $N1$ minimum. Moreover, we have also described that the percentage of
coexistence between the minima $N8$ and $N1$ is zero for $A=0$. All the above
mentioned SPs become saddle points in a region where $N1$ is a minimum. We
have realized this statement through our numerical analysis, but otherwise, we
do not have an analytical proof for this.

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