# A note about majority colorings of countable DAGs

Bartłomiej Bosek∗<EMAIL_ADDRESS>and  Aleksander Katan
<EMAIL_ADDRESS>Jagiellonian University
Faculty of Mathematics and Computer Science
Theoretical Computer Science Department
ul. Łojasiewicza 6, Kraków, Poland

###### Abstract.

A majority coloring of an undirected graph is a vertex coloring in which for
each vertex there are at least as many bi-chromatic edges containing that
vertex as monochromatic ones. It is known that for every countable graph a
majority 3-coloring always exists. The Unfriendly Partition Conjecture states
that every countable graph admits a majority 2-coloring. Since the 3-coloring
result extends to countable DAGs, a variant of the conjecture states that 2
colors are enough to majority color every countable DAG. We show that this is
false by presenting a DAG for which 3 colors are necessary.

Presented construction is strongly based on a StackExchange conversation [BbR]
regarding labellings of infinite graphs.

∗ Research supported by the National Science Center of Poland under grant no.
2020/37/B/ST1/03298.

##  1\. Introduction

Given a finite simple directed graph $G=(V,E)$, a coloring
$\phi:V(G)\rightarrow\mathbb{N}$ is called a majority coloring if for every
vertex $v\in V(G)$

$\\#\\{vu\in E(G):\phi(u)\neq\phi(v)\\}\geqslant\\#\\{vu\in
E(G):\phi(u)=\phi(v)\\}.$

In other words, for every vertex $v$ at most half of the out-edges from $v$
may be monochromatic. A digraph $G$ is majority $k$-colorable if it admits a
majority coloring using at most $k$ colors.

It is a well known fact that every simple finite DAG (directed acyclic graph)
admits a majority 2-coloring. It can be constructed by sorting the graph
topologically, and then coloring the vertices greedily in the reverse
topological order.

The definition extends directly to graphs of any cardinality. In [ABG20]
Anholcer, Bosek and Grytczuk prove that every countable DAG is not only
majority 4-colorable, but majority 4-choosable. In [Has20] Haslegrave improves
their result and shows that every countable DAG is majority 3-choosable, which
implies majority 3-colorability. In both [ABG20] and [Has20] it is conjectured
that every countable DAG is majority 2-colorable. Here, we give a
counterexample to this.

##  2\. The counterexample

###  2.1. Construction of the counterexample

Figure 1. An OR${}^{T}(v_{1},v_{2})$ gadget and an
OR${}^{T}(v_{1},v_{2},v_{3})$ gadget.

Our construction makes use of gadgets called by us the OR gadgets.
Intuitively, given a graph $G$ and a vertex $T\in V(G)$, one can interpret a
2-coloring of $G$ as as a truth assignment to the vertices of $G$ based on
whether the vertex is colored the same color as $T$ or not. Given
$v_{1},v_{2}\in V(G)-\\{T\\}$, an OR${}^{T}(v_{1},v_{2})$ gadget is as defined
as shown in Figure 1.
Let us refer to the vertex $o$ in Figure 1 as an output vertex of the gadget.
Let OR${}^{T}(v_{1},\dots,v_{k})$ denote a chained OR gadget, like exemplified
in Figure 1. Output $o$ of such gadget is defined analogously,

Let us now present the counterexample graph $G$. As shown in Figure 2, it is
constructed by taking an infinite directed path $(v_{1},v_{2},v_{3},\dots)$, a
vertex $T$, for every $i,j\in\mathbb{N}$ such that $2\leq i<j$ an
OR${}^{T}(v_{i},\dots,v_{j})$ gadget (later referred to as ORi,j), and finally
joining the output of ORi,j with an edge starting in $v_{i-1}$.

Figure 2. A countable acyclic digraph $G$ that we claim not to be majority
2-colorable.

###  2.2. Proof that 3 colors are required for any majority coloring of $G$.

An OR${}^{T}(u_{1},\dots,u_{k})$ gadget consisting of vertices $U$ is
considered _valid_ in $H$ if there are no outgoing edges between
$U-\\{T,u_{1},\dots,u_{k}\\}$ and $V(G)-U$ in $H$. Note that every ORi,j is
valid in $G$.
For any 2-coloring $\phi$ of $H$ and any $v,T\in V(G)$, let $\phi^{T}(v)$ be
true iff $\phi(v)=\phi(T)$. We will say that $v$ is colored _true_ in $\phi$.

######  Lemma 1.

Given a graph $H=(V,E)$ of any cardinality and any majority 2-coloring $\phi$
of $G$, if $U$ is a valid OR${}^{T}(u_{1},\dots,u_{k})$ gadget in $H$ with an
output vertex $o$, then
$\phi^{T}(u_{1})\lor\dots\lor\phi^{T}(u_{k})=\phi^{T}(o)$.

###### Proof.

Induction on $i$.

Base case: since there are no outgoing edges from $U-\\{T,u_{1},u_{2}\\}$ to
the rest of the graph, it suffices to check all 4 possible precolorings of
$v_{1}$ and $v_{2}$ in regard to the color of $T$ and see that in all possible
extensions to $U$ the output vertex is colored as stated.

Induction step: note that OR${}^{T}(u_{1},\dots,u_{l})$ is in fact an
OR${}^{T}(o,u_{l})$ on the output $o$ of OR${}^{T}(u_{1},\dots,u_{l-1})$. The
same reasoning applies. ∎

######  Lemma 2.

Let $\phi$ be a majority 2-coloring of $G$. For each $i\in\mathbb{N}_{+}$
$\phi^{T}(v_{i})$ is true if and only if for each $j>i$ $\phi^{T}(v_{j})$ is
not true.

###### Proof.

Let $i\in\mathbb{N}_{+}$.

$\implies$

Contra-position. If there exists $j>i$ such that $\phi^{T}(v_{j})$ is true,
then notice, that the output of ORi+1,k is true for any $k\geqslant j$ as well
by Lemma 1. Therefore, among the out-neighbors of $v_{i}$, infinitely many are
colored true, and finitely many are colored false, thus $\phi^{T}(v_{i})$ must
be false.

$\impliedby$

If all $v_{j}$ are colored false for $j>i$, then all outputs of ORi,j must as
well be colored false by Lemma 1 and therefore $v_{i}$ must be colored true.

∎

######  Theorem 3.

The graph $G$ is a simple countable acyclic digraph that is not majority
2-colorable.

###### Proof.

$G$ is trivially countable.

$G$ is acyclic. Note that the vertex $T$ has out-degree $0$. We will show a
topological sort $\sigma$ of the graph $G-\\{T\\}$. To each vertex $v$ we will
assign a triplet $\sigma(v)=(i,j,k)\in\mathbb{N}^{3}$ so that for every edge
$uv$ of $G$ it is true that $\sigma(u)<_{L}\sigma(v)$, where $<_{L}$ denotes
the lexicographical ordering.

Note that each ORi,j used in the construction is acyclic. Let
$\sigma_{i,j}:V(\text{OR}_{i,j})\rightarrow\mathbb{N}$ denote the topological
sort of ORi,j.

We assign the triplets as follows:

  * •

$\sigma(v_{i})=(i,0,0)$,

  * •

for $v\in V(\text{OR}_{i,j})-\\{v_{i}:i\in\mathbb{N}_{+}\\}$,
$\sigma(v)=(i-1,j,\sigma_{i,j}(v))$.

For every edge type, we show that the inequality holds:

  * •

for $v_{i}v_{i+1}\in E(G-\\{T\\})$, we have

$\sigma(v_{i})=(i,0,0)<_{L}(i+1,0,0)=\sigma(v_{i+1})$,

  * •

for $v_{i}o$ where $o$ is an output of ORi+1,j, we have

$\sigma(v_{i})=(i,0,0)<_{L}(i,j,\sigma_{i+1,j}(o))=\sigma(o)$,

  * •

for $uv_{k}$ where $u\in V(\text{OR}_{i,j})-\\{v_{i}:i\in\mathbb{N}\\}$ and
$k\in[i,j]$, we have

$\sigma(u)=(i-1,j,\sigma_{i,j}(u))<_{L}(k,0,0)=\sigma(v_{k})$,

  * •

for $uw$ where $u,w\in V(\text{OR}_{i,j})-\\{v_{i}:i\in\mathbb{N}\\}$, due to
the definition of $\sigma_{i,j}$ we have

$\sigma(u)=(i-1,j,\sigma_{i,j}(u))<_{L}(i-1,j,\sigma_{i,j}(w))=\sigma(w)$.

Every edge of $G-\\{T\\}$ was considered, and $T$ is a vertex of outdegree
$0$, therefore $G$ is acyclic.

Let us assume that a majority 2-coloring $\phi$ of $G$ exists - we will
eventually reach a contradiction.
From Lemma 2 notice that there is at most one vertex among $v_{1},v_{2},\dots$
colored true. If exactly one vertex $v_{k}$ is colored true, notice that
$v_{k+1}$ also satisfies the constraint of Lemma 2, and therefore should also
be colored true. Thus, all of $v_{1},v_{2},\dots$ must be colored false. But
in this case, $v_{1}$ satisfies the constraint of Lemma 2, and therefore
should be colored true.
This gives us a contradiction to our assumption that a majority 2-coloring of
$G$ exists.

∎

##  3\. Conclusions and open problems

We state a problem that may act like a bridge between this result and the
Unfriendly Partition Conjecture.

By a multigraph, we mean an undirected graph $G=(V,E)$ together with a weight
function $w:E(G)\rightarrow\mathbb{N}$. A coloring
$\phi:V(G)\rightarrow\mathbb{N}$ is called a majority coloring of $G$ if for
every vertex $v\in V(G)$

$\sum_{vu\in E(G)\land\phi(v)\neq\phi(v)}w(u)\geqslant\sum_{vu\in
E(G)\land\phi(v)=\phi(v)}w(u)$

where sigma denotes the limit of a series in the countable case.

Just like finite undirected graphs, by a max-cut argument, finite multi-graphs
are majority 2-colorable. Multi-graphs allow us to emulate DAGs to some degree
by manipulating the weight function. Unfortunately, the presented construction
for directed graphs does not directly transfer to multi-graphs.

Nevertheless, we postulate the following.

######  Conjecture 1.

There exists an undirected countable multi-graph that is not majority
3-colorable.

##  4\. Acknowledgments

We thank Stack Exchange users Jo Bain, bof and Alex Ravsky for providing a
really strong inspiration for this result in this StackExchange conversation
[BbR].

## References

  * [ABG20] Marcin Anholcer, Bartłomiej Bosek, and Jarosław Grytczuk.  Majority choosability of countable graphs, 2020.
  * [BbR] Jo Bain, bof, and Alex Ravsky.  Labelings of countable digraphs.  math.stackexchange.  URL:https://math.stackexchange.com/questions/1271698/labelings-of-infinite-directed-acyclic-graphs (visited on 2024-04-18).
  * [Has20] John Haslegrave.  Countable graphs are majority 3-choosable.  Discussiones Mathematicae Graph Theory, 43(2):499, 2020.


# Scalar Dark Matter and Stability of Higgs Vacuum within a Minimal SO(10) GUT
Model

Nilavjyoti Hazarika111E-mail<EMAIL_ADDRESS>Paramita Deka
<EMAIL_ADDRESS>Kalpana<EMAIL_ADDRESS>Department Of
Physics, Gauhati University, Assam, India

###### Abstract

In this work, we delve to investigate a feasible range of dark matter (DM)
masses within a non-supersymmetric $SO(10)$ Grand Unified Theory (GUT) scalar
dark matter model, in freeze-out scenario. This model includes a singlet
scalar denoted as S and an inert doublet represented by $\phi$. Being part of
SO(10), the quantum numbers of DM particles are assigned and hence we know
their nature. These fields are odd under a discrete $Z_{2}$ matter parity
$(-1)^{3(B-L)}$. The dark matter with mass 300 $\leq M_{DM}\leq$ 1000 GeV
emerges as a mixture of the $Z_{2}$-odd scalar singlet $S$ and the neutral
element of the doublet $\phi$, both residing within a 16-dim scalar
representation of $SO(10)$. Though similar propositions are available in
literature for complex singlet scalar S as DM, in this work we consider a real
scalar S belonging to 16 of SO(10) as DM. We also investigate the one-loop
vacuum stability through the solution of Renormalization Group Equations
(RGEs) for the model’s parameters. Subsequently, we scrutinize the model’s
predictions within the confines of contemporary theoretical and experimental
restrictions. The novelty of this work is that, in addition to achieving
vacuum stability in the SO(10) framework, the DM mass is found to lie in the
previously unaddressed (theoretically) intermediate mass region of
300$\leqslant$ MDM $\leqslant$ 1000 GeV which also adheres to many current
phenomenological constraints, like recent direct experimental bounds from
XENON1T, indirect detection bounds from Fermi-LAT experiment, Higgs invisible
decay and Electroweak Precision Test. The stability of the electroweak vacuum
is seen to be present up to the Planck scale. These model predictions possess
the potential for future validation through dark matter search experiments, as
they are testable in future DM search experiments, along with the added
feature that the model is a part of an elegant grand unified theory.

##  I Introduction

Following the discovery of the Higgs boson by ATLAS and CMS experiments at the
LHC in 2012, the value of the Higgs mass is determined accurately to be around
125 GeV ATLAS:2012yve ; CMS:2012qbp . Subsequently, measurements ATLAS:2013xga
; CMS:2014nkk ; CMS:2018uag ; ATLAS:2019nkf  on its properties such as spin,
parity, and interactions with SM particles validated the SM at energies
accessible to the LHC. Although the standard model (SM) of particle physics is
one of the most successful theories; however, it still has a few unresolved
observations. The shortcomings of the SM are mainly- the strong CP problem,
neutrino mass and mixing, matter-antimatter asymmetry, nature of dark matter
(DM) and dark energy. Astrophysical and cosmological observations indicate
that about 26 $\%$ of the current universe’s energy density is composed of DM
ParticleDataGroup:2018ovx , with a present abundance expressed as
$\Omega_{DM}h^{2}=0.1198\pm 0.0026$ Planck:2018vyg , where $\Omega_{DM}$
represents the DM density parameter and $h=\boldsymbol{H}/100$
$kms^{-1}Mpc^{-1}$, where $\boldsymbol{H}$ denotes the Hubble parameter.

The SM also fails to provide absolute stability for the electroweak (EW)
vacuum up to the Planck scale Isidori:2001bm ; Bezrukov:2012sa ;
Buttazzo:2013uya , with the EW vacuum residing in a metastable state instead
of a stable state within the SM framework. Having known the values of Higgs
mass, the top quark mass to be about 176 GeV CDF:1995wbb  and the value of the
Higgs vacuum expectation value (VEV) to be $v=246$ GeV, the SM vacuum at
energy scale around $10^{10}$ GeV starts to be metastable Elias-Miro:2011sqh ;
Tang:2013bz ; Degrassi:2014hoa ; Rojas:2015yzm . This is due to the fact that
in the SM, the top quark has a large negative contribution in the
renormalisation group equations (RGE) for the Higgs quartic coupling
$\lambda_{h}$. Thus, the Higgs quartic coupling becomes negative at higher
energy scales which in turn makes the vacuum metastable. Given these
limitations, it is necessary to introduce new physics beyond the SM (BSM) as
the observed instability implies limitations in its description of particle
interactions and the behaviour of the Higgs field.

It is known that in the scalar singlet DM model, the only region which is not
yet excluded is a narrow region close to the Higgs resonance
$m_{s}\sim(m_{h}/2)$, others ruled out from different experimental and
theoretical bounds (see Hazarika:2022tlc  and references therein). In the
inert doublet model, the mass region ( 60-80 GeV) and the high-mass region
(heavier than 550 GeV) are allowed. This motivated us to explore a parameter
range in the so far theoretically unaddressed DM mass region, the
intermediate-mass region $M_{W}<M_{DM}<550$ GeV, which we successfully did in
a scotogenic extension of SM with a scalar doublet and scalar singlets
Hazarika:2022tlc .

One possibility to address the issue of vacuum stability is the addition of
scalars, so that the vacuum can become stable up to the Planck scale. It has
been extensively studied in the literature. For instance, extending the SM
with an extra real scalar with $Z_{2}$ symmetry is studied in
Gonderinger:2009jp ; Falkowski:2015iwa , with a complex scalar in Elias-
Miro:2012eoi ; Gonderinger:2012rd ; Athron:2018ipf , Two-Higgs doublet models
(2HDM) in Ferreira:2004yd ; Maniatis:2006fs ; Battye:2011jj ; Kannike:2016fmd
; Xu:2017vpq . Further, the vacuum stability is studied in type-II seesaw
models with $SU(2)_{L}$ -triplet scalars in Gogoladze:2008gf ; Chun:2012jw ;
BhupalDev:2013xol ; Bonilla:2015eha ; Haba:2016zbu ; Dev:2017ouk , in $U(1)$
extensions of SM Datta:2013mta ; Coriano:2014mpa ; Haba:2015rha ; Das:2015nwk
, as well as in the left-right symmetric models Mohapatra:1986pj ;
BhupalDev:2018xya ; Chauhan:2019fji .

Though one can propose many extra BSM fields in the theory, however, the
question remains - from where these fields originate in a single theory? And,
the answer to this question lies with Grand Unified Theories (GUTs), that are
very lucrative proposals, as they can accommodate solutions to many problems,
in a holistic manner. Therefore, in view of the above mentioned issues, in
this work we have undertaken to address the EW vacuum stability problem along
with the existence of DM in intermediate mass region (300$\leq M_{DM}\leq$1000
GeV), as a manifestation of a GUT. In the GUT framework, in addition to
explaining the origin of DM, it is also possible to determine the nature of
the DM particle and constrain its properties, due to its pinpointed quantum
numbers. We know that in a GUT theory, at very high energy, all three
interactions of the SM exhibit identical strength. However, as a result of the
distinct renormalization group evolution with energy driven by the particle
content of the model, they ultimately manifest varying strengths at lower
energy levels. Any prospective GUT candidate must include the SM gauge group
as a subgroup, thereby ensuring that the SM remains applicable at the
electroweak scale. Since the SM’s gauge group has a rank of four, the GUT’s
gauge group must possess a rank greater than or equal to four. These
attributes are naturally realized in SO(10) GUTs Georgi:1974my ;
Fritzsch:1974nn ; Chanowitz:1977ye ; Georgi:1979ga ; Mohapatra:1982aq ;
Babu:1992ia , that offer numerous appealing characteristics, including:

  * •

First, it allows for the incorporation of all SM quarks, leptons, and right-
handed neutrinos within 16 representations of SO(10).

  * •

Second, it naturally resolves the issue of anomaly cancellation in the SM, as
it is devoid of anomalies.

  * •

Third, achieves partial unification at an intermediate mass scale within
SO(10) which results in improved gauge coupling unification Georgi:1979ga ;
Masiero:1980dd  and improved fermion mass ratios Georgi:1979ga ;
Lazarides:1980nt .

  * •

Fourth - It can accommodate DM particle.

In SO(10) GUT models, stable DM could be incorporated in a straightforward way
Frigerio:2009wf ; Kadastik:2009cu ; Kadastik:2009dj ; Mambrini:2013iaa ;
Mambrini:2015vna ; Nagata:2015dma . Additionally, SO(10) incorporates an extra
$U(1)$ symmetry, which is assumed to undergo breaking at the intermediate
scale. Multiple breaking chains for breaking SO(10) down into the SM are
possible, aligning with experimental observations while also offering unique
predictions. SO(10) introduces new particles beyond the SM, and may modify the
running of the SM Higgs quartic coupling $\lambda_{H}$ through the addition of
new particles at an intermediate scale below $10^{10}$ GeV, ensuring that
$\lambda_{H}$ remains positive all the way up to the Planck scale
Mambrini:2016dca . In earlier works it has been shown that the existence of DM
is a possible consequence of SO(10) GUT symmetry breaking. Gauge coupling
unification and dark matter production through non-equilibrium thermal
processes in the context of SO(10) models was presented in Mambrini:2013iaa .
In Frigerio:2009wf , TeV scale fermion isotriplet with zero hypercharge
belonging to a 45 (or larger) representation of SO(10) was considered to be DM
candidate. In Kadastik:2009cu ; Kadastik:2009dj , a minimal non-supersymmetric
SO(10) framework was considered with DM belonging to complex scalar singlet
and doublets both belonging to 16, where it was shown that electroweak
symmetry breaking occurs radiatively due to DM couplings to the SM Higgs
boson. In Mambrini:2015vna , multiple breaking schemes of SO(10) which lead to
gauge coupling unification and stable DM along with production of light
neutrino mass were presented. Scalar and fermion DM candidates in SO(10) GUT
model is discussed in Nagata:2015dma . In Mambrini:2016dca , vacuum stability
in addition to radiative electroweak symmetry breaking was studied in the
SO(10) model, where a dark matter mass region of ($1.35-2$) TeV was found to
be consistent with the theoretical and experimental bounds. In Sahu:2022rwq ,
they considered an extension of SM by a scalar leptoquark and a fermion
triplet embedded in a non-SUSY SO(10) GUT, and the neutral component of
fermion triplet served as DM candidate and unification mass scale along with
the corresponding proton decay constraints were discussed.

The scalar DM models being simple and attractive, in this work, we address two
of the important problems discussed above, that remain unaddressed so far -
the scalar DM is the intermediate mass region and vacuum stability, in a
single theory. We choose a minimal DM model where the DM is an admixture of a
real SM singlet scalar that arises from a 16 representation and an Inert
Doublet $\phi$ that belong to same 16 representation of the SO(10) GUT model.
The novelty of this work is that, in addition to achieving vacuum stability in
the SO(10) framework, the DM mass is found to lie in the intermediate mass
region of 300 $\leq M_{DM}\leq$ 1000 GeV which also adheres to many current
phenomenological constraints, like recent direct experimental bounds from
XENON1T XENON:2017vdw ; XENON:2018voc , indirect detection bounds from Fermi-
LAT experiment Fermi-LAT:2015att , Higgs invisible decay ATLAS:2015ciy  and
Electroweak Precision Test as well Baek:2012uj . All these issues have not
been addressed previously in a single model, and that makes the work
interesting.

The paper is structured as follows: In section II, we describe our model. In
section III, we present the theory for computing DM abundance and the DM
scattering cross-section in the the freeze-out scenario. In section IV, we
discuss the method used to achieve electroweak vacuum stability. For
completeness, we give the expressions for one-loop beta functions used in our
analysis in this section. Section V contains our results, which are
subsequently discussed. Finally, in section VI, we present a summary of our
work and our conclusions.

##  II The Model

As stated earlier, in this work we aim to solve the problem of stability of
vacuum as well to find the DM candidate in so far theoretically unaddressed
intermediate mass range, in SO(10) grand unified theory. Our objective is to
elucidate the nature and origin of DM, as well as constrain the parameter
space based on current experimental limitations. Additionally, we wish to
examine the stability of the electroweak vacuum up to the Planck scale, all
within the SO(10) GUT framework. To stabilize the DM, we must enforce a
discrete $Z_{2}$ symmetry. However, Planck scale operators break global
discrete symmetries, which can be remedied by obtaining a $Z_{2}$ symmetry
through the breaking of a gauged $U(1)$ embedded within a GUT. This could be
achieved by breaking SO(10) GUT group to the SM gauge group and an extra
$U(1)_{X}$ subgroup, where $X$ is orthogonal to the SM hypercharge $Y$.
Therefore, SO(10) can be broken down to the symmetry group of the SM and the
gauged $Z_{2}$ parity Kadastik:2009cu

$P_{X}\equiv P_{M}=(-1)^{3(B-L)}$ (1)

which is the matter parity. We assume that the SO(10) gauge group is
spontaneously broken to an intermediate subgroup $G_{int}$ at the GUT scale
$M_{G}$, and subsequently broken to the SM gauge group $G_{SM}\equiv
SU(3)_{C}\times SU(2)_{L}\times U(1)_{Y}$ and $P_{M}$ at an intermediate EWSB
scale $M_{Z}$ :

$SO(10)\rightarrow G_{int}\rightarrow G_{SM}\times P_{M}$ (2)

Some possible intermediate subgroups $G_{int}$ are listed in the following
table:

$G_{int}$
---
$SU(4)_{C}\times SU(2)_{L}\times SU(2)_{R}$
$SU(4)_{C}\times SU(2)_{L}\times SU(2)_{R}\times D$
$SU(4)_{C}\times SU(2)_{L}\times U(1)_{R}$
$SU(3)_{C}\times SU(2)_{L}\times SU(2)_{R}\times U(1)_{B-L}$
$SU(3)_{C}\times SU(2)_{L}\times SU(2)_{R}\times U(1)_{B-L}\times D$
$SU(3)_{C}\times SU(2)_{L}\times U(1)_{R}\times U(1)_{B-L}$
$SU(5)\times U(1)$
Table 1: Candidates for $G_{int}$ (the intermediate gauge
group).Mambrini:2015vna

In the above table, $D$ denotes the so-called $D$-parity or the left-right
symmetry Frigerio:2009wf , that is, the symmetry with respect to the exchange
of $SU(2)_{L}\leftrightarrow SU(2)_{R}$. $D$-parity can be related to an
element of SO(10) Kuzmin:1980yp ; Kibble:1982dd ; Chang:1983fu ; Chang:1984uy
under which a fermion field transforms into its charge conjugate.

Within the SO(10) GUT framework, the DM is postulated to be a combination of a
$P_{M}$-odd scalar singlet $S$, which belongs to the $SU(2)_{L}\times
U(1)_{Y}$ sector, and the neutral component of the doublet $\phi$, which is
part of a new scalar 16 of SO(10). The grand unified group SO(10) undergoes a
breakdown resulting in $SU(3)_{c}\times SU(2)_{L}\times U(1)_{Y}\times P_{M}$,
where only the SM Higgs boson $H$ belongs to 10, and the DM candidates - a
mixture of the singlet S belonging to 16 and the Inert Doublet $\phi$
belonging to 16 \- are light, while all other particles have masses of the
order of GUT scale $M_{G}$. The SO(10) GUT breaking may occur in one or in
several steps through intermediate subgroup as mentioned in table 1. In this
work we assume these breakings to occur near the GUT scale.

To verify the proposed DM scenario, we analyze the scalar potential of a
minimal SO(10) GUT model consisting of a scalar 16 (i.e., $\phi$) for the DM
and a scalar 10 (i.e., H) for the SM Higgs doublet. The inert doublet and SM
Higgs are as follows:

$H=\begin{pmatrix}0\\\
\frac{1}{\sqrt{2}}(v+h)\end{pmatrix}\quad,\phi=\begin{pmatrix}H^{+}\\\
\frac{1}{\sqrt{2}}(H_{0}+iA_{0})\end{pmatrix}$ (3)

The $SO(10)$ symmetric scalar potential of one 16 and one 10 is
Kadastik:2009cu

$\displaystyle\begin{array}[]{rcl}V&=&\mu_{1}^{2}(\textbf{10}\hskip
2.84544pt\textbf{10})+\mu_{2}^{2}(\textbf{16}\hskip
2.84544pt\textbf{16})+\lambda_{h}(\textbf{10}\hskip
2.84544pt\textbf{10})^{2}+\lambda_{2}(\textbf{16}\hskip
2.84544pt\textbf{16})^{2}+\lambda_{3}((\textbf{10\hskip
2.84544pt10})(\textbf{16}\hskip 2.84544pt\textbf{16})\\\
&&+\lambda_{4}(\textbf{16\hskip 2.84544pt10})(\textbf{16\hskip
2.84544pt10}))+\rho(\textbf{16\hskip 2.84544pt10\hskip
2.84544pt16}+hc)\end{array}$ (6)

For simplicity, we have assumed that all parameters are real. The scalar
potential below $M_{G}$ that we consider is the most general CP-invariant
potential that remains invariant under the transformations $H\rightarrow H$,
$\phi\rightarrow\phi$, and $S\rightarrow-S$.

$\displaystyle\left.\begin{array}[]{rcl}V^{{}^{\prime}}&=&\mu_{1}^{2}(H^{\dagger}H)+\mu_{2}^{2}(\phi^{\dagger}\phi)+\mu_{S}^{2}(S^{2})+\lambda_{h}(H^{\dagger}H)^{2}+\lambda_{2}(\phi^{\dagger}\phi)^{2}\\\
&&+\lambda_{S}(S)^{4}+\lambda_{3}(H^{\dagger}H)(\phi^{\dagger}\phi)+\lambda_{4}(H^{\dagger}\phi)(\phi^{\dagger}H)\\\
&&+\frac{\lambda_{5}}{2}[(H^{\dagger}\phi)^{2}+h.c]+\lambda_{HS}(S^{2})(H^{\dagger}H)+\lambda_{\phi
S}(S^{2})(\phi^{\dagger}\phi)\\\
&&+\rho[(H^{\dagger}\phi)S+h.c]\end{array}\right.$ (11)

where $H$ is the SM Higgs doublet (belonging to 10), and 16 contain the inert
doublet $\phi$ and the singlet $S$. If $M_{G}$ is the GUT unification scale,
the GUT scale boundary conditions can be expressed as, Kadastik:2009cu

$\displaystyle\mu^{2}_{1}(M_{G})>0,\quad\mu^{2}_{2}(M_{G})=\mu^{2}_{S}(M_{G})>0;$
(12) $\displaystyle\lambda_{2}(M_{G})=\lambda_{S}(M_{G})=\lambda_{\phi
S},\quad\lambda_{3}(M_{G})=\lambda_{HS}(M_{G})$

We consider mixing between the real singlet $S$ and Inert doublet $\phi$ and
obtain a $2\times 2$ mass matrix for the lightest neutral $Z_{2}$ odd scalar
as Hazarika:2022tlc

$\displaystyle M=\begin{pmatrix}\mu_{2}^{2}+\lambda_{L}v^{2}&\rho v\\\ \rho
v&2\mu_{S}^{2}+\lambda_{HS}v^{2}\end{pmatrix}=\begin{pmatrix}m_{H_{0}}^{2}&m_{H_{0}S}^{2}\\\
m_{H_{0}S}^{2}&m_{S}^{2}\end{pmatrix}$ (13)

where $\lambda_{L}=\frac{\lambda_{3}+\lambda_{4}+\lambda_{5}}{2}$. The
physical mass eigenstates can be defined as

$\displaystyle\begin{array}[]{l}\chi_{1}=H_{0}\cos\theta+S\sin\theta\\\
\chi_{2}=-H_{0}\sin\theta+S\cos\theta\end{array}$ (16)

Masses of the neutral physical scalars $\chi_{1}$ and $\chi_{2}$ are,

$\displaystyle\begin{array}[]{l}m_{1}^{2}=\frac{m_{H_{0}}^{2}+m_{s}^{2}}{2}-\frac{m_{H_{0}}^{2}-m_{s}^{2}}{2}\sqrt{1+\tan^{2}2\theta},\\\
m_{2}^{2}=\frac{m_{H_{0}}^{2}+m_{s}^{2}}{2}+\frac{m_{H_{0}}^{2}-m_{s}^{2}}{2}\sqrt{1+\tan^{2}2\theta}\end{array}$
(19)

The couplings in terms of the masses are ( see Hazarika:2022tlc )

$\displaystyle\begin{array}[]{l}\lambda_{1}=\dfrac{\mu_{1}^{2}}{2v^{2}},\\\
\lambda_{3}=\frac{2(\lambda_{L}v^{2}+\mu_{H^{\pm}}^{2}-m_{1}^{2}\cos^{2}\theta-
m_{2}^{2}\sin^{2}\theta)}{v^{2}},\\\
\lambda_{4}=\frac{m_{1}^{2}\cos^{2}{\theta}+m_{2}^{2}\sin^{2}{\theta}+\mu_{A_{0}}^{2}-2\mu_{H^{\pm}}^{2}}{v^{2}},\\\
\lambda_{5}=\frac{m_{1}^{2}\cos^{2}{\theta}+m_{2}^{2}\sin^{2}{\theta}-\mu_{A_{0}}^{2}}{v^{2}},\\\
\mu_{S}^{2}=\frac{1}{2}(m_{1}^{2}\sin^{2}{\theta}+m_{2}^{2}\cos^{2}{\theta}-\lambda_{HS}v^{2}),\\\
\rho=\frac{(m_{2}^{2}-m_{1}^{2})\sin{2\theta}}{2v},\\\
\mu_{2}^{2}=m_{1}^{2}\cos^{2}{\theta}+m_{2}^{2}\sin^{2}{\theta}-\lambda_{L}v^{2}\par\end{array}$
(27)

To ensure that the potential in our model remains bounded from below, vacuum
stability necessitates Kadastik:2009cu

$\displaystyle\lambda_{h},\lambda_{\phi},\lambda_{S}>0;\quad\lambda_{3}+2\sqrt{\lambda_{h}\lambda_{2}}>0$
(28)
$\displaystyle\lambda_{3}+\lambda_{4}-|\lambda_{5}|+2\sqrt{\lambda_{h}\lambda_{2}}>0$

##  III Dark Matter

Now, we present a discussion on various constraint checks on dark matter as
done in our analysis.

###  III.1 Relic density constraints

As previously mentioned, the chosen DM candidate in our model is the lightest
component, denoted as $\chi_{1}$, which is a mixture of singlet and doublet
scalar DM fields. To ascertain the thermal relic abundance of this DM
particle, which follows the WIMP archetype, we solve the Boltzmann equation
Kolb:1990vq

$\frac{dn_{DM}}{dt}+3\boldsymbol{H}n_{DM}=-<\sigma
v>[n_{DM}^{2}-(n_{DM}^{eq})^{2}]$ (29)

where $n_{DM}$ is the number density of the DM particle and $n_{DM}^{eq}$ is
the number density of the DM particle when it is in thermal equilibrium.
$\boldsymbol{H}$ is the Hubble rate of expansion of the Universe and $<\sigma
v>$ is the thermally averaged annihilation cross-section of the DM particle.
One can obtain the numerical solution of the Boltzmann equation in terms of
partial expansion $<\sigma v>=a+bv^{2}$ as Scherrer:1985zt

$\Omega_{DM}h^{2}\approx\frac{1.04\times
10^{9}x_{F}}{M_{Pl}\sqrt{g_{\ast}}(a+\frac{3b}{x_{F}})}$ (30)

where $x_{F}=m_{DM}/T_{F}$, $T_{F}$ is the freeze-out temperature, $g_{\ast}$
is the number of relativistic degrees of freedom at the time of freeze-out.
After further simplifications, the above solution takes the form as
Jungman:1995df

$\Omega_{DM}h^{2}\approx\frac{3\times 10^{-27}cm^{3}s^{-1}}{<\sigma v>}$ (31)

###  III.2 DM-nucleon cross section (DD)

The thermal averaged annihilation cross section $<\sigma v>$ is given by
Gondolo:1990dk

$<\sigma
v>=\frac{1}{m_{DM}^{4}TK_{2}^{2}(\frac{m_{DM}}{T})}\int\sigma(s-4m_{DM}^{2})\surd
sK_{1}(\frac{\surd s}{T})ds$ (32)

Here, $K_{i}$ represents the modified Bessel functions of order $i$, $m_{DM}$
is the DM particle’s mass, and $T$ corresponds to the Universe’s temperature.
Moreover, the relevant spin-independent scattering cross-section concerning
the scalar DM $H_{2}$, mediated through the SM Higgs, can be formulated as
follows Barbieri:2006dq :

$\sigma_{SI}=\frac{\lambda_{L}^{2}f^{2}}{4\pi}\frac{\mu^{2}m_{n}^{2}}{m_{h}^{4}m_{DM}^{2}}$
(33)

Here, $m_{n}$ stands for the mass of a nucleon, and
$\mu=\frac{m_{n}m_{DM}}{m_{n}+m_{DM}}$ represents the reduced mass of DM and
nucleons. Similarly,
$\lambda_{L}=\frac{\lambda_{3}+\lambda_{4}+\lambda_{5}}{2}$ signifies the
quartic coupling between DM and the Higgs particle, while $f$ denotes the
coupling between the Higgs and nucleons. According to a recent evaluation, $f$
has been approximated as $0.32$ Giedt:2009mr , although the permissible range
of values spans from $0.26$ to $0.63$ Mambrini:2011ik .

###  III.3  DM-Higgs invisible decay

Moreover, the DM-Higgs coupling $\lambda_{L}$ can be restricted based on the
most recent constraints from the LHC regarding the hidden decay width of the
SM Higgs boson. The invisible Higgs decay width has a connection to the DM-
Higgs coupling $\lambda_{L}$ as explained in ATLAS:2015ciy  as,

$\Gamma(h\rightarrow Invisible)=\frac{\lambda_{L}v^{2}}{64\pi
m_{h}}\sqrt{1-4m_{DM}^{2}\diagup m_{h}^{2}}$ (34)

###  III.4 Indirect detection (ID) and Fermi-LAT

Apart from direct detection experiments, the realm of DM parameters can also
be confined using findings from various indirect detection experiments, such
as those carried out by Fermi-LAT Fermi-LAT:2015att . These experiments focus
on searching for SM particles, which are produced either through the
annihilation of DM or through its decay in the local Universe. The resulting
neutral and stable final products, including photons and neutrinos, can reach
the indirect detection experiments with minimal interference from intermediate
regions. For DM of the WIMP type, these photons fall within the gamma-ray
range, detectable by space-based telescopes like Fermi-LAT through
observations of dwarf spheroidal satellite galaxies (dSphs). The observed
differential gamma-ray flux originating from DM annihilations is
mathematically expressed as follows: Borah:2017dfn

$\dfrac{d\Phi}{dE}(\vartriangle\Omega)=\frac{1}{4\pi}<\sigma
v>\frac{J(\vartriangle\Omega)}{2M_{DM}^{2}}\dfrac{dN}{dE}$ (35)

In this context, the portion of the sky under observation is represented by
the solid angle $\vartriangle\Omega$, $<\sigma v>$ denotes the thermally
averaged cross section for DM annihilation, and $dN/dE$ signifies the average
gamma-ray spectrum produced per annihilation event. The astrophysical factor
$J$ is mathematically expressed as follows: Borah:2017dfn

$J(\vartriangle\Omega)=\int_{\vartriangle\Omega}d\Omega^{\prime}\int_{LOS}dl\rho^{2}(l,\Omega^{\prime})$
(36)

where, $\rho$ represents the density of DM, and LOS stands for the line of
sight. As a result, it becomes feasible to limit the DM annihilation into
various final states, such as
$\mu^{+}\mu^{-},\tau^{+}\tau^{-},W^{+}W^{-},ZZ,b\bar{b}$. By utilizing the
constraints derived from DM annihilation into these specific final states, we
can confine the DM parameters through a comprehensive analysis of observations
from Fermi-LAT Fermi-LAT:2015att  involving dwarf spheroidal satellite
galaxies.

##  IV Vacuum stability

The scalar potential associated with the SM Higgs doublet is

$V_{SM}=\mu^{2}|H|^{2}+\frac{\lambda_{h}}{2}|H|^{4}$ (37)

where, the parameters $\lambda_{h}$ and $\mu$ are influenced by both the Higgs
Vacuum Expectation Value (VEV) and the Higgs particle’s mass, denoted as
$m_{h}$. If we assume the validity of the SM up to a very high energy scale,
approximately $10^{10}$ GeV, an intriguing observation emerges. The quartic
coupling $\lambda_{h}$ undergoes a transition to a negative value at energy
scales below $10^{10}$ GeV as shown by the solid blue curve in figure 1. This
transition holds significant consequences. Specifically, the scalar potential
within the SM becomes unbounded in the downward direction for sufficiently
large values of the Higgs field ($H$). Consequently, the stability of the
vacuum in the electroweak theory is compromised. These findings suggest the
need for new physics to come into play at energy scales below $10^{10}$ GeV.

Figure 1: Running of Higgs quartic coupling $\lambda_{h}$ in the SM

Within the framework of an SO(10) unification model, fresh insights emerge. In
this work we have considered the DM candidate to be a mixture of the singlet
$S$ and neutral component of Inert Doublet $\phi$ belonging to its multiplet
16. We use renormalization group equations (RGEs) for scalar mass parameters
$\mu^{2}_{i}$ and interaction couplings $\lambda_{i}$ below the GUT breaking
scale and study the vacuum stability for those parameters Ferreira:2009jb . We
also impose the GUT boundary conditions as stated in Eq.12. We use
SARAH-4.14.3 package Staub:2013tta  to compute the RGEs. Here, a candidate for
DM and an intermediate scale, lying below the energy scale of $10^{10}$ GeV,
bring forth new particles that extend beyond the SM. The inclusion of new
particles leads to an observation where the introduction of these particles
causes the vacuum to exhibit a positive nature, extending all the way to the
PLANCK scale $M_{Pl}$ as shown by the solid red curve in Figure 2.

Figure 2: Comparison of running of Higgs quartic coupling $\lambda_{h}$ in
SO(10) with SM.

The expressions of renormalization group equations (RGEs) for the model in
Equation 11 have been obtained, and the one-loop beta functions for the scalar
couplings are as follows: Kadastik:2009cu

$\beta_{\lambda_{h}}=24\lambda_{h}^{2}+2\lambda_{3}^{2}+2\lambda_{3}\lambda_{4}+\lambda_{4}^{2}+\lambda_{5}^{2}+\lambda_{SH}^{2}+\frac{3}{8}(3g_{2}^{4}+2g_{2}^{2}g_{1}^{2})+g_{1}^{4}-3\lambda_{1}(3g_{2}^{2}+g_{1}^{2}-4y_{t}^{2})-6y_{t}^{4}$
(38)

$\beta_{\lambda_{2}}=24\lambda_{2}^{2}+2\lambda_{3}^{2}+2\lambda_{3}\lambda_{4}+\lambda_{4}^{2}+\lambda_{5}^{2}+\lambda_{\phi
S}^{2}+\frac{3}{8}(3g_{2}^{4}+2g_{2}^{2}g_{1}^{2})-3\lambda_{2}(3g_{2}^{2}+g_{1}^{2})$
(39)

$\beta_{\lambda_{3}}=4(\lambda_{h}+\lambda_{2})(3\lambda_{3}+\lambda_{4})+4\lambda_{3}^{2}+2\lambda_{4}^{2}+2\lambda_{5}^{2}+2\lambda_{HS}\lambda_{\phi
S}+\frac{3}{4}(3g_{2}^{4}+g_{1}^{4}-2g_{2}^{2}g_{1}^{2})-3\lambda_{3}(3g_{2}^{2}+g_{1}^{2}-4y_{t}^{2})$
(40)

$\beta_{\lambda_{4}}=4(\lambda_{h}+\lambda_{2})\lambda_{4}+8\lambda_{3}\lambda_{4}+4\lambda_{4}^{2}+8\lambda_{5}^{2}+3g_{2}^{2}g_{1}^{2}-3\lambda_{4}(3g_{2}^{2}+g_{1}^{2}-4y_{t}^{2})$
(41)

$\beta_{\lambda_{5}}=4(\lambda_{h}+\lambda_{2}+2\lambda_{3}+3\lambda_{4})\lambda_{5}-3\lambda_{5}(3g_{2}^{2}+g_{1}^{2}-2y_{t}^{2})$
(42)

$\beta_{\lambda_{S}}=20\lambda_{S}^{2}+2\lambda_{S}^{2}+2\lambda_{\phi S}^{2}$
(43)

$\beta_{\lambda_{HS}}=4(3\lambda_{h}+2\lambda_{S}+\lambda_{HS})\lambda_{HS}+(4\lambda_{3}+2\lambda_{4})\lambda_{\phi
S}-\frac{3}{2}\lambda_{SH}(3g_{2}^{2}+g_{1}^{2}-y_{t}^{2})$ (44)

$\beta_{\lambda_{\phi S}}=4(3\lambda_{2}+2\lambda_{S}+\lambda_{\phi
S})\lambda_{\phi
S}+(4\lambda_{3}+2\lambda_{4})\lambda_{HS}-\frac{3}{2}\lambda_{\phi
S}(3g_{2}^{2}+g_{1}^{2})$ (45)

$\beta_{\mu_{1}^{2}}=12\mu_{1}^{2}\lambda_{h}+4\mu_{2}^{2}\lambda_{3}+2\mu_{2}^{2}\lambda_{4}+2\mu_{S}^{2}\lambda_{HS}+\rho^{2}-\frac{3}{2}\mu_{1}^{2}(3g_{2}^{2}+g_{1}^{2}-4y_{t}^{2})$
(46)

$\beta_{\mu_{2}^{2}}=12\mu_{2}^{2}\lambda_{2}+4\mu_{1}^{2}\lambda_{3}+2\mu_{1}^{2}\lambda_{4}+2\mu_{S}^{2}\lambda_{\phi
S}+\rho^{2}-\frac{3}{2}\mu_{\phi S}^{2}(3g_{2}^{2}+g_{1}^{2})$ (47)

$\beta_{\mu_{S}^{2}}=8\mu_{S}^{2}\lambda_{S}+4\mu_{1}^{2}\lambda_{HS}+4\mu_{2}^{2}\lambda_{\phi
S}+2\mu_{S}^{2}\lambda_{\phi S}+\rho^{2}$ (48)

$\beta_{\rho}=2\rho(\lambda_{3}+2\lambda_{4}+\lambda_{HS}+\lambda_{\phi
S})-\frac{3}{2}\rho(3g_{2}^{2}+g_{1}^{2}-2y_{t}^{2})$ (49)

Further we have included the one-loop $\beta$-functions for
$g_{1},g_{2},g_{3}$ and $y_{t}$, given by

$\displaystyle\begin{array}[]{l}\beta_{g_{1}}=7g^{3}_{1}\\\
\beta_{g_{2}}=-3g^{3}_{2}\\\ \beta_{g_{3}}=-7g^{3}_{1}\\\
\beta{y_{t}}=y_{t}(\frac{9}{2}y^{2}_{t}-\frac{17}{12}g^{2}_{1}-\frac{9}{4}g^{2}_{2}-8g^{2}_{3})\end{array}$
(54)

##  V Results and discussions

In this section, we explore how our SO(10) model implications relate to DM and
contribute to the electroweak vacuum. We compare our results with the most
recent limitations on DM relic density (derived from Planck data) and the
spin-independent direct detection cross-section (measured by the XENON1T
experiment). Additionally, we enforce a perturbativity condition
($\lambda_{i}$ $<$ 4$\pi$ across all scales) for all data points. Further, we
enforce the Electroweak Precision Test (EWPT) restriction, as outlined in
Baek:2012uj , which establishes an upper limit on the mixing angles $\theta$
across the entire range of masses. These constraints are uniformly applied to
all parameter points.

As stated earlier, we choose $\chi_{1}$ to serve as the dark matter candidate,
which is a composite of singlet and doublet scalars. Subsequently, we explore
the parameter space that adheres to the necessary constraints on relic
density. We vary the DM mass in the range $5-1000$ GeV and use the software
package micrOMEGA 4.3.2 Belanger:2013oya  to compute both the relic abundance
and the spin-independent cross-section of the DM. The results of our study are
presented in figures (3-6). In figure 3 we have shown the variation of the DM
relic abundance with DM mass. It is seen that there is a significant area of
parameter space that can produce the correct relic density of DM in our model.
Additionally, a distinctive funnel-shaped area emerges around the Higgs
resonance point ($M_{DM}\approx m_{h}/2$), corresponding to the s-channel
annihilation of DM into SM fermions, facilitated by the Higgs boson mediation.

Figure 3: The change in DM relic density $\Omega_{\text{DM}}h^{2}$ with
respect to the mass of the DM is depicted for parameters $\lambda_{i}=0.01$
and $\sin{\theta}=0.2$. The black line corresponds to the existing value of DM
relic density. The region below the black line is the allowed region from
relic density constraint.

Subsequently, we check across viable values of the DM-Higgs coupling
$\lambda_{L}$ for a given mass squared difference $(\bigtriangleup
m=m_{\chi_{1}}^{2}-m_{\chi_{1}}^{2}=50$ GeV), and show the permissible
parameter range in the $\lambda_{L}$ vs $M_{DM}$ plane based on the
requirement of achieving the accurate relic abundance, in Figure 4. The
exclusion lines in black and red correspond to the constraints from the
XENON1TXENON:2017vdw  and LHC limits on Higgs invisible decay, respectively.
The shaded region, lightly coloured in orange, is deemed impermissible due to
the LHC limits ATLAS:2015ciy  with reference to invisible decay and the direct
detection observations from the XENON1T experiment XENON:2018voc . It is
important to note that the latest LHC constraint on the invisible decay width
of the SM Higgs boson is applicable solely for DM masses
$m_{DM}<\frac{m_{h}}{2}$, which implies that the area to the left of the red
line is not allowed. As a result, the parameter space region located to the
right of the red line and situated below the black line successfully adheres
to both LHC and XENON1T limitations. Consequently, it is reasonable to select
smaller values of the coupling $\lambda_{L}\leq 0.1$ in our analysis that
follows.

Figure 4: Points that satisfy the relic density requirements are depicted in
the $\lambda_{L}$ vs $M_{DM}$ plane. The blue points are allowed from correct
relic density constraints. The exclusion lines in black and red correspond to
the constraints from the XENON1T and LHC limits on Higgs invisible decay,
respectively. The area to the right of the red line and below the black curve
are allowed.

Exploring the impact of varying coupling constants adds an intriguing
dimension to our analysis. Thus, in figure 5, we present the correlation
between the spin-independent DM-nucleon cross-section $\sigma_{SI}$ ($cm^{2}$)
and the coupling $\lambda_{L}$ values, which comply with the correct relic
density constraint. We compare these results with the boundaries of the spin-
independent DM-nucleon cross-section as constrained by the latest XENON1T
experiment. As depicted in the spin-independent direct detection cross-section
plot of figure 5, it’s evident that numerous parameter points within the
space, satisfying the correct relic density constraint Planck:2018vyg , also
reside within the permissible range defined by the XENON1T experimental curve
(situated below the black curve). Consequently, we deduce that the DM
candidate with a mass of $M_{DM}\geq 300$ GeV and featuring a modest coupling
$\lambda_{L}$ adheres to both the experimental constraints for the proper
relic density and the DM-nucleon scattering cross-section. Based on the
outcomes illustrated in Figure 5, it becomes evident that our model has the
capability to predict DM candidates that meet both the constraints for relic
density and cross-section. This validation of our model is particularly
notable as it encompasses a wide range of DM masses, in previously unaddressed
(theoretically) intermediate range.

Figure 5: The spin-independent cross-section of DM with nucleons, denoted as
$\sigma_{SI}$ ($cm^{2}$), is presented for the points that satisfy the
accurate relic density constraint. The exclusion curve resulting from the
XENON1T experiment is represented by the black line. The region below the
black curve are allowed from current constraints from XENON1T experiment.

Since the DM in our model consists of a combination of singlet and doublet
scalars, it is important to recognize that aside from the constraints arising
from direct detection experiments, the parameter space for DM can also be
explored through various indirect detection experiments. We place restrictions
on the DM parameters based on the indirect detection bounds derived from a
comprehensive analysis of observations from Fermi-LAT of dwarf spheroidal
satellite galaxies (dSphs) Fermi-LAT:2015att . Figure 6 portrays the cross
section for DM annihilation into final states $\tau^{+}\tau^{-}$ and
$W^{+}W^{-}$, and the outcomes are compared with the most recent indirect
detection constraints from Fermi-LAT Fermi-LAT:2015att . The regions below the
red line are allowed whereas the shaded region above it is disallowed from
recent indirect detection constraints from Fermi-LAT.

(a) DM annihilations into $\tau^{+}\tau^{-}$

(b) DM annihilations into $W^{+}W^{-}$

Figure 6: DM annihilations into $\tau^{+}\tau^{-}$(left), $W^{+}W^{-}$ (right)
compared against the latest indirect detection bounds of Fermi-LAT. The
regions below the red line are allowed.

In Fig. 6, the permitted region lie below the red curve. It’s noticeable that
the DM mass range ($M_{DM}\geq 300$ GeV) could actually yield the accurate
relic abundance within our model and is also compliant with the bounds from DM
annihilation. In regions of lower mass, the primary mechanism for DM
annihilation is the s-channel process through Higgs mediation into SM
fermions. We have identified that the key processes contributing to the relic
abundance of DM include: $\chi_{1}\chi_{1}$ $\rightarrow$
$\tau^{+}\tau^{-},W^{+}W^{-},hh,ZZ,b\bar{b}$. These annihilation channels play
a significant role in generating the correct relic abundance for DM within
this intermediate-mass range of the DM candidate. Consequently, the DM
candidate $\chi_{1}$ in our model effectively satisfies both the constraints
for relic abundance and direct detection cross-section within the mass range
of $M_{DM}\geq 300$ GeV.

Hence, from the above analysis, it becomes evident that a potential DM
candidate within the mass range of $M_{DM}\geq 300$ GeV, introduces new
particles that go beyond the SM. The incorporation of these new particles
results in an interesting observation - their introduction leads to a positive
vacuum behavior, extending even to the PLANCK scale $M_{Pl}$. These additional
particles possess the capacity to induce modifications in the behavior of the
quartic coupling $\lambda_{h}$. More specifically, they may contribute in a
manner that guides $\lambda_{h}$ along a trajectory that maintains its
positivity across the entire energy spectrum, extending up to the PLANCK scale
$M_{Pl}$. This outcome represents a notable departure from the previous
scenario and emphasises the captivating role these newly introduced particles
play in upholding the theory’s stability even at exceptionally high energy
levels. This novel insight stands as a distinctive and significant
contribution to this study.

##  VI Conclusions

Of the many shortcomings of the SM, we revisited SM electroweak vacuum
instability along with addressing the nature of intermediate mass range DM, in
SO(10) GUT in this work. At an energy scale of around $10^{10}$ GeV, the SM
suffers from a vacuum metastability Degrassi:2014hoa ; Rojas:2015yzm . The
addition of extra scalars could provide a way to stablize the vacuum up to the
Planck scale. In this regard, we delved into an exploration of a viable range
of dark matter (DM) masses within a non-supersymmetric SO(10) Grand Unified
Theory (GUT) scalar dark matter model. This framework encompasses a scalar
singlet denoted as $S$, in conjunction with an inert doublet represented by
$\phi$. Notably, these components are odd under discrete $Z_{2}$ matter parity
$(-1)^{3(B-L)}$. The dark matter arises from the combination of the
$Z_{2}$-odd scalar singlet $S$ and the neutral component of the doublet
$\phi$, both belonging to a new scalar 16 representation inherent to SO(10).

We obtained regions in the parameter space which adheres to the observed DM
relic density. Our investigation extends to the assessment of one-loop vacuum
stability, achieved through the solution of Renormalization Group Equations
(RGEs) governing the model’s parameters. Subsequently, we rigorously evaluated
the model’s predictions, consistent with the boundaries of contemporary
theoretical and empirical constraints. We successfully find a parameter range
that concurs with considerations such as relic density, direct search bounds
derived from the latest XENON1T findings, and the persistent stability of the
electroweak vacuum, extending its validity up to the Planck scale. The novelty
of the study is that it highlights a potential dark matter candidate which is
a mixture of scalar singlet $S$ and inert doublet $\phi$ within a new
intermediate mass range of 300$\leq M_{DM}\leq$1000 GeV. This region is shown
to be consistent with the bounds from current direct detection XENON1T
experiment XENON:2018voc  as well as indirect detection bounds of Fermi-LAT
Fermi-LAT:2015att . Additionally we have shown that unlike the SM case, the
introduction of new particles intriguingly yields a positive vacuum behavior,
extending all the way to the PLANCK scale $M_{Pl}$. These additional particles
possess the capacity to induce alterations in the behavior of the quartic
coupling $\lambda_{h}$. Specifically, they can contribute in a way that steers
$\lambda_{h}$ along a trajectory ensuring its positivity across the entire
energy spectrum, reaching up to the PLANCK scale $M_{Pl}$.

Thus, the prediction of mixed scalar DM in the intermediate mass range and
upholding the vacuum stability at high energy from within the same SO(10) GUT
group constitutes a distinctive and substantial contribution of this study.
The predictions stemming from this model hold promise for potential validation
in upcoming highly sensitive experimental searches focused on DM such as
XENONnT XENON:2024wpa . Moreover, these predictions align smoothly with the
beauty found in grand unified theories.

##  VII Acknowledgment

We acknowledge the RUSA and FIST grants of Govt. of India for support in
upgrading computer laboratory of the Physics Department of Gauhati University,
where this work was completed.

##  VIII References

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