#  Pseudo-Goldstone Dark Matter in $SO(10)$

Nobuchika Okada111<EMAIL_ADDRESS>Digesh Raut222<EMAIL_ADDRESS>Qaisar
Shafi333<EMAIL_ADDRESS>and Anil Thapa444<EMAIL_ADDRESS>aDepartment of
Physics and Astronomy,University of Alabama, Tuscaloosa, Alabama 35487, USA
b,c Bartol Research Institute, Department of Physics and Astronomy,University
of Delaware, Newark DE 19716, USA  d Department of Physics, Oklahoma State
University, Stillwater, OK 74078, USA

###### Abstract

We propose a pseudo-Goldstone boson dark matter (pGDM) particle in $SO(10)$
grand unified theory (GUT). Due to its Goldstone nature, this pGDM evades the
direct DM detection experiments which, otherwise, severely constrain the
parameter space of DM models. In $SO(10)$, the pGDM is embedded as a linear
combination of the Standard Model (SM) singlet scalars in ${\bf 16_{H}}$ and
${\bf\overline{126}_{H}}$ representations. We consider two scenarios for the
intermediate route of $SO(10)$ symmetry breaking (SB) to the SM: $SU(5)\times
U(1)_{X}$ and Pati-Salam the $SU(4)_{c}\times SU(2)_{L}\times SU(2)_{R}$
(4-2-2) gauge groups. The vacuum expectation value of
${\bf\overline{126}_{H}}$, which triggers the breaking of $U(1)_{X}$ and 4-2-2
symmetry in the two scenarios, respectively, determines the pGDM lifetime
whose astrophysical lower bound provides one of the most stringent
constraints. For the 4-2-2 route to $SO(10)$, the successful SM gauge coupling
unification requires the 4-2-2 breaking scale to be ${\cal O}(10^{11})$ GeV,
and most of the parameter space is excluded. For the $SU(5)\times U(1)_{X}$
route, on the other hand, the $U(1)_{X}$ breaking scale can be significantly
higher, and a wide range of the parameter space is allowed. Furthermore, the
proton lifetime in the $SU(5)$ case is predicted to be $4.53\times 10^{34}$
years, which lies well within the sensitivity reach of the Hyper-Kamiokande
experiment. We also examine the constraints on the model parameter space from
the Large Hadron Collider and the indirect DM search by Fermi-LAT and MAGIC
experiments.

##  I Introduction

Astrophysical and cosmological observations [1] have provided compelling
evidence for the existence of dark matter (DM), presumably a non-baryonic
particle which constitutes about 85% of the observed matter energy density in
the universe [2]. A (quasi-)stable, electrically neutral weakly interacting
massive particle (WIMP), absent in the Standard Model (SM), is arguably one of
the most attractive DM candidates. The interaction cross-section of WIMPs with
the SM particles is typically subject to stringent bounds from the direct DM
detection experiments for a wide range of the DM mass.

In Ref. [3] the authors have proposed a SM Higgs-portal DM scenario based on a
softly broken global $U(1)$ symmetry, which is called “pseudo-Goldstone DM
(pGDM)”. The model includes a SM singlet complex scalar field $S$ with a
$U(1)$ global charge, whose vacuum expectation value (VEV) spontaneously
breaks the $U(1)$ symmetry, and a soft $U(1)$ symmetry breaking term,

$\displaystyle\mu_{S}^{2}\left(S^{2}+(S^{\dagger})^{2}\right),$ (1)

where $\mu_{S}$ is a real mass parameter. The resulting scalar potential is
invariant under $S\to S^{\dagger}$, so that the model has a ${\bf Z}_{2}$
symmetry under which only the imaginary component of $S$ (called $\chi$) has
an odd-parity, while the real component of $S$ and all the SM fields have
even-parity. Hence, the pseudo-Goldstone scalar $\chi$, which is the Goldstone
boson associated with the spontaneous $U(1)$ symmetry breaking in the limit of
$\mu_{s}\to 0$, is a stable DM candidate. Thanks to its Goldstone nature, the
coupling of the pGDM $\chi$ with the SM Higgs boson is proportional to its
momentum. As a result, the scattering cross-section of the DM particle with a
nucleon through the Higgs boson exchange vanishes in the non-relativistic
limit, thus evading the direct DM detection constraints [3].

An ultraviolet (UV) completion of this pGDM scenario was proposed in Refs. [4,
5]. The model is based on the gauged $B-L$ (Baryon number minus Lepton number)
extended SM [6, 7, 8, 9, 10, 11, 12, 13], in which the accidental global $B-L$
symmetry of the SM is gauged. In this model, the soft breaking term of Eq.
(1), which is crucial to realize the pGDM scenario, is obtained from

$\displaystyle\Lambda\left(\Phi_{A}\Phi_{B}^{2}+{\rm h.c}\right),$ (2)

where $\Phi_{A,B}$ are complex scalars with $B-L$ charges $Q_{A}$ and
$Q_{B}=-Q_{A}/2$, and $\Lambda$ is a mass parameter. The fields $\Phi_{A,B}$
both develop VEVs and break the $B-L$ symmetry. The pGDM is realized as a
linear combination of the imaginary components of $\Phi_{A,B}$. If the
imaginary component of $\Phi_{B}$ dominates, we approximately reproduce Eq.
(1) by replacing $\Phi_{A}\to\langle\Phi_{A}\rangle$ and identifying
$\Lambda\langle\Phi_{A}\rangle$ and $\Phi_{B}$ with $\mu_{S}^{2}$ and $S$,
respectively. This is a key idea for the UV completion of the pGDM scenario.
It is important to note, however, that the $B-L$ gauge interaction explicitly
breaks the ${\bf Z}_{2}$ symmetry responsible for the stability of the pGDM,
and hence the pGDM can decay via the $B-L$ gauge interaction. To satisfy the
lifetime bound for a decaying DM, $\tau_{DM}>10^{26}$ sec, from cosmic ray
observations [14], the $B-L$ symmetry breaking scale is required to be
$v_{BL}\gtrsim 10^{11}$ GeV [5]. In addition, since the Goldstone nature of
the DM particle is lost due to the cubic scalar coupling of Eq. (2), the
direct DM detection cross section does not exactly vanish [5].

In this paper we propose a UV completion of the pGDM scenario in an $SO(10)$
grand unified theory (GUT), or in other words, an embedding of the gauge
extension of pGDM in $SO(10)$. The $SO(10)$ grand unification is an
interesting paradigm for the unification of strong, weak, and electromagnetic
forces into a single force at a high energy scale. Each family of the SM
quarks and leptons belongs in a single irreducible ${\bf 16}$ representation
of $SO(10)$ along with a SM singlet fermion. Electric charge quantization is
incorporated, and the SM singlet fermion in the ${\bf 16}$ representation is
identified with the right-handed neutrino, which paves a natural way for tiny
neutrino masses generation by the seesaw mechanism [15, 16, 17, 18, 19]. We
will show in the following how the pGDM scenario can be successfully
implemented in the $SO(10)$ framework.

The basic idea is as follows. We introduce a scalar field in the ${\bf 16}$
representation (${\bf 16_{H}}$)111 While the ${\bf 16_{H}}$ must acquire a VEV
in the pGDM scenario, various $SO(10)$ DM models for a ${\bf 16_{H}}$ scalar
with a vanishing VEV are discussed in [20, 21, 22].. Along with a Higgs field
in the ${\bf 126}$ representation (${\bf\overline{126}_{H}}$), whose VEV
generates Majorana mass terms for the right-handed neutrinos, consider the
following $SO(10)$ invariant term in the scalar potential:

$\displaystyle\Lambda{\bf\overline{126}_{H}}({\bf 16_{H}})^{2}+{\rm h.c}.$ (3)

Comparing this to Eq. (2), one can identify $\Phi_{A}$ and $\Phi_{B}$ with the
SM singlet scalars contained in ${\bf\overline{126}_{H}}$ and ${\bf 16_{H}}$,
respectively. In the following, we will consider two symmetry breaking
patterns commonly used to obtain the SM from $SO(10)$ via one intermediate
step, namely, $SU(4)_{c}\times SU(2)_{L}\times SU(2)_{R}$ and $SU(5)\times
U(1)_{X}$. The high VEV scale of ${\bf\overline{126}_{H}}$, which is necessary
for the phenomenological viability of the (gauged) pGDM scenario, can be
naturally realized in SO(10) unification. We show a correlation between pGDM
physics and GUT phenomenology, such as a successful gauge coupling unification
and observable proton decay. We carry out detailed phenomenological studies
for the two symmetry breaking patters, taking into account the constraints
from the Large Hadron Collider and the indirect DM detection experiments, in
order to identify the allowed parameter space.

##  II Phenomenology of PGDM

Although our goal is to realize the pGDM scenario in the context of $SO(10)$,
the $B-L$ gauge extended model of the pGDM scenario contains all the essential
features. For the pGDM phenomenology, it is sufficient to consider the
following scalar potential involving $\Phi_{A,B}$ and the SM Higgs doublet
($H$):

$\displaystyle V_{{\rm eff}}$ $\displaystyle=-\mu_{H}^{2}\
H^{\dagger}H-\mu_{A}^{2}\ \Phi_{A}^{\dagger}\Phi_{A}-\mu_{B}^{2}\
\Phi_{B}^{\dagger}\Phi_{B}+\lambda_{H}(H^{\dagger}H)^{2}+\lambda_{HA}(H^{\dagger}H)(\Phi_{A}^{\dagger}\Phi_{A})$
$\displaystyle+\lambda_{HB}(H^{\dagger}H)(\Phi_{B}^{\dagger}\Phi_{B})+\lambda_{AB}(\Phi_{A}^{\dagger}\Phi_{A})(\Phi_{B}^{\dagger}\Phi_{B})+\lambda_{A}(\Phi_{A}^{\dagger}\Phi_{A})^{2}+\lambda_{B}(\Phi_{B}^{\dagger}\Phi_{B})^{2}$
$\displaystyle-\\{\Lambda\ \Phi_{A}\Phi_{B}^{2}+{\rm H.c.}\\}.$ (4)

We represent the Higgs fields as follows:

$H=\begin{pmatrix}G^{+}\\\
\frac{(h+v_{H}+iG^{0})}{\sqrt{2}}\end{pmatrix}\,,\hskip
28.45274pt\Phi_{A,B}=\frac{1}{\sqrt{2}}(\phi_{A,B}+v_{A,B}+i\chi_{A,B}),$ (5)

where $v_{H,A,B}$ are the VEVs of $H$, $\Phi_{A}$, and $\Phi_{B}$ fields,
resepctively, with $v_{H}=246$ GeV. Requiring the scalar potential to exhibit
a minimum around its VEV location leads to the following relations:

$\displaystyle\mu_{H}^{2}$ $\displaystyle=\frac{\lambda_{HA}\
v_{A}^{2}}{2}+\frac{\lambda_{HB}\ v_{B}^{2}}{2}+\lambda_{H}\ v_{H}^{2}\,,$
$\displaystyle\mu_{A}^{2}$
$\displaystyle=\lambda_{A}v_{A}^{2}+\frac{\lambda_{HA}\
v_{H}^{2}}{2}+\frac{\lambda_{AB}\
v_{B}^{2}}{2}-\frac{v_{B}^{2}\Lambda}{\sqrt{2}\ v_{A}}\,,$
$\displaystyle\mu_{B}^{2}$
$\displaystyle=\lambda_{B}v_{B}^{2}+\frac{\lambda_{AB}\
v_{A}^{2}}{2}+\frac{\lambda_{HB}\ v_{H}^{2}}{2}-\sqrt{2}\ v_{A}\Lambda\,.$ (6)

After spontaneous symmetry breaking the Goldstone modes $G^{\pm,0}$, are
absorbed by the SM gauge bosons $W^{\pm}$ and $Z$. The remaining mixing mass
matrices are expressed as

$\displaystyle
V\supset\frac{1}{2}\begin{bmatrix}\chi_{A}&\chi_{B}\end{bmatrix}.{\cal
M}_{{\rm I}}^{2}.\begin{bmatrix}\chi_{A}\\\
\chi_{B}\end{bmatrix}+\frac{1}{2}\begin{bmatrix}h&\phi_{B}&\phi_{A}\end{bmatrix}.{\cal
M}_{{\rm R}}^{2}.\begin{bmatrix}h\\\ \phi_{B}\\\ \phi_{A}\end{bmatrix},$ (7)

where the mass matrices for the $CP$-odd scalars ($\chi_{A},\chi_{B}$) and
$CP$-even scalars ($\phi_{A},\phi_{B}$, $h$) are given by

$M^{2}_{{\rm
I}}=\sqrt{2}\Lambda\begin{pmatrix}\frac{v_{B}^{2}}{2v_{A}}&v_{B}\\\
v_{B}&2v_{A}\\\ \end{pmatrix},M_{{\rm
R}}^{2}=\begin{pmatrix}2\lambda_{H}v_{H}^{2}&\lambda_{HB}v_{B}v_{H}&\lambda_{HA}v_{A}v_{H}\\\
\lambda_{HB}v_{B}v_{H}&2\lambda_{B}v_{B}^{2}&v_{B}(\lambda_{AB}v_{A}-\sqrt{2}\Lambda)\\\
\lambda_{HA}v_{A}v_{H}&v_{B}(\lambda_{AB}v_{A}-\sqrt{2}\Lambda)&2\lambda_{A}v_{A}^{2}+\frac{v_{B}^{2}\Lambda}{\sqrt{2}v_{A}}\end{pmatrix}$
(8)

In the $CP$-odd sector, one combination of $\chi_{A}$ and $\chi_{B}$ is the
would-be Nambu-Goldstone mode ($\tilde{G}$) absorbed by the $B-L$ gauge boson
and the orthogonal combination is the pGDM $\chi$:

${\tilde{G}}=-\cos\theta\;\chi_{A}+\sin\theta\;\chi_{B},\hskip
28.45274pt\chi=\sin\theta\;\chi_{A}+\cos\theta\;\chi_{B},$ (9)

where $\cos\theta=2v_{A}/\sqrt{4v_{A}^{2}+v_{B}^{2}}$,
$\sin\theta=v_{B}/\sqrt{4v_{A}^{2}+v_{B}^{2}}$, and the mass of DM $\chi$ is
given by

$m_{\chi}^{2}=2\sqrt{2}\Lambda v_{A}\left(1+\tan^{2}\theta\right).$ (10)

For the $CP$-even sector, we consider the limiting case for the parameters in
the mass matrix, namely, $v_{A}\gg v_{B},v_{H}$,
$\lambda_{A}v_{A}^{2}\gg\Lambda v_{B}$, and $\lambda_{HA},\lambda_{AB}\to 0$.
In this limit, $\phi_{A}$ decouples from the low energy effective theory, and
we only consider the sub-matrix for $h$ and $\phi_{B}$. This sub-matrix is
diagonalized for the mass eigenstate, $\tilde{h}$ and $\tilde{\phi}_{B}$,
defined as

$\displaystyle\tilde{\phi}_{B}$ $\displaystyle=\cos\theta_{H}\
\phi_{B}+\sin\theta_{H}\ h\,,$ (11) $\displaystyle\tilde{h}$
$\displaystyle=-\sin\theta_{H}\ \phi_{B}+\cos\theta_{H}\ h\,.$ (12)

Their mass eigenvalues are given by

$m_{\tilde{B},\tilde{h}}^{2}=\frac{1}{2}\bigg{(}m_{B}^{2}+m_{h}^{2}\pm\frac{m_{B}^{2}-m_{h}^{2}}{\cos
2\theta_{H}}\bigg{)}\,,$ (13)

where $m_{B}^{2}=2\lambda_{B}v_{B}^{2}$, $m_{h}^{2}=2\lambda_{H}v_{H}^{2}$,
and the mixing angle $\theta_{H}$ is given by

$\tan 2\theta_{H}=\frac{2v_{B}v_{H}\lambda_{HB}}{m_{B}^{2}-m_{h}^{2}}\,.$ (14)

###  II.1 Direct pGDM Detection Amplitude

Let us calculate the direct DM detection amplitude for the elastic scattering
of DM with a nucleon. The scattering occurs at very low energies, so zero
momentum transfer limit, $t\to 0$222For $t\neq 0$ the maximum value of the DM
direct detection cross section is approximately $\theta_{H}^{2}t_{\rm
max}^{2}\times\sigma_{HP}$ [4], where $\sigma_{HP}$ is the Higgs-portal DM
cross section, $\theta_{H}\simeq 0.1$, $|t_{\rm max}|=m_{DM}^{2}v^{2}$ and
$v\simeq 10^{-3}$ is the DM velocity. Hence, realistically we can expect the
DM direct detection cross section to be closer to the neutrino floor. See, for
example, [23]., is a good approximation to evaluate the scattering amplitude.
In the limiting case with $v_{A}\gg v_{B}$, the pGDM $\chi\equiv\chi_{B}$ to
leading order in $\theta$, such that the amplitude in the interaction basis
$(h,\phi_{B},\phi_{A})$ is expressed as

$\displaystyle\mathcal{M}$ $\displaystyle\propto$
$\displaystyle\begin{pmatrix}\frac{1}{2}\lambda_{HB}v_{H}\hskip
8.53581pt&\lambda_{B}v_{B}\hskip
8.53581pt&\frac{1}{2}\lambda_{AB}v_{A}+\frac{1}{\sqrt{2}}\Lambda\end{pmatrix}\,(M_{R}^{2})^{-1}\,\begin{pmatrix}Y_{hf\overline{f}}\\\
0\\\ 0\end{pmatrix}+{\cal O}(\theta^{2})$ (15) $\displaystyle\simeq$
$\displaystyle-\frac{\Lambda\lambda_{HB}}{\sqrt{2}\lambda_{AB}\lambda_{H}v_{A}v_{H}}+{\cal
O}\bigg{(}\frac{v_{B}^{2}}{4v_{A}^{2}}\bigg{)},$

where $Y_{hf\overline{f}}$ is the interaction of $h$ with the SM fermions.
Using Eq. (10), we obtain

$\displaystyle|{\cal M}|^{2}$ $\displaystyle\propto$
$\displaystyle\frac{m_{\chi}^{4}}{v_{A}^{4}v_{H}^{2}}.$ (16)

It is clear that the direct DM detection amplitude is negligibly small.

##  III DM relic density and Indirect Detection

The pGDM can interact with the SM particles through the
$\tilde{h}/\tilde{\phi}_{B}$ portal interaction as follows:

$\mathcal{L}\supset\frac{\lambda_{\tilde{h}}}{2}\
\tilde{h}\chi\chi+\frac{\lambda_{\tilde{B}}}{2}\ \tilde{\phi}_{B}\chi\chi\,,$
(17)

with

$\displaystyle\lambda_{\tilde{h}}=-\frac{m_{\tilde{h}}^{2}}{v_{B}}\
\sin\theta_{H}\,,\hskip
56.9055pt\lambda_{\tilde{B}}=\frac{m_{\tilde{B}}^{2}}{v_{B}}\
\cos\theta_{H}\,.$ (18)

The couplings of $\tilde{h}$/$\tilde{\phi}_{B}$ to the SM fermions are as
follows:

$\mathcal{L}\supset-\
(\tilde{h}\cos\theta_{H}+\tilde{\phi}_{B}\sin\theta_{H})\sum_{f}\frac{m_{f}}{v_{H}}\bar{f}f.$
(19)

Figure 1: The allowed region in the pGDM mass $m_{\chi}$ and VEV ratio
$v_{H}/v_{B}$ plane for three different benchmark values $m_{\tilde{B}}/{\rm
GeV}=80,200,500$, and $\sin\theta_{H}=0.1$ for the mixing between $h$ and
$\phi_{B}$. The thermal relic abundance of $0.094\leq\Omega h^{2}\leq 0.128$
[2] is satisfied along the blue curves. The green shaded regions are excluded
by the Higgs invisible decay constraint [24]. The brown and gray shaded
segments along the each abundance curve is excluded after imposing Fermi-LAT +
MAGIC bound obtained from annihilation of pGDM particles to $b\bar{b}$ and
$W^{+}W^{-}$ (see Fig. 2). For the $SU(5)\times U(1)_{X}$ case, the parameter
region depicted by the orange and the purple band respectively are excluded by
the pGDM lifetime bound, $\tau_{DM}\gtrsim 10^{26}$ sec [14], and
perturbativity bound, namely, for $\lambda_{B}$ to remain perturbative up to
the $SO(10)$ unification scale. Similarly, the orange and purple dashed line
respectively depict the boundary of the excluded region for the 4-2-2 case.

We numerically evaluate the relic abundance of the DM using the software
MicrOmegas [25]. The model is implemented in CalcHEP [26] by using LanHEP
[27]. The DM relic abundance depends on the pGDM pair annihilation cross
section, which is determined by 4 free parameters, $m_{\chi}$, $\theta_{H}$,
$v_{B}$ and $m_{\tilde{B}}$. In the following analysis, we fix
$\sin\theta_{H}=0.1$ as our benchmark and impose the criterion that the DM
relic abundance $\Omega h^{2}=0.120\pm 0.001$ [2] measured by PLANCK
collaboration at 3$\sigma$ confidence level is satisfied.

A pair of DM particles annihilating into the SM particles which subsequently
decay to produce gamma-rays can be searched by indirect DM detection
experiments such as Fermi-LAT [28] and MAGIC [29]. These experiments have set
an upper bound on the cross section of DM pair annihilation into $b\bar{b}$
and $W^{+}W^{-}$ final states. If $m_{\chi}<m_{\tilde{h}}/2$, the SM(-like)
Higgs boson can decay into a pair of pGDM particles. The CMS result provides
an upper bound on the branching ratio of the invisible Higgs boson decay,
$BR({\tilde{h}}\to\chi\chi)\leq 0.16$ [24]. As previously mentioned, the pGDM
gauge interaction explicitly violates the ${\bf Z}_{2}$ symmetry, which would
otherwise forbid the decay of the pGDM. Hence we also impose a lifetime bound
on pGDM, $\tau_{DM}>10^{26}$ sec [14], to further constrain the parameter
space. In the following, we show the results of our analysis, including the
lifetime constraints for both 4-2-2 and $SU(5)\times U(1)_{X}$ scenarios,
leaving out the details of the pGDM decay, which will be presented in the
later sections.

Our overall conclusions is that the allowed region for the 4-2-2 scenario is
significantly smaller than for the $SU(5)\times U(1)_{X}$ case. This is
because the VEV of ${\bf\overline{126}_{H}}$ ($v_{A}$) in the 4-2-2 case
required by the successful SM gauge coupling unification is at the
intermediate scale, $10^{11}$ GeV, which is much smaller than the VEV scale
required in the $SU(5)\times U(1)_{X}$ case. In particular, for the later
case, the pGDM can be as heavy as $m_{\chi}/{\rm GeV}=690,715,860$ for
$m_{\tilde{B}}/{\rm GeV}=80,200,500$, respectively. In contrast, only the
region close to the resonance points survive for the 4-2-2 case with the pGDM
mass $m_{\chi}\gtrsim{\cal O}(10^{2})$ GeV excluded.

In Fig. 1, we show the allowed region consistent with observed relic abundance
as a function of the DM mass $m_{\chi}$ and the ratio of the VEVs
$v_{H}/v_{B}$ for three different choices of $m_{\tilde{B}}=80,200,500$ GeV.
Here, the mixing angle between $h$ and $\phi_{B}$ is taken to be
$\sin\theta_{H}=0.1$. The thermal relic abundance $0.094\leq\Omega h^{2}\leq
0.128$ [2] is satisfied along the blue curve. The brown and gray shaded line
segments on top of the blue curve are excluded by imposing the Fermi-LAT +
MAGIC bound [30] on the DM pair annihilation cross section into $b\bar{b}$ and
$W^{+}W^{-}$ (see Fig. 2). The green region is excluded by the constraint on
the Higgs invisible decays, $BR(\tilde{h}\to\chi\chi)<0.16$, obtained from the
CMS result [24]. The orange band depicts the parameter region excluded in the
$SU(5)\times U(1)_{X}$ case, whereas the region to the right of the dashed
orange line is excluded for the 4-2-2 case in order to satisfy the pGDM
lifetime bound from cosmic ray observations, $\tau_{DM}\gtrsim 10^{26}$ sec
[14].

The value of $\lambda_{B}$ is determined at low energy for fixed values of
$m_{\tilde{B}}$, $v_{B}$ and $\theta_{H}$ in Eq. (13). Using this value, we
have evaluated the renormalization group running of $\lambda_{B}$ up to the
$SO(10)$ unification scale ($v_{GUT}$). Requiring $\lambda_{B}$ to remain
perturbative at $v_{GUT}$, we obtain an upper bound on $v_{H}/v_{B}$ (see Sec.
V and VI for details). For the $SU(5)\times U(1)_{X}$ case, the excluded
region is depicted by the purple band, whereas the region above the dashed
purple line is excluded for the 4-2-2 case.

In Fig. 2, we show the DM annihilation cross section to $b\bar{b}$ (left) and
$W^{+}W^{-}$ (right) for various values of DM mass along with the upper bounds
on the cross section from Fermi-LAT + MAGIC experiments [30]. The pGDM pair
annihilation cross sections are evaluated by using $v_{B}$ and $m_{DM}$ values
that satisfy the relic abundance constraint in Fig. 1. The green, red, and
blue curves show the results for $m_{\tilde{B}}/{\rm GeV}=80,300$, and $500$,
respectively. The purple band represents the exclusion from Fermi-LAT + MAGIC
experiments [30].

Figure 2: The left and right panels respectively show the DM annihilation
cross section to $b\bar{b}$ and $W^{+}W^{-}$ for various values of the pGDM
mass along with the exclusion (purple band) from Fermi-LAT + MAGIC experiments
[30] for various benchmark values of $m_{\tilde{B}}$.

##  IV $SO(10)$ Breaking via $SU(5)\times U(1)_{X}$

A simple way to embed the pGDM scenario in $SO(10)$ GUT is based on the
generalization of the minimal ${B-L}$ scenario to the so-called gauged
$U(1)_{X}$ extended SM [31]. The generalized $U(1)_{X}$ charges ($Q_{X}$) of
each particles are defined to be a linear combination of its $B-L$ charge
($Q_{BL}$) and hypercharge ($Q_{Y}$), $Q_{X}=x_{H}Q_{Y}+Q_{BL}$, where $x_{H}$
is a free parameter. The interesting feature of the $U(1)_{X}$ extended SM is
that for $x_{H}=-4/5$, the SM quarks and lepton are unified in ${\bf 10}$ and
${\bf{\overline{5}}}$ representations of $SU(5)$ with $U(1)_{X}$ charges
$+1/5$ and $-3/5$, respectively [32]. In the $SO(10)$ embedding, the
$U(1)_{X}$ charge is normalized to be $Q_{X}\to\sqrt{5/8}\;Q_{X}$[33].

The Higgs fields involved in breaking $SO(10)$ down to the SM along with their
decomposition under $SU(5)\times U(1)_{X}$ are listed below:

$\displaystyle{\bf 45_{H}}$ $\displaystyle={\bf 1_{H}}_{0}\oplus{\bf
24_{H}}_{0}{\color[rgb]{0.6,0.6,0.6}\oplus{\bf
10_{H}}_{-\sqrt{2/5}}\oplus{\bf\overline{10_{H}}}_{\sqrt{2/5}}}\,,$
$\displaystyle{\bf{126}_{H}}$ $\displaystyle={\bf
1_{H}}_{\sqrt{5/2}}{\color[rgb]{0.6,0.6,0.6}\oplus{\bf
5_{H}}_{\sqrt{1/10}}\oplus{\bf\overline{10_{H}}}_{\sqrt{3/10}}\oplus{\bf
15_{H}}_{-\sqrt{3/10}}\oplus{\bf\overline{45_{H}}}_{-\sqrt{1/10}}\oplus{\bf
50_{H}}_{\sqrt{1/10}}}\,,$ $\displaystyle{\bf 16_{H}}$ $\displaystyle={\bf
1_{H}}_{\sqrt{5/8}}{\color[rgb]{0.6,0.6,0.6}\oplus{\bf\overline{5_{H}}}_{-3/{\sqrt{4}0}}\oplus{\bf
10_{H}}_{1/{\sqrt{4}0}}}\,,$ $\displaystyle{\bf 10_{H}}$ $\displaystyle={\bf
5_{H}}_{-1/{\sqrt{1}0}}{\color[rgb]{0.6,0.6,0.6}\oplus{\bf\overline{5}}_{1/{\sqrt{1}0}}}\,.$
(20)

Here only the $SU(5)\times U(1)_{X}$ multiplets depicted in bold develop
nonzero VEVs to trigger the following symmetry breaking chain:

$\displaystyle SO(10)\ \xrightarrow[v_{GUT}]{\langle{\bf 45_{H}}\rangle}\
SM\times
U(1)_{X}\xrightarrow[v_{A},v_{B}]{\langle{\bf\overline{126}_{H}}\rangle,\langle{\bf
16_{H}}\rangle}\ SM\xrightarrow[v_{EW}]{\langle{\bf
10_{H}}\rangle}SU(3)_{c}\times U(1)_{EM},$ (21)

Here, $v_{GUT}$, $v_{A,B}$, and $v_{EW}$ denote the VEVs of the corresponding
scalars fields involved in the various stages of the symmetry breaking.

The Higgs potential relevant for the discussion of pGDM at low energies is
expressed as

$\displaystyle V$ $\displaystyle\supset-\mu_{1}^{2}({\bf
10_{H}^{\dagger}10_{H}})-\mu_{2}^{2}({\bf\overline{126}_{H}^{\dagger}\overline{126}_{H}})-\mu_{3}^{2}({\bf
16_{H}^{\dagger}16_{H}})+\lambda_{1}({\bf
10_{H}^{\dagger}10_{H}})^{2}+\lambda_{2}({\bf 16_{H}^{\dagger}16_{H}})^{2}$
$\displaystyle+\lambda_{3}({\bf\overline{126}_{H}^{\dagger}\overline{126}_{H}})^{2}+\lambda_{4}({\bf
10_{H}^{\dagger}10_{H}})({\bf\overline{126}_{H}^{\dagger}\overline{126}_{H}})+\lambda_{5}\
({\bf\overline{126}_{H}^{\dagger}\overline{126}_{H}})({\bf
16_{H}^{\dagger}16_{H}})$ $\displaystyle+\lambda_{6}\ ({\bf
10_{H}^{\dagger}10_{H}})({\bf
16_{H}^{\dagger}16_{H}})-\big{(}{\Lambda_{1}{\bf\overline{126}_{H}}({\bf
16_{H}})^{2}+{\rm h.c}}\big{)}.$ (22)

Here we have excluded the $SO(10)$ invariant terms ${\bf 10_{H}(16_{H})}^{2}$
and ${\bf 10_{H}^{\dagger}(16_{H})}^{2}$, since they explicitly violate the
Goldstone nature [3] by allowing a direct coupling between pGDM and SM Higgs
$H\supset{\bf 10_{H}}$. The SM singlet components in ${\bf\overline{126}_{H}}$
and ${\bf 16_{H}}$ are identified to be $\Phi_{A}$ and $\Phi_{B}$, and the SM
Higgs doublet is contained in the ${\bf 5}$-plet of ${\bf 10_{H}}$. With a
suitable choice of $\mu_{2,3}$ to balance the contribution of other scalars to
the ${\bf 16_{H}}$ and ${\bf\overline{126}_{H}}$ masses, the $\Phi_{B}$ and
$\Phi_{A}$ masses can be arranged to be at the TeV scale and GUT scale,
respectively. The electroweak scale mass for the SM Higgs $H$ can be generated
by the well known triplet-doublet splitting mechanism using the ${\bf 10_{H}}$
coupling with the $SU(5)$ adjoint ${\bf 24_{H}}\supset{\bf 45_{H}}$. Hence the
low energy effective scalar potential matches the pGDM potential in Eq. (4).

It is known that the unification of the three SM gauge couplings can be
achieved by introducing new vector-like quark/lepton pairs at certain mass
scales [34, 35, 36, 37, 38, 39, 32, 40, 41]. Let us introduce a new vector-
like fermion pair in ${\bf 16}+{\bf\overline{16}}$ representation under
$SO(10)$ and consider the following Lagrangian [33]:

$\displaystyle{\cal L}$ $\displaystyle\supset$
$\displaystyle{\bf{\overline{\bf 16}}}\big{(}Y{\bf 45_{H}}-M\big{)}{\bf 16}$
(23) $\displaystyle\supset$
$\displaystyle{\bf{\overline{5}}}\big{(}a_{5}Y\langle{\bf
1_{H}}\rangle+b_{5}Y\langle{\bf 24_{H}}\rangle-M\big{)}{\bf
5}+{\bf{\overline{10}}}\big{(}a_{10}Y\langle{\bf
1_{H}}\rangle+b_{10}Y\langle{\bf 24_{H}}\rangle-M\big{)}{\bf 10},$

where $Y$ is a Yukawa coupling constant, $M$ is a Dirac mass, and the second
line denotes the decomposition under the $SU(5)$ sub-group with $a_{i}$ and
$b_{i}$ being the Clebsch-Gordan coefficients. Assuming $\langle{\bf
1_{H}}\rangle={\cal O}(v_{GUT})$, $\langle{\bf 24_{H}}\rangle={\cal
O}(v_{GUT}){\rm diag}(1,1,1,-3/2,-3/2)$ and $M={\cal O}(v_{GUT})$ for the
symmetry breaking of $SO(10)\to SM\times U(1)_{X}$, we obtain the mass
splitting among the vector-like fermions in different SM representations.
Tuning the Yukawa coupling value of $Y$ allows us to make one vector-like
fermion pair light while the others have mass at the GUT scale. We introduce
three vector-like fermion pairs of ${\bf 16}+{\bf\overline{16}}$
representation and tune three Yukawa couplings to provide three vector-like
fermion pairs below the GUT scale, namely, one new down-type quark pair
$(D^{c}+\overline{D^{c}})$ with mass $M_{D^{c}}$, one new quark-doublet pair
$(Q+\overline{Q})$ with mass $M_{Q}$ and one new lepton-doublet pair
$(L+\bar{L})$ with mass $M_{L}$.

Figure 3:  RG running of the SM ($\alpha_{1,2,3}$) and $U(1)_{X}$
($\alpha_{X}$) gauge couplings.

For the energy scale $\mu<v_{GUT}$, the renormlaization group (RG) equations
of the SM and the $U(1)_{X}$ gauge couplings at the 1-loop order are given by

$\displaystyle\mu\frac{d{\alpha}_{1}}{d\mu}$ $\displaystyle=$
$\displaystyle\frac{1}{2\pi}{\alpha}_{1}^{2}\left(\frac{41}{10}+\frac{2}{5}\times\rm{\theta(\mu-
M_{Q})}+\frac{2}{5}\times\rm{\theta(\mu-M_{L})}\right)$
$\displaystyle\mu\frac{d{\alpha}_{2}}{d\mu}$ $\displaystyle=$
$\displaystyle\frac{1}{2\pi}{\alpha}_{2}^{2}\left(-\frac{19}{6}+2\times\rm{\theta(\mu-
M_{Q})}+\frac{2}{3}\times\rm{\theta(\mu-M_{L})}\right),$
$\displaystyle\mu\frac{d{\alpha}_{3}}{d\mu}$ $\displaystyle=$
$\displaystyle\frac{1}{2\pi}{\alpha}_{3}^{2}\left(-7+2\times\rm{\theta(\mu-
M_{Q})}\right),$ $\displaystyle\mu\frac{d{\alpha}_{X}}{d\mu}$ $\displaystyle=$
$\displaystyle\frac{1}{2\pi}{\alpha}_{X}^{2}\left(\frac{169}{60}+\frac{11}{10}\times\rm{\theta(\mu-
M_{Q})}+\frac{3}{5}\times\rm{\theta(\mu-M_{L})}\right),$ (24)

Here $\alpha_{i}\equiv g_{i}^{2}/4\pi$, $g_{2,3}$ respectively are the
$SU(3)_{c}$ and $SU(2)_{L}$ SM gauge couplings, $\alpha_{1}=g_{1}^{2}/4\pi$ is
related to the SM hypercharge gauge coupling by $g_{1}=\sqrt{5/3}\;g_{Y}$, and
we have set $M_{D^{c}}=M_{Q}$ for simplicity. In our RG analysis, we employ
the low energy values of the SM gauge couplings at $\mu=m_{t}=172.44$ GeV
[42]:

$\displaystyle g_{1}(m_{t})=\sqrt{5/3}\times 0.35830,\qquad
g_{2}(m_{t})=0.64779,\qquad g_{3}(m_{t})=1.1666.$ (25)

We find that the $SU(5)$ gauge coupling unification condition,
$\alpha_{1,2,3}(v_{GUT})=\alpha_{GUT}$, is satisfied for benchmark values
$M_{Q}=4.8\times 10^{3}$ GeV and $M_{L}=4.5\times 10^{12}$ GeV, and we obtain
$v_{GUT}=7.58\times 10^{15}$ GeV with $1/\alpha_{GUT}=35.8$. The $SO(10)$
unification requires $\alpha_{X}(v_{GUT})=\alpha_{GUT}$. The RG running of the
SM and $U(1)_{X}$ gauge couplings are shown in Fig. 3. Using this result, the
proton lifetime mediated by the $SO(10)$ GUT gauge bosons can be approximated
as [43]

$\displaystyle\tau_{p}\approx\frac{1}{\alpha_{GUT}^{2}}\frac{v_{GUT}^{4}}{m_{p}^{5}}\approx
9.61\times 10^{34}\;{\rm years},$ (26)

where $m_{p}=0.983$ GeV is the proton mass. This is consistent with the
current bound on proton lifetime obtained by the Super-Kamiokande experiment
$\tau_{p}(p\to\pi^{0}e^{+})\gtrsim 1.6\times 10^{34}$ yr [44]. Importantly,
this is within the expected sensitivity reach of the future Hyper-Kamiokande
experiment, $\tau_{p}\simeq 1.3\times 10^{35}$ yr [45]. Since the RG running
of the three SM couplings only depends on two free parameters, there is a one-
to-one correspondence between the masses $M_{Q}$ and $M_{L}$ to satisfy the
unification of the couplings. As a result, we find that $M_{Q}\gtrsim
3.4\times 10^{3}$ GeV, or equivalently $M_{L}\lesssim 8.0\times 10^{12}$ GeV,
for the proton lifetime to be within the search reach of Hyper-Kamiokande
experiment.

Next let us discuss the gauge interaction of the pGDM which explicitly breaks
the ${\bf Z}_{2}$-parity and thereby enabling it to decay. For instance, the
gauge interaction in the $B-L$ extended model [5] is given by

$\displaystyle{\cal L}\supset-
g_{BL}\Big{(}Q_{B}\phi_{B}\cos\theta+Q_{A}\phi_{A}\sin\theta\Big{)}(\partial_{\mu}\chi){Z_{BL}^{\mu}},$
(27)

where $Q_{A,B}$ is the $B-L$ charge of $\Phi_{A,B}$. Because $\phi_{A}$ is
much heavier than $\phi_{B}$ the dominant decay mode is
$\chi\to{Z^{\prime}}^{*}\phi_{B}^{*}\to{\bar{f}}_{SM}f_{SM}{\bar{f}}_{SM}f_{SM}$.
In the $SO(10)$ model, we can identify $Z_{BL}\to Z_{X}$ with
$Q_{A}=-\sqrt{5/2}$ and $Q_{B}=\sqrt{5/8}$. For the interaction of the $Z_{X}$
gauge boson with the SM $SU(2)_{L}$ singlet quarks ($q$) and leptons ($\ell$)
we obtain

$\displaystyle{\cal L}\supset-
g_{X}\Big{(}Q_{q}\;{\overline{q}}\gamma^{\mu}q+Q_{\ell}\;{\overline{\ell}}\gamma^{\mu}\ell\Big{)}{Z_{X\mu}},$
(28)

where $Q_{q}=-3/\sqrt{40}$ and $Q_{\ell}=+1/\sqrt{40}$. The $Z_{X}$ mass is
given by

$\displaystyle
m_{Z_{X}}=\sqrt{\frac{5}{8}}g_{X}\sqrt{4v_{A}^{2}+v_{B}^{2}}\simeq\sqrt{\frac{5}{2}}g_{X}v_{A},$
(29)

where we have used $v_{B}\ll v_{A}$ in the last expression. Using Eqs. (27)
and (28), we have evaluated the lifetime of the pGDM. We impose the bound on
its lifetime $\tau_{\chi}>10^{26}$ sec from the cosmic ray observations [14]
to obtain a lower bound on the pGDM mass. The results for various benchmark
values of $m_{\tilde{B}}/{\rm GeV}=80,200,500$ are depicted by the yellow band
in Fig. 1. Here, we have fixed $v_{A}=v_{GUT}$.

For the RG equation of the $\Phi_{B}$ quartic coupling
$\lambda_{B}\equiv\lambda_{1}$ in Eq. (22) we obtain

$\displaystyle\mu\frac{d{\lambda_{B}}}{d\mu}$ $\displaystyle=$
$\displaystyle\frac{1}{16\pi^{2}}20\lambda_{B}^{2}.$ (30)

for $\mu<v_{GUT}$. We impose the perturbative bound on
$20\lambda_{B}^{2}/16\pi^{2}<1$ at $\mu=v_{GUT}$, which is extrapolated to the
upper bound on $\lambda_{B}(v_{B})$. Thus, we find the lower bound on $v_{B}$
by using $m_{\tilde{B}}^{2}\simeq 2\lambda_{B}(v_{B})v_{B}^{2}$. For
${\tilde{m}_{B}}/{\rm GeV}=80,200,500$, we obtain an upper bound of
$v_{H}/v_{B}\lesssim 2.0,0.81,0.33$ with
$\lambda_{B}(v_{GUT})=0.21,0.22,0.22$, respectively. These are depicted by the
dashed purple line in Fig. 1.

##  V $SO(10)$ Breaking via $SU(4)_{c}\times SU(2)_{L}\times SU(2)_{R}$
(4-2-2)

The Higgs fields involved in breaking $SO(10)$ down to the SM along with their
decomposition under the 4-2-2 gauge group ( $SU(4)_{c}\times SU(2)_{L}\times
SU(2)_{R}$) are listed below

$\displaystyle{\bf 210_{H}}$ $\displaystyle=$ $\displaystyle({\bf 1},{\bf
1},{\bf 1}){\color[rgb]{0.6,0.6,0.6}\oplus({\bf 15},{\bf 1},{\bf
1})\oplus({\bf 6},{\bf 2},{\bf 2})\oplus({\bf 15},{\bf 3},{\bf 1})\oplus({\bf
15},{\bf 1},{\bf 3})\oplus({\bf 10},{\bf 2},{\bf 2})\oplus(\overline{\bf
10},{\bf 2},{\bf 2})},$ $\displaystyle{{\bf\overline{126}_{H}}}$
$\displaystyle=$ $\displaystyle({\bf 10},{\bf 1},{\bf 3})\oplus({\bf 15},{\bf
2},{\bf 2}){\color[rgb]{0.6,0.6,0.6}\oplus({\overline{\bf 10}},{\bf 3},{\bf
1})\oplus({\bf 6},{\bf 1},{\bf 1})},$ $\displaystyle{\bf 45_{H}}$
$\displaystyle=$ $\displaystyle({\bf 1},{\bf 1},{\bf 3})\oplus({\bf 15},{\bf
1},{\bf 1}){\color[rgb]{0.6,0.6,0.6}\oplus({\bf 1},{\bf 3},{\bf 1})\oplus({\bf
6},{\bf 2},{\bf 2})},$ $\displaystyle{\bf 16_{H}}$ $\displaystyle=$
$\displaystyle({\bf 4},{\bf 1},{\bf
2}){\color[rgb]{0.6,0.6,0.6}\oplus({\overline{\bf 4}},{\bf 2},{\bf 1})},$
$\displaystyle{\bf 10_{H}}$ $\displaystyle=$ $\displaystyle({\bf 1},{\bf
2},{\bf 2}){\color[rgb]{0.6,0.6,0.6}\oplus({\bf 6},{\bf 1},{\bf 1})}.$ (31)

Here only the 4-2-2 multiplets depicted in bold develop a nonzero VEV, which
results in the following symmetry breaking pattern:

$\displaystyle SO(10)\ \xrightarrow[v_{GUT}]{\langle{\bf 210_{H}}\rangle}\
{\rm PS}\
\xrightarrow[v_{A},v_{A},v_{B}]{\langle{\bf\overline{126}_{H}}\rangle,\langle{\bf
45_{H}}\rangle,\langle{\bf 16_{H}}\rangle}\ SM\xrightarrow[V_{H}]{\langle{\bf
10_{H}}\rangle}SU(3)_{c}\times U(1)_{EM},$ (32)

where $v_{GUT}$, $v_{A,B}$, and $v_{EW}$ denote the VEVs of the corresponding
scalars fields involved at the various stages of the symmetry breaking.

In the fermion sector, the ${\bf 10_{H}}$ and ${\bf\overline{126}_{H}}$ Higgs
field generate realistic fermion masses [46, 47, 48, 49, 50, 51, 52, 53, 54,
55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73],
while the SM singlet component in ${\bf(\overline{10},1,3)}$ of
${\bf\overline{126}_{H}}$ generates intermediate scale Majorana masses for the
right-handed neutrinos in the ${\bf 16}$-plet fermion.

There are a total of four Higgs doublets in $({\bf 1},{\bf 2},{\bf 2})$ of
${\bf 10_{H}}$ and $({\bf 15},{\bf 2},{\bf 2})$ of ${\overline{\bf 126_{H}}}$.
We assume all of them develop nonzero VEV at the electroweak scale such that
only one of the four linear combination of the doublets is light (doublet-
doublet Higgs mass splitting), which is identified as the SM Higgs doublet
$H$. The rest of the linear combinations are assumed to have masses of order
$v_{A}$. The SM singlets components in $({\bf 10},{\bf 1},{\bf 3})$ of
${\overline{\bf 126_{H}}}$ and $({\bf 4},{\bf 1},{\bf 2})$ of ${\bf 16_{H}}$
are identified to be $\Phi_{A}$ and $\Phi_{B}$, respectively. The inclusion of
${\bf 45}_{H}$ is crucial to realize the electroweak scale mass for $\Phi_{B}$
while making sure that all of the remaining charged scalars masses in $({\bf
4},{\bf 1},{\bf 2})$ are of the order $v_{A}$. Without ${\bf 45}_{H}$, all
charged scalars will have the same mass as $\Phi_{B}$, which is inconsistent
with the null observation of any elementary light-charged scalar. To
illustrate the mass splitting, consider the following coupling:

$\displaystyle V$ $\displaystyle\supset$ $\displaystyle{\bf
16_{H}}^{\dagger}{\bf 45_{H}}{\bf 16_{H}}$ (33) $\displaystyle\supset$
$\displaystyle(a_{3}V_{T}-a_{15}V_{15})(\Phi_{A}^{\dagger}\Phi_{A})+(y_{3}V_{T}+y_{15}V_{15})(\Phi_{Y}^{\dagger}\Phi_{Y})+(C_{3}V_{T}+C_{15}V_{15})(\Phi_{C}^{\dagger}\Phi_{C}).$

Here $\Phi_{Y}$($\Phi_{C}$) is the component that is only charged under
hypercharge ($SU(3)_{c}$), $a_{i},y_{i}$ and $C_{i}$ are Clebsch-Gordan
coefficients which are of the same order, and $V_{T}$ and $V_{15}$ are
respectively the VEVs of $({\bf 1},{\bf 1},{\bf 3})$ and $({\bf 15},{\bf
1},{\bf 3})$ in ${\bf 45_{H}}$ which are of order $v_{A}$. By appropriately
fixing $V_{T}\simeq V_{15}$, $\Phi_{B}$ can be light while the others fields
are heavy. The $\Phi_{A}\supset({\bf 10},{\bf 1},{\bf 3})$ has mass compatible
to the scale $v_{A}$. As discussed before, we do not include the terms ${\bf
10_{H}(16_{H})}^{2}$ and ${\bf 10_{H}^{\dagger}(16_{H})}^{2}$ which explicitly
violate the Goldstone nature of the pGDM [3]. The resulting scalar potential
matches that of the low energy effective potential of pGDM in Eq. (4).

Let us now examine the RG evolution of the SM and 4-2-2 gauge couplings by
solving their RG equations at the 1-loop level. For this analysis, we fix the
masses of $({\bf 1},{\bf 1},{\bf 3})$ and $({\bf 15},{\bf 1},{\bf 3})$ in
${\bf 45_{H}}$333If $({\bf 1},{\bf 1},{\bf 3})$ and $({\bf 15},{\bf 1},{\bf
3})$ masses are lighter, the scale $v_{A}$ can be reduced, which can
significantly lower the upper bound on the pGDM mass in Fig. 1. and all the
multiplets highlighted in gray in Eq. (31) have masses of order $v_{GUT}$. For
energy scale $\mu$ below the 4-2-2 scale ($\mu<v_{A}$), the RG equations of
the SM gauge couplings are given by

$\displaystyle\mu\frac{d\alpha_{1}}{d\mu}$ $\displaystyle=$
$\displaystyle\frac{\alpha_{1}^{2}}{2\pi}\left(\frac{41}{10}\right),$
$\displaystyle\mu\frac{d\alpha_{2}}{d\mu}$ $\displaystyle=$
$\displaystyle\frac{\alpha_{2}^{2}}{2\pi}\left(-\frac{19}{6}\right),$
$\displaystyle\mu\frac{d\alpha_{3}}{d\mu}$ $\displaystyle=$
$\displaystyle\frac{\alpha_{3}^{2}}{2\pi}\left(-7\right).$ (34)

Their low energy values are fixed to match the measured values at
$\mu=m_{t}=172.44$ GeV [42]:

$\displaystyle g_{1}(m_{t})=\sqrt{5/3}\times 0.35830,\qquad
g_{2}(m_{t})=0.64779,\qquad g_{3}(m_{t})=1.1666.$ (35)

For $v_{A}<\mu<v_{GUT}$, our theory is based on the 4-2-2 gauge group. The
relationships between the SM and the 4-2-2 gauge couplings at $\mu=v_{A}$ are
given by the tree-level matching conditions:

$\displaystyle\alpha_{2}(v_{A})={\alpha}_{L}(v_{A}),\qquad\alpha_{3}(v_{A})={\alpha}_{4}(v_{A}),\qquad\alpha_{1}^{-1}(v_{A})=\frac{3}{5}{\alpha}_{R}^{-1}(v_{A})+\frac{2}{5}{\alpha}_{4}^{-1}(v_{A}),$
(36)

Here $\alpha_{4,L,R}$ are $SU(4){{}_{c}}$, $SU(2)_{L}$ and $SU(2)_{R}$ gauge
couplings and their RG equations are given by

$\displaystyle\mu\frac{d{\alpha}_{4}}{d\mu}$ $\displaystyle=$
$\displaystyle\frac{1}{2\pi}{\alpha}_{4}^{2}\left(+1\right),$
$\displaystyle\mu\frac{d{\alpha}_{L}}{d\mu}$ $\displaystyle=$
$\displaystyle\frac{1}{2\pi}{\alpha}_{L}^{2}\left(4\right),$
$\displaystyle\mu\frac{d{\alpha}_{R}}{d\mu}$ $\displaystyle=$
$\displaystyle\frac{1}{2\pi}{\alpha}_{R}^{2}\left(\frac{32}{3}\right).$ (37)

Figure 4:  The RG running of the SM gauge couplings ($\alpha_{1,2,3}$) for
$\mu<v_{A}$ and 4-2-2 gauge couplings ($\alpha_{4,L,R}$) for
$v_{A}<\mu<v_{GUT}$.

With the initial values of the SM gauge couplings fixed at $\mu=m_{t}$, the RG
equations for the SM and 4-2-2 gauge couplings can be solved analytically.
After imposing the unification conditions

$\displaystyle\alpha_{4}(v_{GUT})=\alpha_{L}(v_{GUT})=\alpha_{R}(v_{GUT})\equiv\alpha_{GUT},$
(38)

$v_{GUT}$ and $v_{A}$ are uniquely determined. We find $v_{A}\simeq 3.6\times
10^{11}$ GeV and $v_{GUT}\simeq 2.3\times 10^{16}$ GeV, with
$1/\alpha_{GUT}=37.2$. Furthermore, using Eq. (26), the proton lifetime is
predicted to be $8.8\times 10^{36}$ yr. The RG running of the SM and 4-2-2
gauge couplings are shown in Fig. 3.

Let us now estimate the lifetime of the pGDM. As discussed in the $SU(5)\times
U(1)_{X}$ scenario, we are interested in the gauge interaction of pGDM $\chi$
with $\phi_{B}$ and the heavy neutral gauge boson $Z_{R}$ from the 4-2-2
symmetry breaking. This channel contributes dominantly to the pGDM decay rate.
The heavy neutral gauge boson mass eigenstate is a linear combination of the
gauge field associated with the diagonal generators of $SU(4)_{c}$ and
$SU(2)_{R}$ in the flavor basis, and its mass is given by

$\displaystyle m_{Z_{R}}=\frac{v_{A}}{2}\sqrt{3g_{4}^{2}+2g_{R}^{2}}$ (39)

The relevant interactions for the pGDM decay, $\chi\to
Z_{R}^{*}\phi_{B}^{*}\to\bar{f}_{SM}f_{SM}\bar{f}_{SM}f_{SM}$, are expressed
as

$\displaystyle{\cal
L}\supset\Big{(}\frac{1}{2}\kappa\phi_{B}\cos\theta-\kappa\phi_{A}\sin\theta\Big{)}(\partial_{\mu}\chi){Z_{R}^{\mu}}+\Big{(}Q_{q}\;{\overline{q}}\gamma^{\mu}q+Q_{\ell}\;{\overline{\ell}}\gamma^{\mu}\ell\Big{)}{Z_{R\mu}},$
(40)

with $\kappa=\sqrt{3g_{4}^{2}+2g_{R}^{2}}/2=\sqrt{2}m_{Z_{R}}/v_{A}$, and the
effective $Z_{R}$ charges for the $SU(2)_{L}$ singlet SM quarks ($q$) and
$SU(2)_{L}$ doublet leptons ($\ell$) given by

$\displaystyle Q_{q}$ $\displaystyle=$
$\displaystyle\frac{1}{12}\Big{(}{\sqrt{6}}g_{4}\sin\delta+6\cos\delta
g_{R}\Big{)},$ $\displaystyle Q_{\ell}$ $\displaystyle=$
$\displaystyle\frac{1}{4}\Big{(}{\sqrt{6}}g_{4}\sin\delta-2\cos\delta
g_{R}\Big{)}.$ (41)

Here $\delta$ is the mixing between the $SU(4)_{c}$ and $SU(2)_{R}$ components
of $Z_{R}$ defined by $\tan\delta=\sqrt{3/2}(g_{4}/g_{R})$. In Fig. 5 we show
the pGDM lifetime as a function of it mass and fixed mass of $\phi_{B}$
($m_{\tilde{B}}$). The parameter region excluded by requiring the pGDM
lifetime $\tau_{\chi}>10^{26}$ sec, is depicted by purple band. For
$m_{\tilde{B}}/{\rm GeV}=80,200,500$, the lifetime lower bound excludes the
pGDM mass below $m_{\chi}/{\rm GeV}\simeq 66,100,153$, respectively. As shown
in Fig. 1, this significantly reduces the allowed parameter space for the pGDM
due to the smallness of $v_{A}$ in the 4-2-2 case compared to the GUT scale
$v_{A}$ value in the $SU(5)\times U(1)_{X}$ case.

Figure 5: Lifetime of pGDM as a function of its mass ($m_{\chi}$). Green, red,
and blue curves denote three benchmark scenarios with $m_{\tilde{B}}/{\rm
GeV}=80,200,500$, respectively. The purple band is excluded in order to
satisfy the DM lifetime bound $\tau_{\chi}\gtrsim 10^{26}$ from cosmic ray
observations [14].

For the $\Phi_{B}$ quartic coupling $\lambda_{B}$ we obtain the following RG
equation

$\displaystyle\mu\frac{d{\lambda_{B}}}{d\mu}$ $\displaystyle=$
$\displaystyle\frac{1}{16\pi^{2}}\Big{(}20\lambda_{B}^{2}+\theta(\mu-
v_{A})\Big{(}48\lambda_{B}^{2}+\frac{151875}{2048}g_{4}^{4}+\frac{243}{128}g_{R}^{4}-\Big{(}\frac{675}{64}g_{4}^{2}+\frac{27}{16}g_{R}^{2}\Big{)}\lambda_{B}\Big{)}\Big{)}.$
(42)

Using the solution for the running of the 4-2-2 gauge couplings in Fig. 4, we
can numerically evaluate the RG running of $\lambda_{B}$ together with
${\tilde{m}_{B}}^{2}\simeq 2\lambda_{B}v_{B}^{2}$ which fixes the initial
value of $\lambda_{B}$ at $\mu=v_{B}$. For fixed values of ${\tilde{m}_{B}}$
and $v_{B}$ we impose the perturbativity bound $68\lambda_{B}^{2}/16\pi^{2}<1$
at $\mu=v_{GUT}$. For ${\tilde{m}_{B}}/{\rm GeV}=80,200,500$, we obtain an
upper bound of $v_{H}/v_{B}\lesssim 1.3,0.53,0.22$ with
$\lambda_{B}(v_{GUT})/10^{-2}\simeq 9.3,9.4,9.5$, respectively. These are the
depicted by dashed purple line in Fig. 1.

##  VI Intermediate and GUT Monopoles

Inflation can resolve the monopole problem in GUTs if the Hubble parameter
during inflation ($H_{inf}$) is lower in magnitude than the monopole producing
symmetry breaking scale. For instance, in $SO(10)$ breaking via 4-2-2,
inflation driven by a gauge singlet with minimal coupling to gravity can, in
principle, take care of both the GUT monopole as well as the intermediate mass
monopole associated with the 4-2-2 breaking to the SM if the inflationary
Hubble parameter is of order $10^{13}-10^{14}$ GeV [74]. A similar situation
holds for quartic inflation driven by a gauge singlet field with non-minimal
coupling to gravity [75, 76]. However, in our case with a 4-2-2 breaking scale
significantly lower than $10^{13}$ GeV, it would be more appropriate to use
the 4-2-2 breaking scalar field as the inflaton. This scenario will be
discussed in detail elsewhere [77]. Since there are only heavy GUT monpoles in
the $SU(5)\times U(1)_{X}$ scenario, the non-minimal inflation by using a
$SO(10)$ singlet field with Hubble parameter is of order $10^{13}-10^{14}$ GeV
[74] nicely works to resolve the monopole problem.

##  VII Conclusion

The pGDM scenario is an interesting WIMP DM scenario which can avoid the very
severe constraints from the direct DM search experiments. Because of the
Goldstone nature of the pGDM, its coupling with the Higgs bosons vanishes in
the limit of zero momentum transfer. In this paper, we have proposed an UV
completion of the pGDM in the $SO(10)$ GUT framework. For this completion, it
is crucial to introduce the $SO(10)$ invariant cubic scalar coupling
${\bf\overline{126}_{H}}{\bf 16_{H}}^{2}$ and the pGDM arises as a linear
combination of the SM singlet scalars in ${\bf 16_{H}}$ and
${\bf\overline{126}_{H}}$.

We have considered two scenarios for the intermediate breaking patterns of
$SO(10)$ to the SM: $SU(5)\times U(1)$ and the 4-2-2 gauge group. Since the
${\bf 16_{H}}$ and ${\bf\overline{126}_{H}}$ Higgs fields are involved in
breaking $SO(10)$ down to the SM, there is a direct correlation between dark
matter and GUT physics. Specifically, the $SU(2)_{R}$ gauge boson in the 4-2-2
case and $U(1)_{X}$ gauge boson in the $SU(5)\times U(1)_{X}$ case allow pGDM
to decay. Hence, to satisfy the cosmic ray bound on the pGDM lifetime,
$\tau_{DM}\gtrsim 10^{26}$ s, we have shown that the vacuum expectation value
of ${\bf\overline{126}_{H}}$, which triggers the 4-2-2 and $U(1)_{X}$ symmetry
breaking, must be much greater than $10^{11}$ GeV in order to maximize the
allowed parameter space for the pGDM. The 4-2-2 symmetry breaking scale occurs
at the intermediate scale, $M_{PS}\simeq 2.3\times 10^{11}$ GeV, which
excludes most of the model parameter space. However, in the $SU(5)\times
U(1)_{X}$ scenario, the $U(1)_{X}$ breaking scale can be much higher such that
a significantly larger portion of the model parameter space is still allowed.
Furthermore, in the $SU(5)$ scenario, the proton lifetime is predicted to be
$4.53\times 10^{34}$ years, which lies well within the sensitivity range of
the future Hyper-Kamiokande experiment. We have also examined the constraints
on the model parameter space from the Large Hadron Collider and indirect DM
searches by Fermi-LAT and MAGIC experiments.

## Note Added

During the write up of our manuscript we came across a recent paper [78] which
discusses the same topic with $SO(10)$ broken via the gauge symmetry
$SU(4)_{c}\times SU(2)_{L}\times SU2)_{R}$. Where there is overlap our results
are in agreement. The $SO(10)$ symmetry breaking via $SU(5)\times U(1)_{X}$
has not been discussed in the above mentioned paper.

## Acknowledgments

This work is supported in part by the United States Department of Energy grant
DE-SC0012447 (N. Okada), DE-SC0013880 (D. Raut and Q. Shafi), and DE-SC0016013
(A. Thapa).

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