#  IPMU19-0186 OU-HET-1040 Proton Decay and Axion Dark Matter
in $SO(10)$ Grand Unification
via Minimal Left-Right Symmetry

Yuta Hamada,111E-mail<EMAIL_ADDRESS>Masahiro Ibe,222E-mail:
<EMAIL_ADDRESS>Yu Muramatsu,333E-mail<EMAIL_ADDRESS>

Kin-ya Oda,444E-mail<EMAIL_ADDRESS>and Norimi
Yokozaki555E-mail<EMAIL_ADDRESS>

∗Université de Paris, CNRS, Astroparticule et Cosmologie, F-75013 Paris,
France
†ICRR, University of Tokyo, Kashiwa, Chiba 277-8582, Japan
†Kavli IPMU (WPI), UTIAS, The University of Tokyo, Kashiwa, Chiba 277-8583,
Japan
‡Institute of Particle Physics and Key Laboratory of Quark and Lepton Physics
(MOE), Central China Normal University, Wuhan, Hubei 430079, People’s Republic
of China
§Department of Physics, Osaka University, Osaka 560-0043, Japan
¶Department of Physics, Tohoku University, Sendai, Miyagi 980-8578, Japan

###### Abstract

We study the proton lifetime in the $SO(10)$ Grand Unified Theory (GUT), which
has the left-right (LR) symmetric gauge theory below the GUT scale. In
particular, we focus on the minimal model without the bi-doublet Higgs field
in the LR symmetric model, which predicts the LR-breaking scale at around
$10^{10\text{--}12}$ GeV. The Wilson coefficients of the proton decay
operators turn out to be considerably larger than those in the minimal $SU(5)$
GUT model especially when the Standard Model Yukawa interactions are generated
by integrating out extra vector-like multiplets. As a result, we find that the
proton lifetime can be within the reach of the Hyper-Kamiokande experiment
even when the GUT gauge boson mass is in the $10^{16\text{--}17}$ GeV range.
We also show that the mass of the extra vector-like multiplets can be
generated by the Peccei-Quinn symmetry breaking in a consistent way with the
axion dark matter scenario.

##  1 Introduction

The Grand Unified Theory (GUT) [1] is one of the most attractive candidates
for physics beyond the Standard Model (SM), which provides an explanation of
the charge quantization. In particular, the $SO(10)$ gauge group [2] is one of
the most attractive candidates for the unification group as it not only
unifies all the gauge interactions in the SM but also unifies a generation of
the SM fermions into one representation. Furthermore, it also predicts the
existence of the right-handed neutrinos, which naturally explains the light
active neutrino masses through the seesaw mechanism [3]. This feature is a
great advantage compared to the $SU(5)$ GUT.

Another interesting feature of the $SO(10)$ GUT is that the rank of $SO(10)$
is larger than the SM. Accordingly, the $SO(10)$ GUT allows various symmetry
breaking paths to the SM gauge groups, such as the Left-Right (LR) symmetric
groups [4]. Among these possibilities, the minimal model based on the
$SU(3)_{\text{C}}\times SU(2)_{\text{L}}\times SU(2)_{\text{R}}\times
U(1)_{B-L}$ gauge group without a bi-doublet of $SU(2)_{\text{L}}\times
SU(2)_{\text{R}}$ uniquely predicts an intermediate breaking scale of the LR
symmetry to be around $10^{10\mbox{--}12}$ GeV [5, 6]; see also [7, 8]. This
model also gets renewed attention as it can explain the small Higgs quartic
coupling constant at a high energy scale while solving the strong CP problem
simultaneously [9, 10]. In this class of models, all the SM Yukawa
interactions are generated by integrating out extra vector-like multiplets at
around the LR-breaking scale.

In this paper, we discuss the proton lifetime in this scenario with the
simplest possibility of the extra matter content.666 Hall and Harigaya have
extensively studied various possibilities of the extra matter contents [9,
10]. In this paper we only study the simplest one among them, which has also
been partly discussed in a different context of the axion dark matter scenario
[11]. As we will see, the preferred GUT scale $\lesssim 10^{17}$ GeV is lower
than expected in Refs. [5, 6] by a factor a few or so, due to the effects of
the extra matter multiplets on the renormalization group running.777 Recall
that the proton decay rate $\Gamma$ is inversely proportional to the fourth
power of the GUT gauge-boson mass $M_{X}$, and hence the slight change of
$M_{X}$ significantly affects $\Gamma$. We also find that the Wilson
coefficients of the proton decay operators are considerably larger than those
in the minimal $SU(5)$ GUT model due to the larger gauge coupling below the
GUT scale as well as the $SU(2)_{\text{R}}$ gauge interaction at the
intermediate scale. As a result, the proton decay rate is enhanced and a
parameter region consistent with the gauge coupling unification in the
$10^{16\text{--}17}$ GeV range can be tested by the Hyper-Kamiokande (Hyper-K)
experiment. We also discuss a possibility to generate the mass of the extra
vector-like multiplet by the Peccei-Quinn (PQ) symmetry breaking in a
consistent way with the axion dark matter scenario.

The organization of the paper is as follows. In Sec. 2, we summarize the
$SO(10)$ model which has the minimal LR-symmetric gauge group at the
intermediate stage. In Sec. 3, we discuss the gauge coupling unification in
the minimal LR symmetric model. In Sec. 4, we study the proton lifetime. In
Sec. 5, we discuss the mass generation of the extra vector-like multiplets by
the PQ symmetry breaking. We give a summary of our discussion in the final
section.

##  2 The minimal setup of the $SO(10)$ GUT model

In this paper, we discuss $SO(10)$ GUT with the following chain of symmetry
breaking:

$\displaystyle SO(10)\underset{M_{\text{GUT}}}{\longrightarrow}G_{\text{LR}}$
$\displaystyle\equiv SU(3)_{\text{C}}\times SU(2)_{\text{L}}\times
SU(2)_{\text{R}}\times U(1)_{B-L}$
$\displaystyle\underset{M_{\text{R}}}{\longrightarrow}G_{\text{SM}}$
$\displaystyle\equiv SU(3)_{\text{C}}\times SU(2)_{\text{L}}\times U(1)_{Y}.$
(1)

To ensure this chain and subsequent SM symmetry breaking, we introduce an
$SO(10)$ adjoint Higgs $H_{45}$ and an $SO(10)$ spinor-representation Higgs
$H_{16}$. $H_{16}$ contains the doublet Higgs bosons of $SU(2)_{\text{R}}$ and
$SU(2)_{\text{L}}$, respectively.888It should be noted that $H_{45}$ and
$H_{16}$ do not contain the bi-doublet Higgs of $SU(2)_{\text{R}}\times
SU(2)_{\text{L}}$. First, the vacuum expectation value (VEV) of $H_{45}$
breaks down the $SO(10)$ symmetry at the GUT scale $M_{\text{GUT}}$. Second,
the VEV of the $SU(2)_{\text{R}}$ doublet Higgs breaks down the LR symmetry at
$M_{\text{R}}$ which we call the LR symmetry breaking scale. In this setup
there is no bi-doublet Higgs. Below the LR symmetry breaking scale, the
$U(1)_{Y}$ gauge symmetry in the SM is obtained by,

$\displaystyle Q_{Y}=\frac{1}{2}Q_{B-L}-T_{3}^{\text{R}},$ (2)

where $T_{3}^{\text{R}}$ is the third generator of $SU(2)_{\text{R}}$. As will
be discussed, the typical values of the GUT and the LR symmetry breaking
scales are $M_{\text{GUT}}={\mathcal{O}}(10^{16\text{--}17})$ GeV and
$M_{R}={\mathcal{O}}(10^{10\text{--}12})$ GeV, respectively. Throughout this
paper, we assume these minimal contents for the Higgs sector, and assume that
only the doublet Higgs bosons of $SU(2)_{\text{R}}$ and $SU(2)_{\text{L}}$
remain massless below the GUT scale.

| $G_{\text{LR}}$ | $G_{\text{SM}}$
---|---|---
up-type Yukawa | $\displaystyle y_{u}\frac{\left({Q_{\text{R}}}^{c}H_{\text{R}}^{*}\right)\left(Q_{\text{L}}H_{\text{L}}\right)}{M_{\text{extra}}}$ | $\displaystyle y_{u}\frac{\langle H_{\text{R}}\rangle}{M_{\text{extra}}}{u_{\text{R}}}^{c}\left(q_{\text{L}}h_{\text{L}}\right)$
down-type Yukawa | $\displaystyle y_{d}\frac{\left({Q_{\text{R}}}^{c}H_{\text{R}}\right)\left(H_{\text{L}}^{*}Q_{\text{L}}\right)}{M_{\text{extra}}}$ | $\displaystyle y_{d}\frac{\langle H_{\text{R}}\rangle}{M_{\text{extra}}}{d_{\text{R}}}^{c}\left(h_{\text{L}}^{*}q_{\text{L}}\right)$
charged Lepton Yukawa | $\displaystyle y_{d}\frac{\left({L_{\text{R}}}^{c}H_{\text{R}}\right)\left(H_{\text{L}}^{*}L_{\text{L}}\right)}{M_{\text{extra}}}$ | $\displaystyle y_{d}\frac{\langle H_{\text{R}}\rangle}{M_{\text{extra}}}{e_{\text{R}}}^{c}\left(h_{\text{L}}^{*}l_{\text{L}}\right)$
neutrino Dirac Yukawa | $\displaystyle y_{u}\frac{\left({L_{\text{R}}}^{c}H_{\text{R}}^{*}\right)\left(L_{\text{L}}H_{\text{L}}\right)}{M_{\text{extra}}}$ | $\displaystyle y_{u}\frac{\langle H_{\text{R}}\rangle}{M_{\text{extra}}}{\nu_{\text{R}}}^{c}\left(l_{\text{L}}h_{\text{L}}\right)$
neutrino Majorana mass | $\displaystyle y_{u}\frac{\left({L_{\text{R}}}^{c}H_{\text{R}}^{*}\right)\left({L_{\text{R}}}^{c}H_{\text{R}}^{*}\right)}{M_{\text{extra}}}$ | $\displaystyle y_{u}\frac{\langle H_{\text{R}}\rangle^{2}}{M_{\text{extra}}}{\nu_{\text{R}}}^{c}{\nu_{\text{R}}}^{c}$
Table 1: The Yukawa interactions which come from the higher dimensional
operators in Eq. (5).

In the minimal $SO(10)$ GUT model, each generation of the quarks and the
leptons of the SM forms an $SO(10)$-spinor $F_{16}$, which is decomposed into
the $G_{\text{LR}}$ and $G_{\text{SM}}$ representations as

$\displaystyle F_{16}$
$\displaystyle\underset{M_{\text{GUT}}}{\longrightarrow}Q_{\text{L}}(3,2,1)_{1\over
3}+{Q_{\text{R}}}^{c}(\overline{3},1,2)_{-{1\over
3}}+L_{\text{L}}(1,2,1)_{-1}+{L_{\text{R}}}^{c}(1,1,2)_{1}$
$\displaystyle\underset{M_{\text{R}}}{\longrightarrow}q_{\text{L}}(3,2)_{1\over
6}+\left({d_{\text{R}}}^{c}(\overline{3},1)_{1\over
3}+{u_{\text{R}}}^{c}(\overline{3},1)_{-{2\over
3}}\right)+l_{\text{L}}(1,2)_{-{1\over
2}}+\bigl{(}{e_{\text{R}}}^{c}(1,1)_{1}+{\nu_{\text{R}}}^{c}(1,1)_{0}\bigr{)},$
(3)

where the subscript is the charges of $B-L$ and $Y$, respectively. To embed
$U(1)_{B-L}$ into $SO(10)$, we renormalize the charges so that the $U(1)$
gauge couplings are given by999With these normalizations, the Dynkin index of
the $U(1)$ gauge group of $F_{16}$ becomes $2$.

$\displaystyle\alpha_{Y}$ $\displaystyle={3\over 5}\alpha_{1}\quad\text{(below
$M_{\text{R}}$)},$ $\displaystyle\alpha_{B-L}$ $\displaystyle={3\over
8}\alpha_{1}\quad\text{(above $M_{\text{R}}$)}.$ (4)

In the LR symmetric model with only $SU(2)_{L,R}$ doublet Higgs bosons, the
Yukawa interactions in the SM are given by the higher dimensional operators in
Table 1. In the $SO(10)$ notation, they correspond to

$\mathcal{L}_{Y}=\frac{y_{u\,ij}}{\Lambda}\left(F_{16\,i}H^{*}_{16}\right)\left(F_{16\,j}H^{*}_{16}\right)+\frac{y_{d\,ij}}{\Lambda}\left(F_{16\,i}H_{16}\right)\left(F_{16\,j}H_{16}\right)+\text{h.c},$
(5)

where $i,j=1,2,3$ is the flavor index. $\Lambda$ is the cutoff scale.
Hereafter, we suppress the gauge and the flavor indices unless otherwise
stated. After the LR symmetry breaking, these operators contribute to the
Yukawa interactions: $y_{u}$ contributes to the up-type and neutrino ones,
while $y_{d}$ to the down-type and charged-lepton ones. In Table 1, the second
and the third columns represents the Yukawa interactions from the higher
dimensional operators in Eq. (5) in the representations of $G_{\text{LR}}$ and
$G_{\text{SM}}$, respectively.

Obviously, these contributions are too small to realize the observed masses of
the heavy flavor fermions in the SM for $\Lambda=M_{\text{GUT}}$, for example.
In fact, since the LR symmetry breaking scale $M_{\text{R}}$ is around
$10^{10}$–$10^{11}$ GeV, while $M_{\text{GUT}}=10^{16}$–$10^{17}$ GeV, the
coefficient of these operators are $\sim
M_{\text{R}}/\Lambda=10^{-7}$–$10^{-5}$, and hence we cannot realize the
Yukawa couplings for the second and third generations. To reproduce the
observed quark and lepton masses, we need to introduce extra vector-like
multiplets with masses of ${M_{\text{R}}}$ so that the terms in Eq. (5) are
generated by integrating out those extra multiplets.

In this paper, we assume that all the SM Yukawa interactions are generated by
integrating out extra vector-like multiplets. In this case, the minimal extra
vector-like fermions consist of three flavors of the fundamental
representation of $SO(10)$, $E_{10}$, and three flavors of the adjoint
representation of $SO(10)$, $E_{45}$.101010See [9, 10] for other
possibilities.. When the Yukawa interactions of the first generation are
provided by the $M_{\text{GUT}}$ suppressed operators, two flavors of $E_{10}$
and $E_{45}$ are enough to reproduce the SM Yukawa interactions. As discussed
in Sec. 5, however, the three flavor model is advantageous as the masses of
the extra vector-like fermions can be interrelated to the PQ symmetry
breaking. In what follows, we denote the number of extra particle flavors by
$N_{E}$.

With these extra matter multiplets, the origin of the Yukawa interactions in
Eq. (5) are obtained from the renormalizable interactions,

$\mathcal{L}_{\text{extra}}=y^{\prime}_{d}F_{16}E_{10}H_{16}+y^{\prime}_{u}F_{16}E_{45}H^{*}_{16}+M_{\text{extra}}E_{10}E_{10}+M_{\text{extra}}E_{45}E_{45}+\text{h.c,}$
(6)

where $M_{\text{extra}}$ is the extra particle mass. We assume that the mass
parameters for $E_{10}$ and $E_{45}$ are the same for simplicity. $E_{10}$ and
$E_{45}$ are decomposed into the $G_{\text{LR}}$ representations as

$\displaystyle E_{10}$
$\displaystyle\underset{M_{\text{GUT}}}{\longrightarrow}D^{(10)}(3,1,1)_{-\frac{2}{3}}+\overline{D}^{(10)}(\overline{3},1,1)_{\frac{2}{3}}+L_{\text{LR}}^{(10)}(1,2,2)_{0},$
(7) $\displaystyle E_{45}$
$\displaystyle\underset{M_{\text{GUT}}}{\longrightarrow}W_{L}^{(45)}(1,3,1)_{0}+W_{\text{R}}^{(45)}(1,1,3)_{0}+G^{(45)}(8,1,1)_{0}+N^{(45)}(1,1,1)_{0}$
$\displaystyle\phantom{\underset{M_{\text{GUT}}}{\longrightarrow}}+U^{(45)}(3,1,1)_{\frac{4}{3}}+\overline{U}^{(45)}(\overline{3},1,1)_{-\frac{4}{3}}+Q_{\text{LR}}^{(45)}(3,2,2)_{-\frac{2}{3}}+\overline{Q}_{\text{LR}}^{(45)}(\overline{3},2,2)_{\frac{2}{3}}.$
(8)

Here and hereafter, the overline on the extra fields denotes a charge
conjugation rather than a Dirac adjoint. By using the $G_{\text{LR}}$
representations, the Eq. (6) is decomposed as

$\displaystyle\mathcal{L}_{\text{extra}}$ $\displaystyle\supset
y^{\prime}_{d}Q_{\text{L}}H_{\text{L}}^{*}\overline{D}^{(10)}+y^{\prime}_{d}{L_{\text{R}}}^{c}H_{\text{L}}^{*}L_{\text{LR}}^{(10)}$
$\displaystyle\quad+y^{\prime}_{d}{Q_{\text{R}}}^{c}H_{\text{R}}D^{(10)}+y^{\prime}_{d}L_{\text{L}}H_{\text{R}}L_{\text{LR}}^{(10)}$
$\displaystyle\quad+y^{\prime}_{u}Q_{\text{L}}H_{\text{L}}\overline{U}^{(45)}+y^{\prime}_{u}L_{\text{L}}H_{\text{L}}N^{(45)}$
$\displaystyle\quad+y^{\prime}_{u}{Q_{\text{R}}}^{c}H_{\text{R}}^{*}U^{(45)}+y^{\prime}_{u}{L_{\text{R}}}^{c}H_{\text{R}}^{*}N^{(45)}.$
(9)

When we integrate out the extra particles, these contributions become the
higher dimensional operators which are summarized in Table 1. The resultant
Yukawa coupling constants in the SM are proportional to
$M_{\text{R}}/M_{\text{extra}}$, and hence, the top Yukawa coupling requires
the extra particle masses should be around LR symmetry breaking scale.

Several comments of the above minimal setup is summarized as follow; see also
[9, 10].

  * •

The large difference between the top mass and the other third generation one
is the most serious problem in realization of the observed fermion masses in
generic $SO(10)$ GUT models. This is not a matter in our model because there
are two origins of the Yukawa interactions.

  * •

Small difference between down-type quark masses and charged-lepton masses is
introduced by higher dimensional operators that come from $SO(10)$ breaking
effects [12].

  * •

The right-handed neutrino masses are around LR-breaking scale $M_{\text{R}}$.
As we assume that the LR-breaking scale is around $10^{10}$–$10^{12}$ GeV, the
masses of the active neutrinos generated by the seesaw mechanism [3] tend to
be much heavier than the observed ones. This is because the Dirac neutrino
Yukawa coupling for third-generation is $O(1)$ since it is unified with the
top Yukawa coupling.

  * •

When we assume that large mixings in the MNS matrix are realized, the CKM
matrix also should be a large mixing matrix because of the unification.
However, this does not satisfy experimental results.

We can solve the above problems by cancellation between the contributions from
the operators in Eq. (5) with some other higher-dimensional operators which
include the GUT breaking effects. The latter operators are suppressed by a
factor of $\langle H_{45}\rangle/\Lambda$. However, in this model, the
suppression factor is not so small even if $\Lambda$ is around the Planck
scale, $M_{\mathrm{Pl}}\sim 2.4\times 10^{18}$ GeV, as $\langle H_{45}\rangle$
can be as large as around $10^{17}$ GeV. By the cancellation, the small
neutrino Yukawa coupling can be achieved even for the $O(1)$ top Yukawa
coupling, and hence the active neutrino masses satisfy the experimental
results. The mixing matrices of the quarks and the neutrinos can also be
consistent with each other by cancellation.

##  3 Gauge Coupling Unification

In the previous section, we introduce extra fermions to achieve the Yukawa
interactions of the SM. In this section, we consider the renormalization group
(RG) flow of the gauge couplings including the contributions of those extra
matter multiplets as well as the $SU(2)_{\text{R}}$ doublet Higgs boson. We
assume for simplicity that the masses of the extra fermions and
$SU(2)_{\text{R}}$ doublet Higgs are $M_{\text{R}}$. As the extra fermions
makes the gauge coupling constants become rather strong at around the GUT
scale, it is important to take into account the two-loop contributions of the
gauge coupling constants to the RG flow; see e.g. Refs. [13]. The extra Yukawa
interactions to the two-loop RGE may slightly affect the precision of the
unification and the GUT gauge boson mass. Since those effects depend on the
detailed mass spectrum of the extra fermions, we neglect those contributions
in this paper.

The $\beta$ function of the gauge coupling $g_{a}$ is given by

$\displaystyle\beta_{g_{a}}$
$\displaystyle=\frac{1}{16\pi^{2}}a_{a}g_{a}^{3}+\frac{1}{(16\pi^{2})^{2}}b_{ab}g_{a}^{3}g_{b}^{2},$
(10)

where $a,b$ take values $1,2,3$ which refer to $U(1)_{\text{Y}}$,
$SU(2)_{\text{L}}$, and $SU(3)_{\text{C}}$ below $M_{\text{R}}$ and take
values 1, $2\text{L}$, $2\text{R}$, and 3 which refer to $U(1)_{B-L}$,
$SU(2)_{\text{L}}$, $SU(2)_{\text{R}}$, and $SU(3)_{\text{C}}$ above
$M_{\text{R}}$, respectively:

  * •

Above $M_{\text{R}}$, the coefficients of the gauge coupling beta functions
are

$\displaystyle a_{a}$
$\displaystyle=\left(a_{0}\right)_{a}+N_{E}\left(a_{10}\right)_{a}+N_{E}\left(a_{45}\right)_{a},$
$\displaystyle b_{ab}$
$\displaystyle=\left(b_{0}\right)_{ab}+N_{E}\left(b_{10}\right)_{ab}+N_{E}\left(b_{45}\right)_{ab},$
(11)

where each of $a_{0}$ and $b_{0}$ contains contributions from the SM particles
and the $SU(2)_{\text{R}}$ doublet Higgs; $a_{10}$ and $b_{10}$ from $E_{10}$;
$a_{45}$ and $b_{45}$ from $E_{45}$; and $N_{E}$ is the number of extra
particle pairs. These coefficients above $M_{R}$ are given by111111 There were
minor errors in the two-loop coefficients in the first arXiv versions of our
work and of Ref. [10]. The coefficients of the current version have been
cross-checked by the authors of Ref. [10].

$\displaystyle\left(a_{0}\right)_{a}$
$\displaystyle=\begin{pmatrix}\frac{9}{2}\\\ -\frac{19}{6}\\\ -\frac{19}{6}\\\
-7\end{pmatrix},$ $\displaystyle\left(a_{10}\right)_{a}$
$\displaystyle=\begin{pmatrix}\frac{2}{3}\\\ \frac{2}{3}\\\ \frac{2}{3}\\\
\frac{2}{3}\end{pmatrix},$ $\displaystyle\left(a_{45}\right)_{a}$
$\displaystyle=\begin{pmatrix}\frac{16}{3}\\\ \frac{16}{3}\\\ \frac{16}{3}\\\
\frac{16}{3}\end{pmatrix},$ (12) $\displaystyle\left(b_{0}\right)_{ab}$
$\displaystyle=\begin{pmatrix}\frac{23}{4}&\frac{27}{4}&\frac{27}{4}&4\\\
\frac{9}{4}&\frac{35}{6}&0&12\\\ \frac{9}{4}&0&\frac{35}{6}&12\\\
\frac{1}{2}&\frac{9}{2}&\frac{9}{2}&-26\end{pmatrix},$
$\displaystyle\left(b_{10}\right)_{ab}$
$\displaystyle=\begin{pmatrix}\frac{1}{3}&0&0&\frac{8}{3}\\\
0&\frac{49}{6}&\frac{3}{2}&0\\\ 0&\frac{3}{2}&\frac{49}{6}&0\\\
\frac{1}{3}&0&0&\frac{38}{3}\end{pmatrix},$
$\displaystyle\left(b_{45}\right)_{ab}$
$\displaystyle=\begin{pmatrix}\frac{20}{3}&6&6&\frac{64}{3}\\\
2&\frac{211}{3}&9&16\\\ 2&9&\frac{211}{3}&16\\\
\frac{8}{3}&6&6&\frac{334}{3}\end{pmatrix}.$ (13)

We set the $SU(2)_{\text{L}}$ and $SU(2)_{\text{R}}$ gauge couplings equal:
$g_{2\text{L}}=g_{2\text{R}}\equiv g_{2}$.

  * •

Below $M_{\text{R}}$, on the other hand, they are given by

$\displaystyle a_{a}$ $\displaystyle=\begin{pmatrix}\frac{41}{10}\\\
-\frac{19}{6}\\\ -7\end{pmatrix},$ $\displaystyle b_{ab}$
$\displaystyle=\begin{pmatrix}\frac{199}{50}&\frac{27}{10}&\frac{44}{5}\\\
\frac{9}{10}&\frac{35}{6}&12\\\ \frac{11}{10}&\frac{9}{2}&-26\end{pmatrix},$
(14)

which come only from the SM particle contribution.121212 Some of our two-loop
coefficients differ from those in Ref. [10] but consistent with those in Ref.
[13].

To calculate the RG flow for the gauge couplings, we consider the one-loop
matching condition at the renormalization scale,

$\left.\frac{1}{\alpha_{1}(M_{R})}\right|_{\text{Below
$M_{\text{R}}$}}=\left.\frac{3}{5}\frac{1}{\alpha_{2\text{R}}(M_{\text{R}})}+\frac{2}{5}\frac{1}{\alpha_{1}(M_{\text{R}})}\right|_{\text{Above
$M_{\text{R}}$}}-\frac{1}{2\pi}\frac{1}{10}.$ (15)

Recall that the value of gauge coupling for $SU(2)_{\text{R}}$ group is the
same as that for $SU(2)_{\text{L}}$ group above $M_{\text{R}}$:
$\alpha_{2\text{R}}=\alpha_{2\text{L}}\equiv\alpha_{2}$. As we are taking the
$\overline{\text{MS}}$ renormalization scheme, there is a mass independent
threshold correction in the right-hand side [14].131313For the
$\overline{\text{DR}}$ renormalization scheme, the mass independent threshold
correction is absent. In the following, we assume that the massive gauge boson
of $SU(2)_{\text{R}}\times U(1)_{B-L}$ and the extra matter multiplets
$E_{10,45}$ have the same mass of $M_{\text{R}}$ for simplicity. The
contributions of the extra matter do not affect the quality of the unification
significantly as long as they have $SO(10)$ consistent masses.141414The mass
scales of the extra vector-like matter affect the size of the GUT gauge
coupling constant. The mass splitting within the extra $SO(10)$ multiplets
also affect the precision of the gauge coupling unification at the GUT scale.

Figure 1: The RG flow of the gauge couplings for $M_{R}=10^{10}$ GeV,
$M_{R}=10^{11}$ GeV and $M_{\text{R}}=10^{12}$ GeV form left to right. Below
the LR symmetry breaking scale $M_{\text{R}}$, the purple, light-blue, and
blue lines refer to gauge couplings of $U(1)_{Y}$, $SU(2)_{\text{L}}$, and
$SU(3)_{\text{C}}$ gauge groups, respectively. Above $M_{\text{R}}$, the
purple line refers to the gauge coupling of the $U(1)_{B-L}$ group. $N_{E}=3$
(solid), $N_{E}=2$ (dashed), $N_{E}=2$ (dashed-dotted), and $N_{\text{E}}=0$
(dotted) pairs of the extra fermion are introduced.

In Figure 1, the RG flow of the gauge couplings is shown. The input values for
the RG flow are taken to be the central values of the experimental
measurements in [15]:

$\alpha(M_{W})$ | $\alpha_{3}(M_{Z})$ | $\sin^{2}\theta_{W}(M_{Z})$ |  $M_{Z}$ [GeV]
---|---|---|---
1/128 | 0.1181 | 0.23122 | 91.1876

Below the LR symmetry breaking scale $M_{\text{R}}$, the purple, light-blue,
and the blue lines refer to the gauge couplings for the $U(1)_{\text{Y}}$, the
$SU(2)_{\text{L}}$, and the $SU(3)_{\text{C}}$ groups, respectively. Above
$M_{\text{R}}$, the purple line refers to the gauge coupling for the
$U(1)_{B-L}$ group. $N_{E}=3$ (solid), $N_{E}=2$ (dashed), $N_{E}=1$ (dashed-
dotted), and $N_{E}=0$ (dotted) of the extra fermions are introduced. From the
left to right, we take the LR symmetry breaking scale, $M_{\text{R}}=10^{10}$
GeV, $M_{\text{R}}=10^{11}$ GeV, and $M_{\text{R}}=10^{12}$ GeV, respectively.

The figure shows that the gauge couplings of the LR symmetric model become
close with each other at around $10^{17}$–$10^{18}$ GeV for
$M_{\text{R}}=10^{10}$ GeV for $N_{E}\leq 2$. The three pairs of the extra
multiplets at $M_{\text{R}}=10^{10}$ GeV, on the other hand, leads to the
Landau pole before unification. The gauge couplings for $M_{\text{R}}=10^{11}$
GeV, on the other hand, meet well together before they hit the Landau pole.
There, we see that the two-loop contributions are not negligible with which
the RG flow becomes non-linear. The results for $M_{\text{R}}=10^{12}$ GeV
also show that the gauge couplings become close with each other moderately at
around $M_{\text{R}}=10^{15}$ GeV.

To quantify the quality of the unification, let us consider the matching
conditions between the gauge coupling constants in the LR symmetric model and
the $SO(10)$ gauge coupling, $\alpha_{G}=g_{G}^{2}/4\pi$:

$\displaystyle\frac{1}{\alpha_{1}(\mu,M_{R})}=\frac{1}{\alpha_{G}(\Lambda)}-\frac{1}{2\pi}\left(a_{1}\log\frac{\mu}{\Lambda}-{14}\log\frac{M_{X}}{\Lambda}-{14}\log\frac{M_{X^{\prime}}}{\Lambda}\right)-\frac{1}{2\pi}\frac{4}{3}+\frac{1}{2\pi}\Delta_{1},$
(16)
$\displaystyle\frac{1}{\alpha_{2}(\mu,M_{R})}=\frac{1}{\alpha_{G}(\Lambda)}-\frac{1}{2\pi}\left(a_{2}\log\frac{\mu}{\Lambda}-{21}\log\frac{M_{X}}{\Lambda}\right)-\frac{1}{2\pi}+\frac{1}{2\pi}\Delta_{2},$
(17)
$\displaystyle\frac{1}{\alpha_{3}(\mu,M_{R})}=\frac{1}{\alpha_{G}(\Lambda)}-\frac{1}{2\pi}\left(a_{3}\log\frac{\mu}{\Lambda}-{14}\log\frac{M_{X}}{\Lambda}-\frac{7}{2}\log\frac{M_{X^{\prime}}}{\Lambda}\right)-\frac{1}{2\pi}\frac{5}{6}+\frac{1}{2\pi}\Delta_{3}.$
(18)

The parameters $\mu$ and $\Lambda$ are the renormalization scale and the
cutoff scale at around the GUT scale. The mass parameter $M_{X}$ and
$M_{X^{\prime}}$ denotes the mass of the gauge boson in the $(3,2,2)_{-2/3}$
and $(3,1,1)_{-4/3}$ representations, respectively. For the symmetry breaking
path $SO(10)\to SU(3)\times SU(2)_{\text{L}}\times SU(2)_{\text{R}}\times
U(1)_{B-L}$ by the VEV of the Higgs boson in the 45 representation, it is
predicted that $M_{X^{\prime}}=2M_{X}$. The mass independent threshold
corrections are due to the $\overline{MS}$ renormalization scheme, which are
absent in the $\overline{DR}$ renormalization scheme. The parameters
${\Delta}_{1,2,3}$ represent the threshold corrections from some particles at
the GUT scale other than the GUT gauge bosons, although we do not specify them
in this paper.151515The parameters $\Delta_{1,2,3}$ also get contributions
from higher dimensional operators $\langle H_{45}\rangle/M_{\mathrm{Pl}}$,
although we assume that $\langle H_{45}\rangle/M_{\mathrm{Pl}}\ll\order{1}$.
See [9, 10] for various contributions of the GUT multiplets to
$\Delta_{1,2,3}$.

Figure 2: The quality of the unification $\bar{\Delta}$ as a function of
$(M_{X},\alpha_{G}^{-1})$. The upper and the lower panels are for $N_{E}=3$
and $N_{E}=2$, respectively. The quality of the unification is reasonably high
in the blue shaded region ($\bar{\Delta}<5$), while it is moderate in the
light-blue shaded region ($\bar{\Delta}<10$). The parameter $\bar{\Delta}$
gets contribution not only from the mass splittings of the GUT multiplets but
also from the mass difference between the GUT particles and $M_{X}$. The pink
shade region is excluded as $M_{X}$ is above the Landau pole of
$\alpha_{1,2,3}(\mu,M_{\text{R}})$. We confine ourselves to the region with
$M_{X}\ll M_{\mathrm{Pl}}$, so that the effective field theory without gravity
is valid.

As a measure of the quality of the unification, we define,

$\displaystyle\bar{\Delta}\equiv\max_{a=1,2,3}\left[\Delta_{a}\right],$ (19)

where we take $\mu=\Lambda=M_{X}$. The definition of $\bar{\Delta}$ is
different from the unification measure $\Delta$ defined in [9, 10]. The
parameter $\bar{\Delta}$ gets contribution not only from the mass splittings
of the GUT multiplets but also from the mass difference between the GUT
particles and $M_{X}$, while $\Delta$ in [9, 10] purely measures the precision
of the unification.

In Fig. 2, we show $\bar{\Delta}$ as a function of $(M_{X},\alpha_{G}^{-1})$
for a given $M_{\text{R}}$. The quality of the unification is reasonably high
in the blue shaded region ($\bar{\Delta}<5$), while it is moderate in the
light-blue shaded region ($\bar{\Delta}<10$). The figure shows that a
reasonable unification, i.e. $\bar{\Delta}<5$, is not possible for
$M_{\text{R}}={\mathcal{O}}(10^{10})\,$GeV due to the Landau pole for
$N_{E}=3$. The figure also shows that the unification is possible for a wide
range of the GUT gauge boson mass, $M_{X}=10^{15}$–$10^{17}$ GeV. These
results should be compared with the previous analyses of the gauge coupling
unification in the LR symmetric model which preferred
$M_{\text{R}}={\mathcal{O}}({10^{10}})$ GeV and $M_{X}={\mathcal{O}}(10^{17})$
GeV [6]. The difference of the results stem from the explicit inclusion of the
three flavors of the extra multiplet into the analysis of the RG flow.

For comparison, we also show $\bar{\Delta}$ for $N_{E}=2$. In this case, the
Landau pole is at the very high energy scale and does not exclude the
parameter region significantly. For $N_{E}=2$, more precise unification is
achieved for a lower $M_{\text{R}}$ and a higher $M_{X}$ than the case of
$N_{E}=3$. In such a parameter region, however, there is a tension with the
possibility to obtain the first generation Yukawa couplings as the higher
dimensional operators suppressed by $M_{\text{GUT}}$.

##  4 Proton Lifetime

In the present model, the exchanges of the massive gauge boson in the
$(3,2,2)_{-2/3}$ representation, i.e. the $X$-type gauge bosons, induce the
proton decay. Incidentally, the each of the $SU(2)_{\text{R}}$ doublet
component of the $X$ gauge boson belongs to the adjoint representation 24 and
the anti-symmetric representation 10 of the minimal $SU(5)$ GUT gauge
symmetry, respectively. In general setup of the $SO(10)$ GUT, they have
different masses (see e.g. [17]), while they are common in the LR symmetric
model. The massive gauge boson in the $(3,1,1)_{4/3}$ representation, on the
other hand, does not lead to the proton decay.

$G_{\text{LR}}$ | $G_{\text{SM}}$ | $SU(3)_{\text{C}}\times U(1)_{em}$
---|---|---
$\mathcal{O}^{(1)}\equiv(\overline{{L_{\text{L}}}^{c}}Q_{\text{L}})(\overline{{Q_{\text{R}}}^{c}}Q_{\text{R}})$ | $2(\overline{{l_{\text{L}}}^{c}}q_{\text{L}})(\overline{{u_{\text{R}}}^{c}}d_{\text{R}})$ | $\mathcal{O}^{1}\equiv 2(\overline{{e_{\text{L}}}^{c}}u_{\text{L}})(\overline{{u_{\text{R}}}^{c}}d_{\text{R}})$
$\mathcal{O}^{2}\equiv
2(\overline{{\nu_{\text{L}}}^{c}}d_{\text{L}})(\overline{{u_{\text{R}}}^{c}}d_{\text{R}})$
$\mathcal{O}^{(2)}\equiv(\overline{{L_{\text{R}}}^{c}}Q_{\text{R}})(\overline{{Q_{\text{L}}}^{c}}Q_{\text{L}})$ | $(\overline{{e_{\text{R}}}^{c}}u_{\text{R}})(\overline{{q_{\text{L}}}^{c}}q_{\text{L}})$ | $\mathcal{O}^{3}\equiv 2(\overline{{e_{\text{R}}}^{c}}u_{\text{R}})(\overline{{u_{\text{L}}}^{c}}d_{\text{L}})$
Table 2: The $B$ and $L$ violating operators mediated by the $X$ gauge boson.

After integrating out the $X$ gauge boson, the gauge interaction of the matter
field $F_{16}$ results in the $B$ and $L$ breaking operators
$\mathcal{O}^{(1,2)}$ in Table 2. Those operators are reduced to

$\displaystyle\mathcal{L}_{\text{eff}}=\frac{g_{G}^{2}}{M_{X}^{2}}\left\\{\left(\overline{{e_{\text{R}}}^{c}}u_{\text{R}}\right)\left(\overline{{q_{\text{L}}}^{c}}q_{\text{L}}\right)+\left(\overline{{l_{\text{L}}}^{c}}q_{\text{L}}\right)\left(\overline{{u_{\text{R}}}^{c}}d_{\text{R}}\right)\right\\}+\frac{g_{G}^{2}}{M_{X}^{2}}\left(\overline{{l_{\text{L}}}^{c}}q_{\text{L}}\right)\left(\overline{{u_{\text{R}}}^{c}}d_{\text{R}}\right),$
(20)

in terms of the $G_{\text{SM}}$ fields [18]; see also [19]. Below the
electroweak symmetry breaking scale, we may decompose it into the proton decay
operators in terms of the $SU(3)_{\text{C}}\times U(1)_{\text{em}}$ fields
such that $\mathcal{L}_{\text{eff}}=C^{I}\mathcal{O}^{I}$ as in Table 2. In
Eq. (20), we do not take account of the effects of the quark mixing angles
[20].161616In the present model, an SM fermion is a linear combination of the
spinor $F_{16}$ and the extra particles $E_{10}$ and $E_{45}$ [9, 10], and
therefore we should consider the proton decay operators which come from the
gauge interactions of the extra particles too, strictly speaking. However, we
have introduced the extra particles to realize the large Yukawa couplings,
while the Yukawa couplings of the first two generations are small. Therefore,
we expect that the contributions from extra particles are small for the first
generation, and thus we do not consider contribution from extra particles in
this paper. The proton decay operators which come from the gauge interaction
of the extra particle $E_{10}$ are summarized in Ref. [16].

The partial decay widths for the $p\to\pi^{0}e^{+}$ is given by

$\Gamma(p\to\pi^{0}e^{+})\simeq\frac{m_{p}}{32\pi}\left\\{1-\left(\frac{m_{\pi^{0}}}{m_{p}}\right)^{2}\right\\}^{2}\sum_{I=1,3}\left|C^{I}(m_{p})W_{0}^{I}\right|^{2},$
(21)

where $m_{p}$ and $m_{\pi^{0}}$ are the proton and the neutral pion masses,
respectively, and $W_{0}^{I}$ are the proton form factor. We may safely
approximate as
$\sum_{I=1,3}\left|C^{I}(m_{p})W_{0}^{I}\right|^{2}=\sum_{I=1,3}\left|C^{I}(m_{p})\right|^{2}W_{0}^{2}$.
In this calculation, $W_{0}$ for $p\to\pi^{0}e^{+}$ decay mode is $-0.131$
GeV2 which have been obtained by lattice simulation [21].

To calculate the coefficients of the proton decay operators at the proton mass
scale $m_{p}$, we have to consider the renormalization factor $A$. In this
paper, we consider the one-loop level renormalization factor from gauge
interactions. Here, we divide the energy region into two parts. The first
region is between the GeV scale and the LR-breaking scale $M_{\text{R}}$,
where the renormalization factor is written as $A_{\text{long}}$. The second
region is between the LR-breaking scale $M_{\text{R}}$ and the GUT scale
$M_{\text{GUT}}$, where renormalization factor is written as
$A_{\text{short}}$. The total renormalization factor $A$ is given by the
product of these factors, $A=A_{\text{long}}\times A_{\text{short}}$. We
calculate this renormalization factor for each of the proton decay operators
$\mathcal{O}^{(1)}$ and $\mathcal{O}^{(2)}$.

The one-loop level renormalization factor for each gauge group is given by

$A_{a}=\left(\frac{\alpha_{a}(M_{\text{start}})}{\alpha_{a}(M_{\text{end}})}\right)^{-\frac{C_{a}}{a_{a}}},$
(22)

where $M_{\text{end}}>M_{\text{start}}$; $a_{a}$ is the coefficient for
$\beta$ function for each gauge coupling which are shown in Eqs. (11) and
(14). $C_{a}$ is the factor appearing in the anomalous dimension $\gamma_{a}$
of the $a$-th gauge interaction for an each proton decay operator:

$\gamma_{a}=-2C_{a}\frac{g_{a}^{2}}{(4\pi)^{2}}.$ (23)

The coefficient $C_{a}$ is summarized in Ref. [22], with which the
renormalization factors are given by171717We use the six flavor RG equations
even below the electroweak scale. If we instead use the three flavor RG below
the electroweak scale, the $A_{\mathrm{long}}^{(1,2)}$ are slightly enhanced
by about $10$%.

$\displaystyle A_{\text{long}}^{(1)}$
$\displaystyle=\left(\frac{\alpha_{3}(1\,\text{GeV})}{\alpha_{3}(M_{\text{R}})}\right)^{-2/a_{3}}\left(\frac{\alpha_{2}(M_{Z})}{\alpha_{2}(M_{\text{R}})}\right)^{-\frac{9}{4}/a_{2}}\left(\frac{\alpha_{1}(M_{Z})}{\alpha_{1}(M_{\text{R}})}\right)^{-\frac{11}{12}/a_{1}},$
(24) $\displaystyle A_{\text{long}}^{(2)}$
$\displaystyle=\left(\frac{\alpha_{3}(1\,\text{GeV})}{\alpha_{3}(M_{\text{R}})}\right)^{-2/a_{3}}\left(\frac{\alpha_{2}(M_{Z})}{\alpha_{2}(M_{\text{R}})}\right)^{-\frac{9}{4}/a_{2}}\left(\frac{\alpha_{1}(M_{Z})}{\alpha_{1}(M_{\text{R}})}\right)^{-\frac{23}{12}/a_{1}},$
(25) $\displaystyle A_{\text{short}}^{(1)}=A_{\text{short}}^{(2)}$
$\displaystyle=\left(\frac{\alpha_{3}(M_{\text{R}})}{\alpha_{3}(M_{\text{GUT}})}\right)^{-2/a_{3}}\left(\frac{\alpha_{2}(M_{\text{R}})}{\alpha_{2}(M_{\text{GUT}})}\right)^{-2\cdot\frac{9}{4}/a_{2}}\left(\frac{\alpha_{1}(M_{\text{R}})}{\alpha_{1}(M_{\text{GUT}})}\right)^{-\frac{1}{4}/a_{1}}.$
(26)

In Eq. (26), we double $SU(2)_{\text{L}}$ contribution to include the
contribution from the $SU(2)_{\text{R}}$ gauge interaction. For $M_{R}\simeq
10^{11}$ GeV, $M_{X}\simeq 10^{16.5}$ GeV and $N_{E}=3$, for example, we find
that the renormalization factors are given by,181818For this choice, we find
$\alpha^{-1}_{1}(M_{X},M_{\text{R}})\simeq 6.9$,
$\alpha^{-1}_{2}(M_{X},M_{\text{R}})\simeq 7.3$, and
$\alpha^{-1}_{3}(M_{X},M_{\text{R}})\simeq 6.0$, respectively.

$\displaystyle A^{(1)}=A_{\text{long}}^{(1)}A_{\text{short}}^{(1)}\simeq
5.9,\quad A^{(2)}=A_{\text{long}}^{(2)}A_{\text{short}}^{(2)}\simeq 6.0.$ (27)

Figure 3: The black solid and black dashed lines are proton decay constraints
on the $p\to\pi^{0}e^{+}$ decay mode from current SK limit and the future HK
prospect. The grey shaded region is excluded by the current SK limit and the
region between black solid line and black dashed line will be explored by the
HK experiment.

In Figure 3, we overlay the current limit and the future prospects on the
proton lifetime for $p\to\pi^{0}e^{+}$ decay mode on Figure 2. The current
limit is the 90%CL exclusion limit by Super-kamiokande (SK) experiment,
$1.6\times 10^{34}$ years [23], which is shown as the black solid line. The
future prospects is the expected exclusion limit at $90$%C.L. of the Hyper-K
(HK) experiment, $1.3\times 10^{35}$ years [24], which is shown as the black
dashed line. The figure shows that some of the parameter region with moderate
coupling unification have been excluded by the current SK limit for
$M_{R}\gtrsim 10^{11.5}$ GeV ($N_{E}=3$). The figure also shows that the HK
experiment has a sensitivity to test large portion of the parameter space with
moderate coupling unification for $M_{\text{R}}=O(10^{11})$ GeV for
$N_{E}=2,3$.

##  5 Model with Peccei-Quinn Symmetry

In the minimal setup with $N_{E}=3$, we assume that all the SM Yukawa
interactions are generated by integrating out the extra vector-like multiplets
with masses around the LR-breaking scale. In this section, we briefly discuss
a possibility to generate those masses by the PQ symmetry breaking. The PQ
mechanism is one of the most successful solution to the Strong CP problem [25,
26].191919Alternatively, the strong CP problem can be solved in the LR
symmetric model by imposing space-time parity appropriately; see [9, 10] and
references therein.  There, the effective $\theta$-angle of QCD is canceled by
the VEV of the pseudo-Nambu-Goldstone boson, axion $a$, which is associated
with the spontaneous breaking of the PQ symmetry [27, 28]. The axion model not
only solve the strong CP problem, but also provides a good candidate for cold
dark matter [29, 30, 31, 32]; see also Ref. [33]. In fact, the axion dark
matter model is successful when the PQ breaking scale is of
$10^{11\text{--}12}$ GeV, which is close to the LR-breaking scale discussed in
this paper; see [34] for review. This coincidence motivates us to see how it
is successful to the mass scale of the extra vector-like fermions with the PQ
breaking scale.

For this purpose, let us introduce a gauge singlet complex scalar field, $P$,
which breaks the PQ symmetry at an intermediate scale. The PQ charge of $P$ is
defined to be $1$. Below the PQ breaking scale, the axion appears as a phase
component of $P$,

$\displaystyle P=\frac{1}{\sqrt{2}}f_{a}e^{ia/f_{a}},$ (28)

where $f_{a}$ is the decay constant of the axion. The PQ symmetry is realized
by the shift of $a$,

$\displaystyle{a\over f_{a}}\to{a^{\prime}\over f_{a}}={a\over
f_{a}}+\alpha,\quad(\alpha\in{\mathbb{R}})$ (29)

where the domain of the axion is given by $a/f_{a}=[-\pi,\pi)$.

To generate the extra fermion masses at the PQ scale, we assume that $P$
couples to $E_{10,45}$ via,

$\displaystyle{\cal L}=k_{10}PE_{10}E_{10}+k_{45}PE_{45}E_{45}+\text{h.c,}$
(30)

where $k_{10,45}$ are the coupling constants. Here, we assume that the PQ
charges of $E_{10}$ and $E_{45}$ are $-1/2$. In this case, the interaction
terms in Eq. (6) impose that the PQ charges of $F_{16}$ are $1/2$, while that
of $H_{16}$ is vanishing.202020We may consider a model in which $E_{10}$ and
$E_{45}$ have the opposite PQ charges. In this case the PQ charge of $F_{16}$
is vanishing, although the domain wall number is again $N_{\text{DW}}=-21$.

With these charge assignments, we find that the anomalous axion coupling to
QCD is given by,

$\displaystyle\mathcal{L}=\frac{g_{3}^{2}}{32\pi^{2}}N_{\text{DW}}\frac{a}{f_{a}}G\tilde{G},$
$\displaystyle N_{\text{DW}}=(2N_{F_{16}}-N_{E_{10}}-8N_{E_{45}})=-21.$ (31)

Here, $G$ and $\tilde{G}$ are the QCD field strength and its hodge dual,
respectively. $N_{F_{16}}=3$ is the number of generation of the SM fermions,
and $N_{E_{10}}=N_{E_{45}}=N_{E}=3$. The Lorentz and color indices are
suppressed. Below the QCD scale the anomalous coupling of the axion to QCD in
Eq. (31) leads to a non-vanishing axion potential and the axion settles down
to its minimum which solves the strong CP problem.212121Here, the origin of
the axion field space is taken to be the one at which the effective
$\theta$-angle of QCD is vanishing without loss of generality. As both the
extra fermions as well as the SM fermions possesses the PQ charges, this model
is in between the KSVZ [35, 36] and DFSZ [37, 38] invisible axion models, and
is in principle distinguishable from these models.

The coherent oscillation of the axion turns into the dark matter density [39],

$\displaystyle\Omega_{a}h^{2}\simeq
0.18\left(\frac{{\mit\Delta}a_{i}}{F_{\text{eff}}}\right)^{2}\left(\frac{F_{\text{eff}}}{10^{12}\,\text{GeV}}\right)^{1.19},$
(32)

where we have defined $F_{\text{eff}}=f_{a}/N_{\text{DW}}$.
${\mit\Delta}{a_{i}}/{F_{\text{eff}}}\in[-\pi,\pi)$ denotes the initial
misalignment angle of the axion from the $N_{\text{DW}}$ degenerate CP
conserving vacua. Therefore, the axion dark matter scenario is successful for
$F_{\text{eff}}\sim 10^{11\text{--}12}$ GeV for a typical initial misalignment
angle. In this present model, the PQ breaking scale is given by
$f_{a}=N_{\text{DW}}F_{\text{eff}}$, the axion dark matter prefers the PQ
breaking scale at $f_{a}\sim 10^{12\text{--}13}$ GeV. Accordingly, we find
that the extra multiplet masses at $M_{R}\sim 10^{11}$ GeV can be provided for
$k_{10,45}\sim 10^{-(1\text{--}2)}$ consistently with the axion dark matter
scenario. It should be emphasized that this scenario does not work for
$N_{E}=2$ since the higher dimensional operator to generate the SM Yukawa
interactions of the first generation explicitly break the PQ symmetry.222222We
may consider the PQ symmetry which is spontaneously broken at the cutoff scale
even for $N_{E}=2$. In such a case, however, the axion dark matter scenario is
not successful.

We argue that the axion in our setup is within the reach of future detection.
Due to the non-vanishing axion potential, the axion get a mass given by [40]

$\displaystyle m_{a}\simeq 5.7\,\mu{\rm eV}\left(\frac{10^{12}\,{\rm
GeV}}{F_{\rm eff}}\right).$ (33)

The axion also couples to photons through the electromagnetic anomaly
$N_{\mathrm{QED}}$ and thorough the mixing with neutral mesons. Many on-going
and future axion search experiments utilize the axion-photon coupling, which
is parameterized as

$\displaystyle\mathcal{L}\supset\frac{g_{a\gamma\gamma}}{4}aF\tilde{F},$ (34)

with [41] 232323The ratio $N_{\mathrm{QED}}/N_{\mathrm{DW}}=8/3$ is a generic
feature of the GUT consistent PQ charge assignment.

$\displaystyle g_{a\gamma\gamma}=\frac{\alpha_{\rm EM}}{2\pi F_{\rm
eff}}\left(\frac{N_{\mathrm{QED}}}{N_{\mathrm{DW}}}-1.92(4)\right)=\frac{\alpha_{\rm
EM}}{2\pi F_{\rm eff}}\left(\frac{8}{3}-1.92(4)\right).$ (35)

Note that $g_{a\gamma\gamma}$ in our model is equivalent to that in the DFSZ
axion model [37, 38], which is already excluded by the current ADMX experiment
for $m_{a}\simeq 2.7$ \- $3.3$ $\mu$eV [42, 43]. The higher mass range of
$m_{a}$ up to 400 $\mu$eV (corresponding to $F_{\rm eff}\sim 10^{11}$ GeV) is
expected to be covered by future cavity haloscopes such as ADMX [42], CULTASK
[44] and MADMAX [45]; see also Ref. [46, 47].

Several comments are in order. The axion potential induced by the anomalous
QCD coupling in Eq. (31) possesses ${\mathbb{Z}}_{N_{\text{DW}}}$ discrete
symmetry in the domain of the axion $a/f_{a}\in[-\pi,\pi)$, or equivalently in
$a/F_{\text{eff}}\in N_{\text{DW}}\times[-\pi,\pi)$. The discrete symmetry is
spontaneously broken by the VEV of the axion. Thus, the domain wall formation
takes place after the onset of the coherent oscillation of the axion, if the
initial misalignment angle in each Hubble volume of the Universe at that time
is random. Once the domain walls are formed, they immediately dominate the
Universe, which conflicts with the Standard Cosmology. To avoid this problem,
we need to assume that the PQ symmetry breaking takes place before inflation
and never gets restored after inflation. Under this assumption, the initial
misalignment angle of the axion is uniform in the entire Universe, and hence
the axion sits in the same sub-domain and evades the formation of the domain
wall.

We mention that the large domain wall number, $N_{\mathrm{DW}}=-21$, is
advantageous to avoid the PQ-symmetry restoration, since the actual PQ
breaking scale is an order of magnitude larger than the effective decay
constant $F_{\text{eff}}$ appropriate for the axion dark matter scenario, i.e.
$F_{\text{eff}}\sim 10^{11\text{--}12}$ GeV. Therefore, the present model can
be consistent with a cosmological scenario with higher reheating temperature
than in the conventional axion dark matter models. In this sense, the present
model can be more easily consistent with the thermal leptogenesis scenario
[48] which requires a rather high reheating temperature, $T_{R}\gtrsim
10^{9\mbox{-}10}$ GeV [49, 50, 51].242424For a given reheating temperature
$T_{R}$, the maximal temperature of the Universe of the thermal plasma during
the inflaton dominated era is in general much higher than $T_{R}$ up to
$T_{\rm Max}\sim(T_{R}^{2}H_{I}M_{\text{Pl}})^{1/4}$ [52, 53, 54]. Here,
$H_{I}$ is the Hubble parameter during inflation and $M_{\text{P}}$ is the
reduced Planck scale.

As another comment, the massless axion fluctuates quantum mechanically during
inflation, which leads to the isocurvature fluctuation of the axion dark
matter density when the PQ symmetry breaking takes place before inflation. The
dark matter isocurvature fluctuation have been severely constrained by the
precise measurements of the cosmic microwave background [55]. The amplitude of
the isocurvature fluctuation is proportional to the Hubble parameter during
inflation, $H_{I}$. As a result, $H_{I}$ is constrained from above as
$H_{I}\lesssim 10^{7\mbox{--}8}$ GeV to avoid the current constraint; see e.g.
Ref. [34, 56]. Therefore, the present scenario with the axion dark matter can
be refuted if the primordial $B$-mode polarization in the cosmic microwave
background is discovered in near future; see, e.g. Ref. [57, 59].

Finally, let us comment on the origin of the PQ symmetry. By definition, the
$U(1)$ PQ symmetry cannot be an exact symmetry as it is explicitly broken by
the QCD anomaly. Besides, it is also argued that any global symmetries are
broken by quantum gravity effects [58, 59, 60, 61, 62, 63]. When explicit
breaking terms exist, the effective $\theta$ angle of QCD is non-vanishing
even in the presence of the axion, which spoils the PQ mechanism. For example,
if the PQ symmetry is completely broken by the quantum gravity effects, it is
expected that there should be a PQ breaking term at least,

$\displaystyle{\cal L}_{\text{PQ--
breaking}}=\frac{P^{5}}{M_{\text{Pl}}}+\text{h.c.}$ (36)

which drastically affects the axion potential and spoils the PQ mechanism.

In the present model, however, we may regard that the discrete
$\mathbb{Z}_{2N_{\rm DW}}$ symmetry to be a discrete gauge symmetry as it can
satisfy the anomaly free conditions [64].252525Here, we normalize the charges
of the discrete symmetry so that the extra vector-like multiplets have a
charge $-1$. Accordingly, the PQ breaking field $P$ possesses the discrete
charge $2$. If ${\mathbb{Z}}_{2N_{\rm DW}}$ symmetry is a gauge symmetry, the
lowest dimensional operator which breaks the $U(1)$ PQ symmetry but is
invariant under the $\mathbb{Z}_{2N_{\rm DW}}$ gauge symmetry is given by,

$\displaystyle{\cal L}_{\text{PQ--
breaking}}=\frac{P^{21}}{M_{\text{Pl}}^{17}}+\text{h.c.}$ (37)

which is highly suppressed and does not spoil the PQ mechanism; see e.g. Ref.
[65]. This argument strengthens the PQ mechanism in the present model.262626We
discus the domain wall problem in Appendix A. As another possible
justification of the $U(1)$ PQ symmetry, we may consider a $U(1)$ gauge
symmetry with an accidental global $U(1)$ PQ symmetry [66], where $E_{10}$ and
$E_{45}$ couple to a different PQ charged complex scalars: see Refs. [67, 68,
69] for details.

##  6 Summary

In this paper, we have investigated the proton lifetime in the $SO(10)$ GUT
which is broken down by the VEV of $H_{45}$ to the minimal LR-symmetric gauge
group $SU(3)_{\text{C}}\times SU(2)_{\text{L}}\times SU(2)_{\text{R}}\times
U(1)_{Y}$, which is in turn broken at the intermediate LR-breaking scale
$M_{\text{R}}$ by the $SU(2)_{\text{R}}$ doublet Higgs that is a part of
$H_{16}$. The $SU(2)_{\text{L}}$ doublet component of the same $H_{16}$ field
eventually plays the role of the SM Higgs doublet. Due to the absence of the
bi-doublet Higgs boson, the LR-breaking scale is determined to be at around
$10^{10\text{--}12}$ GeV in order to achieve the gauge coupling unification.

As a notable feature of the model, it requires extra vector-like fermions to
generate the SM Yukawa interactions. Such extra multiplets affect the RG flow,
and lower the unification scale down to $M_{X}\lesssim 10^{17}$ GeV from that
expected in Refs. [5, 6] by a factor a few or so. We have also found that the
Wilson coefficients of the proton decay operators are considerably larger than
those in the minimal $SU(5)$ GUT model. As a result, the proton decay rate is
enhanced and we find that some portion of the parameter space consistent with
the gauge coupling unification can be tested by the Hyper-K experiment
thorough the proton decay search even when the GUT gauge boson mass is in the
range $10^{16\text{--}17}$ GeV.

We also discussed a possibility to generate the mass of the extra vector-like
multiplets by the PQ symmetry breaking. We found that the axion dark matter
scenario and the present model can be successfully combined for the model with
$N_{E}=3$. This combination can be tested by the proton decay search, the
axion search and the search for the primordial $B$-mode fluctuation in the
cosmic microwave background.

## Acknowledgments

We thank Shigeki Matsumoto, Kyohei Mukaida, and Kenichi Saikawa for useful
discussion. We are grateful to Keisuke Harigaya for valuable comments and
cross-checks. This work is supported in part by JSPS KAKENHI Grant Nos.
15H05889, 16H03991, 17H02878, 18H05542 (M.I.), 19H01899 (K.O.) 15H05889,
15K21733, and 17H02875 (N.Y.); World Premier International Research Center
Initiative (WPI Initiative), MEXT, Japan (M.I.); National Natural Science
Foundation of China (NNSFC) under Contracts Nos. 11675061, 11775092, 11521064,
and 11435003 (Y.M.); and the European Union via the Advanced ERC grant SM-
grav, No 669288 (Y.H.).

## Appendix

##  Appendix A Discrete Gauge Symmetry and the Domain Wall Problem

In Sec. 5, we considered a discrete gauge symmetry which explains the origin
of the approximate global $U(1)$ PQ symmetry. In this appendix, we briefly
comment on the domain wall problem in the presence of the discrete gauge
symmetry behind the PQ symmetry. In this set up, the $N_{\mathrm{DW}}$ axion
domains in $a/F_{\mathrm{eff}}=N_{\mathrm{DW}}\times[-\pi,\pi)$ are gauge
equivalent with each other, and hence, the axion domain wall configurations
which connects different domains are not completely stable. As we will see,
however, the axion domain wall problem remains even in the model with the
discrete gauge symmetry.

To make our discussion concrete, let us assume that the discrete
$\mathbb{Z}_{N_{DW}}$ gauge symmetry originates from a $U(1)$ gauge symmetry
broken by the VEV of a complex scalar $\Phi$ whose gauge charge is large,
$N_{\mathrm{DW}}\gg 1$. Note that this $U(1)$ gauge symmetry is different from
the global $U(1)$ PQ symmetry. The $U(1)$ gauge charge of the PQ breaking
field $P$ is $1$ as in Sec. 5.272727To make the $U(1)$ gauge symmetry anomaly
free, we need to introduce additional SM charged fermions (see e.g. [67, 68,
69]), although they do not affect the following discussion. Here, we only pay
attention to the complex scalars where $\mathbb{Z}_{N_{\mathrm{DW}}}$ in stead
of $\mathbb{Z}_{2N_{\mathrm{DW}}}$ is good enough for the following
discussion.  The VEV of the PQ breaking field $P$ eventually breaks the
$\mathbb{Z}_{N_{\mathrm{DW}}}$ symmetry.

In this model, the stable topological defect is not the domain wall but the
local strings which are associated with the spontaneous $U(1)$ gauge symmetry
breaking. For example, a cosmic local string around which the phase of $\Phi$
winds from $0$–$2\pi$ are expected to be formed when $\Phi$ obtains a VEV at a
very high energy scale. The phase of the PQ breaking field $P$ is changed by
$2\pi/N_{\mathrm{DW}}$ under the parallel transport around this local string,
which corresponds to the Aharanov-Bohem effect.

Now let us assume that the spontaneous symmetry breaking of the $U(1)$ gauge
symmetry by the VEV of $\Phi$ takes place well before inflation, while the
approximate global PQ symmetry breaking occurs after inflation. In this case,
the cosmic local strings that are formed when $\Phi$ obtains a VEV have been
diluted away by cosmic inflation. After inflation, the cosmic temperature
decreases below the PQ breaking scale. Then, associated with the spontaneous
breaking of the approximate global $U(1)$ PQ symmetry, a few cosmic global
strings are expected to be formed in each Hubble volume. Note that these
global strings are different from the ones diluted away during the inflation.
When we turn around the global string, the phase of the PQ field $P$ takes
values from $0$ to $2\pi$ when the winding number is one, and hence, the axion
field takes values from $0$ to $f_{a}\times 2\pi=N_{\mathrm{DW}}\times
F_{\mathrm{eff}}\times 2\pi$.

Around the global string, the $[0,N_{\mathrm{DW}}\times F_{\mathrm{eff}}\times
2\pi)$ region has $N_{\text{DW}}$ domains that are gauge equivalent under the
$\mathbb{Z}_{N_{\text{DW}}}$. Since the approximate $U(1)$ PQ symmetry is
highly protected by the $\mathbb{Z}_{N_{\text{DW}}}$ symmetry, the tension of
the domain walls connecting the $N_{\text{DW}}$ domains is negligibly small.
Therefore, we have no domain wall problem associated with the
$\mathbb{Z}_{N_{\text{DW}}}$ symmetry breaking by $\langle P\rangle\neq 0$.
When the cosmic temperature decreases further, the cosmic global string
networks follow the so-called scaling solution where the number of the cosmic
global strings in each Hubble volume at that time remains of $\order{1}$; see
e.g. [70]. When the axion potential is generated at around the QCD scale
$\Lambda_{\mathrm{QCD}}$, potential barriers appear around each global string
which result in $N_{\mathrm{DW}}$ domain walls whose boundary is the global
string.

As mentioned earlier, each domain wall attached to the global string connects
different domains which are gauge equivalent under the discrete
$\mathbb{Z}_{N_{\mathrm{DW}}}$ symmetry. Therefore, this domain wall is not
completely stable. In fact, each wall can be punctured by a loop of the
earlier mentioned local string, around which the phase of $\Phi$ winds from
$0$–$2\pi$, since this local string connects the different axion domains
without potential barrier. Once the domain wall is punctured, the loop of
local string expands on the domain wall, and the domain wall disappears
eventually. The rate of such a puncturing process, however, is highly
suppressed, since the formation of the loop of the local string is suppressed
by
$e^{-|\langle{\Phi}\rangle|^{4}/\Lambda_{\mathrm{QCD}}^{2}F_{\mathrm{eff}}T}$
at a temperature below the QCD scale: $T\lesssim\Lambda_{\mathrm{QCD}}$.282828
For a loop of a radius $\ell$, the removed wall energy is $\sim\ell^{2}\times
m_{a}F_{\text{eff}}^{2}$, while the string energy cost is
$\sim\ell\times|\langle\Phi\rangle|^{2}$. Therefore we need at least
$\ell\gtrsim|\langle\Phi\rangle|^{2}/m_{a}F_{\text{eff}}^{2}$. Multiplying the
string tension $|\langle\Phi\rangle|^{2}$ and putting
$m_{a}\sim\Lambda_{\text{QCD}}^{2}/F_{\text{eff}}$, the lowest mass of the
string loop which can puncture the domain wall is
$|\langle\Phi\rangle|^{4}/\Lambda_{\text{QCD}}^{2}F_{\text{eff}}$.  As a
result, the domain wall is virtually stable below the QCD scale and they
immediately dominate over the energy density of the universe, which causes the
domain wall problem.292929For a study of cosmic string formation when both
$\Phi$ and $P$ obtain VEVs after inflation, see Ref. [71].

## References

  * [1] H. Georgi and S. L. Glashow, Phys. Rev. Lett. 32, 438 (1974).
  * [2] H. Georgi, AIP Conf. Proc. 23, 575 (1975); H. Fritzsch and P. Minkowski, Annals Phys. 93, 193 (1975).
  * [3] P. Minkowski, Phys. Lett. B67, 421 (1977); T. Yanagida, in Proceedings of the Workshop on Unified Theory and Baryon Number of the Universe, eds. O. Sawada and A. Sugamoto (KEK, 1979) p.95; M. Gell-Mann, P. Ramond, and R. Slansky, in Supergravity, eds. P. van Niewwenhuizen and D. Freedman (North Holland, Amsterdam, 1979); S. L. Glashow, in Proceedings of the Cargése Summer Institute on Quarks and Leptons, Cargése, July 9–29, 1979, eds. M. Lévy et al. (Plenum, 1980, New York), p. 707; R. N. Mohapatra and G. Senjanovic, Phys. Rev. Lett. 44 (1980) 912.
  * [4] J. C. Pati and A. Salam, Phys. Rev. D 10, 275 (1974), Erratum: [Phys. Rev. D 11, 703 (1975)]; R. N. Mohapatra and J. C. Pati, Phys. Rev. D 11, 566 (1975); R. N. Mohapatra and J. C. Pati, Phys. Rev. D 11, 2558 (1975); G. Senjanovic and R. N. Mohapatra, Phys. Rev. D 12, 1502 (1975); R. N. Mohapatra, F. E. Paige and D. P. Sidhu, Phys. Rev. D 17, 2462 (1978); G. Senjanovic, Nucl. Phys. B 153, 334 (1979); R. N. Mohapatra and R. E. Marshak, Phys. Rev. Lett. 44, 1316 (1980), Erratum: [Phys. Rev. Lett. 44, 1643 (1980)].
  * [5] D. Chang, R. N. Mohapatra, J. Gipson, R. E. Marshak and M. K. Parida, Phys. Rev. D 31, 1718 (1985).
  * [6] F. Siringo, Phys. Part. Nucl. Lett. 10, no. 2, 94 (2013).
  * [7] S. Rajpoot, Phys. Rev. D 22, 2244 (1980).
  * [8] M. Fukugita and T. Yanagida, Berlin, Germany: Springer (2003) 593 p.
  * [9] L. J. Hall and K. Harigaya, JHEP 1810, 130 (2018).
  * [10] L. J. Hall and K. Harigaya, JHEP 1911 (2019) 033.
  * [11] Yuta Hamada, Masahiro Ibe, Shigeki Matsumoto, Kyohei Mukaida, Kin-ya Oda, Kenichi Saikawa, and Norimi Yokozaki, “LR-UNIFICATION,” YITP Workshop on “LHC vs Beyond the Standard Model—Frontier of particle physics,” 19–25 March 2013, Yukawa Institute, Japan.
  * [12] J. R. Ellis and M. K. Gaillard, Phys. Lett. 88B, 315 (1979).
  * [13] D. R. T. Jones, Phys. Rev. D 25, 581 (1982); M. E. Machacek and M. T. Vaughn, Nucl. Phys. B 222, 83 (1983); H. Arason, D. J. Castano, B. Keszthelyi, S. Mikaelian, E. J. Piard, P. Ramond and B. D. Wright, Phys. Rev. D 46, 3945 (1992).
  * [14] S. Weinberg, Phys. Lett. 91B, 51 (1980).
  * [15] M. Tanabashi et al. [Particle Data Group], Phys. Rev. D 98, no. 3, 030001 (2018).
  * [16] N. Maekawa and Y. Muramatsu, Phys. Rev. D 88, no. 9, 095008 (2013); N. Maekawa and Y. Muramatsu, Phys. Lett. B 767, 398 (2017).
  * [17] N. Haba, Y. Mimura and T. Yamada, JHEP 1907, 155 (2019).
  * [18] S. Bertolini, L. Di Luzio and M. Malinsky, Phys. Rev. D 87, no. 8, 085020 (2013).
  * [19] P. Langacker, Phys. Rept. 72 (1981) 185.
  * [20] J. R. Ellis, M. K. Gaillard, D. V. Nanopoulos and S. Rudaz, Nucl. Phys. B 176, 61 (1980).
  * [21] Y. Aoki, T. Izubuchi, E. Shintani and A. Soni, Phys. Rev. D 96, no. 1, 014506 (2017).
  * [22] W. E. Caswell, J. Milutinovic and G. Senjanovic, Phys. Rev. D 26, 161 (1982).
  * [23] K. Abe et al. [Super-Kamiokande Collaboration], Phys. Rev. D 95, no. 1, 012004 (2017).
  * [24] K. Abe et al., arXiv:1109.3262 [hep-ex].
  * [25] R. D. Peccei and H. R. Quinn, Phys. Rev. Lett. 38, 1440 (1977).
  * [26] R. D. Peccei and H. R. Quinn, Phys. Rev. D 16, 1791 (1977).
  * [27] S. Weinberg, Phys. Rev. Lett. 40, 223 (1978).
  * [28] F. Wilczek, Phys. Rev. Lett. 40, 279 (1978).
  * [29] M. Dine and W. Fischler, Phys. Lett. 120B, 137 (1983).
  * [30] L. F. Abbott and P. Sikivie, Phys. Lett. 120B, 133 (1983).
  * [31] J. Preskill, M. B. Wise and F. Wilczek, Phys. Lett. 120B, 127 (1983).
  * [32] J. Preskill, S. P. Trivedi, F. Wilczek and M. B. Wise, Nucl. Phys. B 363, 207 (1991).
  * [33] M. I. Vysotsky, Y. B. Zeldovich, M. Y. Khlopov and V. M. Chechetkin, Pisma Zh. Eksp. Teor. Fiz. 27 (1978) 533 [JETP Lett. 27 (1978) 502]; Z. G. Berezhiani, M. Y. Khlopov and R. R. Khomeriki, Sov. J. Nucl. Phys. 52 (1990) 65 [Yad. Fiz. 52 (1990) 104]; Z. G. Berezhiani and M. Y. Khlopov, Z. Phys. C 49 (1991) 73; Z. G. Berezhiani, A. S. Sakharov and M. Y. Khlopov, Sov. J. Nucl. Phys. 55 (1992) 1063 [Yad. Fiz. 55 (1992) 1918]; A. S. Sakharov and M. Y. Khlopov, Phys. Atom. Nucl. 57 (1994) 485 [Yad. Fiz. 57 (1994) 514]; A. S. Sakharov, D. D. Sokoloff and M. Y. Khlopov, Phys. Atom. Nucl. 59 (1996) 1005 [Yad. Fiz. 59N6 (1996) 1050]; M. Y. Khlopov, A. S. Sakharov and D. D. Sokoloff, Nucl. Phys. Proc. Suppl. 72 (1999) 105.
  * [34] M. Kawasaki and K. Nakayama, Ann. Rev. Nucl. Part. Sci. 63, 69 (2013).
  * [35] J. E. Kim, Phys. Rev. Lett. 43, 103 (1979).
  * [36] M. A. Shifman, A. I. Vainshtein and V. I. Zakharov, Nucl. Phys. B 166, 493 (1980).
  * [37] A. R. Zhitnitsky, Sov. J. Nucl. Phys. 31, 260 (1980) [Yad. Fiz. 31, 497 (1980)].
  * [38] M. Dine, W. Fischler and M. Srednicki, Phys. Lett. 104B, 199 (1981).
  * [39] M. S. Turner, Phys. Rev. D 33, 889 (1986).
  * [40] M. Gorghetto and G. Villadoro, JHEP 1903, 033 (2019).
  * [41] G. Grilli di Cortona, E. Hardy, J. Pardo Vega and G. Villadoro, JHEP 1601, 034 (2016).
  * [42] N. Du et al. [ADMX Collaboration], Phys. Rev. Lett. 120, no. 15, 151301 (2018).
  * [43] T. Braine et al. [ADMX Collaboration], arXiv:1910.08638 [hep-ex].
  * [44] E. Petrakou [CAPP/IBS Collaboration], EPJ Web Conf. 164, 01012 (2017).
  * [45] A. Caldwell et al. [MADMAX Working Group], Phys. Rev. Lett. 118, no. 9, 091801 (2017).
  * [46] P. W. Graham, I. G. Irastorza, S. K. Lamoreaux, A. Lindner and K. A. van Bibber, Ann. Rev. Nucl. Part. Sci. 65, 485 (2015).
  * [47] M. Lawson, A. J. Millar, M. Pancaldi, E. Vitagliano and F. Wilczek, Phys. Rev. Lett. 123 (2019) no.14, 141802.
  * [48] M. Fukugita and T. Yanagida, Phys. Lett. B 174, 45 (1986).
  * [49] G. F. Giudice, A. Notari, M. Raidal, A. Riotto and A. Strumia, Nucl. Phys. B 685, 89 (2004).
  * [50] W. Buchmuller, R. D. Peccei and T. Yanagida, Ann. Rev. Nucl. Part. Sci. 55, 311 (2005).
  * [51] S. Davidson, E. Nardi and Y. Nir, Phys. Rept. 466, 105 (2008).
  * [52] E. W. Kolb and M. S. Turner, Front. Phys. 69, 1 (1990).
  * [53] J. Yokoyama, Phys. Rev. D 70, 103511 (2004).
  * [54] K. Harigaya and K. Mukaida, JHEP 1405, 006 (2014).
  * [55] Y. Akrami et al. [Planck Collaboration], arXiv:1807.06211 [astro-ph.CO].
  * [56] M. Kawasaki, E. Sonomoto and T. T. Yanagida, Phys. Lett. B 782, 181 (2018).
  * [57] K. N. Abazajian et al. [CMB-S4 Collaboration], arXiv:1610.02743 [astro-ph.CO].
  * [58] S. W. Hawking, Phys. Lett. B 195, 337 (1987).
  * [59] G. V. Lavrelashvili, V. A. Rubakov and P. G. Tinyakov, JETP Lett. 46, 167 (1987) [Pisma Zh. Eksp. Teor. Fiz. 46, 134 (1987)].
  * [60] S. B. Giddings and A. Strominger, Nucl. Phys. B 307, 854 (1988).
  * [61] S. R. Coleman, Nucl. Phys. B 310, 643 (1988).
  * [62] G. Gilbert, Nucl. Phys. B 328, 159 (1989).
  * [63] T. Banks and N. Seiberg, Phys. Rev. D 83, 084019 (2011).
  * [64] C. Csaki and H. Murayama, Nucl. Phys. B 515, 114 (1998).
  * [65] L. M. Carpenter, M. Dine and G. Festuccia, Phys. Rev. D 80, 125017 (2009).
  * [66] S. M. Barr and D. Seckel, Phys. Rev. D 46, 539 (1992).
  * [67] H. Fukuda, M. Ibe, M. Suzuki and T. T. Yanagida, Phys. Lett. B 771, 327 (2017).
  * [68] H. Fukuda, M. Ibe, M. Suzuki and T. T. Yanagida, JHEP 1807, 128 (2018).
  * [69] M. Ibe, M. Suzuki and T. T. Yanagida, JHEP 1808, 049 (2018).
  * [70] A. Vilenkin and E. P. S. Shellard, Cambridge University Press.
  * [71] T. Hiramatsu, M. Ibe and M. Suzuki, arXiv:1910.14321 [hep-ph].