#  HUPD-2215 Mass Relations, Unification, and Proton Decay
in the Mirror GUT Model

Yusuke Shimizu<EMAIL_ADDRESS>and Shonosuke Takeshita
<EMAIL_ADDRESS>

1Physics Program, Graduate School of Advanced Science
and Engineering, Hiroshima University,
Higashi-Hiroshima 739-8526, Japan
2Core of Research for the Energetic Universe, Hiroshima University,
Higashi-Hiroshima 739-8526, Japan

( Abstract
We propose an SU(5)$\times$U(1)${}_{\text{X}}\times$U(1)${}_{\text{PQ}}$ model
without SUSY because the SUSY particles have not been observed yet. The
$\mathrm{U(1)_{X}}$ gauge symmetry is the generalization of the
$\mathrm{U(1)_{B-L}}$ gauge symmetry and $\mathrm{U(1)_{PQ}}$ symmetry is the
global Peccei-Quinn symmetry. We introduce three mirror families in order to
unify the SM gauge couplings and avoid the restriction of proton lifetime. In
order to obtain the difference of the masses for the down-type quarks and
charged leptons, we also introduce the $\mathbf{45}$ representation Higgs in
addition to the $\mathbf{5}$ representation one. In this paper, we discuss the
mass relations between the SM and mirror particles and identify the mass
scales of the mirror particles. Since the new particles exist in the
intermediate energy scale and contribute to the renormalization group
equation, the SM gauge couplings unify successfully at high energy. Our model
can be tested by the future proton decay search, e.g., the Hyper-Kamiokande
experiment expected as $\tau_{p}(p\to\pi^{0}{e^{+}})<1.0\times 10^{35}$ years.
Also, by using the mass relations between the active and mirror neutrinos, we
estimate the lower bound of the heavy neutrino masses. )

##  1 Introduction

The standard model (SM) is the successful one which includes
$\mathrm{SU(3)_{C}}$, $\mathrm{SU(2)_{L}}$, and $\mathrm{U(1)_{Y}}$ gauge
symmetries. The SM has been completed with the discovery of the Higgs boson in
2012 [1, 2]. However, the SM cannot be explained the origins of the charges
and flavor structures for the SM particles and so on. Then, we need new
physics beyond the SM to solve these problems. One of the attractive theories
is the grand unified theory (GUT). The main feature of the GUT is the
unification of strong, weak, and electromagnetic forces. The minimal model of
the GUT is the SU(5) one [3]. In the minimal SU(5) model, the SM matter fields
and gauge bosons are unified into ${\bf\bar{5}}$, ${\bf 10}$, and ${\bf 24}$
adjoint representations, respectively. Thanks to introducing the $\mathbf{45}$
representation Higgs field [4, 5] in this model, we obtain the difference of
the masses for the down-type quarks and charged leptons [6]. In addition to
the unification of the SM particles and gauge interactions, the GUT predicts
proton decay. Then, the GUT can be tested by the proton decay search. However,
there are several problems with the GUT. First, the proton lifetime expected
by the GUT is inconsistent with the current proton decay search results, for
example, the Super-Kamiokande experiment reported
$\tau_{p}(p\to\pi^{0}{e^{+}})>2.4\times 10^{34}$ years [7]. Especially, the
minimal SU(5) GUT model expects the proton lifetime such as
$\tau_{p}(p\to\pi^{0}{e^{+}})\approx 10^{30}$-$10^{31}$ years [8]. On the
other hand, by solving the renormalization group equations (RGE), we can get
the SM gauge couplings at the high energy scale. However, the SM gauge
couplings do not unify successfully at high energy. This is another problem
with the GUT. Considering the supersymmetry (SUSY) in the theory, the SM gauge
couplings unify successfully at high energy [9, 10, 11, 12, 13, 14, 15]. Since
the SUSY particles have not been observed yet in the experiments, we need some
extensions of the GUT model except for SUSY.

In this paper, we propose an
SU(5)$\times$U(1)${}_{\text{X}}\times$U(1)${}_{\text{PQ}}$ model without SUSY.
The $\mathrm{U(1)_{X}}$ gauge symmetry [16] is the generalization of the
$\mathrm{U(1)_{B-L}}$ (baryon minas lepton number) gauge symmetry [17, 18, 19,
20, 21, 22, 23]. The definition of the $\mathrm{U(1)_{X}}$ charge is
$Q_{\mathrm{X}}=x_{H}~{}Q_{\mathrm{Y}}+Q_{\mathrm{B-L}}$, where
$Q_{\mathrm{Y}}$ is hypercharge, $Q_{\mathrm{B-L}}$ is B-L charge, and $x_{H}$
is a free parameter [24]. For $x_{H}=-4/5$ [25], the $\mathrm{U(1)_{X}}$
charge can be assigned to the ${\bf\bar{5}}$ and ${\bf 10}$ representations
successfully. The three gauge singlet right-handed Majorana neutrinos cancel
all the $\mathrm{U(1)_{X}}$ related anomalies. The $\mathrm{U(1)_{PQ}}$
symmetry is the global Peccei-Quinn (PQ) symmetry [26, 27]. The PQ symmetry
can solve the strong CP problem [28] and the pseudo-Nambu Goldstone boson from
the PQ symmetry breaking called the axion is the dark matter (DM) candidate
[29, 30]. In previous work [31], they added one family of the same
representation of the SM particles and one mirror family. The mirror family is
the conjugate of the representation for the SM particles. In our model, we
introduce only three mirror families [32, 33, 34] in order to unify the SM
gauge couplings successfully and avoid the restriction of proton lifetime. In
order to obtain the difference of the masses for the down-type quarks and
charged leptons, we also introduce the $\mathbf{45}$ representation Higgs in
addition to the $\mathbf{5}$ representation one. Then, we discuss the mass
relations between the SM and mirror particles and identify the mass scales of
the mirror particles. We set the benchmark for the mass scales of the mirror
particles in our analysis. Since the new particles exist in the intermediate
energy scale and contribute to the RGE, the SM gauge couplings unify
successfully at $M_{\mathrm{GUT}}\approx{7.28}\times 10^{15}$ GeV and
$\alpha_{\mathrm{GUT}}=\alpha_{1}=\alpha_{2}=\alpha_{3}\approx{1/31.7}$. Our
model expects the proton lifetime as
$\tau_{p}(p\to\pi^{0}{e^{+}})\approx{8.07\times 10^{34}}~{}\mathrm{years}$ and
can be tested by the future proton decay search, for example, the Hyper-
Kamiokande experiment expected as $\tau_{p}(p\to\pi^{0}{e^{+}})<1.0\times
10^{35}$ years [35]. Also, by using the mass relations between the active and
mirror neutrinos, we estimate the lower bound of the heavy neutrino masses.
The lowest bound of the heavy neutrino masses is
${m_{\mathrm{heavy}}}>104.34~{}\mathrm{GeV}$ and it is testable for the future
experiment about the sterile neutrinos [36].

This paper is organized as follows. In section 2, we propose an
SU(5)$\times$U(1)${}_{X}\times$U(1)${}_{\text{PQ}}$ model and discuss the
scenario of the spontaneous symmetry breaking with vacuum expectation values
(VEVs) of the scalar fields. We also discuss the Yukawa couplings for the SM
and mirror particles after spontaneous symmetry breaking. In section 3, we
derive the mass relations between the SM and mirror particles. Section 4 is
shown the identification of the new particles, the SM gauge unification, and
proton decay in our model. In section 5, we obtain the mass relations between
the active and mirror neutrinos and estimate the heavy neutrino masses.
Section 6 is devoted to summary. In appendix A, we show the unification of the
SM particles into the SU(5) representations. Appendix B is shown the RGE form
and the beta coefficients of the relevant particles.

##  2 SU(5)$\times$U(1)${}_{\text{X}}\times$U(1)${}_{\text{PQ}}$ model

In this section, we propose an
SU(5)$\times$U(1)${}_{\text{X}}\times$U(1)${}_{\text{PQ}}$ model without SUSY.
We introduce two symmetries, the $\mathrm{U(1)_{X}}$ gauge symmetry [16] and
$\mathrm{U(1)_{PQ}}$ symmetry [26, 27], in addition to the SU(5) symmetry. The
$\mathrm{U(1)_{X}}$ gauge symmetry is the generalization of the
$\mathrm{U(1)_{B-L}}$ gauge symmetry [17, 18, 19, 20, 21, 22, 23]. The
definition of the $\mathrm{U(1)_{X}}$ charge is
$Q_{\mathrm{X}}=x_{H}~{}Q_{\mathrm{Y}}+Q_{\mathrm{B-L}}$, where
$Q_{\mathrm{Y}}$ is hypercharge, $Q_{\mathrm{B-L}}$ is B-L charge, and $x_{H}$
is a free parameter [24]. For $x_{H}=-4/5$ [25], the $\mathrm{U(1)_{X}}$
charge can be assigned to the ${\bf\bar{5}}$ and ${\bf 10}$ representations
successfully. The $\mathrm{U(1)_{PQ}}$ symmetry is the global Peccei-Quinn
(PQ) symmetry. The PQ symmetry can solve the strong CP problem [28] and the
axion which is the pseudo-Nambu Goldstone boson from the PQ symmetry breaking
is the dark matter (DM) candidate [29, 30]. In the SU(5) model, the SM matter
fields are unified into a $\mathbf{\bar{5}}$ and $\mathbf{10}$ representations
defined as $\psi_{\bar{5}}^{i}$ and $\psi_{10}^{i}$, respectively ($i=1$-$3$
generations). In appendix A, we describe the unification of the SM particles
into the SU(5) representations in detail. In order to unify the SM gauge
couplings successfully and avoid the restriction of proton lifetime, we
introduce three mirror families which are defined as $\tilde{\psi}_{5}^{i}$
and $\tilde{\psi}_{\overline{10}}^{i}$ ($i=1$-$3$) for $\mathbf{5}$ and
$\mathbf{\overline{10}}$ representations in SU(5), respectively. The mirror
family is the conjugate of the representation for the SM particles. In our
model, three mirror families have the $\mathrm{U(1)_{PQ}}$ charge [37, 38,
39]. We also introduce three gauge singlet right-handed Majorana neutrinos
$(N^{c})^{i}$ ($i=1$-$3$) in order to derive the left-handed Majorana neutrino
masses through the type-I seesaw mechanism [40, 41, 42, 43, 44] and cancel all
the U(1)${}_{\text{X}}$ related anomalies. Here, $``c"$ means charge
conjugation. In addition to the $\mathbf{5}$ representation Higgs particle
which is defined as $H$, we introduce $\mathbf{24}$ and $\mathbf{45}$
representation Higgs fields [4, 5] defined as $\Sigma$ and $\chi$,
respectively. The $\chi$ satisfies the following conditions as
$\chi^{ab}_{c}=-\chi^{ba}_{c},~{}{\chi}^{ab}_{a}=0~{}(a,b,c=1$-$5)$. Thanks to
the $\mathbf{45}$ representation Higgs field $\chi$, we obtain the difference
of the masses for the down-type quarks and charged leptons [6]. The gauge
singlet scalar field $\Phi$ is introduced in order to occur inflation and
break the U(1)${}_{\text{X}}$ gauge symmetry. The assignments of the particles
for SU(5)$\times$U(1)${}_{\text{X}}\times$U(1)${}_{\text{PQ}}$ symmetry are
summarized in Table 1.

| SU(5) | $\mathrm{U(1)_{X}}$ | $\mathrm{U(1)_{PQ}}$
---|---|---|---
$\psi_{\bar{5}}^{i}$ | $\mathbf{\bar{5}}$ | $-3/5$ | 0
$\psi_{10}^{i}$ | $\mathbf{10}$ | $1/5$ | 0
$\tilde{\psi}_{5}^{i}$ | $\mathbf{5}$ | $3/5$ | 1
$\tilde{\psi}_{\overline{10}}^{i}$ | $\mathbf{\overline{10}}$ | $-1/5$ | 1
$(N^{c})^{i}$ | $\mathbf{1}$ | +1 | 0
$\Sigma$ | $\mathbf{24}$ | 0 | $-1$
$\chi$ | $\mathbf{45}$ | $-2/5$ | 0
$\Phi$ | $\mathbf{1}$ | $-2$ | 0
$H$ | $\mathbf{5}$ | $-2/5$ | 0
Table 1: The assignments of the particles for
SU(5)$\times$U(1)${}_{\text{X}}\times$U(1)${}_{\text{PQ}}$ symmetry.

In our model, the scenario of spontaneous symmetry breaking is following three
steps. At first, the $\mathbf{24}$ representation Higgs $\Sigma$ breaks SU(5)
and U(1)${}_{\text{PQ}}$ symmetries by taking the VEVs as
$\langle\Sigma\rangle=v_{\Sigma}/(2\sqrt{15})\mathrm{Diag}\left(-2,-2,-2,3,3\right)$.
Next, the gauge singlet scalar field $\Phi$ breaks U(1)${}_{\text{X}}$
symmetry after taking VEV as $\langle\Phi\rangle=v_{\Phi}/\sqrt{2}$. Finally,
the $\mathbf{5}$ and $\mathbf{45}$ representation Higgs, $H$ and $\chi$, break
the SM gauge symmetries by taking VEVs as $\langle
H\rangle=\left(0~{}0~{}0~{}0~{}v_{H}/\sqrt{2}\right)^{T}$,
$\langle{\chi}\rangle^{15}_{1}=\langle{\chi}\rangle^{25}_{2}=\langle{\chi}\rangle^{35}_{3}=v_{\chi}/\sqrt{2}$,
and $\langle{\chi}\rangle^{45}_{4}=-3v_{\chi}/\sqrt{2}$, respectively.

Let us discuss the Yukawa interactions which are invariant under the
SU(5)$\times$U(1)${}_{\text{X}}\times$U(1)${}_{\text{PQ}}$ symmetry. In our
model, we can write the following Lagrangian which includes three types of
Yukawa interactions:

$\mathcal{L}_{Y}\supset\mathcal{L}_{\mathrm{SM}}+\mathcal{L}_{\mathrm{mirror}}+\mathcal{L}_{\mathrm{neutrino}}~{},$
(1)

where the relevant SM particles are written as

$\mathcal{L}_{\mathrm{SM}}\supset\sum_{i,j=1}^{3}[(Y^{ij}_{1}H+Y^{ij}_{2}\chi)\psi_{10}^{i}\psi_{10}^{j}]+\sum_{i,j=1}^{3}[(Y^{ij}_{3}H^{*}+Y^{ij}_{4}\chi)\psi_{\bar{5}}^{i}\psi_{10}^{j}],$
(2)

and the Lagrangian for the SM and mirror particles is obtained as

$\mathcal{L}_{\mathrm{mirror}}\supset\sum_{i,j=1}^{3}\tilde{Y}_{5}^{ij}\Sigma\psi_{\bar{5}}^{i}\tilde{\psi}_{5}^{j}+\operatorname{Tr}\left[\sum_{i,j=1}^{3}\tilde{Y}_{10}^{ij}\Sigma\psi_{10}^{i}\tilde{\psi}_{\overline{10}}^{j}\right],$
(3)

and the Lagrangian for the Dirac neutrinos and right-handed Majorana neutrinos
is given as

$\mathcal{L}_{\mathrm{neutrino}}\supset-\sum_{i,j=1}^{3}Y_{D}^{ij}H\psi_{\bar{5}}^{i}(N^{c})^{j}-\left(\frac{1}{2}\sum_{i=1}^{3}Y_{M}^{i}\Phi(N^{c})^{i}{(N^{c})^{i}}+\mathrm{h.c.}\right).$
(4)

After spontaneous symmetry breaking, we obtain the mass matrices from Eq. (1)
as follows:

$\displaystyle\mathcal{M}_{U}$
$\displaystyle=\begin{pmatrix}2\sqrt{2}Y^{ij}_{1}{v_{H}}&-\tilde{Y}_{10}^{ij}\frac{v_{\Sigma}}{4\sqrt{15}}\\\
\tilde{Y}_{10}^{ij}\frac{4v_{\Sigma}}{\sqrt{15}}&0\\\ \end{pmatrix},$
$\displaystyle\mathcal{M}_{D}$
$\displaystyle=\begin{pmatrix}\frac{1}{2}[Y^{ji}_{3}{v_{H}^{*}}+2Y^{ji}_{4}v^{*}_{\chi}]&-\tilde{Y}_{10}^{ij}\frac{v_{\Sigma}}{4\sqrt{15}}\\\
-\tilde{Y}_{5}^{ij}\frac{v_{\Sigma}}{2\sqrt{15}}&0\\\ \end{pmatrix},$ (5)
$\displaystyle\mathcal{M}_{E}$
$\displaystyle=\begin{pmatrix}\frac{1}{2}[Y^{ij}_{3}{v_{H}^{*}}-6Y^{ij}_{4}v^{*}_{\chi}]&\tilde{Y}_{5}^{ij}\frac{3v_{\Sigma}}{4\sqrt{15}}\\\
-\tilde{Y}_{10}^{ij}\frac{3v_{\Sigma}}{2\sqrt{15}}&0\\\ \end{pmatrix},$
$\displaystyle\mathcal{M}_{\nu}$
$\displaystyle=\begin{pmatrix}0&\tilde{M}_{D}&M_{D}\\\
\tilde{M}_{D}^{T}&0&0\\\ M_{D}^{T}&0&M_{R}\\\ \end{pmatrix},$

where $\mathcal{M}_{U}$, $\mathcal{M}_{D}$, and $\mathcal{M}_{E}$ are the
$6\times 6$ mass matrices which include the SM and mirror particles for the
up- and down-type quark and charged lepton sectors, respectively. On the other
hand, $\mathcal{M}_{\nu}$ is the $9\times 9$ mass matrix which includes the
Dirac, mirror, and right-handed Majorana neutrinos. By using Eq. (5), we
obtain the mass eigenvalues and derive the mass relations between the SM and
mirror particles in the next section. Note that mass relations among neutrinos
are shown in section 5.

##  3 Mass relations for the SM and mirror particles

In the previous section, we have discussed the Yukawa interactions for the SM
and mirror particles and have obtained the $6\times 6$ mass matrices for the
up- and down-type quarks and charged leptons. Let us discuss the mass
relations for the SM and mirror particles in this section.

First, we obtain the mass eigenvalues for the up- and down-type quark and
charged lepton sectors. In our analysis, we assume the diagonal matrices for
each particle for simplicity. By using the $6\times 6$ mass matrices
$\mathcal{M}_{U}$ in Eq. (5), we analyze the up-type quark mass eigenvalues
$m_{ui}$ ($i=1$-$6$). The up-type quark squared mass eigenvalues are written
just below:

$\displaystyle m^{2}_{u1}=\frac{1}{480}$
$\displaystyle\Bigg{[}17{\big{(}v_{\Sigma}\tilde{Y}_{10}^{11}\big{)}}^{2}+1920{\big{(}v_{H}Y_{1}^{11}\big{)}}^{2}$
$\displaystyle-\sqrt{-64{\big{(}v_{\Sigma}\tilde{Y}_{10}^{11}\big{)}}^{4}+\Big{(}17{\big{(}v_{\Sigma}\tilde{Y}_{10}^{11}\big{)}}^{2}+1920{\big{(}v_{H}Y_{1}^{11}\big{)}}^{2}\Big{)}^{2}}\Bigg{]},$
$\displaystyle m^{2}_{u2}=\frac{1}{480}$
$\displaystyle\Bigg{[}17{\big{(}v_{\Sigma}\tilde{Y}_{10}^{11}\big{)}}^{2}+1920{\big{(}v_{H}Y_{1}^{11}\big{)}}^{2}$
$\displaystyle+\sqrt{-64{\big{(}v_{\Sigma}\tilde{Y}_{10}^{11}\big{)}}^{4}+\Big{(}17{\big{(}v_{\Sigma}\tilde{Y}_{10}^{11}\big{)}}^{2}+1920{\big{(}v_{H}Y_{1}^{11}\big{)}}^{2}\Big{)}^{2}}\Bigg{]},$
$\displaystyle m^{2}_{u3}=\frac{1}{480}$
$\displaystyle\Bigg{[}17{\big{(}v_{\Sigma}\tilde{Y}_{10}^{22}\big{)}}^{2}+1920{\big{(}v_{H}Y_{1}^{22}\big{)}}^{2}$
$\displaystyle-\sqrt{-64{\big{(}v_{\Sigma}\tilde{Y}_{10}^{22}\big{)}}^{4}+\Big{(}17{\big{(}v_{\Sigma}\tilde{Y}_{10}^{22}\big{)}}^{2}+1920{\big{(}v_{H}Y_{1}^{22}\big{)}}^{2}\Big{)}^{2}}\Bigg{]},$
$\displaystyle m^{2}_{u4}=\frac{1}{480}$
$\displaystyle\Bigg{[}17{\big{(}v_{\Sigma}\tilde{Y}_{10}^{22}\big{)}}^{2}+1920{\big{(}v_{H}Y_{1}^{22}\big{)}}^{2}$
(6)
$\displaystyle+\sqrt{-64{\big{(}v_{\Sigma}\tilde{Y}_{10}^{22}\big{)}}^{4}+\Big{(}17{\big{(}v_{\Sigma}\tilde{Y}_{10}^{22}\big{)}}^{2}+1920{\big{(}v_{H}Y_{1}^{22}\big{)}}^{2}\Big{)}^{2}}\Bigg{]},$
$\displaystyle m^{2}_{u5}=\frac{1}{480}$
$\displaystyle\Bigg{[}17{\big{(}v_{\Sigma}\tilde{Y}_{10}^{33}\big{)}}^{2}+1920{\big{(}v_{H}Y_{1}^{33}\big{)}}^{2}$
$\displaystyle-\sqrt{-64{\big{(}v_{\Sigma}\tilde{Y}_{10}^{33}\big{)}}^{4}+\Big{(}17{\big{(}v_{\Sigma}\tilde{Y}_{10}^{33}\big{)}}^{2}+1920{\big{(}v_{H}Y_{1}^{33}\big{)}}^{2}\Big{)}^{2}}\Bigg{]},$
$\displaystyle m^{2}_{u6}=\frac{1}{480}$
$\displaystyle\Bigg{[}17{\big{(}v_{\Sigma}\tilde{Y}_{10}^{33}\big{)}}^{2}+1920{\big{(}v_{H}Y_{1}^{33}\big{)}}^{2}$
$\displaystyle+\sqrt{-64{\big{(}v_{\Sigma}\tilde{Y}_{10}^{33}\big{)}}^{4}+\Big{(}17{\big{(}v_{\Sigma}\tilde{Y}_{10}^{33}\big{)}}^{2}+1920{\big{(}v_{H}Y_{1}^{33}\big{)}}^{2}\Big{)}^{2}}\Bigg{]}.$

The up-type quark mass eigenvalues include the $\tilde{Y}_{10}$ matrix
diagonal elements. Then, the $\tilde{Y}_{10}$ diagonal elements can be
expressed in terms of the up-type quark mass eigenvalues. On the other hand,
the down-type quark and charged lepton mass eigenvalues are related to
$\tilde{Y}_{10}$ matrix diagonal elements. Therefore, we substitute these
values for the down-type quark and charged lepton mass eigenvalues and
represent the down-type quark $m_{di}$ and charged lepton mass eigenvalues
$m_{ei}$ in terms of the up-type quark mass eigenvalues $(i=1$-$6)$. We show
the down-type quark squared mass eigenvalues just below:

$\displaystyle m^{2}_{d1}=$
$\displaystyle\frac{1}{120}\Bigg{[}15m_{u1}m_{u2}+15\big{(}v_{H}Y_{3}^{11}+2v_{\chi}Y_{4}^{11}\big{)}^{2}+{\big{(}{v_{\Sigma}}\tilde{Y}_{5}^{11}\big{)}}^{2}$
$\displaystyle-\sqrt{-60m_{u1}m_{u2}{\big{(}{v_{\Sigma}}\tilde{Y}_{5}^{11}\big{)}}^{2}+\Big{(}15m_{u1}m_{u2}+15\big{(}v_{H}Y_{3}^{11}+2v_{\chi}Y_{4}^{11}\big{)}^{2}+{\big{(}{v_{\Sigma}}\tilde{Y}_{5}^{11}\big{)}}^{2}\Big{)}^{2}}\Bigg{]},$
$\displaystyle m^{2}_{d2}=$
$\displaystyle\frac{1}{120}\Bigg{[}15m_{u1}m_{u2}+15\big{(}v_{H}Y_{3}^{11}+2v_{\chi}Y_{4}^{11}\big{)}^{2}+{\big{(}{v_{\Sigma}}\tilde{Y}_{5}^{11}\big{)}}^{2}$
$\displaystyle+\sqrt{-60m_{u1}m_{u2}{\big{(}{v_{\Sigma}}\tilde{Y}_{5}^{11}\big{)}}^{2}+\Big{(}15m_{u1}m_{u2}+15\big{(}v_{H}Y_{3}^{11}+2v_{\chi}Y_{4}^{11}\big{)}^{2}+{\big{(}{v_{\Sigma}}\tilde{Y}_{5}^{11}\big{)}}^{2}\Big{)}^{2}}\Bigg{]},$
$\displaystyle m^{2}_{d3}=$
$\displaystyle\frac{1}{120}\Bigg{[}15m_{u3}m_{u4}+15\big{(}v_{H}Y_{3}^{22}+2v_{\chi}Y_{4}^{22}\big{)}^{2}+{\big{(}{v_{\Sigma}}\tilde{Y}_{5}^{22}\big{)}}^{2}$
$\displaystyle-\sqrt{-60m_{u3}m_{u4}{\big{(}{v_{\Sigma}}\tilde{Y}_{5}^{22}\big{)}}^{2}+\Big{(}15m_{u3}m_{u4}+15\big{(}v_{H}Y_{3}^{22}+2v_{\chi}Y_{4}^{22}\big{)}^{2}+{\big{(}{v_{\Sigma}}\tilde{Y}_{5}^{22}\big{)}}^{2}\Big{)}^{2}}\Bigg{]},$
$\displaystyle m^{2}_{d4}=$
$\displaystyle\frac{1}{120}\Bigg{[}15m_{u3}m_{u4}+15\big{(}v_{H}Y_{3}^{22}+2v_{\chi}Y_{4}^{22}\big{)}^{2}+{\big{(}{v_{\Sigma}}\tilde{Y}_{5}^{22}\big{)}}^{2}$
(7)
$\displaystyle+\sqrt{-60m_{u3}m_{u4}{\big{(}{v_{\Sigma}}\tilde{Y}_{5}^{22}\big{)}}^{2}+\Big{(}15m_{u3}m_{u4}+15\big{(}v_{H}Y_{3}^{22}+2v_{\chi}Y_{4}^{22}\big{)}^{2}+{\big{(}{v_{\Sigma}}\tilde{Y}_{5}^{22}\big{)}}^{2}\Big{)}^{2}}\Bigg{]},$
$\displaystyle m^{2}_{d5}=$
$\displaystyle\frac{1}{120}\Bigg{[}15m_{u5}m_{u6}+15\big{(}v_{H}Y_{3}^{33}+2v_{\chi}Y_{4}^{33}\big{)}^{2}+{\big{(}{v_{\Sigma}}\tilde{Y}_{5}^{33}\big{)}}^{2}$
$\displaystyle-\sqrt{-60m_{u5}m_{u6}{\big{(}{v_{\Sigma}}\tilde{Y}_{5}^{33}\big{)}}^{2}+\Big{(}15m_{u5}m_{u6}+15\big{(}v_{H}Y_{3}^{33}+2v_{\chi}Y_{4}^{33}\big{)}^{2}+{\big{(}{v_{\Sigma}}\tilde{Y}_{5}^{33}\big{)}}^{2}\Big{)}^{2}}\Bigg{]},$
$\displaystyle m^{2}_{d6}=$
$\displaystyle\frac{1}{120}\Bigg{[}15m_{u5}m_{u6}+15\big{(}v_{H}Y_{3}^{33}+2v_{\chi}Y_{4}^{33}\big{)}^{2}+{\big{(}{v_{\Sigma}}\tilde{Y}_{5}^{33}\big{)}}^{2}$
$\displaystyle+\sqrt{-60m_{u5}m_{u6}{\big{(}{v_{\Sigma}}\tilde{Y}_{5}^{33}\big{)}}^{2}+\Big{(}15m_{u5}m_{u6}+15\big{(}v_{H}Y_{3}^{33}+2v_{\chi}Y_{4}^{33}\big{)}^{2}+{\big{(}{v_{\Sigma}}\tilde{Y}_{5}^{33}\big{)}}^{2}\Big{)}^{2}}\Bigg{]},$

and the charged lepton squared mass eigenvalues are written as

$\displaystyle m^{2}_{e1}=$
$\displaystyle\frac{1}{160}\Bigg{[}720m_{u1}m_{u2}+20\big{(}v_{H}Y_{3}^{11}-6v_{\chi}Y_{4}^{11}\big{)}^{2}+3{\big{(}{v_{\Sigma}}\tilde{Y}_{5}^{11}\big{)}}^{2}$
$\displaystyle-\sqrt{-8640m_{u1}m_{u2}{({v_{\Sigma}}\tilde{Y}_{5}^{11})}^{2}+\Big{(}720m_{u1}m_{u2}+20\big{(}v_{H}Y_{3}^{11}-6v_{\chi}Y_{4}^{11}\big{)}^{2}+{3\big{(}{v_{\Sigma}}\tilde{Y}_{5}^{11}\big{)}}^{2}\Big{)}^{2}}\Bigg{]},$
$\displaystyle m^{2}_{e2}=$
$\displaystyle\frac{1}{160}\Bigg{[}720m_{u1}m_{u2}+20\big{(}v_{H}Y_{3}^{11}-6v_{\chi}Y_{4}^{11}\big{)}^{2}+3{\big{(}{v_{\Sigma}}\tilde{Y}_{5}^{11}\big{)}}^{2}$
$\displaystyle+\sqrt{-8640m_{u1}m_{u2}{({v_{\Sigma}}\tilde{Y}_{5}^{11})}^{2}+\Big{(}720m_{u1}m_{u2}+20\big{(}v_{H}Y_{3}^{11}-6v_{\chi}Y_{4}^{11}\big{)}^{2}+{3\big{(}{v_{\Sigma}}\tilde{Y}_{5}^{11}\big{)}}^{2}\Big{)}^{2}}\Bigg{]},$
$\displaystyle m^{2}_{e3}=$
$\displaystyle\frac{1}{160}\Bigg{[}720m_{u3}m_{u4}+20\big{(}v_{H}Y_{3}^{22}-6v_{\chi}Y_{4}^{22}\big{)}^{2}+3{\big{(}{v_{\Sigma}}\tilde{Y}_{5}^{22}\big{)}}^{2}$
$\displaystyle-\sqrt{-8640m_{u3}m_{u4}{({v_{\Sigma}}\tilde{Y}_{5}^{22})}^{2}+\Big{(}720m_{u3}m_{u4}+20\big{(}v_{H}Y_{3}^{22}-6v_{\chi}Y_{4}^{22}\big{)}^{2}+{3\big{(}{v_{\Sigma}}\tilde{Y}_{5}^{22}\big{)}}^{2}\Big{)}^{2}}\Bigg{]},$
$\displaystyle m^{2}_{e4}=$
$\displaystyle\frac{1}{160}\Bigg{[}720m_{u3}m_{u4}+20\big{(}v_{H}Y_{3}^{22}-6v_{\chi}Y_{4}^{22}\big{)}^{2}+3{\big{(}{v_{\Sigma}}\tilde{Y}_{5}^{22}\big{)}}^{2}$
(8)
$\displaystyle+\sqrt{-8640m_{u3}m_{u4}{({v_{\Sigma}}\tilde{Y}_{5}^{22})}^{2}+\Big{(}720m_{u3}m_{u4}+20\big{(}v_{H}Y_{3}^{22}-6v_{\chi}Y_{4}^{22}\big{)}^{2}+{3\big{(}{v_{\Sigma}}\tilde{Y}_{5}^{22}\big{)}}^{2}\Big{)}^{2}}\Bigg{]},$
$\displaystyle m^{2}_{e5}=$
$\displaystyle\frac{1}{160}\Bigg{[}720m_{u5}m_{u6}+20\big{(}v_{H}Y_{3}^{33}-6v_{\chi}Y_{4}^{33}\big{)}^{2}+3{\big{(}{v_{\Sigma}}\tilde{Y}_{5}^{33}\big{)}}^{2}$
$\displaystyle-\sqrt{-8640m_{u5}m_{u6}{({v_{\Sigma}}\tilde{Y}_{5}^{33})}^{2}+\Big{(}720m_{u5}m_{u6}+20\big{(}v_{H}Y_{3}^{33}-6v_{\chi}Y_{4}^{33}\big{)}^{2}+{3\big{(}{v_{\Sigma}}\tilde{Y}_{5}^{33}\big{)}}^{2}\Big{)}^{2}}\Bigg{]},$
$\displaystyle m^{2}_{e6}=$
$\displaystyle\frac{1}{160}\Bigg{[}720m_{u5}m_{u6}+20\big{(}v_{H}Y_{3}^{33}-6v_{\chi}Y_{4}^{33}\big{)}^{2}+3{\big{(}{v_{\Sigma}}\tilde{Y}_{5}^{33}\big{)}}^{2}$
$\displaystyle+\sqrt{-8640m_{u5}m_{u6}{({v_{\Sigma}}\tilde{Y}_{5}^{33})}^{2}+\Big{(}720m_{u5}m_{u6}+20\big{(}v_{H}Y_{3}^{33}-6v_{\chi}Y_{4}^{33}\big{)}^{2}+{3\big{(}{v_{\Sigma}}\tilde{Y}_{5}^{33}\big{)}}^{2}\Big{)}^{2}}\Bigg{]}.$

Now we obtain the mass eigenvalues for the up- and down-type quark and charged
lepton sectors. Next, by using these mass eigenvalues, we derive the mass
relations for the SM and mirror particles. The down-type quark and charged
lepton mass eigenvalues are represented in terms of the up-type quark mass
eigenvalues. Then, we can derive the mass relations between the up-type quark
and the down-type quark mass eigenvalues and between the up-type quark and the
charged lepton mass eigenvalues,

$\displaystyle m^{2}_{d1}m^{2}_{d2}$
$\displaystyle=\frac{m_{u1}m_{u2}\big{(}{v_{\Sigma}}\tilde{Y}_{5}^{11}\big{)}^{2}}{240},\quad{m^{2}_{e1}m^{2}_{e2}}=\frac{27m_{u1}m_{u2}\big{(}{v_{\Sigma}}\tilde{Y}_{5}^{11}\big{)}^{2}}{80},$
(9) $\displaystyle m^{2}_{d3}m^{2}_{d4}$
$\displaystyle=\frac{m_{u3}m_{u4}\big{(}{v_{\Sigma}}\tilde{Y}_{5}^{22}\big{)}^{2}}{240},\quad{m^{2}_{e3}m^{2}_{e4}}=\frac{27m_{u3}m_{u4}\big{(}{v_{\Sigma}}\tilde{Y}_{5}^{22}\big{)}^{2}}{80},$
$\displaystyle m^{2}_{d5}m^{2}_{d6}$
$\displaystyle=\frac{m_{u5}m_{u6}\big{(}{v_{\Sigma}}\tilde{Y}_{5}^{33}\big{)}^{2}}{240},\quad{m^{2}_{e5}m^{2}_{e6}}=\frac{27m_{u5}m_{u6}\big{(}{v_{\Sigma}}\tilde{Y}_{5}^{33}\big{)}^{2}}{80}.$

By using these mass relations, we identify the mass scale of the mirror
particles in the next section.

##  4 Gauge unification and proton decay

In this section, we discuss the contribution of the new particles to the RGE
and proton lifetime. In order to obtain the RGE including the contribution of
the new particles, we identify the mass scales of the new particles by using
the mass relations in Eq. (9). Once, we assume that $m_{d1}$ is bottom quark
(4.18 GeV) and $m_{e1}$ is electron (0.511 MeV) [45]. Then, from the mass
relations in Eq. (9), we can identify $m_{d2}=\mathcal{O}(10^{3})$ GeV,
$m_{e2}=7.36\times{10}^{4}\times{m_{d2}}$ GeV. The others are the SM or
$\mathcal{O}(M_{\mathrm{GUT}})$ scale particles that satisfied the mass
relations. We show the fermion mass setup in Table 2.

$m_{u1}$ | SM particle | $m_{d1}$ | bottom quark | $m_{e1}$ | electron
---|---|---|---|---|---
$m_{u2}$ | $\mathcal{O}(M_{\mathrm{GUT}})$ | $m_{d2}$ |  $\mathcal{O}(10^{3})$ GeV | $m_{e2}$ |  $7.36\times 10^{4}\times{m_{d2}}$ GeV
$m_{u3}$ | SM particle | $m_{d3}$ | SM particle | $m_{e3}$ | SM particle
$m_{u4}$ | $\mathcal{O}(M_{\mathrm{GUT}})$ | $m_{d4}$ | $\mathcal{O}(M_{\mathrm{GUT}})$ | $m_{e4}$ | $\mathcal{O}(M_{\mathrm{GUT}})$
$m_{u5}$ | SM particle | $m_{d5}$ | SM particle | $m_{e5}$ | SM particle
$m_{u6}$ | $\mathcal{O}(M_{\mathrm{GUT}})$ | $m_{d6}$ | $\mathcal{O}(M_{\mathrm{GUT}})$ | $m_{e6}$ | $\mathcal{O}(M_{\mathrm{GUT}})$
Table 2: The mass setup of the SM and mirror particles.

In addition to the fermion masses, we identify the scalar mass scale in order
to unify the SM gauge couplings. The $\mathbf{45}$ representation Higgs $\chi$
is composed of

$\displaystyle\chi\sim\Phi_{1}(8,2,\frac{1}{2})\oplus\Phi_{2}(\bar{6},1,-\frac{1}{3})$
$\displaystyle\oplus\Phi_{3}(3,3,-\frac{1}{3})$
$\displaystyle\oplus\Phi_{4}(\bar{3},2,-\frac{7}{6})\oplus\Phi_{5}(3,1,-\frac{1}{3})\oplus\Phi_{6}(\bar{3},1,\frac{4}{3})\oplus{H_{2}}(1,2,\frac{1}{2}).$
(10)

We assume that the scalar $\phi_{1}$ mass is $M_{1}=\mathcal{O}(10^{3})$ GeV,
the scalar $\phi_{3}$ mass is $M_{3}=\mathcal{O}(10^{9})$ GeV, and the
remained scalar masses are $\mathcal{O}(M_{\mathrm{GUT}})$ scale. Therefore,
four new particles, $m_{d2}$, $m_{e2}$, $\phi_{1}$, and $\phi_{3}$, exist in
the intermediate energy scale and contribute to the achieving of unifying the
SM gauge couplings successfully.

Next, we evaluate the unifying of the SM gauge couplings and proton lifetime
including the contribution of the new fermion and scalar masses. The SM gauge
couplings for $\mathrm{U(1)_{Y}},~{}\mathrm{SU(2)_{L}},$ and
$\mathrm{SU(3)_{C}}$ are defined as $\alpha_{i}=g_{i}/4\pi$ $(i=1$-$3)$,
respectively. In our analysis, we assume $m_{d2}$ and $M_{1}$ are the same
scale for simplicity. We show the relations between the scalar $\phi_{1}$ mass
($M_{1}$) and proton lifetime ($\tau_{p}$), the scalar $\phi_{3}$ mass
($M_{3}$), the scale of unifying the SM gauge couplings ($M_{\mathrm{GUT}}$),
and the inverse of the GUT coupling (1/$\alpha_{\mathrm{GUT}}$) in Figure 1.
The yellow shaded region shows the bound of the vector-like quark mass,
$m_{d2}>1660$ GeV [46], and the purple shaded region shows the bound of the
color octet scalar mass, $M_{1}>1$ TeV [47, 48]. The green (blue) shaded
region denotes the range of unifying the SM gauge couplings with an accuracy
of 1 % (3 %) or less. We define the accuracy of the unification as a
percentage difference between the energy scale of unifying the SM gauge
couplings $\alpha_{1},~{}\alpha_{2}$ and $\alpha_{2},~{}\alpha_{3}$. In figure
1, the gray shaded region shows the excluded one from the Super-Kamiokande
experiment, $\tau_{p}(p\to\pi^{0}{e^{+}})>2.4\times 10^{34}$ years [7], and
the red dashed line depicts the expected proton lifetime limit from the Hyper-
Kamiokande experiment, $\tau_{p}(p\to\pi^{0}{e^{+}})<1.0\times 10^{35}$ years
[35]. If experiments about the vector-like quarks or the color octet scalar
are more precise, we find that the region within an accuracy of 1 % can be
tested by the Hyper-Kamiokande experiment is more extended. In figure 1, we
denote the region that is allowed by the Super-Kamiokande experiment and can
be tested by the Hyper-Kamiokande one.

We set the benchmark that $m_{d2}=M_{1}=5$ TeV, $m_{e2}=3.68\times 10^{8}$
GeV, and $M_{3}=2.5\times 10^{9}$ GeV. By using this benchmark, we estimate
$M_{\mathrm{GUT}}$ and proton lifetime $\tau_{p}$. We show the unification of
the SM gauge couplings and the running of the SM Higgs quartic coupling in
Figure 2. The black dashed line depicts the RGE solutions of only the SM
particles and the red dashed line depicts these including the new particle
contributions. In the case that includes the new particle contributions, the
SM gauge couplings unify successfully at the
$M_{\mathrm{GUT}}\approx{7.28}\times 10^{15}$ GeV and
$\alpha_{\mathrm{GUT}}=\alpha_{1}=\alpha_{2}=\alpha_{3}\approx{1/31.7}$. The
SM Higgs quartic coupling is the positive value at all energy scales including
the new particle contributions and the SM Higgs potential is stabilized. Also,
by using these values, we estimate the proton lifetime approximately [49],

$\tau_{p}(p\to\pi^{0}{e^{+}})\approx{\frac{1}{\alpha^{2}_{\mathrm{GUT}}}}\frac{M^{4}_{\mathrm{GUT}}}{m^{5}_{p}}\approx{8.07\times
10^{34}}~{}\mathrm{years},$ (11)

where $m_{p}=0.938$ GeV is the proton mass [45]. The current proton lifetime
limit from the Super-Kamiokande experiment is
$\tau_{p}(p\to\pi^{0}{e^{+}})>2.4\times 10^{34}$ years [7] and the expected
proton lifetime limit from the Hyper-Kamiokande experiment is
$\tau_{p}(p\to\pi^{0}{e^{+}})<1.0\times 10^{35}$ years [35]. Therefore, it is
consistent with the current proton lifetime limit from the Super-Kamiokande
experiment and can be tested by the Hyper-Kamiokande experiment. The proton
decay can be mediated by the color triplet scalar field contained in the
$\mathbf{5}$ representation Higgs and the $\mathbf{45}$ representation Higgs.
The Super-Kamiokande experiment excludes the colored scalar mass that is
lighter than $\mathcal{O}(10^{11})$ GeV [49]. Considering the cross terms for
the $H$ and $\Sigma$, for example,
$H^{\dagger}{H}\operatorname{Tr}[\Sigma^{\dagger}{\Sigma}]$, the color triplet
scalar field contained in the $\mathbf{5}$ representation Higgs is greater
than $\mathcal{O}(10^{11})$ GeV. Since the couplings which induce the proton
decay by mediating the $\phi_{3}$ are absent, the $\phi_{3}$ cannot induce the
proton decay and thus it can be light.

(a) |  (b)
---|---
(c) |  (d)
Figure 1: In the above four figures, the horizontal axis is the scalar
$\phi_{1}$ mass ($M_{1}$). The vertical axis is (1): proton lifetime
($\tau_{p}$), (1): scalar $\phi_{3}$ mass ($M_{3}$), (1): the scale of
unifying the SM gauge couplings ($M_{\mathrm{GUT}}$), and (1): the inverse of
the GUT coupling (1/$\alpha_{\mathrm{GUT}}$), respectively. The yellow shaded
region shows the bound of the vector-like quark mass, $m_{d2}>1660$ GeV [46],
and the purple shaded region shows the bound of the color octet scalar mass,
$M_{1}>1$ TeV [47, 48]. The green (blue) shaded region denotes the range of
unifying the SM gauge couplings with an accuracy of 1 % (3 %) or less. In
figure (1), the gray shaded region shows the excluded one from the Super-
Kamiokande experiment, $\tau_{p}(p\to\pi^{0}{e^{+}})>2.4\times 10^{34}$ years
[7], and the red dashed line depicts the expected proton lifetime limit from
the Hyper-Kamiokande experiment, $\tau_{p}(p\to\pi^{0}{e^{+}})<1.0\times
10^{35}$ years [35].

(a)

(b)

Figure 2: The figure (2): the running of the SM gauge couplings. The figure
(2): the running of the SM Higgs quartic coupling. In both figures, the black
dashed line depicts the RGE solutions of only the SM particles and the red
dashed line depicts these including the new particle contributions. In figure
(2), the SM gauge couplings unify successfully at the
$M_{\mathrm{GUT}}\approx{7.28}\times 10^{15}$ GeV and
$\alpha_{\mathrm{GUT}}=\alpha_{1}=\alpha_{2}=\alpha_{3}\approx{1/31.7}$. In
figure (2), the horizontal green line depicts $\lambda=0$. Thanks to the new
particle contributions, the SM Higgs quartic coupling is the positive value at
all energy scales and the SM Higgs potential is stabilized.

##  5 Estimate of heavy neutrino masses

In this section, we obtain the mass relations among the neutrinos and estimate
the heavy neutrino masses. First, we analyze the neutrino mass eigenvalues by
using the $9\times 9$ mass matrix $\mathcal{M}_{\nu}$ in Eq. (5):

$\mathcal{M}_{\nu}=\begin{pmatrix}0&\tilde{M}_{D}&M_{D}\\\
\tilde{M}_{D}^{T}&0&0\\\ M_{D}^{T}&0&M_{R}\\\ \end{pmatrix},$ (12)

where the $3\times 3$ mass matrices are defined as

$\displaystyle\tilde{M}_{D}$
$\displaystyle=\tilde{Y}_{5}^{ij}\frac{3v_{\Sigma}}{4\sqrt{15}},$
$\displaystyle M_{D}$ $\displaystyle=Y_{D}^{ij}\frac{v_{H}}{\sqrt{2}},$ (13)
$\displaystyle M_{R}$ $\displaystyle=Y_{M}^{i}\frac{v_{\Phi}}{\sqrt{2}}.$

In our analysis of neutrino masses, we also assume the diagonal matrix for
simplicity. By using the type-I seesaw mechanism [40, 41, 42, 43, 44], the Eq.
(12) is rewritten as

$\mathcal{M}_{\nu}=\begin{pmatrix}M_{\nu}&\tilde{M}_{D}&0\\\
\tilde{M}_{D}^{T}&0&0\\\ 0&0&\sim{M_{R}}\\\ \end{pmatrix}.$ (14)

Then, we diagonalize $\mathcal{M}_{\nu}$ in Eq. (14) and get the neutrino mass
eigenvalues,

$\mathcal{M}_{\nu}=\begin{pmatrix}m_{\nu}&0&0\\\ 0&\tilde{m}_{\nu}&0\\\
0&0&\sim{M_{R}}\\\ \end{pmatrix},$ (15)

where $m_{\nu}$ is the left-handed Majorana neutrino mass matrix,
$\tilde{m}_{\nu}$ is the mirror neutrino mass matrix, and $M_{R}$ is the
right-handed Majorana neutrino mass matrix. The left-handed Majorana and the
mirror neutrino mass eigenvalues are related to $\tilde{Y}_{5}$ matrix
diagonal elements in Eq. (5). The down-type quark mass eigenvalues in Eq. (7)
also include the $\tilde{Y}_{5}$ matrix diagonal elements. Since the
$\tilde{Y}_{5}$ diagonal elements include the up- and down-type quark mass
eigenvalues, the left-handed Majorana and the mirror neutrino mass eigenvalues
are represented in terms of the up- and down-type quark mass eigenvalues. We
show the left-handed Majorana and the mirror neutrino mass eigenvalues
$m_{i}~{}(i=1$-$6)$ just below:

$\displaystyle m_{1}$
$\displaystyle=\frac{\big{(}v_{H}{Y_{D}^{11}}\big{)}^{2}-\sqrt{\Big{[}\big{(}{v_{H}}Y_{D}^{11}\big{)}^{4}+\frac{72m_{d1}^{2}m_{d2}^{2}}{m_{u1}m_{u2}}\big{(}{v_{\Phi}}{Y_{M}^{1}}\big{)}^{2}\Big{]}}}{2\sqrt{2}v_{\Phi}{Y_{M}^{1}}},$
(16) $\displaystyle m_{2}$
$\displaystyle=\frac{\big{(}v_{H}{Y_{D}^{11}}\big{)}^{2}+\sqrt{\Big{[}\big{(}{v_{H}}Y_{D}^{11}\big{)}^{4}+\frac{72m_{d1}^{2}m_{d2}^{2}}{m_{u1}m_{u2}}\big{(}{v_{\Phi}}{Y_{M}^{1}}\big{)}^{2}\Big{]}}}{2\sqrt{2}v_{\Phi}{Y_{M}^{1}}},$
$\displaystyle m_{3}$
$\displaystyle=\frac{\big{(}v_{H}{Y_{D}^{22}}\big{)}^{2}-\sqrt{\Big{[}\big{(}{v_{H}}Y_{D}^{22}\big{)}^{4}+\frac{72m_{d3}^{2}m_{d4}^{2}}{m_{u3}m_{u4}}\big{(}{v_{\Phi}}{Y_{M}^{2}}\big{)}^{2}\Big{]}}}{2\sqrt{2}v_{\Phi}{Y_{M}^{2}}},$
$\displaystyle m_{4}$
$\displaystyle=\frac{\big{(}v_{H}{Y_{D}^{22}}\big{)}^{2}+\sqrt{\Big{[}\big{(}{v_{H}}Y_{D}^{22}\big{)}^{4}+\frac{72m_{d3}^{2}m_{d4}^{2}}{m_{u3}m_{u4}}\big{(}{v_{\Phi}}{Y_{M}^{2}}\big{)}^{2}\Big{]}}}{2\sqrt{2}v_{\Phi}{Y_{M}^{2}}},$
$\displaystyle m_{5}$
$\displaystyle=\frac{\big{(}v_{H}{Y_{D}^{33}}\big{)}^{2}-\sqrt{\Big{[}\big{(}{v_{H}}Y_{D}^{33}\big{)}^{4}+\frac{72m_{d5}^{2}m_{d6}^{2}}{m_{u5}m_{u6}}\big{(}{v_{\Phi}}{Y_{M}^{3}}\big{)}^{2}\Big{]}}}{2\sqrt{2}v_{\Phi}{Y_{M}^{3}}},$
$\displaystyle m_{6}$
$\displaystyle=\frac{\big{(}v_{H}{Y_{D}^{33}}\big{)}^{2}+\sqrt{\Big{[}\big{(}{v_{H}}Y_{D}^{33}\big{)}^{4}+\frac{72m_{d5}^{2}m_{d6}^{2}}{m_{u5}m_{u6}}\big{(}{v_{\Phi}}{Y_{M}^{3}}\big{)}^{2}\Big{]}}}{2\sqrt{2}v_{\Phi}{Y_{M}^{3}}}.$

By using these eigenvalues, we obtain the mass relations among the neutrinos,

$\displaystyle m_{1}m_{2}$
$\displaystyle=\frac{9m_{d1}^{2}m_{d2}^{2}}{m_{u1}m_{u2}},$ (17)
$\displaystyle m_{3}m_{4}$
$\displaystyle=\frac{9m_{d3}^{2}m_{d4}^{2}}{m_{u3}m_{u4}},$ $\displaystyle
m_{5}m_{6}$ $\displaystyle=\frac{9m_{d5}^{2}m_{d6}^{2}}{m_{u5}m_{u6}}.$

Now, we assume that $m_{1}$, $m_{3}$, and $m_{5}$ are the left-handed Majorana
neutrino masses. Since the neutrino masses are related to the up- and down-
type quark masses, we can get the product of two neutrino masses. Then, we
estimate the heavy neutrino masses by using Eq. (17). There are two patterns
about the heavy neutrino masses.

  * 1

. The $m_{u3}$ is the SM particle, $m_{u4}$ is
$\mathcal{O}(M_{\mathrm{GUT}})$, $m_{d3}$ is the SM particle, and $m_{d4}$ is
$\mathcal{O}(M_{\mathrm{GUT}})$.

In this case, the neutrino mass relation is

$m_{3}m_{4}=\mathcal{O}(M_{\mathrm{GUT}}).$ (18)

Then, the heavy neutrino mass is the GUT scale. The mass relation between
$m_{5}$ and $m_{6}$ is the same result.

  * 2

. The $m_{u1}$ is the SM particle, $m_{u2}$ is
$\mathcal{O}(M_{\mathrm{GUT}})$, $m_{d1}$ is bottom quark, and $m_{d2}$ is
$\mathcal{O}(10^{3})$ GeV.

By using the benchmark that $m_{d1}=4.18$ GeV, $m_{d2}=5$ TeV, and
$m_{u2}=M_{\mathrm{GUT}}={7.28}\times 10^{15}$ GeV, the neutrino mass relation
is

$m_{1}m_{2}\approx\frac{5.40\times{10^{-7}}}{m_{u1}}~{}\mathrm{(GeV)}^{2}.\\\
$ (19)

In Eq. (19), $m_{u1}$ is three patterns such as the up quark, charm quark, and
top quark. Then, we get three neutrino mass relations:

$m_{1}m_{2}=2.50\times{10}^{14}~{}\mathrm{(eV)^{2}},\quad{m_{1}m_{2}}=4.25\times{10}^{11}~{}\mathrm{(eV)^{2}},\quad{m_{1}m_{2}}=3.13\times{10}^{9}~{}\mathrm{(eV)^{2}}.$
(20)

By using Eq. (20), we estimate the heavy neutrino masses. There are two
neutrino mass hierarchies, normal hierarchy (NH) and inverted hierarchy (IH).
We can derive the upper bound of the lightest neutrino masses from the
experimental results of the neutrino mass squared differences [50, 51] and the
sum of neutrino masses [52]. In the NH, the upper bound of the lightest
neutrino mass is 0.03 eV. Then, the lower bound of the heavy neutrino masses
is obtained just below:

$m_{\mathrm{heavy}}>8.33\times{10}^{6}~{}\mathrm{GeV},\quad{m_{\mathrm{heavy}}}>1.42\times{10}^{4}~{}\mathrm{GeV},\quad{m_{\mathrm{heavy}}}>104.34~{}\mathrm{GeV}.$
(21)

On the other hand, in the IH, the upper bound of the lightest neutrino mass is
0.016 eV. Then, the lower bound of the heavy neutrino masses is obtained just
below:

$m_{\mathrm{heavy}}>1.56\times{10}^{7}~{}\mathrm{GeV},\quad{m_{\mathrm{heavy}}}>2.66\times{10}^{4}~{}\mathrm{GeV},\quad{m_{\mathrm{heavy}}}>195.63~{}\mathrm{GeV}.$
(22)

Therefore, the lower bound of the heavy neutrino masses in the top quark case
is $\mathcal{O}(10^{2})$ GeV and it is testable for the future experiment
about the sterile neutrinos[36].

##  6 Summary and Discussions

We have proposed an SU(5)$\times$U(1)${}_{\text{X}}\times$U(1)${}_{\text{PQ}}$
model without SUSY. The $\mathrm{U(1)_{X}}$ gauge symmetry is the
generalization of the $\mathrm{U(1)_{B-L}}$ gauge symmetry. The
$\mathrm{U(1)_{PQ}}$ symmetry is the global PQ symmetry. The PQ symmetry can
solve the strong CP problem and the pseudo-Nambu Goldstone boson from the PQ
symmetry breaking called the axion is the DM candidate. In previous work, they
added one family of the same representation for the SM particles and one
mirror family. The mirror family is the conjugate of the representation for
the SM particles. We have introduced only three mirror families in order to
unify the SM gauge couplings and avoid the restriction of proton lifetime. The
SM matter fields are unified into $\mathbf{\bar{5}}$+$\mathbf{10}$
representation of SU(5) and the mirror family into
$\mathbf{5}$+$\mathbf{\overline{10}}$ representation. We also have introduced
three gauge singlet right-handed Majorana neutrinos to derive the left-handed
Majorana neutrino masses through the type-I seesaw mechanism and cancel all
the U(1)${}_{\text{X}}$ related anomalies. In order to obtain the difference
of the masses for the down-type quarks and charged leptons, we have introduced
the $\mathbf{45}$ representation Higgs in addition to the $\mathbf{5}$
representation and $\mathbf{24}$ one. In order to occur the inflation and
break the U(1)${}_{\text{X}}$ gauge symmetry, the gauge singlet scalar field
is introduced. Then, we have discussed the mass relations between the SM and
mirror particles and identified the mass scales of the mirror particles. We
have set the benchmark for the mass scales of the mirror particles in our
analysis. Since the new particles exist in the intermediate energy scale and
contribute to the RGE, the SM gauge couplings unify successfully at
$M_{\mathrm{GUT}}\approx{7.28}\times 10^{15}$ GeV and
$\alpha_{\mathrm{GUT}}=\alpha_{1}=\alpha_{2}=\alpha_{3}\approx{1/31.7}$. Our
model expects the proton lifetime as
$\tau_{p}(p\to\pi^{0}{e^{+}})\approx{8.07\times 10^{34}}~{}\mathrm{years}$ and
can be tested by the future proton decay search, e.g., the Hyper-Kamiokande
experiment expected as $\tau_{p}(p\to\pi^{0}{e^{+}})<1.0\times 10^{35}$ years.
Also, by using the mass relations between the active and mirror neutrinos, we
have estimated the lower bound of the heavy neutrino masses. The lowest bound
of the heavy neutrino masses is ${m_{\mathrm{heavy}}}>104.34~{}\mathrm{GeV}$
and it is testable for the future experiment about the sterile neutrinos.

In our model, the heavy neutrinos have the potential to be the candidate for
DM in addition to the axion. Since the SU(5) symmetry breaking and the
$\mathrm{U(1)_{PQ}}$ symmetry breaking occur at the same energy scale, the
scenarios of inflation are restricted by the axion domain wall, axion DM
isocurvature [53, 54], and SU(5) monopole problems [55, 56, 57, 58]. Then, we
will discuss the scenarios of dark matter and inflation in future work.

Acknowledgement

We would like to thank Prof. N. Okada for useful discussions and comments.

## Appendix

##  Appendix A The relevant representations in SU(5)

We show the unification of the SM particles into the SU(5) representations.
The SM matter fields are unified into $\mathbf{\bar{5}}+\mathbf{10}$ just
below:

$\displaystyle\psi_{\bar{5}}$ $\displaystyle=\begin{pmatrix}d^{c}_{1}\\\
d^{c}_{2}\\\ d^{c}_{3}\\\ e\\\ -\nu\\\
\end{pmatrix}_{L},\quad\psi_{10}=\frac{1}{\sqrt{2}}\begin{pmatrix}0&u_{3}^{c}&-u_{2}^{c}&-u^{1}&-d^{1}\\\
-u_{3}^{c}&0&u_{1}^{c}&-u^{2}&-d^{2}\\\
u_{2}^{c}&-u_{1}^{c}&0&-u^{3}&-d^{3}\\\ u^{1}&u^{2}&u^{3}&0&-e^{+}\\\
d^{1}&d^{2}&d^{3}&e^{+}&0\\\ \end{pmatrix}_{L}.$ (23)

Here, $``c"$ means charge conjugation, and the SM gauge bosons are unified
into $\mathbf{24}$ just below:

$\mathbf{24}=\begin{pmatrix}G_{\mu}-\frac{1}{\sqrt{15}}B_{\mu}&V^{\dagger}_{\mu}\\\
V_{\mu}&W_{\mu}+\frac{3}{2\sqrt{15}}B_{\mu}\\\ \end{pmatrix},$ (24)

with quantum numbers as

$G_{\mu}\sim(8,1,0),\quad{W_{\mu}}\sim(1,3,0),\quad{B_{\mu}}\sim(1,1,0),\quad{V}^{\mu}\sim(3,2,-5/6).$
(25)

On the other hand, the $\mathbf{5}$ representation Higgs $H$ and $\mathbf{24}$
representation Higgs $\Sigma$ are unified as follows:

$H=\begin{pmatrix}H_{1}\\\ H_{2}\\\ H_{3}\\\ H^{+}\\\ H^{0}\\\
\end{pmatrix},\quad\Sigma=\begin{pmatrix}\Sigma_{8}-\frac{2}{\sqrt{30}}\Sigma_{0}&\Sigma_{(\bar{3},2)}\\\
\Sigma_{(3,2)}&\Sigma_{3}+\frac{3}{\sqrt{30}}\Sigma_{0}\end{pmatrix},\\\ $
(26)

where $H_{1}$, $H_{2}$, and $H_{3}$ are colored Higgs and $H^{+}$ and $H^{0}$
are the SM Higgs doublet. The component of the $\mathbf{24}$ representation
Higgs have the quantum numbers just below:

$\Sigma_{8}\sim(8,1,0),\quad\Sigma_{3}\sim(1,3,0),\quad\Sigma_{0}\sim(1,1,0),\quad\Sigma_{(3,2)}\sim(3,2,-\frac{5}{6}),\quad\Sigma_{(\bar{3},2)}\sim(\bar{3},2,\frac{5}{6}),\\\
$ (27)

and the $\mathbf{45}$ representation Higgs $\chi$ has the following
components,

$\displaystyle\chi\sim\Phi_{1}(8,2,\frac{1}{2})\oplus\Phi_{2}(\bar{6},1,-\frac{1}{3})$
$\displaystyle\oplus\Phi_{3}(3,3,-\frac{1}{3})$
$\displaystyle\oplus\Phi_{4}(\bar{3},2,-\frac{7}{6})\oplus\Phi_{5}(3,1,-\frac{1}{3})\oplus\Phi_{6}(\bar{3},1,\frac{4}{3})\oplus{H_{2}}(1,2,\frac{1}{2}).$
(28)

##  Appendix B The RGE and beta coefficients

We show the RGE form,

$\mu\frac{dg_{i}}{d\mu}=g_{i}^{3}[\beta_{gi}(\mathrm{SM})+\beta_{gi}(\mathrm{NEW})],$
(29)

where $g_{i}$ $(i=1\mathrm{-}3)$ are the SM gauge couplings,
$\beta_{gi}(\mathrm{SM})$ are the contributions of the SM particles, and
$\beta_{gi}(\mathrm{NEW})$ are the new particle contributions, respectively.
In our analysis, we consider the contributions of the SM particles at 2-loop
level [59, 60, 61] and the new particle contributions at 1-loop level such as

$\beta_{gi}(\mathrm{NEW})=\frac{1}{16\pi^{2}}[b_{i}\times\theta(\mu-M)],$ (30)

where $M$ is the mass of each field and the beta coefficients $b_{i}$ for each
field are listed in Table 3. Here, $\theta(\mu-M)$ is a step function and we
add the new particle contributions as the step function for each particle.

Fields | $b_{1}$ | $b_{2}$ | $b_{3}$
---|---|---|---
$\Phi_{1}$ | 4/5 | 4/3 | 2
$\Phi_{2}$ | 2/15 | 0 | 5/6
$\Phi_{3}$ | 1/5 | 2 | 1/2
$\Phi_{4}$ | 49/30 | 1/2 | 1/3
$\Phi_{5}$ | 1/15 | 0 | 1/6
$\Phi_{6}$ | 16/15 | 0 | 1/6
$H_{2}$ | 1/10 | 1/6 | 0
$m_{ui}$ | 17/30 | 1/2 | 2/3
$m_{di}$ | 1/6 | 1/2 | 2/3
$m_{ei}$ | 3/5 | 1/3 | 0
Table 3: The beta coefficients for each field.

## References

  * [1] G. Aad et al. [ATLAS], Phys. Lett. B 716 (2012), 1-29 [arXiv:1207.7214 [hep-ex]].
  * [2] S. Chatrchyan et al. [CMS], Phys. Lett. B 716 (2012), 30-61 [arXiv:1207.7235 [hep-ex]].
  * [3] H. Georgi and S. L. Glashow, Phys. Rev. Lett. 32 (1974), 438-441
  * [4] P. Kalyniak and J. N. Ng, Phys. Rev. D 26 (1982), 890
  * [5] P. Eckert, J. M. Gerard, H. Ruegg and T. Schucker, Phys. Lett. B 125 (1983), 385-388
  * [6] H. Georgi and C. Jarlskog, Phys. Lett. B 86 (1979), 297-300
  * [7] A. Takenaka et al. [Super-Kamiokande], Phys. Rev. D 102 (2020) no.11, 112011 [arXiv:2010.16098 [hep-ex]].
  * [8] H. Georgi, H. R. Quinn and S. Weinberg, Phys. Rev. Lett. 33 (1974), 451-454
  * [9] S. Dimopoulos, S. Raby and F. Wilczek, Phys. Rev. D 24 (1981), 1681-1683
  * [10] W. J. Marciano and G. Senjanovic, Phys. Rev. D 25 (1982), 3092
  * [11] M. B. Einhorn and D. R. T. Jones, Nucl. Phys. B 196 (1982), 475-488
  * [12] U. Amaldi, W. de Boer and H. Furstenau, Phys. Lett. B 260 (1991), 447-455
  * [13] P. Langacker and M. x. Luo, Phys. Rev. D 44 (1991), 817-822
  * [14] J. R. Ellis, S. Kelley and D. V. Nanopoulos, Phys. Lett. B 260 (1991), 131-137
  * [15] C. Giunti, C. W. Kim and U. W. Lee, Mod. Phys. Lett. A 6 (1991), 1745-1755
  * [16] T. Appelquist, B. A. Dobrescu and A. R. Hopper, Phys. Rev. D 68 (2003), 035012 [arXiv:hep-ph/0212073 [hep-ph]].
  * [17] A. Davidson, Phys. Rev. D 20 (1979), 776
  * [18] R. N. Mohapatra and R. E. Marshak, Phys. Rev. Lett. 44 (1980), 1316-1319 [erratum: Phys. Rev. Lett. 44 (1980), 1643]
  * [19] R. E. Marshak and R. N. Mohapatra, Phys. Lett. B 91 (1980), 222-224
  * [20] C. Wetterich, Nucl. Phys. B 187 (1981), 343-375
  * [21] A. Masiero, J. F. Nieves and T. Yanagida, Phys. Lett. B 116 (1982), 11-15
  * [22] R. N. Mohapatra and G. Senjanovic, Phys. Rev. D 27 (1983), 254
  * [23] W. Buchmuller, C. Greub and P. Minkowski, Phys. Lett. B 267 (1991), 395-399
  * [24] S. Oda, N. Okada and D. s. Takahashi, Phys. Rev. D 92 (2015) no.1, 015026 [arXiv:1504.06291 [hep-ph]].
  * [25] N. Okada, S. Okada and D. Raut, Phys. Lett. B 780 (2018), 422-426 [arXiv:1712.05290 [hep-ph]].
  * [26] R. D. Peccei and H. R. Quinn, Phys. Rev. Lett. 38 (1977), 1440-1443
  * [27] R. D. Peccei and H. R. Quinn, Phys. Rev. D 16 (1977), 1791-1797
  * [28] R. D. Peccei, Lect. Notes Phys. 741 (2008), 3-17 [arXiv:hep-ph/0607268 [hep-ph]].
  * [29] S. Weinberg, Phys. Rev. Lett. 40 (1978), 223-226
  * [30] F. Wilczek, Phys. Rev. Lett. 40 (1978), 279-282
  * [31] N. Okada, D. Raut and Q. Shafi, Phys. Rev. D 104 (2021) no.5, 055041 [arXiv:2106.10073 [hep-ph]].
  * [32] S. I. Blinnikov and M. Y. Khlopov, Sov. J. Nucl. Phys. 36 (1982), 472 ITEP-11-1982.
  * [33] S. I. Blinnikov and M. Khlopov, Sov. Astron. 27 (1983), 371-375
  * [34] M. Y. Khlopov, G. M. Beskin, N. E. Bochkarev, L. A. Pustylnik and S. A. Pustylnik, Sov. Astron. 35 (1991), 21 FERMILAB-PUB-89-193-A.
  * [35] T. Dealtry [Hyper-Kamiokande], [arXiv:1904.10206 [hep-ex]].
  * [36] S. Alekhin, W. Altmannshofer, T. Asaka, B. Batell, F. Bezrukov, K. Bondarenko, A. Boyarsky, K. Y. Choi, C. Corral and N. Craig, et al. Rept. Prog. Phys. 79 (2016) no.12, 124201 [arXiv:1504.04855 [hep-ph]].
  * [37] A. Davidson and K. C. Wali, Phys. Rev. Lett. 48 (1982), 11
  * [38] A. Davidson, V. P. Nair and K. C. Wali, Phys. Rev. D 29 (1984), 1504
  * [39] A. Davidson, V. P. Nair and K. C. Wali, Phys. Rev. D 29 (1984), 1513
  * [40] P. Minkowski, Phys. Lett. B 67 (1977), 421-428
  * [41] T. Yanagida, in Proceedings of the Workshop on Unified Theories and Baryon Number in the Universe, eds. O. Sawada and A. Sugamoto (KEK report 79-18, 1979).
  * [42] M. Gell-Mann, P. Ramond and R. Slansky, Conf. Proc. C 790927 (1979), 315-321 [arXiv:1306.4669 [hep-th]].
  * [43] R. N. Mohapatra and G. Senjanovic, Phys. Rev. Lett. 44 (1980), 912
  * [44] J. Schechter and J. W. F. Valle, Phys. Rev. D 22 (1980), 2227
  * [45] R. L. Workman et al. [Particle Data Group], PTEP 2022 (2022), 083C01
  * [46] A. M. Sirunyan et al. [CMS], Eur. Phys. J. C 79 (2019), 90 [arXiv:1809.08597 [hep-ex]].
  * [47] A. Hayreter and G. Valencia, Phys. Rev. D 96 (2017) no.3, 035004 [arXiv:1703.04164 [hep-ph]].
  * [48] V. Miralles and A. Pich, Phys. Rev. D 100 (2019) no.11, 115042 [arXiv:1910.07947 [hep-ph]].
  * [49] P. Nath and P. Fileviez Perez, Phys. Rept. 441 (2007), 191-317 [arXiv:hep-ph/0601023 [hep-ph]].
  * [50] I. Esteban, M. C. Gonzalez-Garcia, M. Maltoni, T. Schwetz and A. Zhou, JHEP 09 (2020), 178 [arXiv:2007.14792 [hep-ph]].
  * [51] M. C. Gonzalez-Garcia, M. Maltoni and T. Schwetz, Universe 7 (2021) no.12, 459 [arXiv:2111.03086 [hep-ph]].
  * [52] N. Aghanim et al. [Planck], Astron. Astrophys. 641 (2020), A6 [erratum: Astron. Astrophys. 652 (2021), C4] [arXiv:1807.06209 [astro-ph.CO]].
  * [53] M. Kawasaki and K. Nakayama, Ann. Rev. Nucl. Part. Sci. 63 (2013), 69-95 [arXiv:1301.1123 [hep-ph]].
  * [54] N. Okada, D. Raut and Q. Shafi, Eur. Phys. J. C 80 (2020) no.11, 1056 [arXiv:2002.07110 [hep-ph]].
  * [55] G. ’t Hooft, Nucl. Phys. B 79, 276 (1974)
  * [56] A. M. Polyakov, JETP Lett. 20 (1974), 194-195 PRINT-74-1566 (LANDAU-INST).
  * [57] P. Langacker and S. Y. Pi, Phys. Rev. Lett. 45 (1980), 1
  * [58] J. Preskill, Ann. Rev. Nucl. Part. Sci. 34 (1984), 461-530
  * [59] M. E. Machacek and M. T. Vaughn, Nucl. Phys. B 222 (1983), 83-103
  * [60] M. E. Machacek and M. T. Vaughn, Nucl. Phys. B 236 (1984), 221-232
  * [61] M. E. Machacek and M. T. Vaughn, Nucl. Phys. B 249 (1985), 70-92


$\displaystyle\frac{d\mathcal{K}}{d\lambda}=-\frac{2\Sigma_{p}}{\Delta_{p}}\left(\varpi_{p}^{2}\mathcal{B}_{p}a_{t}+a\mathcal{B}_{p}a_{\phi}+\Delta_{p}^{2}u_{r}a_{r}\right).$
(66c)

To find the partial derivatives $\partial r_{p}/\partial\mathcal{J}_{j}$, make
use of the value of radial potential $V_{r}$ defined as

$\begin{split}V_{r}(r)&=\mathcal{B}^{2}-\Delta\left(r^{2}+\mathcal{K}\right)\\\
&=-(1-\mathcal{E}^{2})(r-r_{1})(r-r_{2})(r-r_{3})(r-r_{4})\\\
&=-(1-\mathcal{E}^{2})(r-r_{p})^{2}(r-r_{3})(r-r_{4}).\end{split}$ (67)

where $r_{1},r_{2},r_{3},r_{4}$ are the roots of $V_{r}$ and in the spherical
case: $r_{1}=r_{2}=r_{p}$. We then define the derivative of $V_{r}$ with
respect to $r$ as

$\begin{split}\kappa\coloneqq&\frac{dV_{r}}{dr}\\\
=&4\mathcal{E}\mathcal{B}r-2\Delta r-2(r-M)(r^{2}+K)\\\
=&-2(1-\mathcal{E}^{2})\bigg{[}(r-r_{3})(r-r_{4})\\\
&+(r-r_{p})(r-r_{4})+(r-r_{p})(r-r_{3})\bigg{]}.\end{split}$ (68)

If we take the derivative of $\kappa$ with respect to $r$ and evaluate at
$r=r_{p}$, we obtain

$\frac{d\kappa_{p}}{dr}=-2(1-\mathcal{E}^{2})(r_{p}-r_{3})(r_{p}-r_{4}).$ (69)

However, if we now take the partial derivatives of $\kappa$ with respect to
$\vec{\mathcal{J}}$ and evaluate at $r=r_{p}$ we are left with

$\begin{split}\frac{\partial\kappa_{p}}{\partial\mathcal{J}_{j}}&=2(1-\mathcal{E}^{2})(r_{p}-r_{3})(r_{p}-r_{4})\frac{\partial
r_{p}}{\partial\mathcal{J}_{j}}\\\ &=-\frac{d\kappa_{p}}{dr}\frac{\partial
r_{p}}{\partial\mathcal{J}_{j}}.\end{split}$ (70)

Subbing the above result into Eq. (65), explicitly calculating
$\partial\kappa/\partial\mathcal{J}_{j}$ using Eq. (68), evaluating at
$r=r_{p}$ and simplifying gives us our final result

$\displaystyle\begin{split}\frac{dr_{p}}{d\lambda}&=-4r_{p}\frac{(\mathcal{B}_{p}+\varpi_{p}^{2}\mathcal{E})\frac{dE}{d\lambda}-a\mathcal{E}\frac{d{\mathcal{L}_{z}}}{d\lambda}+\frac{M-r_{p}}{2r_{p}}\frac{d\mathcal{K}}{d\lambda}}{\frac{d\kappa_{p}}{dr}}\\\
&\coloneqq F_{r}^{(1)}\end{split}$ (71)

The above relationship is finite for all bound stable spherical orbits and
only becomes ill-defined when $d\kappa_{p}/dr=0$, which, only occurs when
$r_{p}=r_{3}$ , i.e., when $r_{p}=r_{\text{ISSO}}$.

##  Appendix B Equivalence between $\vec{\varphi}$ and waveform voices

In this appendix, we wish to establish a relationship between the solutions
for the inspiral quantities and the waveform. We start by assigning the origin
of our coordinate scheme to be the position of the primary MBH. We then set
the Cartesian coordinates of our observer to be $x=(t_{\text{ret}},\vec{x})$.
It is useful to express the observer’s coordinates in terms of spherical
coordinates $(R,\theta,\Phi)$, where $R$ is the distance from the observer to
origin given by $R^{2}=\vec{x}\cdot\vec{x}$, $\Theta$ is the observer’s
latitude and $\Phi$ is the observer’s azimuth. We assign the Cartesian
coordinates of the secondary, which we model as a point particle, to be
$x_{p}=(t_{p},\vec{x}_{p})$. As such, we can express the retarded time as
measured by the observer in terms of Boyer-Lindquist coordinate time $t$ via
$t_{\text{ret}}=t-|\vec{x}-\vec{x}_{p}|\approx t-R$ in the limit where $R$ is
large. The complex waveform strain can be decomposed onto a basis of spin-
weighted spherical harmonics ${}_{-2}Y_{lm}$ given by:

$h(t_{\text{ret}})=h_{+}-ih_{\times}=\frac{1}{R}\sum_{lm}H_{lm}(t_{\text{ret}})_{-2}Y_{lm}(\Theta,\Phi).$
(72)

If we assume the secondary is moving on a geodesic, the waveform modes
$H_{lm}$ exhibit a discrete frequency spectrum and so can be Fourier
decomposed into:

$H_{lm}(t_{\text{ret}})=\sum_{(n,k)}A_{lmnk}(\vec{P})e^{-i(m\Omega_{\phi}^{(0)}+n\Omega_{r}^{(0)}+k\Omega_{z}^{(0)})t_{\text{ret}}}$
(73)

where the complex amplitudes $A_{lmnk}$ can be related to non-homogeneous
Teukolsky amplitudes, which can be precomputed for a given set of orbital
elements $\vec{P}$ [14]. Such waveforms are known as “snapshot” waveforms, as
they only capture a small section of the total waveform [112].

For a full EMRI waveform, one needs to account for the fact that the
frequencies and the orbital elements will slowly evolve with time, resulting
in a continuous frequency spectrum. As such, this Fourier mode decomposition
becomes a “multi-voice” decomposition [14]:

$H_{lm}(t_{\text{ret}})=\sum_{(n,k)}A_{lmnk}(\vec{P}(t_{\text{ret}}))e^{-i\varPhi_{mnk}(t_{\text{ret}})},$
(74)

where the waveform “voices” $\varPhi_{mnk}$ are given by

$\varPhi_{mnk}=m\int_{0}^{t}\Omega_{\phi}(t)dt+n\int_{0}^{t}\Omega_{r}(t)dt+k\int_{0}^{t}\Omega_{z}(t)dt.$
(75)

Let us recall the equations of motion for the orbital phases obtains after
performing the NIT and the additional transformation such that our solutions
are in terms of Boyer-Lindquist time (46). When we express this in integral
form, one obtains

$\displaystyle\begin{split}\varphi_{\alpha}&=\int_{0}^{t}\left(\Omega^{(0)}_{\alpha}(t)+\epsilon\Omega^{(1)}_{\alpha}(t)+\mathcal{O}(\epsilon^{2})\right)dt\\\
&=\int_{0}^{t}\Omega_{\alpha}(t)dt+\mathcal{O}(\epsilon)\end{split}$ (76)

As such we can see that to leading order in $\epsilon$, these phases are
directly related to the waveform voices:

$\varPhi_{mnk}=m\varphi_{\phi}+n\varphi_{r}+k\varphi_{z}+\mathcal{O}(\epsilon)$
(77)

This is further supported by Ref. [82], where a relationship between the
waveform phases and the NIT phases $\vec{\tilde{q}}$ was independently
derived. When expressed in our notation with polar motion included, this
relationship is given by:

$\begin{split}\varPhi_{mnk}&=m\tilde{\phi}+n\tilde{q}_{r}+k\tilde{q}_{z}\\\
&\quad+(m\Omega_{\phi}^{(0)}+n\Omega_{r}^{(0)}+k\Omega_{z}^{(0)})(t-\tilde{t})+\mathcal{O}(\epsilon)\end{split}$
(78)

Using the relationship between $t$ and $\tilde{t}$ given by Eq. (42c), the
result for $\Delta t$ given by Eq. (49a), and the relationship between the NIT
action angles given by Eq. (46b), one can obtain to leading order in
$\epsilon$:

$\begin{split}\varPhi_{mnk}&=m(\tilde{\phi}-\Omega^{(0)}_{\phi}(t-\tilde{t}))+n(\tilde{q}_{r}-\Omega^{(0)}_{r}(t-\tilde{t}))+k(\tilde{q}_{z}-\Omega_{z}(t-\tilde{t}))+\mathcal{O}(\epsilon)\\\
&=m(\tilde{\phi}-\Omega^{(0)}_{\phi}Z_{t}^{(0)})+n(\tilde{q}_{r}-\Omega^{(0)}_{r}Z_{t}^{(0)})+k(\tilde{q}_{z}-\Omega^{(0)}_{z}Z_{t}^{(0)}+\mathcal{O}(\epsilon))\\\
&=m(\tilde{\phi}+\Omega^{(0)}_{\phi}\Delta
t)+n(\tilde{q}_{r}+\Omega^{(0)}_{r}\Delta
t)+k(\tilde{q}_{z}+\Omega^{(0)}_{z}\Delta t)+\mathcal{O}(\epsilon)\\\
&=m(\tilde{\phi}+\Delta\varphi_{\phi})+n(\tilde{q}_{r}+\Delta\varphi_{r})+k(\tilde{q}_{z}+\Delta\varphi_{z})+\mathcal{O}(\epsilon)\\\
&=m\varphi_{\phi}+n\varphi_{r}+k\varphi_{z}+\mathcal{O}(\epsilon)\end{split}$
(79)

One can also freely replace the dependence of the evolving orbital elements
$\vec{P}(t)$ with either $\vec{\tilde{P}}(t)$ or $\vec{\mathcal{P}}(t)$, as at
leading order in $\epsilon$ they are identical. As a result, we can now relate
a solution to the EMRI’s inspiral trajectory to its associated Teukolsky based
waveform.

##  Appendix C Gauge completion integrands

Here we give the explicit integrands for the gauge completion integrals (18)
and (19),

$\mathcal{I}_{\alpha}=\begin{aligned}
&\mathcal{E}^{2}r_{p}\left(1-z_{p}^{2}\right)\Big{[}3r_{p}^{10}+3r_{p}^{9}\left(1-z_{p}^{2}\right)-6r_{p}^{8}a^{2}\left(1+z_{p}^{2}-3z_{p}^{4}\right)-2r_{p}^{7}a^{2}\left(5+4z_{p}^{2}+3z_{p}^{4}\right)\\\
&\qquad-
r_{p}^{6}a^{2}\left(3a^{2}-(48-31a^{2})z_{p}^{2}+(24-47a^{2})z_{p}^{4}-3a^{2}z_{p}^{6}\right)-r_{p}^{5}a^{4}\left(5-9z_{p}^{2}+41z_{p}^{4}+3z_{p}^{6}\right)\\\
&\qquad-2r_{p}^{4}a^{4}z_{p}^{2}\left(2(1+2a^{2})-(4+21a^{2})z_{p}^{2}-(a^{2}+12)z_{p}^{4}\right)+2r_{p}^{3}a^{6}z_{p}^{2}\left(11-16z_{p}^{2}+z_{p}^{4}\right)\\\
&\qquad-
r_{p}^{2}a^{6}z_{p}^{2}\left(10a^{2}-15a^{2}z_{p}^{2}+5a^{2}z_{p}^{4}+8z_{p}^{4}\right)-r_{p}a^{8}z_{p}^{4}\left(5-13z_{p}^{2}\right)-2a^{10}z_{p}^{4}\left(1-2z_{p}^{2}\right)\left(1-z_{p}^{2}\right)\Big{]}\\\
&+({\mathcal{L}_{z}}-a\mathcal{E})^{2}\left(1-z_{p}^{2}\right)^{2}\Big{[}3r_{p}^{8}-3r_{p}^{7}\left(4-a^{2}+5a^{2}z_{p}^{2}\right)-6r_{p}^{6}a^{2}\left(3+z_{p}^{2}\right)\\\
&\qquad+r_{p}^{5}a^{2}\left(4+6a^{2}+3(12-13a^{2})z_{p}^{2}-3a^{2}z_{p}^{4}\right)-r_{p}^{4}a^{4}\left(13-12z_{p}^{2}+9z_{p}^{4}\right)\\\
&\qquad+3r_{p}^{3}a^{4}\left(a^{2}-(4+11a^{2})z_{p}^{2}-2a^{2}z_{p}^{4}\right)+2r_{p}^{2}a^{6}z_{p}^{2}\left(9-z_{p}^{2}\right)-3r_{p}a^{8}z_{p}^{2}\left(3+z_{p}^{2}\right)-a^{8}z_{p}^{4}\Big{]}\\\
&-{\mathcal{L}_{z}}^{2}r_{p}\Big{[}3r_{p}^{8}\left(1+z_{p}^{2}\right)^{2}-3r_{p}^{7}\left(1+3z_{p}^{2}\right)\left(3+z_{p}^{2}\right)-r_{p}^{6}\left(a^{2}-(48-2a^{2})z_{p}^{2}-47a^{2}z_{p}^{4}+12a^{2}z_{p}^{6}\right)\\\
&\qquad-
r_{p}^{5}a^{2}\left(5-8z_{p}^{2}+65z_{p}^{4}+18z_{p}^{6}\right)-r_{p}^{4}a^{2}z_{p}^{2}\left(16+26a^{2}-57a^{2}z_{p}^{2}-(48-4a^{2})z_{p}^{4}+3a^{2}z_{p}^{6}\right)\\\
&\qquad+r_{p}^{3}a^{4}z_{p}^{2}\left(22-21z_{p}^{2}-8z_{p}^{4}-9z_{p}^{6}\right)-r_{p}^{2}a^{4}z_{p}^{2}\left(10a^{2}-15a^{2}z_{p}^{2}+8(2+a^{2})z_{p}^{4}-3a^{2}z_{p}^{6}\right)\\\
&\qquad-
r_{p}a^{6}z_{p}^{4}\left(1-5z_{p}^{2}\right)\left(5-z_{p}^{2}\right)+2a^{8}z_{p}^{4}\left(1-4z_{p}^{2}+z_{p}^{4}\right)\Big{]}\\\
&-r_{p}\left(3r_{p}^{2}-a^{2}\right)\left(1-z_{p}^{4}\right)\left(a^{2}z_{p}^{2}-\mathcal{Q}\right)\Delta_{p}^{2}\Sigma_{p},\end{aligned}$
(80)

and

$\mathcal{I}_{\beta}=\begin{aligned}
&\mathcal{E}^{2}a^{2}r_{p}\left(1-z_{p}^{2}\right)\Big{[}r_{p}^{9}\left(1-z_{p}^{2}\right)\left(1-9z_{p}^{2}\right)-r_{p}^{8}\left(6-(18+a^{2})z_{p}^{2}+(12-a^{2})z_{p}^{4}\right)\\\
&\qquad-
r_{p}^{7}a^{2}\left(2+35z_{p}^{2}-32z_{p}^{4}+3z_{p}^{6}\right)-r_{p}^{6}a^{2}\left(6-(46+a^{2})z_{p}^{2}+2(22-a^{2})z_{p}^{4}-(12+a^{2})z_{p}^{6}\right)\\\
&\qquad+r_{p}^{5}a^{4}\left(1-28z_{p}^{2}+33z_{p}^{4}-14z_{p}^{6}\right)-r_{p}^{4}a^{4}z_{p}^{2}\left(1-z_{p}^{2}\right)\left(4+a^{2}+(26+a^{2})z_{p}^{2}\right)\\\
&\qquad+r_{p}^{3}a^{6}z_{p}^{2}\left(1+18z_{p}^{2}-11z_{p}^{4}\right)-r_{p}^{2}a^{6}z_{p}^{2}\left(a^{2}+2(3+a^{2})z_{p}^{2}+(2+a^{2})z_{p}^{4}\right)+(4r_{p}-a^{2})a^{8}z_{p}^{4}\left(1+z_{p}^{2}\right)\Big{]}\\\
&+({\mathcal{L}_{z}}-a\mathcal{E})^{2}\left(1-z_{p}^{2}\right)^{2}\Big{[}3(r_{p}-2)r_{p}^{9}+3r_{p}^{8}a^{2}\left(4-3z_{p}^{2}\right)-2r_{p}^{7}a^{2}\left(11-6z_{p}^{2}\right)+r_{p}^{6}a^{4}\left(7-33z_{p}^{2}\right)\\\
&\qquad-2r_{p}^{5}a^{4}\left(4-18z_{p}^{2}+3z_{p}^{4}\right)+r_{p}^{4}a^{6}\left(2-27z_{p}^{2}-z_{p}^{4}\right)-6r_{p}^{3}a^{6}z_{p}^{4}-r_{p}^{2}a^{8}z_{p}^{2}\left(3-2z_{p}^{2}\right)-z_{p}^{4}a^{10}\Big{]}\\\
&-{\mathcal{L}_{z}}^{2}r_{p}\Big{[}3r_{p}^{9}\left(1-z_{p}^{2}\right)^{2}-r_{p}^{8}\left(6-a^{2}-2(6+a^{2})z_{p}^{2}+(6-a^{2})z_{p}^{4}\right)+r_{p}^{7}a^{2}\left(2-29z_{p}^{2}+20z_{p}^{4}-9z_{p}^{6}\right)\\\
&\qquad-
r_{p}^{6}a^{2}\left(6+a^{2}-(40+3a^{2})z_{p}^{2}+5(6-a^{2})z_{p}^{4}-(12+a^{2})z_{p}^{6}\right)+r_{p}^{5}a^{4}\left(1-14z_{p}^{2}+25z_{p}^{4}-28z_{p}^{6}\right)\\\
&\qquad-2r_{p}^{4}a^{4}z_{p}^{2}\left(1-z_{p}^{2}\right)\left(2+2a^{2}+(17+2a^{2})z_{p}^{2}-3z_{p}^{4}\right)+r_{p}^{3}a^{6}z_{p}^{2}\left(2z_{p}^{6}-19z_{p}^{4}+32z_{p}^{2}+1\right)\\\
&\qquad-
r_{p}^{2}a^{6}z_{p}^{2}\left(a^{2}+(6+5a^{2})z_{p}^{2}+(4+3a^{2})z_{p}^{4}+(6-a^{2})z_{p}^{6}\right)+(4r_{p}-a^{2})a^{8}z_{p}^{4}\left(1+z_{p}^{2}\right)^{2}\Big{]}\\\
&-a^{2}r_{p}\left(r_{p}^{2}-a^{2}\right)\left(1-z_{p}^{4}\right)(a^{2}z_{p}^{2}-\mathcal{Q})\Delta_{p}^{2}\Sigma_{p}.\end{aligned}$
(81)