Conditions for Suppressing
Dimension-five Proton Decay
in Renormalizable SUSY $SO(10)$ GUT

Naoyuki Habaa and Toshifumi Yamadab

a Department of Physics, Osaka Metropolitan University, Osaka 558-8585, Japan
b Department of Physics, Yokohama National University, Yokohama 240-8501,
Japan

Abstract

The SUSY $SO(10)$ GUT is in severe tension with the experimental bounds on
proton partial lifetimes because proton decay mediated by colored Higgsinos
(dimension-five proton decay) is too rapid. In this paper, we pursue the
possibility that a texture of the Yukawa coupling matrices in a renormalizable
SUSY $SO(10)$ GUT model suppresses dimension-five proton decay. We focus on a
general renormalizable SUSY $SO(10)$ GUT model which contains ${\bf 10}+{\bf
126}+{\bf\overline{126}}+{\bf 120}$ representation fields and where the Yukawa
coupling matrices of the 16 matter fields with the ${\bf 10}$,
${\bf\overline{126}}$, ${\bf 120}$ fields, $Y_{10},Y_{126},Y_{120}$, provide
the quark and lepton Yukawa couplings and Majorana mass of the singlet
neutrinos. We find that if components in certain flavor bases,
$(Y_{10})_{u_{R}d_{R}}$, $(Y_{126})_{u_{R}d_{R}}$, $(Y_{10})_{u_{R}s_{R}}$,
$(Y_{126})_{u_{R}s_{R}}$, $(Y_{10})_{u_{L}d_{L}}$, $(Y_{126})_{u_{L}d_{L}}$,
$(Y_{10})_{u_{L}s_{L}}$, $(Y_{126})_{u_{L}s_{L}}$, $(Y_{10})_{u_{L}u_{L}}$,
$(Y_{126})_{u_{L}u_{L}}$, are all on the order of the up quark Yukawa
coupling, dimension-five proton decay can be suppressed while the Yukawa
coupling matrices still reproduce the realistic quark and lepton masses and
flavor mixings. We numerically obtain specific Yukawa coupling matrices
satisfying the above conditions, calculate proton partial lifetimes from them
and evaluate how dimension-five proton decay is suppressed when these
conditions are met.

##  1 Introduction

The $SO(10)$ grand unified theory (GUT) [1, 2] is a viable extension of the
Standard Model (SM) for its attractive features such as the embedding of the
SM gauge groups into an anomaly-free group, the unification of one generation
of the matter fields into a 16 representation field, and the automatic
realization of the seesaw mechanism [3, 4, 5, 6] that naturally explains the
tiny neutrino mass. The supersymmetric (SUSY) $SO(10)$ GUT can further
alleviate the gauge hierarchy problem, and achieve the gauge coupling
unification without intermediate scale. A drawback of the SUSY $SO(10)$ GUT is
that proton decay mediated by colored Higgsinos (dimension-five proton decay)
[7, 8] is too rapid to be consistent with the current experimental bounds on
proton partial lifetimes. In particular, since the unification of the top and
bottom quark Yukawa couplings implies $\tan\beta\sim 50$, the contribution of
$E^{c}U^{c}U^{c}D^{c}$ operators [9] to the $p\to K^{+}\bar{\nu}_{\tau}$ decay
is enhanced, and since simultaneous cancellations of $E^{c}U^{c}U^{c}D^{c}$
and $QQQL$ operators’ contributions to $p\to K^{+}\bar{\nu}_{\tau}$, and
$QQQL$ operators’ contributions to $p\to K^{+}\bar{\nu}_{\mu}$ are difficult
to realize, the SUSY $SO(10)$ GUT is in severe tension with the experimental
bound on the $p\to K^{+}\bar{\nu}$ partial lifetime [10].

However, there is a possibility that a texture of the Yukawa coupling matrices
suppresses the troublesome dimension-five proton decay, because the decay
amplitudes are proportional to bi-products of Yukawa couplings. In this paper,
we pursue the above possibility and find conditions for a texture of the
Yukawa coupling matrices suppressing dimension-five proton decay. We further
obtain specific Yukawa coupling matrices that satisfy the conditions. We focus
on a general renormalizable $SO(10)$ GUT model which contains ${\bf 10}+{\bf
126}+{\bf\overline{126}}+{\bf 120}$ representation fields from which the Higgs
fields of the minimal SUSY SM (MSSM) originate (a broader class of
renormalizable SUSY $SO(10)$ GUT models have been studied in Refs. [11]-[37]).
In the model, the 16 matter fields have Yukawa couplings with the ${\bf
10},{\bf\overline{126}},{\bf 120}$ fields, which provide the quark and lepton
Yukawa couplings and Majorana mass of the singlet neutrinos. Note that the
Yukawa couplings with the ${\bf 10},{\bf\overline{126}},{\bf 120}$ fields are
most general, since ${\bf 16}\times{\bf 16}={\bf 10}+{\bf 126}+{\bf 120}$ and
they are the only allowed renormalizable couplings involving a pair of 16
matter fields.

In the main body of the paper, we identify those components of the Yukawa
coupling matrices that are involved in dimension-five proton decay and that
can be on the order of the up quark Yukawa coupling without contradicting that
they give the realistic quark and lepton Yukawa couplings. Here the up quark
Yukawa coupling is considered as the smallest scale of the components of the
Yukawa coupling matrices because it is a specially small Yukawa coupling in
the SUSY $SO(10)$ GUT where $\tan\beta\sim 50$. That the components identified
above be on the order of the up quark Yukawa coupling, is the desired
conditions for a texture suppressing dimension-five proton decay. Next, we
obtain specific Yukawa coupling matrices satisfying these conditions, by
fitting the experimental data of quark and lepton masses and flavor mixings
with the Yukawa coupling matrices of the ${\bf 10},{\bf\overline{126}},{\bf
120}$ fields under the constraint that the components identified above be on
the order of the up quark Yukawa coupling. Then we calculate proton partial
lifetimes from these Yukawa coupling matrices, and compare them with those
calculated from Yukawa coupling matrices that do not necessarily satisfy the
conditions. Thereby we evaluate how dimension-five proton decay is suppressed
owing to the conditions.

Previously, suppression of dimension-five proton decay by a texture of the
Yukawa coupling matrices in the SUSY $SO(10)$ GUT has been studied in Refs.
[23, 25, 37]. Those papers deal with the case when the active neutrino mass is
dominated by the contribution of the Type-2 seesaw mechanism coming from a
tiny VEV of the $SU(2)_{L}$-triplet component of the ${\bf\overline{126}}$
field. However, the dominance of the Type-2 seesaw contribution is not a
general situation, since it requires a fine-tuning of a mass term, coupling
constants and VEVs of GUT-breaking fields [37, 38] so as not to spoil the
gauge coupling unification. Thus, the present paper considers the case when
the active neutrino mass is generated solely by the Type-1 seesaw mechanism,
with singlet neutrinos coming from the 16 matter fields and their Majorana
mass from the GUT-breaking VEV of the ${\bf\overline{126}}$ field.

This paper is organized as follows: In Section 2, we review the general
renormalizable SUSY $SO(10)$ GUT model containing ${\bf 10}+{\bf
126}+{\bf\overline{126}}+{\bf 120}$ fields, and write the formulas for partial
widths of dimension-five proton decay. In Section 3, we derive conditions for
a texture of the Yukawa coupling matrices suppressing dimension-five proton
decay. It will turn out that not only the texture of the Yukawa coupling
matrices, but also a certain texture of the colored Higgs mass matrix is
needed to suppress dimension-five proton decay. The latter texture is studied
in Section 4. In Section 5, we numerically obtain specific Yukawa coupling
matrices satisfying the conditions found in Section 3. We further calculate
proton partial lifetimes from them and evaluate how dimension-five proton
decay is suppressed when these conditions are met. Section 6 summarizes the
paper.

##  2 General Renormalizable SUSY $SO(10)$ GUT Model

###  2.1 Model Description

We consider a SUSY $SO(10)$ GUT model that contains single 10, single pair of
${\bf 126}+{\bf\overline{126}}$, single 120 fields, denoted by
$H,\Delta+\overline{\Delta},\Sigma$, respectively. The matter fields of MSSM
and a singlet neutrino of each generation are unified into a ${\bf 16}$
representation field, denoted by ${\bf 16}^{i}$ with $i$ being the flavor
index. The Yukawa couplings are given by

$\displaystyle W\ =\ (\tilde{Y}_{10})_{ij}\,{\bf 16}^{i}H{\bf
16}^{j}+(\tilde{Y}_{126})_{ij}\,{\bf 16}^{i}\overline{\Delta}{\bf
16}^{j}+(\tilde{Y}_{120})_{ij}\,{\bf 16}^{i}\Sigma{\bf 16}^{j},$ (1)

where $\tilde{Y}_{10}$, $\tilde{Y}_{126}$, $\tilde{Y}_{120}$ are Yukawa
coupling matrices in the flavor space, and $\tilde{Y}_{10}$, $\tilde{Y}_{126}$
are complex symmetric and $\tilde{Y}_{120}$ is complex antisymmetric. The
quark and lepton Yukawa couplings are assumed to arise solely from Eq. (1).

Additionally, we introduce single 210 and single 45 fields, denoted by
$\Phi,A$, respectively. The $\Phi,A$ develop vacuum expectation values (VEVs)
to break $SU(5)$ subgroup of the $SO(10)$ while $\Delta+\overline{\Delta}$
develop VEVs to break $U(1)$ subgroup.

When the $SO(10)$ is broken into the SM gauge groups $SU(3)_{C}\times
SU(2)_{L}\times U(1)_{Y}$, the $({\bf 1},{\bf 2},\pm\frac{1}{2})$ components
of $H,\Delta,\overline{\Delta},\Sigma,\Phi$ yield the Higgs fields of MSSM.
Accordingly, $\tilde{Y}_{10},\tilde{Y}_{126},\tilde{Y}_{120}$ give the up-type
quark, down-type quark, charged lepton and neutrino Dirac Yukawa coupling
matrices, $Y_{u},Y_{d},Y_{e},Y_{D}$, as

$\displaystyle W_{\rm eff}\ =\
(Y_{u})_{ij}\,Q_{i}H_{u}U_{j}^{c}+(Y_{d})_{ij}\,Q_{i}H_{d}D_{j}^{c}+(Y_{e})_{ij}\,L_{i}H_{d}E_{j}^{c}+(Y_{D})_{ij}\,L_{i}H_{u}N_{j}^{c},$
(2)

where the Yukawa coupling matrices satisfy at a GUT breaking scale $\mu_{\rm
GUT}$ the following relations:

$\displaystyle Y_{u}\ =\ Y_{10}+r_{2}\,Y_{126}+r_{3}\,Y_{120},$ (3)
$\displaystyle Y_{d}\ =\ r_{1}\left(Y_{10}+Y_{126}+Y_{120}\right),$ (4)
$\displaystyle Y_{e}\ =\ r_{1}\left(Y_{10}-3Y_{126}+r_{e}\,Y_{120}\right),$
(5) $\displaystyle Y_{D}\ =\ Y_{10}-3r_{2}\,Y_{126}+r_{D}\,Y_{120},$ (6)

where $Y_{10}\propto\tilde{Y}_{10}$, $Y_{126}\propto\tilde{Y}_{126}$,
$Y_{120}\propto\tilde{Y}_{120}$, and $r_{1},r_{2},r_{3},r_{e},r_{D}$ are
complex numbers determined from the mass matrix of the $({\bf 1},{\bf
2},\pm\frac{1}{2})$ components. Hereafter we perform a phase redefinition of
fields to make $r_{1}$ real positive.

The GUT-breaking VEV of $\overline{\Delta}$, denoted by $\overline{v}_{R}$,
provides the singlet neutrinos with Majorana mass as

$\displaystyle W_{\rm Majorana}\ =\
\frac{1}{2}(M_{N})_{ij}\,N_{i}^{c}\,N_{j}^{c},\ \ \ \ \
M_{N}\propto\tilde{Y}_{126}\,\overline{v}_{R}.$ (7)

Integrating out $N_{i}^{c}$’s, we get the Weinberg operator

$\displaystyle W_{\rm eff}\ =\
\frac{1}{2}(C_{\nu})_{ij}\,L_{i}H_{u}L_{j}H_{u},$ (8)

where the Wilson coefficient $C_{\nu}$ satisfies at the scale of the singular
values of $M_{N}$,

$\displaystyle C_{\nu}\ =\ -Y_{D}M_{N}^{-1}Y_{D}^{T}.$ (9)

Eq. (8) gives rise to the Type-1 seesaw contribution to the active neutrino
mass. In this paper, we assume that the VEV of the $({\bf 1},{\bf 3},1)$
component of $\overline{\Delta}$ is so small that the Type-2 seesaw
contribution is negligible compared to the Type-1 seesaw one.

###  2.2 Dimension-five Proton Decay

The $({\bf 3},{\bf 1},-\frac{1}{3})+({\bf\overline{3}},{\bf 1},\frac{1}{3})$
components of $H$, $\Delta$, $\overline{\Delta}$, $\Sigma$, $\Phi$, which we
call colored Higgs fields and denote by $H_{C}^{A},\overline{H}_{C}^{B}$
($A,B$ are labels), induce dimension-five operators responsible for proton
decay. After the GUT breaking, the colored Higgs fields have GUT-scale mass
${\cal M}_{H_{C}}$ and Yukawa couplings with matter fields as

$\displaystyle W_{\rm colored\,Higgs}$ $\displaystyle=\sum_{A,B}\
\overline{H}_{C}^{A}({\cal M}_{H_{C}})_{AB}H_{C}^{B}$ $\displaystyle+\sum_{A}\
\frac{1}{2}(Y_{L}^{A})_{ij}Q_{i}H_{C}^{A}Q_{j}+(\overline{Y}^{A}_{L})_{ij}Q_{i}\overline{H}^{A}_{C}L_{j}+(Y_{R}^{A})_{ij}E^{c}_{i}H_{C}^{A}U^{c}_{j}+(\overline{Y}_{R}^{A})_{ij}U^{c}_{i}\overline{H}_{C}^{A}D^{c}_{j}.$
(10)

Here $Y_{L}^{A},\overline{Y}_{L}^{A},\,Y_{R}^{A},\,\overline{Y}^{A}_{R}\propto
Y_{10}$ when $H_{C}^{A},\overline{H}^{A}_{C}$ are components of $H$, and they
are proportional to $Y_{126}$ when $H_{C}^{A},\overline{H}^{A}_{C}$ are
components of $\overline{\Delta}$. When $H_{C}^{A},\overline{H}^{A}_{C}$ are
components of $\Sigma$, Yukawa couplings
$\overline{Y}_{L}^{A},\,Y_{R}^{A},\,\overline{Y}^{A}_{R}$ are proportional to
$Y_{120}$ while $Y_{L}^{A}$ vanishes because $Y_{L}^{A}$ must be symmetric. In
the other cases, all the Yukawa couplings vanish. By integrating out the
colored Higgs fields, we obtain dimension-five operators responsible for
proton decay

$\displaystyle W_{5}\ =\
-\frac{1}{2}C_{5L}^{ijkl}\,(Q_{k}Q_{l})(Q_{i}L_{j})-C_{5R}^{ijkl}\,E^{c}_{k}U^{c}_{l}U^{c}_{i}D^{c}_{j},$
(11)

where isospin indices are summed in each bracket in the first term. The Wilson
coefficients satisfy at the scale of the singular values of ${\cal
M}_{H_{C}}$,

$\displaystyle C_{5L}^{ijkl}\ $ $\displaystyle=\ \sum_{A,B}({\cal
M}_{H_{C}}^{-1})_{AB}\left\\{(Y_{L}^{A})_{kl}(\overline{Y}^{B}_{L})_{ij}-\frac{1}{2}(Y_{L}^{A})_{li}(\overline{Y}^{B}_{L})_{kj}-\frac{1}{2}(Y_{L}^{A})_{ik}(\overline{Y}^{B}_{L})_{lj}\right\\},$
(12) $\displaystyle C_{5R}^{ijkl}\ $ $\displaystyle=\ \sum_{A,B}({\cal
M}_{H_{C}}^{-1})_{AB}\left\\{(Y_{R}^{A})_{kl}(\overline{Y}_{R}^{B})_{ij}-(Y_{R}^{A})_{ki}(\overline{Y}_{R}^{B})_{lj}\right\\}.$
(13)

We write the partial widths of the $p\to K^{+}\bar{\nu}_{\tau}$, $p\to
K^{+}\bar{\nu}_{\mu}$, $p\to K^{+}\bar{\nu}_{e}$, $p\to K^{0}\mu^{+}$, $p\to
K^{0}e^{+}$ decays induced by dimension-five operators (other decay modes will
be commented on in the last paragraph of Section 3). The partial widths read,
for $\beta=e,\mu$,

$\displaystyle\Gamma(p\to K^{+}\bar{\nu}_{\tau})=$
$\displaystyle\frac{m_{N}}{64\pi}\left(1-\frac{m_{K}^{2}}{m_{N}^{2}}\right)^{2}\left|\beta_{H}(\mu_{\rm
had})\frac{1}{f_{\pi}}\left\\{\left(1+\frac{D}{3}+F\right)C_{LL}^{s\tau\,ud}(\mu_{\rm
had})+\frac{2D}{3}C_{LL}^{d\tau\,us}(\mu_{\rm had})\right\\}\right.$
$\displaystyle\left.\ \ \ \ \ \ \ \ \ \ \ \ +\alpha_{H}(\mu_{\rm
had})\frac{1}{f_{\pi}}\left\\{\left(1+\frac{D}{3}+F\right)C_{RL}^{ud\,\tau
s}(\mu_{\rm had})+\frac{2D}{3}C_{RL}^{us\,\tau d}(\mu_{\rm
had})\right\\}\right|^{2},$ (14) $\displaystyle\Gamma(p\to
K^{+}\bar{\nu}_{\beta})=$
$\displaystyle\frac{m_{N}}{64\pi}\left(1-\frac{m_{K}^{2}}{m_{N}^{2}}\right)^{2}\left|\beta_{H}(\mu_{\rm
had})\frac{1}{f_{\pi}}\left\\{\left(1+\frac{D}{3}+F\right)C_{LL}^{s\beta\,ud}(\mu_{\rm
had})+\frac{2D}{3}C_{LL}^{d\beta\,us}(\mu_{\rm had})\right\\}\right|^{2},$
(15) $\displaystyle\Gamma(p\to K^{0}\beta^{+})=$
$\displaystyle\frac{m_{N}}{64\pi}\left(1-\frac{m_{K}^{2}}{m_{N}^{2}}\right)^{2}\left|\beta_{H}(\mu_{\rm
had})\frac{1}{f_{\pi}}\left(1-D+F\right)\overline{C}_{LL}^{u\beta\,us}(\mu_{\rm
had})\right|^{2},$ (16)

where the Wilson coefficients of dimension-six operators
$C_{RL},C_{LL},\overline{C}_{LL}$ satisfy111 By writing
$C_{5L}^{s\alpha\,ud}$, we mean that $Q_{i}$ is in the flavor basis where the
down-type quark Yukawa coupling is diagonalized and that the down-type quark
component of $Q_{i}$ is exactly $s$ quark. Likewise, $Q_{k}$ is in the flavor
basis where the up-type quark Yukawa coupling is diagonalized and its up-type
component is exactly $u$ quark, and $Q_{l}$ is in the flavor basis where the
down-type quark Yukawa coupling is diagonalized and its down-type quark
component is exactly $d$ quark. The same rule applies to other Wilson
coefficients. , for $\alpha=e,\mu,\tau$,

$\displaystyle C_{RL}^{ud\,\tau s}(\mu_{\rm had})=A_{RL}(\mu_{\rm
had},\mu_{\rm
SUSY})\frac{\mu_{H}}{m_{\tilde{t}_{R}}^{2}}\frac{1}{16\pi^{2}}{\cal
F}^{\prime}(V^{\rm ckm}_{ts})^{*}\,y_{t}y_{\tau}\,C_{5R}^{ud\tau
t}|_{\mu=\mu_{\rm SUSY}},$ (17) $\displaystyle C_{RL}^{us\,\tau d}(\mu_{\rm
had})=A_{RL}(\mu_{\rm had},\mu_{\rm
SUSY})\frac{\mu_{H}}{m_{\tilde{t}_{R}}^{2}}\frac{1}{16\pi^{2}}{\cal
F}^{\prime}(V^{\rm ckm}_{td})^{*}\,y_{t}y_{\tau}\,C_{5R}^{us\tau
t}|_{\mu=\mu_{\rm SUSY}}$ (18) $\displaystyle C_{LL}^{s\alpha\,ud}(\mu_{\rm
had})=A_{LL}(\mu_{\rm had},\mu_{\rm
SUSY})\frac{M_{\widetilde{W}}}{m_{\tilde{q}}^{2}}\frac{1}{16\pi^{2}}{\cal
F}\,g_{2}^{2}\left(C_{5L}^{s\alpha\,ud}-C_{5L}^{u\alpha\,ds}\right)|_{\mu=\mu_{\rm
SUSY}},$ (19) $\displaystyle C_{LL}^{d\alpha\,us}(\mu_{\rm
had})=A_{LL}(\mu_{\rm had},\mu_{\rm
SUSY})\frac{M_{\widetilde{W}}}{m_{\tilde{q}}^{2}}\frac{1}{16\pi^{2}}{\cal
F}\,g_{2}^{2}\left(C_{5L}^{d\alpha\,us}-C_{5L}^{u\alpha\,ds}\right)|_{\mu=\mu_{\rm
SUSY}},$ (20) $\displaystyle\overline{C}_{LL}^{u\beta\,us}(\mu_{\rm
had})=A_{LL}(\mu_{\rm had},\mu_{\rm
SUSY})\frac{M_{\widetilde{W}}}{m_{\tilde{q}}^{2}}\frac{1}{16\pi^{2}}{\cal
F}\,g_{2}^{2}\left(-C_{5L}^{u\beta\,us}+C_{5L}^{s\beta\,uu}\right)|_{\mu=\mu_{\rm
SUSY}},$ (21)

and the Wilson coefficients of dimension-five operators satisfy

$\displaystyle C_{5R}^{ud\tau t}(\mu_{\rm SUSY})=A_{R}^{\tau t}(\mu_{\rm
SUSY},\mu_{\rm GUT})\sum_{A,B}({\cal
M}_{H_{C}}^{-1})_{AB}\left.\left\\{(Y_{R}^{A})_{\tau
t}(\overline{Y}^{B}_{R})_{ud}-(Y_{R}^{A})_{\tau
u}(\overline{Y}^{B}_{R})_{td}\right\\}\right|_{\mu=\mu_{\rm GUT}},$ (22)
$\displaystyle C_{5R}^{us\tau t}(\mu_{\rm SUSY})=A_{R}^{\tau t}(\mu_{\rm
SUSY},\mu_{\rm GUT})\sum_{A,B}({\cal
M}_{H_{C}}^{-1})_{AB}\left.\left\\{(Y_{R}^{A})_{\tau
t}(\overline{Y}^{B}_{R})_{us}-(Y_{R}^{A})_{\tau
u}(\overline{Y}_{R}^{B})_{ts}\right\\}\right|_{\mu=\mu_{\rm GUT}},$ (23)
$\displaystyle C_{5L}^{s\alpha\,ud}(\mu_{\rm
SUSY})-C_{5L}^{u\alpha\,ds}(\mu_{\rm SUSY})$
$\displaystyle=A_{L}^{\alpha}(\mu_{\rm SUSY},\mu_{\rm GUT})\sum_{A,B}({\cal
M}_{H_{C}}^{-1})_{AB}\,\left.\frac{3}{2}\left\\{(Y_{L}^{A})_{ud}(\overline{Y}_{L}^{B})_{s\alpha}-(Y_{L}^{A})_{ds}(\overline{Y}_{L}^{B})_{u\alpha}\right\\}\right|_{\mu=\mu_{\rm
GUT}},$ (24) $\displaystyle C_{5L}^{d\alpha\,us}(\mu_{\rm
SUSY})-C_{5L}^{u\alpha\,ds}(\mu_{\rm SUSY})$
$\displaystyle=A_{L}^{\alpha}(\mu_{\rm SUSY},\mu_{\rm GUT})\sum_{A,B}({\cal
M}_{H_{C}}^{-1})_{AB}\,\left.\frac{3}{2}\left\\{(Y_{L}^{A})_{us}(\overline{Y}_{L}^{B})_{d\alpha}-(Y_{L}^{A})_{ds}(\overline{Y}_{L}^{B})_{u\alpha}\right\\}\right|_{\mu=\mu_{\rm
GUT}},$ (25) $\displaystyle C_{5L}^{u\beta\,us}(\mu_{\rm
SUSY})-C_{5L}^{s\beta\,uu}(\mu_{\rm SUSY})$
$\displaystyle=A_{L}^{\beta}(\mu_{\rm SUSY},\mu_{\rm GUT})\sum_{A,B}({\cal
M}_{H_{C}}^{-1})_{AB}\,\left.\frac{3}{2}\left\\{(Y_{L}^{A})_{us}(\overline{Y}_{L}^{B})_{u\beta}-(Y_{L}^{A})_{uu}(\overline{Y}_{L}^{B})_{s\beta}\right\\}\right|_{\mu=\mu_{\rm
GUT}},$ (26)

where $\mu_{\rm had}$ denotes a hadronic scale, $\mu_{\rm SUSY}$ a soft SUSY
breaking scale and $\mu_{\rm GUT}$ a GUT- breaking scale. Here
$\alpha_{H},\beta_{H}$ denote the hadronic matrix elements, $D,F$ are
parameters of the baryon chiral Lagrangian, and
$C_{LL},\overline{C}_{LL},C_{RL}$ are the Wilson coefficients of the effective
Lagrangian where the SUSY particles are integrated out, $-{\cal
L}_{6}=C_{LL}^{ijkl}(\psi_{u_{L}k}\psi_{d_{L}l})(\psi_{d_{L}i}\psi_{\nu_{L}j})+\overline{C}_{LL}^{ijkl}(\psi_{u_{L}k}\psi_{d_{L}l})(\psi_{u_{L}i}\psi_{e_{L}j})+C_{RL}^{ijkl}(\psi_{\nu_{L}k}\psi_{d_{L}l})(\psi_{u_{R}^{c}i}\psi_{d_{R}^{c}j})$
($\psi$ denote SM Weyl spinors, and spinor indices are summed in each
bracket). In Eqs. (17)-(21), $y_{t},y_{\tau},g_{2}$ denote the top quark
Yukawa, tau lepton Yukawa and weak gauge couplings in MSSM, respectively, and
$V^{\rm ckm}_{ij}$ denotes $(i,j)$-component of CKM matrix. ${\cal
F}^{\prime},{\cal F}$ are loop functions defined as ${\cal
F}^{\prime}=\frac{1}{x-y}(\frac{x}{1-x}\log x-\frac{y}{1-y}\log y)$ and ${\cal
F}=\frac{1}{z-w}(\frac{z}{1-z}\log z-\frac{w}{1-w}\log
w)+\frac{1}{z-1}(\frac{z}{1-z}\log z+1)$, where
$x=|\mu_{H}|^{2}/m_{\tilde{t}_{R}}^{2}$,
$y=m_{\tilde{\tau}_{R}}^{2}/m_{\tilde{t}_{R}}^{2}$,
$z=|M_{\widetilde{W}}|^{2}/m_{\tilde{q}}^{2}$,
$w=m_{\tilde{\ell}_{\alpha}}^{2}/m_{\tilde{q}}^{2}$, and
$\mu_{H},m_{\tilde{t}_{R}},m_{\tilde{\tau}_{R}},M_{\widetilde{W}},m_{\tilde{\ell}_{\alpha}},m_{\tilde{q}}$
denote the pole masses of Higgsinos, isospin-singlet top squark, isospin-
singlet tau slepton, Winos, isospin-doublet slepton of flavor $\alpha$, and
1st and 2nd generation isospin-doublet squarks, respectively. $A_{LL},A_{RL}$
account for corrections from renormalization group (RG) evolutions in SM from
scale $\mu_{\rm SUSY}$ to $\mu_{\rm had}$. Here RG corrections involving SM
Yukawa couplings other than the top quark one are neglected and thus
$A_{LL},A_{RL}$ are flavor-universal. $A_{R}^{\tau t},A_{L}^{\alpha}$ account
for corrections from RG evolutions in MSSM from scale $\mu_{\rm GUT}$ to
$\mu_{\rm SUSY}$.

We rewrite the flavor-dependent part of Eqs. (22)-(26) with the GUT Yukawa
coupling matrices $Y_{10},Y_{126},Y_{120}$ as ($\alpha=e,\mu,\tau$;
$\beta=e,\mu$)

$\displaystyle\sum_{A,B}({\cal M}_{H_{C}}^{-1})_{AB}\left\\{(Y_{R}^{A})_{\tau
t}(\overline{Y}^{B}_{R})_{ud}-(Y_{R}^{A})_{\tau
u}(\overline{Y}^{B}_{R})_{td}\right\\}$
$\displaystyle=\frac{1}{M_{H_{C}}}\left[a\left\\{(Y_{10})_{\tau_{R}t_{R}}(Y_{10})_{u_{R}d_{R}}-(Y_{10})_{\tau_{R}u_{R}}(Y_{10})_{t_{R}d_{R}}\right\\}+b\left\\{(Y_{10})_{\tau_{R}t_{R}}(Y_{126})_{u_{R}d_{R}}-(Y_{10})_{\tau_{R}u_{R}}(Y_{126})_{t_{R}d_{R}}\right\\}\right.$
$\displaystyle+c\left\\{(Y_{10})_{\tau_{R}t_{R}}(Y_{120})_{u_{R}d_{R}}-(Y_{10})_{\tau_{R}u_{R}}(Y_{120})_{t_{R}d_{R}}\right\\}$
$\displaystyle+d\left\\{(Y_{126})_{\tau_{R}t_{R}}(Y_{10})_{u_{R}d_{R}}-(Y_{126})_{\tau_{R}u_{R}}(Y_{10})_{t_{R}d_{R}}\right\\}+e\left\\{(Y_{126})_{\tau_{R}t_{R}}(Y_{126})_{u_{R}d_{R}}-(Y_{126})_{\tau_{R}u_{R}}(Y_{126})_{t_{R}d_{R}}\right\\}$
$\displaystyle+f\left\\{(Y_{126})_{\tau_{R}t_{R}}(Y_{120})_{u_{R}d_{R}}-(Y_{126})_{\tau_{R}u_{R}}(Y_{120})_{t_{R}d_{R}}\right\\}$
$\displaystyle+g\left\\{(Y_{120})_{\tau_{R}t_{R}}(Y_{10})_{u_{R}d_{R}}-(Y_{120})_{\tau_{R}u_{R}}(Y_{10})_{t_{R}d_{R}}\right\\}+h\left\\{(Y_{120})_{\tau_{R}t_{R}}(Y_{126})_{u_{R}d_{R}}-(Y_{120})_{\tau_{R}u_{R}}(Y_{126})_{t_{R}d_{R}}\right\\}$
$\displaystyle\left.+j\left\\{(Y_{120})_{\tau_{R}t_{R}}(Y_{120})_{u_{R}d_{R}}-(Y_{120})_{\tau_{R}u_{R}}(Y_{120})_{t_{R}d_{R}}\right\\}\right],$
$\displaystyle\sum_{A,B}({\cal M}_{H_{C}}^{-1})_{AB}\left\\{(Y_{R}^{A})_{\tau
t}(\overline{Y}^{B}_{R})_{us}-(Y_{R}^{A})_{\tau
u}(\overline{Y}^{B}_{R})_{ts}\right\\}={\rm(Above\ expression\ with\
exchange}\ d_{R}\leftrightarrow s_{R}),$ (28) $\displaystyle\sum_{A,B}({\cal
M}_{H_{C}}^{-1})_{AB}\left\\{(Y_{L}^{A})_{ud}(\overline{Y}^{B}_{L})_{s\alpha}-(Y_{L}^{A})_{ds}(\overline{Y}^{B}_{L})_{u\alpha}\right\\}$
$\displaystyle=\frac{1}{M_{H_{C}}}\left[a\left\\{(Y_{10})_{u_{L}d_{L}}(Y_{10})_{s_{L}\alpha_{L}}-(Y_{10})_{d_{L}s_{L}}(Y_{10})_{u_{L}\alpha_{L}}\right\\}+b\left\\{(Y_{10})_{u_{L}d_{L}}(Y_{126})_{s_{L}\alpha_{L}}-(Y_{10})_{d_{L}s_{L}}(Y_{126})_{u_{L}\alpha_{L}}\right\\}\right.$
$\displaystyle+c\left\\{(Y_{10})_{u_{L}d_{L}}(Y_{120})_{s_{L}\alpha_{L}}-(Y_{10})_{d_{L}s_{L}}(Y_{120})_{u_{L}\alpha_{L}}\right\\}$
$\displaystyle+d\left\\{(Y_{126})_{u_{L}d_{L}}(Y_{10})_{s_{L}\alpha_{L}}-(Y_{126})_{d_{L}s_{L}}(Y_{10})_{u_{L}\alpha_{L}}\right\\}+e\left\\{(Y_{126})_{u_{L}d_{L}}(Y_{126})_{s_{L}\alpha_{L}}-(Y_{126})_{d_{L}s_{L}}(Y_{126})_{u_{L}\alpha_{L}}\right\\}$
$\displaystyle\left.+f\left\\{(Y_{126})_{u_{L}d_{L}}(Y_{120})_{s_{L}\alpha_{L}}-(Y_{126})_{d_{L}s_{L}}(Y_{120})_{u_{L}\alpha_{L}}\right\\}\right],$
$\displaystyle\sum_{A,B}({\cal
M}_{H_{C}}^{-1})_{AB}\left\\{(Y_{L}^{A})_{us}(\overline{Y}^{B}_{L})_{d\alpha}-(Y_{L}^{A})_{ds}(\overline{Y}^{B}_{L})_{u\alpha}\right\\}={\rm(Above\
expression\ with\ exchange}\ d_{L}\leftrightarrow s_{L}),$ (30)
$\displaystyle\sum_{A,B}({\cal
M}_{H_{C}}^{-1})_{AB}\left\\{(Y_{L}^{A})_{us}(\overline{Y}^{B}_{L})_{u\beta}-(Y_{L}^{A})_{uu}(\overline{Y}^{B}_{L})_{s\beta}\right\\}$
$\displaystyle=\frac{1}{M_{H_{C}}}\left[a\left\\{(Y_{10})_{u_{L}s_{L}}(Y_{10})_{u_{L}\beta_{L}}-(Y_{10})_{u_{L}u_{L}}(Y_{10})_{s_{L}\beta_{L}}\right\\}+b\left\\{(Y_{10})_{u_{L}s_{L}}(Y_{126})_{u_{L}\beta_{L}}-(Y_{10})_{u_{L}u_{L}}(Y_{126})_{s_{L}\beta_{L}}\right\\}\right.$
$\displaystyle+c\left\\{(Y_{10})_{u_{L}s_{L}}(Y_{120})_{u_{L}\beta_{L}}-(Y_{10})_{u_{L}u_{L}}(Y_{120})_{s_{L}\beta_{L}}\right\\}$
$\displaystyle+d\left\\{(Y_{126})_{u_{L}s_{L}}(Y_{10})_{u_{L}\beta_{L}}-(Y_{126})_{u_{L}u_{L}}(Y_{10})_{s_{L}\beta_{L}}\right\\}+e\left\\{(Y_{126})_{u_{L}s_{L}}(Y_{126})_{u_{L}\beta_{L}}-(Y_{126})_{u_{L}u_{L}}(Y_{126})_{s_{L}\beta_{L}}\right\\}$
$\displaystyle\left.+f\left\\{(Y_{126})_{u_{L}s_{L}}(Y_{120})_{u_{L}\beta_{L}}-(Y_{126})_{u_{L}u_{L}}(Y_{120})_{s_{L}\beta_{L}}\right\\}\right],$

where $M_{H_{C}}$ is the scale of the singular values of ${\cal M}_{H_{C}}$,
and $a,b,c,d,e,f,g,h,j$ are $O(1)$ numbers determined from ${\cal M}_{H_{C}}$
as Ref. [39]-[44]. Here $(Y_{10})_{\tau_{R}t_{R}}$ denotes the component of
$Y_{10}$ in the term $(Y_{10})_{ij}\,\Psi_{i}H\Psi_{j}$ that involves the
right-handed tau lepton component of $\Psi_{i}$ and the right-handed top quark
component of $\Psi_{j}$. Other symbols are defined analogously.

##  3 Conditions for a Texture of the Yukawa Coupling Matrices suppressing
Dimension-five Proton Decay

We identify those components of the Yukawa coupling matrices
$Y_{10},Y_{126},Y_{120}$ that are involved in dimension-five proton decay,
namely, appear in Eqs. (LABEL:yyr1)-(LABEL:yyl3), and that can be on the order
of the up quark Yukawa coupling $y_{u}$. Here the up quark Yukawa coupling is
considered as the smallest scale of the components of the Yukawa coupling
matrices because it is a specially small Yukawa coupling in the SUSY $SO(10)$
GUT where $\tan\beta\sim 50$. That the components identified above be on the
order of the up quark Yukawa coupling, is the desired conditions for a texture
suppressing dimension-five proton decay.

  * •

We focus on the first term in each {…} of Eqs. (LABEL:yyr1),(28). Components
$(Y_{10})_{\tau_{R}t_{R}}$ and $(Y_{126})_{\tau_{R}t_{R}}$ are almost
$(3,3)$-components of the symmetric matrices $Y_{10},Y_{126}$ and hence always
on the order of the top quark Yukawa coupling $y_{t}$. For other components,
at most two of $(Y_{10})_{u_{R}d_{R}}$, $(Y_{126})_{u_{R}d_{R}}$,
$(Y_{120})_{u_{R}d_{R}}$ and at most two of $(Y_{10})_{u_{R}s_{R}}$,
$(Y_{126})_{u_{R}s_{R}}$, $(Y_{120})_{u_{R}s_{R}}$ can be on the order of the
up quark Yukawa coupling $y_{u}$. However, all of them cannot be so because of
the following equalities that result from Eq. (4):

$\displaystyle(Y_{10})_{u_{R}d_{R}}+(Y_{126})_{u_{R}d_{R}}+(Y_{120})_{u_{R}d_{R}}\
$ $\displaystyle=\ \frac{1}{r_{1}}(Y_{d})_{u_{R}d_{R}}$ $\displaystyle\simeq\
\frac{y_{t}}{y_{b}}y_{d}\times(d_{L}\mathchar 45\relax u_{R}\ {\rm part\ of\
the\ mixing\ matrix}),$ (32)
$\displaystyle(Y_{10})_{u_{R}s_{R}}+(Y_{126})_{u_{R}s_{R}}+(Y_{120})_{u_{R}s_{R}}\
$ $\displaystyle=\ \frac{1}{r_{1}}(Y_{d})_{u_{R}s_{R}}$ $\displaystyle\simeq\
\frac{y_{t}}{y_{b}}y_{s}\times(s_{L}\mathchar 45\relax u_{R}\ {\rm part\ of\
the\ mixing\ matrix}),$ (33)

where $r_{1}$ is estimated to be $y_{b}/y_{t}$ so that the ratio of the top
and bottom quark Yukawa couplings is reproduced. $d_{L}\mathchar 45\relax
u_{R}\ {\rm part\ of\ the\ mixing\ matrix}$ is estimated to be about 1 and we
get
$(Y_{10})_{u_{R}d_{R}}+(Y_{126})_{u_{R}d_{R}}+(Y_{120})_{u_{R}d_{R}}\simeq\frac{y_{t}}{y_{b}}y_{d}$,
which is much greater than $y_{u}$. Also, $s_{L}\mathchar 45\relax u_{R}\ {\rm
part\ of\ the\ mixing\ matrix}$ is estimated to be the Cabibbo angle
$\lambda=0.22$ and hence we get
$(Y_{10})_{u_{R}s_{R}}+(Y_{126})_{u_{R}s_{R}}+(Y_{120})_{u_{R}s_{R}}\simeq
0.22\times\frac{y_{t}}{y_{b}}y_{s}$, which is much greater than $y_{u}$.
Therefore, to make the entire Eqs. (LABEL:yyr1),(28) proportional to $y_{u}$,
we have to tune the colored Higgs mass matrix such that some of
$a,b,c,d,e,f,g,h,j$ are much smaller than 1. The most economical choice is to
tune the colored Higgs mass matrix to make

$\displaystyle c=f=0$ (34)

and at the same time consider the following texture:

$\displaystyle(Y_{10})_{u_{R}d_{R}}=O(y_{u}),\ \ \ \ \
(Y_{126})_{u_{R}d_{R}}=O(y_{u}),\ \ \ \ \
(Y_{120})_{u_{R}d_{R}}\simeq\frac{y_{t}}{y_{b}}y_{d},$ (35)
$\displaystyle(Y_{10})_{u_{R}s_{R}}=O(y_{u}),\ \ \ \ \
(Y_{126})_{u_{R}s_{R}}=O(y_{u}),\ \ \ \ \ (Y_{120})_{u_{R}s_{R}}\simeq
0.22\times\frac{y_{t}}{y_{b}}y_{s}.$ (36)

The terms $j(Y_{120})_{\tau_{R}t_{R}}(Y_{120})_{u_{R}d_{R}}$ and
$j(Y_{120})_{\tau_{R}t_{R}}(Y_{120})_{u_{R}s_{R}}$ appear to be not
proportional to $y_{u}$. However, since $Y_{120}$ is antisymmetric in the
flavor space, $(Y_{120})_{\tau_{R}t_{R}}$, being a nearly diagonal component,
is so small that the above terms are suppressed compared to the other
remaining terms.

  * •

We focus on the second term in each {…} of Eq. (LABEL:yyr1). Each term is
estimated to be $\sin^{2}\theta^{\rm ckm}_{13}\,y_{t}^{2}$ ($\theta^{\rm
ckm}_{13}$ is the $(1,3)$-mixing angle of CKM matrix), since each term
contains two 1st-3rd generation flavor mixings. The value of
$\sin^{2}\theta^{\rm ckm}_{13}\,y_{t}^{2}$ is numerically close to
$y_{u}\,y_{t}$. Thus, these terms are always on the same order as first terms
$(Y_{10})_{\tau_{R}t_{R}}(Y_{10})_{u_{R}d_{R}}$,
$(Y_{10})_{\tau_{R}t_{R}}(Y_{126})_{u_{R}s_{R}}$,
$(Y_{126})_{\tau_{R}t_{R}}(Y_{10})_{u_{R}d_{R}}$,
$(Y_{126})_{\tau_{R}t_{R}}(Y_{126})_{u_{R}s_{R}}$ with the texture of Eq.
(35).

Likewise the second term in each {…} of Eq. (28) is estimated to be
$\sin\theta^{\rm ckm}_{13}\sin\theta^{\rm ckm}_{23}\,y_{t}^{2}$. These terms
contribute to the proton decay amplitude always by a similar amount to the
second terms in {…}’s of Eq. (LABEL:yyr1) because they enter the proton decay
amplitude in the form $V_{ts}^{\rm
ckm}(Y_{A})_{\tau_{R}u_{R}}(Y_{B})_{t_{R}d_{R}}$ and $V_{td}^{\rm
ckm}(Y_{A})_{\tau_{R}u_{R}}(Y_{B})_{t_{R}s_{R}}$ and the CKM matrix components
satisfy $|V_{ts}^{\rm ckm}|\sim\sin\theta^{\rm ckm}_{23}$ and $|V_{td}^{\rm
ckm}|\sim\sin\theta^{\rm ckm}_{13}$.

  * •

We proceed to Eqs. (LABEL:yyl1),(30). It is impossible to suppress
$(Y_{A})_{s_{L}\alpha_{L}}$, $(Y_{A})_{d_{L}\alpha_{L}}$,
$(Y_{A})_{u_{L}\alpha_{L}}$ for all flavors $\alpha=e,\mu,\tau$ to the order
of $y_{u}$. Therefore, we do not consider a texture where some of them are
$O(y_{u})$. For other components, at least one of $(Y_{10})_{u_{L}s_{L}}$,
$(Y_{10})_{d_{L}s_{L}}$, $(Y_{126})_{u_{L}s_{L}}$, $(Y_{126})_{d_{L}s_{L}}$ is
on the order of $|V^{\rm ckm}_{cd}|\frac{y_{t}}{y_{b}}y_{s}$ because of two
equalities below,

$\displaystyle(Y_{10})_{s_{L}c_{L}}+(Y_{126})_{s_{L}c_{L}}+(Y_{120})_{s_{L}c_{L}}\
\simeq\ \frac{y_{t}}{y_{b}}y_{s}\times(c_{L}\mathchar 45\relax s_{R}\ {\rm
part\ of\ the\ mixing\ matrix)},$ (37)
$\displaystyle(Y_{10})_{d_{L}s_{L}}+(Y_{126})_{d_{L}s_{L}}-V^{\rm
ckm}_{ud}\left\\{(Y_{10})_{u_{L}s_{L}}+(Y_{126})_{u_{L}s_{L}}\right\\}$
$\displaystyle\ \ \ \ \ \ \ \ =\ V^{\rm
ckm}_{cd}\left\\{(Y_{10})_{c_{L}s_{L}}+(Y_{126})_{c_{L}s_{L}}\right\\}+V^{\rm
ckm}_{td}\left\\{(Y_{10})_{t_{L}s_{L}}+(Y_{126})_{t_{L}s_{L}}\right\\},$ (38)

and by the facts that $c_{L}\mathchar 45\relax s_{R}\ {\rm part\ of\ the\
mixing\ matrix}$ is about 1 because $c_{L}$ and $s_{R}$ are 2nd generation
flavors, and that $(Y_{120})_{s_{L}c_{L}}$ is suppressed compared to
$(Y_{10})_{c_{L}s_{L}},(Y_{126})_{c_{L}s_{L}}$ because it is nearly
$(2,2)$-component of the antisymmetric matrix $Y_{120}$.222 The situation that
the term $V^{\rm
ckm}_{td}\left\\{(Y_{10})_{t_{L}s_{L}}+(Y_{126})_{t_{L}s_{L}}\right\\}$
cancels the term $V^{\rm
ckm}_{cd}\left\\{(Y_{10})_{c_{L}s_{L}}+(Y_{126})_{c_{L}s_{L}}\right\\}$ is
incompatible with the correct quark Yukawa couplings. As a result, at least
one of $(Y_{10})_{u_{L}s_{L}}$, $(Y_{10})_{d_{L}s_{L}}$,
$(Y_{126})_{u_{L}s_{L}}$, $(Y_{126})_{d_{L}s_{L}}$ cannot be on the order of
$y_{u}$. Still, it is possible to make the entire Eqs. (LABEL:yyl1),(30)
proportional to $y_{u}$ by tuning the colored Higgs mass matrix such that
$a,b,d,e$ satisfy

$\displaystyle a\,(Y_{10})_{d_{L}s_{L}}+d\,(Y_{126})_{d_{L}s_{L}}=0,\ \ \ \ \
\ \ b\,(Y_{10})_{d_{L}s_{L}}+e\,(Y_{126})_{d_{L}s_{L}}=0$ (39)

and at the same time considering the following texture:

$\displaystyle(Y_{10})_{u_{L}s_{L}}=O(y_{u}),\ \ \ \ \
(Y_{126})_{u_{L}s_{L}}=O(y_{u}).$ (40)

  * •

Finally, we focus on Eq. (LABEL:yyl3). The only components that do not appear
in Eqs. (LABEL:yyl1),(30) are $(Y_{10})_{u_{L}u_{L}}$,
$(Y_{126})_{u_{L}u_{L}}$, and there is no obstacle in considering the
following texture:

$\displaystyle(Y_{10})_{u_{L}u_{L}}=O(y_{u}),\ \ \ \ \
(Y_{126})_{u_{L}u_{L}}=O(y_{u}).$ (41)

To summarize, dimension-five proton decay is suppressed if the components
below are all on the order of the up quark Yukawa coupling $y_{u}$,

$\displaystyle(Y_{10})_{u_{R}d_{R}},\ (Y_{126})_{u_{R}d_{R}},\
(Y_{10})_{u_{R}s_{R}},\ (Y_{126})_{u_{R}s_{R}},$
$\displaystyle(Y_{10})_{u_{L}d_{L}},\ (Y_{126})_{u_{L}d_{L}},\
(Y_{10})_{u_{L}s_{L}},\ (Y_{126})_{u_{L}s_{L}},\ (Y_{10})_{u_{L}u_{L}},\
(Y_{126})_{u_{L}u_{L}},$ (42)

and at the same time the colored Higgs mass matrix is tuned such that
$a,b,c,d,e,f,g,h,j$ in Eqs. (LABEL:yyr1)-(LABEL:yyl3) satisfy

$\displaystyle c=f=0,$ (43) $\displaystyle
a\,(Y_{10})_{d_{L}s_{L}}+d\,(Y_{126})_{d_{L}s_{L}}=0,\ \ \ \ \ \
b\,(Y_{10})_{d_{L}s_{L}}+e\,(Y_{126})_{d_{L}s_{L}}=0.$ (44)

That the components of Eq. (42) be on the order of the up quark Yukawa
coupling, is the desired conditions for a texture of the Yukawa coupling
matrices suppressing dimension-five proton decay.

We comment on nucleon decay modes other than Eqs. (14)-(16). The partial
widths of the $N\to\pi\beta^{+}$ and $p\to\eta\beta^{+}$ decays
($\beta=e,\mu$) involve the same Yukawa coupling components as the $p\to
K^{0}\beta^{+}$ except that $s_{L}$ is replaced by $d_{L}$. Hence, once we
consider the texture where $(Y_{10})_{u_{L}u_{L}}$, $(Y_{126})_{u_{L}u_{L}}$
are on the order of the up quark Yukawa coupling, the $N\to\pi\beta^{+}$ and
$p\to\eta\beta^{+}$ decays are also suppressed. Constraints on the rest of the
decay modes are relatively weak [45] and are not in tension with the SUSY
$SO(10)$ GUT.

##  4 Texture of the Colored Higgs Mass Matrix

We present a texture of the colored Higgs mass matrix that gives $c=f=0$ and
$a/d=b/e$. Here $a/d=b/e$ is a necessary condition for Eq. (44). We utilize
the result of Ref. [41], and use the same notation of fields, coupling
constants and VEVs except that 120 field is written as $\Sigma$ in our paper.
The definitions of the couplings, coupling constants and masses are summarized
in Appendix A.

The desired texture of the colored Higgs mass matrix is obtained from the
following relations of the coupling constants and VEVs:

$\displaystyle\lambda_{18}\ $ $\displaystyle=\ 0,\ \ \ \ \ \lambda_{20}\ =\
0,\ \ \ \ \ \frac{\lambda_{21}}{\lambda_{19}}\ =\
3\frac{\lambda_{17}}{\lambda_{16}},$ $\displaystyle i\,A_{1}\ $
$\displaystyle=\ -\frac{1}{6}\frac{\lambda_{21}}{\lambda_{19}}\Phi_{3},\ \ \ \
\ i\,A_{2}\ =\ -\frac{\sqrt{3}}{6}\frac{\lambda_{21}}{\lambda_{19}}\Phi_{2}.$
(45)

The above relations are obtained by fine-tuning, which is natural at the
quantum level thanks to the non-renormalization theorem. The VEV configuration
of Eq. (45) can be consistent with the $F$-flatness conditions of the six SM-
gauge-singlet components $\Phi_{1},\Phi_{2},\Phi_{3},A_{1},A_{2},v_{R}$ when
six model parameters, for example
$\lambda_{5}/\lambda_{2},\,\lambda_{6}/\lambda_{2},\,\lambda_{7}/\lambda_{2},\,m_{1},m_{2},m_{4}$,
are tuned appropriately.

When Eq. (45) holds, the colored Higgs mass matrix has the following texture:

$\displaystyle W_{\rm
colored\,Higgs}\supset\begin{pmatrix}H^{(\overline{3},1,\frac{1}{3})}&\overline{\Delta}^{(\overline{3},1,\frac{1}{3})}_{(6,1,1)}&\Delta^{(\overline{3},1,\frac{1}{3})}_{(6,1,1)}&\Delta^{(\overline{3},1,\frac{1}{3})}_{(\overline{10},1,3)}&\Phi^{(\overline{3},1,\frac{1}{3})}&\Sigma^{(\overline{3},1,\frac{1}{3})}_{(6,1,3)}&\Sigma^{(\overline{3},1,\frac{1}{3})}_{(\overline{10},1,1)}\end{pmatrix}{\cal
M}_{H_{C}}\begin{pmatrix}H^{(3,1,-\frac{1}{3})}\\\
\Delta^{(3,1,-\frac{1}{3})}_{(6,1,1)}\\\
\overline{\Delta}^{(3,1,-\frac{1}{3})}_{(6,1,1)}\\\
\overline{\Delta}^{(3,1,-\frac{1}{3})}_{(10,1,3)}\\\
\Phi^{(3,1,-\frac{1}{3})}\\\ \Sigma^{(3,1,-\frac{1}{3})}_{(6,1,3)}\\\
\Sigma^{(3,1,-\frac{1}{3})}_{(10,1,1)}\end{pmatrix},$ (46)

where

$\displaystyle{\cal
M}_{H_{C}}=\begin{pmatrix}m_{3}&\frac{\lambda_{3}\Phi_{2}}{\sqrt{30}}-\frac{\lambda_{3}\Phi_{1}}{\sqrt{10}}&-\frac{\lambda_{4}\Phi_{1}}{\sqrt{10}}-\frac{\lambda_{4}\Phi_{2}}{\sqrt{30}}&-\sqrt{\frac{2}{15}}\lambda_{4}\Phi_{3}&\frac{\lambda_{4}\overline{v}_{R}}{\sqrt{5}}&0&0\\\
\frac{\lambda_{4}\Phi_{2}}{\sqrt{30}}-\frac{\lambda_{4}\Phi_{1}}{\sqrt{10}}&m_{2}+i\frac{\lambda_{21}}{\lambda_{19}}\frac{\lambda_{6}\Phi_{2}}{30\sqrt{2}}&0&0&0&0&0\\\
-\frac{\lambda_{3}\Phi_{1}}{\sqrt{10}}-\frac{\lambda_{3}\Phi_{2}}{\sqrt{30}}&0&m_{2}-i\frac{\lambda_{21}}{\lambda_{19}}\frac{\lambda_{6}\Phi_{2}}{30\sqrt{2}}&\frac{\lambda_{2}\Phi_{3}}{15\sqrt{2}}&-\frac{\lambda_{2}\overline{v}_{R}}{10\sqrt{3}}&0&0\\\
-\sqrt{\frac{2}{15}}\lambda_{3}\Phi_{3}&0&\frac{\lambda_{2}\Phi_{3}}{15\sqrt{2}}&m_{66}&-\frac{\lambda_{2}\overline{v}_{R}}{5\sqrt{6}}&0&0\\\
\frac{\lambda_{3}v_{R}}{\sqrt{5}}&0&-\frac{\lambda_{2}v_{R}}{10\sqrt{3}}&-\frac{\lambda_{2}v_{R}}{5\sqrt{6}}&m_{77}&0&0\\\
-\lambda_{17}\frac{\Phi_{3}}{\sqrt{3}}&0&\lambda_{21}\frac{\Phi_{3}}{6\sqrt{5}}&\lambda_{21}\frac{\Phi_{2}}{3\sqrt{5}}&\frac{\lambda_{21}\overline{v}_{R}}{2\sqrt{15}}&m_{22}&\frac{2\lambda_{15}\Phi_{3}}{9}\\\
-\sqrt{\frac{2}{3}}\lambda_{17}\Phi_{2}&0&\lambda_{21}\frac{\Phi_{2}}{3\sqrt{10}}&\lambda_{21}\frac{\Phi_{3}}{3\sqrt{10}}&\frac{\lambda_{21}\overline{v}_{R}}{2\sqrt{15}}&\frac{2\lambda_{15}\Phi_{3}}{9}&m_{33}\end{pmatrix}$
(47) $\displaystyle
m_{66}=m_{2}+\lambda_{2}(\frac{\Phi_{1}}{10\sqrt{6}}+\frac{\Phi_{2}}{30\sqrt{2}})-i\frac{\lambda_{21}}{\lambda_{19}}\frac{\lambda_{6}\Phi_{2}}{30\sqrt{2}}$
(48) $\displaystyle
m_{77}=m_{1}+\lambda_{1}(\frac{\Phi_{1}}{\sqrt{6}}+\frac{\Phi_{2}}{3\sqrt{2}}+\frac{2\Phi_{3}}{3})-i\frac{\lambda_{21}}{\lambda_{19}}\frac{\sqrt{2}\lambda_{7}\Phi_{2}}{15}$
(49) $\displaystyle
m_{22}=m_{6}+\frac{1}{3}\sqrt{\frac{2}{3}}\lambda_{15}\Phi_{1}$ (50)
$\displaystyle m_{33}=m_{6}+\frac{\sqrt{2}}{9}\lambda_{15}\Phi_{2}.$ (51)

The dimension-five operators responsible for proton decay Eq. (11) satisfy at
the scale of the colored Higgs mass,

$\displaystyle W_{5}=$
$\displaystyle-\begin{pmatrix}(\tilde{Y}_{10})_{ij}&0&(\tilde{Y}_{126})_{ij}&(\tilde{Y}_{126})_{ij}&0&(\tilde{Y}_{120})_{ij}&(\tilde{Y}_{120})_{ij}\end{pmatrix}{\cal
M}_{H_{C}}^{-1}\begin{pmatrix}(\tilde{Y}_{10})_{kl}\\\
(\tilde{Y}_{126})_{kl}\\\ 0\\\ 0\\\ 0\\\ (\tilde{Y}_{120})_{kl}\\\
(\tilde{Y}_{120})_{kl}\end{pmatrix}E_{i}^{c}U_{j}^{c}U_{k}^{c}D_{l}^{c}$
$\displaystyle-\begin{pmatrix}(\tilde{Y}_{10})_{ij}&0&(\tilde{Y}_{126})_{ij}&(\tilde{Y}_{126})_{ij}&0&0&0\end{pmatrix}{\cal
M}_{H_{C}}^{-1}\begin{pmatrix}(\tilde{Y}_{10})_{kl}\\\
(\tilde{Y}_{126})_{kl}\\\ 0\\\ 0\\\ 0\\\ (\tilde{Y}_{120})_{kl}\\\
(\tilde{Y}_{120})_{kl}\end{pmatrix}(Q_{i}Q_{j})(Q_{k}L_{l}),$ (52)

where the inverse of the colored Higgs mass ${\cal M}_{H_{C}}^{-1}$ has the
following properties resulting from Eq. (47):

  * •

The upper-right $5\times 2$ part of ${\cal M}_{H_{C}}^{-1}$ is zero because
the upper-right $5\times 2$ part of ${\cal M}_{H_{C}}$ is zero. Thus, the
coefficients proportional to $(\tilde{Y}_{10})_{ij}(\tilde{Y}_{120})_{kl}$ or
$(\tilde{Y}_{126})_{ij}(\tilde{Y}_{120})_{kl}$ are zero, namely, we get
$c=f=0$ in Eqs. (LABEL:yyr1)-(LABEL:yyl3).

  * •

The upper-left $5\times 5$ part of ${\cal M}_{H_{C}}^{-1}$ is given by the
inverse of the upper-left $5\times 5$ part of ${\cal M}_{H_{C}}$, since the
upper-right $5\times 2$ part of ${\cal M}_{H_{C}}$ is zero. Then the
equalities $({\cal M}_{H_{C}})_{32}=({\cal M}_{H_{C}})_{42}=({\cal
M}_{H_{C}})_{52}=0$ lead to the relation $({\cal M}_{H_{C}}^{-1})_{11}:({\cal
M}_{H_{C}}^{-1})_{31}:({\cal M}_{H_{C}}^{-1})_{41}=({\cal
M}_{H_{C}}^{-1})_{12}:({\cal M}_{H_{C}}^{-1})_{32}:({\cal
M}_{H_{C}}^{-1})_{42}$. Since the coefficients proportional to
$(\tilde{Y}_{10})_{ij}(\tilde{Y}_{10})_{kl}$,
$(\tilde{Y}_{10})_{ij}(\tilde{Y}_{126})_{kl}$,
$(\tilde{Y}_{126})_{ij}(\tilde{Y}_{10})_{kl}$ or
$(\tilde{Y}_{126})_{ij}(\tilde{Y}_{126})_{kl}$ are given by

$\displaystyle({\cal M}_{H_{C}}^{-1})_{11}\
(\tilde{Y}_{10})_{ij}(\tilde{Y}_{10})_{kl}+\left\\{({\cal
M}_{H_{C}}^{-1})_{31}+({\cal
M}_{H_{C}}^{-1})_{41}\right\\}(\tilde{Y}_{126})_{ij}(\tilde{Y}_{10})_{kl}$
$\displaystyle+({\cal M}_{H_{C}}^{-1})_{12}\
(\tilde{Y}_{10})_{ij}(\tilde{Y}_{126})_{kl}+\left\\{({\cal
M}_{H_{C}}^{-1})_{32}+({\cal
M}_{H_{C}}^{-1})_{42}\right\\}(\tilde{Y}_{126})_{ij}(\tilde{Y}_{126})_{kl},$
(53)

the equality $({\cal M}_{H_{C}}^{-1})_{11}/\left\\{({\cal
M}_{H_{C}}^{-1})_{31}+({\cal M}_{H_{C}}^{-1})_{41}\right\\}=({\cal
M}_{H_{C}}^{-1})_{12}/\left\\{({\cal M}_{H_{C}}^{-1})_{32}+({\cal
M}_{H_{C}}^{-1})_{42}\right\\}$ leads to the desired relation $a/d=b/e$ in
Eqs. (LABEL:yyr1)-(LABEL:yyl3).

We still have to check that Eq. (45) is consistent with the situation that
$a/d$ satisfies $a\,(Y_{10})_{d_{L}s_{L}}+d\,(Y_{126})_{d_{L}s_{L}}=0$, that
only one pair of (${\bf 1}$, ${\bf 2}$, $\pm\frac{1}{2}$) fields
(corresponding to the MSSM Higgs fields) have almost zero mass compared to the
GUT scale, and that $r_{1},r_{2},r_{3},r_{e},r_{D}$ take values that reproduce
the realistic quark and lepton masses and flavor mixings. With Eq. (45), and
with the tuning of
$\lambda_{5}/\lambda_{2},\,\lambda_{6}/\lambda_{2},\,\lambda_{7}/\lambda_{2},\,m_{1},m_{2},m_{4}$
to satisfy the $F$-flatness conditions, the colored Higgs mass matrix and the
mass matrix of the $({\bf 1},{\bf 2},\pm\frac{1}{2})$ components still have
free parameters $\Phi_{1},\Phi_{2},\Phi_{3},v_{R}$, $m_{3}$,
$\lambda_{21}/\lambda_{19},\,\lambda_{21}/\lambda_{2}$,
$\lambda_{1}/\lambda_{2},\,\lambda_{3}/\lambda_{2},\,\lambda_{4}/\lambda_{2},\,\lambda_{15}/\lambda_{2},\,\lambda_{17}/\lambda_{2}$.
These free parameters are sufficient to make one pair of (${\bf 1}$, ${\bf
2}$, $\pm\frac{1}{2}$) fields nearly massless and realize any values of $a/d$,
$r_{1},r_{2},r_{3},r_{e},r_{D}$.

##  5 Yukawa Coupling Matrices satisfying the Conditions

###  5.1 Procedures of the Analysis

We will obtain specific Yukawa coupling matrices $Y_{10}$, $Y_{126}$,
$Y_{120}$ which give the realistic quark and lepton masses and flavor mixings
and which satisfy the conditions found in Section 3 that the components of Eq.
(42) be on the order of the up quark Yukawa coupling $y_{u}$. To this end, we
fit the experimental values of the quark and lepton masses and flavor mixings
with $Y_{10}$, $Y_{126}$, $Y_{120}$ and numbers
$r_{1},r_{2},r_{3},r_{e},r_{D}$ based on Eqs. (3)-(9) and at the same time
minimize the following quantity:

$\displaystyle\frac{1}{y_{u}^{2}}\sum_{A=10,126}\
\left|(Y_{A})_{u_{R}d_{R}}\right|^{2}+\left|(Y_{A})_{u_{R}s_{R}}\right|^{2}+\left|(Y_{A})_{u_{L}d_{L}}\right|^{2}+\left|(Y_{A})_{u_{L}s_{L}}\right|^{2}+\left|(Y_{A})_{u_{L}u_{L}}\right|^{2}.$
(54)

The procedures are as follows:

We adopt the following experimental values of the quark and charged lepton
masses and quark flavor mixings: We use the results of lattice calculations of
the isospin-averaged quark mass and the strange quark mass in $\overline{{\rm
MS}}$ scheme in Refs. [46]-[51], which read $\frac{1}{2}(m_{u}+m_{d})(2~{}{\rm
GeV})=3.373(80)~{}{\rm MeV}$ and $m_{s}(2~{}{\rm GeV})=92.0(2.1)~{}{\rm MeV}$,
and the estimate on the up/down quark mass ratio in Ref. [52],
$m_{u}/m_{d}=0.46(3)$. We use the results of QCD sum rule calculations of the
charm and bottom quark masses in $\overline{{\rm MS}}$ scheme in Ref. [53],
which read $m_{c}(3~{}{\rm
GeV})=0.986-9(\alpha_{s}^{(5)}(M_{Z})-0.1189)/0.002\pm 0.010~{}{\rm GeV}$ and
$m_{b}(m_{b})=4.163+7(\alpha_{s}^{(5)}(M_{Z})-0.1189)/0.002\pm 0.014~{}{\rm
GeV}$. We use the top quark pole mass measured by CMS in Ref. [54], which
reads $M_{t}=170.5\pm 0.8$ GeV. We calculate the CKM mixing angles and CP
phase from the Wolfenstein parameters in Ref. [55]. The lepton pole masses and
$W$, $Z$, Higgs boson pole masses are taken from Particle Data Group [45], and
the QCD and QED gauge couplings are set as $\alpha_{s}^{(5)}(M_{Z})=0.1181$
and $\alpha^{(5)}(M_{Z})=1/127.95$. The above data are translated into the
values of the quark and lepton Yukawa coupling matrices and the gauge
couplings at scale $\mu=M_{Z}$ in $\overline{{\rm MS}}$ scheme with the help
of the code [56] based on Refs. [57]-[61].

We calculate the two-loop RG equations [62]-[64] of SM from scale $\mu=M_{Z}$
to the soft SUSY breaking scale $\mu_{\rm SUSY}$. The results are matched to
the Yukawa coupling matrices and gauge couplings of MSSM in $\overline{{\rm
DR}}$ scheme. Then we calculate the two-loop RG equations of MSSM from scale
$\mu_{\rm SUSY}$ to the GUT scale $\mu_{\rm GUT}$. We assume a degenerate SUSY
particle mass spectrum where the pole masses of SUSY particles and $\tan\beta$
are given by

$\displaystyle m_{\rm sfermion}=m_{H^{0}}=m_{H^{\pm}}=m_{A}=1500~{}{\rm TeV},$
$\displaystyle|M_{\widetilde{g}}|=|M_{\widetilde{W}}|=|\mu_{H}|=1500~{}{\rm
TeV},\ \ \ \ \ \ \tan\beta=50.$ (55)

We set $\mu_{\rm SUSY}=1500$ TeV and $\mu_{\rm GUT}=2\cdot 10^{16}$ GeV. The
values of the Yukawa coupling matrices at scale $\mu=\mu_{\rm GUT}$ are shown
in Table 1, in the form of the singular values of the matrices and the
parameters of the CKM matrix at this scale. The errors of the quark Yukawa
couplings, propagated from the experimental errors of the corresponding
masses, and the maximal errors of the CKM parameters, obtained by assuming
maximal correlation of the experimental errors of the Wolfenstein parameters,
are also displayed.

Table 1: The singular values of the Yukawa coupling matrices and the CKM mixing angles and CP phase in MSSM at $\mu=\mu_{\rm GUT}=2\cdot 10^{16}$ GeV. Also shown are the errors of the quark Yukawa couplings, propagated from the experimental errors of the corresponding masses, and the maximal errors of the CKM parameters, obtained by assuming maximal correlation of the experimental errors of the Wolfenstein parameters. | Value with Eq. (55)
---|---
$y_{u}$ | 2.79(14)$\times 10^{-6}$
$y_{c}$ | 0.001435(15)
$y_{t}$ | 0.4718(58)
$y_{d}$ | 0.0003081(97)
$y_{s}$ | 0.00614(14)
$y_{b}$ | 0.3497(20)
$y_{e}$ | 0.0001258
$y_{\mu}$ | 0.02655
$y_{\tau}$ | 0.5081
$\cos\theta_{13}^{\rm ckm}\sin\theta_{12}^{\rm ckm}$ | 0.22475(25)
$\cos\theta_{13}^{\rm ckm}\sin\theta_{23}^{\rm ckm}$ | 0.0430(11)
$\sin\theta_{13}^{\rm ckm}$ | 0.00380(23)
$\delta_{\rm km}$ (rad) | 1.147(33)

Also, we evaluate one-loop RG corrections to the Wilson coefficient of the
Weinberg operator. We write the Weinberg operator in MSSM as Eq. (8) and that
in SM as $-{\cal
L}=\frac{1}{2}(C^{\prime}_{\nu})_{ij}\,\psi_{L_{i}}\psi_{L_{j}}HH$ where
$\psi_{L_{i}}$ denote the lepton doublets and $H$ the Higgs field. We express
the one-loop RG corrections in MSSM and SM as
$C_{\nu}(\mu)=R(\mu)C_{\nu}(\mu_{\rm SUSY})R^{T}(\mu)$,
$C^{\prime}_{\nu}(\mu)=R^{\prime}(\mu)C^{\prime}_{\nu}(M_{Z})R^{\prime
T}(\mu)$, respectively, and perform the matching as $C_{\nu}(\mu_{\rm
SUSY})=C^{\prime}_{\nu}(\mu_{\rm SUSY})$, since $\tan\beta\gg 1$. We solve the
one-loop RG equations and calculate the product of $R(\mu_{\rm GUT})$ and
$R^{\prime}(\mu_{\rm SUSY})$. Here we approximate the scale of the Majorana
mass $(M_{N})_{ij}$ to be $\mu_{\rm GUT}$. The product of $R(\mu_{\rm GUT})$
and $R^{\prime}(\mu_{\rm SUSY})$ in the flavor basis where the lepton doublets
have a diagonal Yukawa coupling matrix, is calculated as

$\displaystyle R(\mu_{\rm GUT})R^{\prime}(\mu_{\rm
SUSY})=\begin{pmatrix}1.09&0&0\\\ 0&1.09&0\\\ 0&0&1.14\\\ \end{pmatrix}.$ (56)

We fit the quark and charged lepton Yukawa couplings and the CKM parameters at
$\mu=\mu_{\rm GUT}$ in Table 1 with the Yukawa coupling matrices
$Y_{10},Y_{126},Y_{120}$ and numbers $r_{1},r_{2},r_{3},r_{e}$ based on Eqs.
(3)-(5). Also, we calculate the neutrino mass matrix up to the overall
constant, which is proportional to $C^{\prime}_{\nu}(M_{Z})$, from
$Y_{10},Y_{126},Y_{120}$ and $r_{2},r_{D}$ using Eqs. (6)-(9),(56), and with
it we fit the neutrino oscillation data in NuFIT 5.1 [65, 66]. Meanwhile, we
minimize Eq. (54).

We restrict the parameter space to the region with $r_{3}=0$, since it is
easier to minimize Eq. (54) when $r_{3}=0$. This is because when $r_{3}=0$,
$Y_{u}=Y_{10}+r_{2}Y_{126}$ holds and we get
$|(Y_{10})_{u_{L}i}+r_{2}(Y_{126})_{u_{L}i}|=|(Y_{u})_{u_{L}i}|\leq y_{u}$ and
$|(Y_{10})_{u_{R}i}+r_{2}(Y_{126})_{u_{R}i}|=|(Y_{u})_{u_{R}i}|\leq y_{u}$ for
any flavor index $i$. Under the restriction of $r_{3}=0$, we parametrize the
Yukawa coupling matrices as follows: We go to the flavor basis where the
isospin-doublet down-type quark components of ${\bf 16}^{i}$ matter fields
have a diagonal Yukawa coupling matrix. Since $Y_{u}=Y_{10}+r_{2}Y_{126}$ is
symmetric, $Y_{u}$ in this basis can be written as

$\displaystyle Y_{u}=V_{\rm CKM}^{T}\begin{pmatrix}y_{u}&0&0\\\
0&y_{c}\,e^{2i\,d_{2}}&0\\\ 0&0&y_{t}\,e^{2i\,d_{3}}\\\ \end{pmatrix}V_{\rm
CKM},$ (57)

where $V_{\rm CKM}$ denotes the CKM matrix and $d_{2},d_{3}$ are undetermined
phases. In the same flavor basis, $Y_{d}$ can be written as

$\displaystyle Y_{d}=\begin{pmatrix}y_{d}&0&0\\\ 0&y_{s}&0\\\ 0&0&y_{b}\\\
\end{pmatrix}V_{dR},$ (58)

where $V_{dR}$ is an undetermined unitary matrix. From Eqs. (3)-(6), $Y_{e}$
and $Y_{D}$ are written in terms of $Y_{u},Y_{d}$ as

$\displaystyle\frac{1}{r_{1}}Y_{e}=Y_{u}-(3+r_{2})Y_{126}+r_{e}\frac{1}{r_{1}}\frac{1}{2}\left(Y_{d}-Y_{d}^{T}\right),$
(59) $\displaystyle
Y_{D}=Y_{u}-4r_{2}Y_{126}+r_{D}\frac{1}{r_{1}}\frac{1}{2}\left(Y_{d}-Y_{d}^{T}\right)$
(60)

with

$\displaystyle
Y_{126}=\frac{1}{1-r_{2}}\left\\{\frac{1}{r_{1}}\frac{1}{2}\left(Y_{d}+Y_{d}^{T}\right)-Y_{u}\right\\}.$
(61)

The Majorana mass Eq. (7) is found to satisfy

$\displaystyle M_{N}\propto Y_{126}.$ (62)

The analysis of fitting and minimization proceeds with the above
parameterization as follows: We fix $y_{u},y_{c},y_{t}$ and the parameters of
the CKM matrix at the central values in Table 1. Then we randomly generate
$y_{d}/r_{1},\,y_{s}/r_{1},\,y_{b}/r_{1}$, phases $d_{2},d_{3}$, unitary
matrix $V_{dR}$, and complex numbers $r_{2},r_{e},r_{D}$, and calculate the
singular values of $\frac{1}{r_{1}}Y_{e}$. We determine $r_{1}$ by requiring
that the smallest singular value of $Y_{e}$ equal the value of $y_{e}$ in
Table 1. Then we require that the values of $y_{d},y_{s},y_{b}$ be within the
3$\sigma$ ranges in Table 1, and the first and second largest singular values
of $Y_{e}$ respectively be within $\pm$0.1% ranges of the values of
$y_{\tau},y_{\mu}$ 333Since the experimental errors of the charged lepton
Yukawa couplings are negligibly small, here we loosen the fitting criteria..
Because the active neutrino mass matrix $M_{\nu}$ is proportional to
$C^{\prime}_{\nu}(M_{Z})$, we calculate $M_{\nu}$ up to the overall constant
as

$\displaystyle M_{\nu}\ \propto\ \left(R(\mu_{\rm GUT})R^{\prime}(\mu_{\rm
SUSY})\right)^{-1}Y_{D}Y_{126}^{-1}Y_{D}^{T}\left(R(\mu_{\rm
GUT})R^{\prime}(\mu_{\rm SUSY})\right)^{T\,-1}.$ (63)

Then we calculate from Eq. (63) the three neutrino mixing angles
$\sin^{2}\theta_{12}$, $\sin^{2}\theta_{23}$, $\sin^{2}\theta_{13}$ and the
ratio of the neutrino mass squared differences $\Delta m_{21}^{2}/\Delta
m_{31}^{2}$, and require them to be within the 3$\sigma$ ranges of NuFIT5.1
results (with SK atmospheric data). Here we assume the normal mass hierarchy,
because it is almost impossible to realize the inverted mass hierarchy from
$Y_{D},Y_{126}$ as these matrices have hierarchical structures. Finally, we
select sets of values of $y_{d}/r_{1},\,y_{s}/r_{1},\,y_{b}/r_{1}$,
$d_{2},d_{3}$, $V_{dR}$, $r_{2},r_{e},r_{D}$ that meet the above fitting
criteria, calculate Eq. (54) from the sets, and minimize the value of Eq.
(54).

###  5.2 Result

From the analysis of Section 5.1, we have obtained the values of the Yukawa
coupling matrices $Y_{10},Y_{126},Y_{120}$ and numbers
$r_{1},r_{2},r_{e},r_{D}$ in Appendix B. There, components of $Y_{10},Y_{126}$
satisfy

$\displaystyle|(Y_{10})_{u_{R}d_{R}}|/y_{u}=0.77,\ \ \
|(Y_{126})_{u_{R}d_{R}}|/y_{u}=1.0,\ \ \ |(Y_{10})_{u_{R}s_{R}}|/y_{u}=0.28,\
\ \ |(Y_{126})_{u_{R}s_{R}}|/y_{u}=0.43,$
$\displaystyle|(Y_{10})_{u_{L}d_{L}}|/y_{u}=1.7,\ \ \
|(Y_{126})_{u_{L}d_{L}}|/y_{u}=1.5,\ \ \ |(Y_{10})_{u_{L}s_{L}}|/y_{u}=0.42,\
\ \ |(Y_{126})_{u_{L}s_{L}}|/y_{u}=0.49,$
$\displaystyle|(Y_{10})_{u_{L}u_{L}}|/y_{u}=0.85,\ \ \
|(Y_{126})_{u_{L}u_{L}}|/y_{u}=0.90.$ (64)

Clearly, the conditions that the components of Eq. (42) be on the order of the
up quark Yukawa coupling are satisfied.

We evaluate how dimension-five proton decay is suppressed when the conditions
are met. To this end, we compare “minimal proton partial lifetimes” calculated
from the Yukawa coupling matrices of Appendix B, with those calculated from
results of “fitting without minimizing Eq. (54)” where we only fit the quark
and charged lepton Yukawa couplings, CKM parameters and neutrino oscillation
data as Section 5.1 but do not minimize Eq. (54) so that the conditions are
not necessarily satisfied. Here the “minimal proton partial lifetimes”,
$1/\Gamma_{\rm max}(p\to K^{+}\bar{\nu})$, $1/\Gamma_{\rm max}(p\to
K^{0}\mu^{+})$, $1/\Gamma_{\rm max}(p\to K^{0}e^{+})$, are defined as
($\beta=e,\mu$)

$\displaystyle\Gamma_{\rm max}(p\to
K^{+}\bar{\nu})=\frac{m_{N}}{64\pi}\left(1-\frac{m_{K}^{2}}{m_{N}^{2}}\right)^{2}$
$\displaystyle\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \times\left(|A_{\rm max}(p\to
K^{+}\bar{\nu}_{\tau})|^{2}+|A_{\rm max}(p\to
K^{+}\bar{\nu}_{\mu})|^{2}+|A_{\rm max}(p\to K^{+}\bar{\nu}_{e})|^{2}\right),$
(65) $\displaystyle\Gamma_{\rm max}(p\to
K^{0}\beta^{+})=\frac{m_{N}}{64\pi}\left(1-\frac{m_{K}^{2}}{m_{N}^{2}}\right)^{2}|A_{\rm
max}(p\to K^{0}\beta^{+})|^{2},$ (66)

where

$\displaystyle A_{\rm max}(p\to K^{+}\bar{\nu}_{\tau})$ $\displaystyle=|A_{\rm
max}(p\to K^{+}\bar{\nu}_{\tau})_{{\rm from}\ C_{5R}}|+|A_{\rm max}(p\to
K^{+}\bar{\nu}_{\tau})_{{\rm from}\ C_{5L}}|$ (67)

and

$\displaystyle A_{\rm max}(p\to K^{+}\bar{\nu}_{\tau})_{{\rm from}\
C_{5R}}=\alpha_{H}(\mu_{\rm had})\frac{1}{f_{\pi}}A_{RL}(\mu_{\rm
had},\mu_{\rm
SUSY})\frac{\mu_{H}}{m_{\tilde{t}_{R}}^{2}}\frac{1}{16\pi^{2}}{\cal
F}^{\prime}\,y_{t}y_{\tau}\,A_{R}^{\tau t}(\mu_{\rm SUSY},\mu_{\rm
GUT})\frac{1}{M_{H_{C}}}$ $\displaystyle\ \ \ \ \ \ \ \
\times\max_{A,B}\left\\{\left|\left(1+\frac{D}{3}+F\right)(V^{\rm
ckm}_{ts})^{*}\left((Y_{A})_{\tau_{R}t_{R}}(Y_{B})_{u_{R}d_{R}}-(Y_{A})_{\tau_{R}u_{R}}(Y_{B})_{t_{R}d_{R}}\right)\right.\right.$
$\displaystyle\left.\left.\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\ \ \ \ \ \ \ \ \ \ +\frac{2D}{3}(V^{\rm
ckm}_{td})^{*}\left((Y_{A})_{\tau_{R}t_{R}}(Y_{B})_{u_{R}s_{R}}-(Y_{A})_{\tau_{R}u_{R}}(Y_{B})_{t_{R}s_{R}}\right)\right|\right\\},$
(68) $\displaystyle A_{\rm max}(p\to K^{+}\bar{\nu}_{\tau})_{{\rm from}\
C_{5L}}=\beta_{H}(\mu_{\rm had})\frac{1}{f_{\pi}}A_{LL}(\mu_{\rm had},\mu_{\rm
SUSY})\frac{M_{\widetilde{W}}}{m_{\tilde{q}}^{2}}\frac{1}{16\pi^{2}}{\cal
F}\,g_{2}^{2}\,A_{L}^{\tau}(\mu_{\rm SUSY},\mu_{\rm GUT})\frac{1}{M_{H_{C}}}$
$\displaystyle\ \ \ \ \ \
\times\max_{A^{\prime},B^{\prime}}\left\\{\left|\left(1+\frac{D}{3}+F\right)(Y_{A^{\prime}})_{u_{L}d_{L}}(Y_{B^{\prime}})_{s_{L}\tau_{L}}+\frac{2D}{3}(Y_{A^{\prime}})_{u_{L}s_{L}}(Y_{B^{\prime}})_{d_{L}\tau_{L}}\right|\right\\},$
(69) $\displaystyle A_{\rm max}(p\to
K^{+}\bar{\nu}_{\beta})=\beta_{H}(\mu_{\rm
had})\frac{1}{f_{\pi}}A_{LL}(\mu_{\rm had},\mu_{\rm
SUSY})\frac{M_{\widetilde{W}}}{m_{\tilde{q}}^{2}}\frac{1}{16\pi^{2}}{\cal
F}\,g_{2}^{2}\,A_{L}^{\beta}(\mu_{\rm SUSY},\mu_{\rm GUT})\frac{1}{M_{H_{C}}}$
$\displaystyle\ \ \ \ \ \
\times\max_{A^{\prime},B^{\prime}}\left\\{\left|\left(1+\frac{D}{3}+F\right)(Y_{A^{\prime}})_{u_{L}d_{L}}(Y_{B^{\prime}})_{s_{L}\beta_{L}}+\frac{2D}{3}(Y_{A^{\prime}})_{u_{L}s_{L}}(Y_{B^{\prime}})_{d_{L}\beta_{L}}\right|\right\\},$
(70) $\displaystyle A_{\rm max}(p\to K^{0}\beta^{+})=\beta_{H}(\mu_{\rm
had})\frac{1}{f_{\pi}}A_{LL}(\mu_{\rm had},\mu_{\rm
SUSY})\frac{M_{\widetilde{W}}}{m_{\tilde{q}}^{2}}\frac{1}{16\pi^{2}}{\cal
F}\,g_{2}^{2}\,A_{L}^{\beta}(\mu_{\rm SUSY},\mu_{\rm GUT})\frac{1}{M_{H_{C}}}$
$\displaystyle\ \ \ \ \ \ \times(1-D+F)\
\max_{A^{\prime},B^{\prime}}\left\\{\left|(Y_{A^{\prime}})_{u_{L}s_{L}}(Y_{B^{\prime}})_{u_{L}\beta_{L}}-(Y_{A^{\prime}})_{u_{L}u_{L}}(Y_{B^{\prime}})_{s_{L}\beta_{L}}\right|\right\\},$
(71)

where $A,B$ in Eq. (68) run as $(A,B)=(10,10)$, (10,126), (126,10), (126,126),
(120,10), (120,126), (120,120), and $A^{\prime},B^{\prime}$ in Eqs. (69)-(71)
run as $(A^{\prime},B^{\prime})=(10,10)$, (10,126), (126,10), (126,126). We
assume the SUSY particle spectrum of Eq. (55) and take $M_{H_{C}}=2\cdot
10^{16}$ GeV in the calculation. Note that Eqs. (68)-(71) take into account
the texture of the colored Higgs mass matrix satisfying Eqs. (43),(44). An
implication of Eqs. (67)-(71) is that we estimate the maximal values of the
amplitudes without specifying the $O(1)$ numbers $a,b,d,e,g,h,j$ in Eqs.
(LABEL:yyr1)-(LABEL:yyl3) and the relative phase between the Higgsino mass and
Wino mass. The “minimal proton partial lifetimes” calculated from the Yukawa
coupling matrices of Appendix B are 444 These values are consistent with the
current experimental bounds on the $p\to K^{+}\bar{\nu}$ partial lifetime [10]
and on the $p\to K^{0}\mu^{+}/e^{+}$ partial lifetimes [70, 71], which
justifies our choice of the benchmark SUSY particle mass spectrum Eq. (55).

$\displaystyle 1/\Gamma_{\rm max}(p\to K^{+}\bar{\nu})=6.3\times
10^{33}~{}{\rm years},$ (72) $\displaystyle 1/\Gamma_{\rm max}(p\to
K^{0}\mu^{+})=7.7\times 10^{36}~{}{\rm years},$ (73) $\displaystyle
1/\Gamma_{\rm max}(p\to K^{0}e^{+})=2.4\times 10^{39}~{}{\rm years}.$ (74)

On the other hand, the “minimal proton partial lifetimes” calculated from
multiple results of “fitting without minimizing Eq. (54)” are distributed as
Fig. 1.

Figure 1:  Distributions of the “minimal proton partial lifetimes” calculated
from multiple results of “fitting without minimizing Eq. (54)”. The upper,
lower-left and lower-right panels correspond to the distributions of
$1/\Gamma_{\rm max}(p\to K^{+}\bar{\nu})$, $1/\Gamma_{\rm max}(p\to
K^{0}\mu^{+})$ and $1/\Gamma_{\rm max}(p\to K^{0}e^{+})$, respectively. In the
upper panel, the blue dot-dashed line indicates the current bound on the $p\to
K^{+}\bar{\nu}$ partial lifetime [10], the red solid line the value of
$1/\Gamma_{\rm max}(p\to K^{+}\bar{\nu})$ calculated from the Yukawa coupling
matrices of Appendix B in Eq. (72), and the blue dashed line the $3\sigma$
discovery reach of 2 years running of Hyper-Kamiokande [72]. In the lower-left
panel, the blue dot-dashed line indicates the current bound on the $p\to
K^{0}\mu^{+}$ partial lifetime [70] (part of the line is hidden behind the
histogram), and the red solid line the value of $1/\Gamma_{\rm max}(p\to
K^{0}\mu^{+})$ calculated from the Yukawa coupling matrices of Appendix B in
Eq. (73). In the lower-right panel, the blue dot-dashed line indicates the
current bound on the $p\to K^{0}e^{+}$ partial lifetime [71], and the red
solid line the value of $1/\Gamma_{\rm max}(p\to K^{0}e^{+})$ calculated from
the Yukawa coupling matrices of Appendix B in Eq. (74).

In Fig. 1, we overlay the values calculated from the Yukawa coupling matrices
of Appendix B in Eqs. (72)-(74). Also, the current bounds on the $p\to
K^{+}\bar{\nu}$ [10], $p\to K^{0}\mu^{+}$ [70], $p\to K^{0}e^{+}$ [71] partial
lifetimes, and the $3\sigma$ discovery reach of 2 years running of Hyper-
Kamiokande [72] are shown.

Comparing Eqs. (72)-(74) with Fig. 1, we see that the proton partial lifetimes
calculated from the Yukawa coupling matrices of Appendix B are on the upper
edge of the distributions of proton partial lifetimes calculated from results
of “fitting without minimizing Eq. (54)”. This confirms that the texture of
the Yukawa coupling matrices satisfying the conditions that the components of
Eq. (42) be on the order of the up quark Yukawa coupling, contributes to
suppressing dimension-five proton decay. Specifically, the benchmark SUSY
particle mass spectrum Eq. (55), where the SUSY particle masses are all at
1500 TeV and $\tan\beta=50$, is consistent with all the experimental bounds on
proton partial lifetimes if the above conditions are satisfied. On the other
hand, if these conditions are not met, this benchmark almost always violates
the bound on the $p\to K^{+}\bar{\nu}$ partial lifetime. We also see that for
this benchmark mass spectrum, when the above conditions are satisfied, we
expect to discover the $p\to K^{+}\bar{\nu}$ decay with 2 years running of
Hyper-Kamiokande.

We comment that the Yukawa coupling matrices and coefficients in Appendix B
give a prediction on poorly or not measured neutrino parameters, which are the
Dirac CP phase of the neutrino mixing matrix, the sum of the neutrino mass,
and the effective neutrino mass for neutrinoless double $\beta$ decay. The
prediction is shown in Appendix C.

##  6 Summary

We have pursued the possibility that dimension-five proton decay is suppressed
by a texture of the Yukawa coupling matrices in the general renormalizable
SUSY $SO(10)$ GUT model where Yukawa coupling matrices of ${\bf 16}$
representation matter fields with ${\bf 10},{\bf\overline{126}},{\bf 120}$
fields $Y_{10},Y_{126},Y_{120}$ give the quark and lepton Yukawa couplings and
Majorana mass of the singlet neutrinos. We have derived conditions for a
texture of the Yukawa coupling matrices suppressing dimension-five proton
decay, which state that components $(Y_{10})_{u_{R}d_{R}}$,
$(Y_{126})_{u_{R}d_{R}}$, $(Y_{10})_{u_{R}s_{R}}$, $(Y_{126})_{u_{R}s_{R}}$,
$(Y_{10})_{u_{L}d_{L}}$, $(Y_{126})_{u_{L}d_{L}}$, $(Y_{10})_{u_{L}u_{L}}$,
$(Y_{126})_{u_{L}u_{L}}$, $(Y_{10})_{u_{L}s_{L}}$, $(Y_{126})_{u_{L}s_{L}}$
should be all on the order of the up quark Yukawa coupling $y_{u}$.
Additionally, the colored Higgs mass matrix should satisfy Eqs. (43),(44). We
have obtained the values of the Yukawa coupling matrices that satisfy the
above conditions and that are consistent with the experimental data of quark
and lepton masses and flavor mixings. By comparing the “minimal proton partial
lifetimes” calculated from the Yukawa coupling matrices that meet the
conditions and those that do not necessarily so, we have confirmed that the
texture of the Yukawa coupling matrices satisfying the conditions contributes
to suppressing dimension-five proton decay. Specifically, we have found that a
SUSY particle mass spectrum where the SUSY particle masses are all at 1500 TeV
and $\tan\beta=50$ is consistent with all the experimental bounds on proton
decay if the above conditions are satisfied. Also, for this mass spectrum,
when the conditions are met, we expect to discover the $p\to K^{+}\bar{\nu}$
decay with 2 years running of Hyper-Kamiokande.

## Acknowledgement

This work is partially supported by Scientific Grants by the Ministry of
Education, Culture, Sports, Science and Technology of Japan, No. 21H00076 (NH)
and No. 19K147101 (TY).

## Appendix A: Superpotential

We review our definition of the coupling constants and masses for $H$,
$\Delta$, $\overline{\Delta}$, $\Sigma$, $\Phi$, $A$ fields in ${\bf 10}$,
${\bf 126}$, ${\bf\overline{126}}$, ${\bf 120}$, ${\bf 210}$, ${\bf 45}$
representations, which follows Eq. (2) of Ref. [41]. The couplings are defined
in the same way as Eq. (3) of Ref. [41]. Note that ${\bf 120}$ representation
field is written as $D$ in Ref. [41], while we write it as $\Sigma$. The
coupling constants are defined as

$\displaystyle W$ $\displaystyle=$
$\displaystyle\frac{1}{2}m_{1}\Phi^{2}+m_{2}\overline{\Delta}\Delta+\frac{1}{2}m_{3}H^{2}$
(75) $\displaystyle+$
$\displaystyle\frac{1}{2}m_{4}A^{2}+\frac{1}{2}m_{6}\Sigma^{2}$
$\displaystyle+$
$\displaystyle\lambda_{1}\Phi^{3}+\lambda_{2}\Phi\overline{\Delta}\Delta+(\lambda_{3}\Delta+\lambda_{4}\overline{\Delta})H\Phi$
$\displaystyle+$
$\displaystyle\lambda_{5}A^{2}\Phi-i\lambda_{6}A\overline{\Delta}\Delta+\frac{\lambda_{7}}{120}\varepsilon
A\Phi^{2}$ $\displaystyle+$ $\displaystyle\lambda_{15}\Sigma^{2}\Phi$
$\displaystyle+$
$\displaystyle\Sigma\\{\lambda_{16}HA+\lambda_{17}H\Phi+(\lambda_{18}\Delta+\lambda_{19}\overline{\Delta})A+(\lambda_{20}\Delta+\lambda_{21}\overline{\Delta})\Phi\\}$

where $\varepsilon$ denotes the antisymmetric tensor in $SO(10)$ space.

## Appendix B: Values of $Y_{10}$, $Y_{126}$, $Y_{120}$ and
$r_{1},r_{2},r_{e},r_{D}$

We present the values of the Yukawa coupling matrices $Y_{10}$, $Y_{126}$,
$Y_{120}$ and numbers $r_{1},r_{2},r_{e},r_{D}$ obtained from the analysis of
Section 5.1. $Y_{10}$, $Y_{126}$, $Y_{120}$ are shown repeatedly in three
different flavor bases. For reference, the central value of the up quark
Yukawa coupling at scale $\mu=\mu_{\rm GUT}=2\cdot 10^{16}$ GeV in
$\overline{{\rm DR}}$ scheme is $y_{u}=2.79\cdot 10^{-6}$.

$\displaystyle\begin{pmatrix}(Y_{10})_{u_{R}d_{R}}&(Y_{10})_{u_{R}s_{R}}&(Y_{10})_{u_{R}b_{R}}\\\
(Y_{10})_{c_{R}d_{R}}&(Y_{10})_{c_{R}s_{R}}&(Y_{10})_{c_{R}b_{R}}\\\
(Y_{10})_{t_{R}d_{R}}&(Y_{10})_{t_{R}s_{R}}&(Y_{10})_{t_{R}b_{R}}\\\
\end{pmatrix}$ $\displaystyle=\begin{pmatrix}2.14\cdot
10^{-6}\,e^{3.14\,i}&0.793\cdot 10^{-6}\,e^{-0.78\,i}&0.00164\,e^{-0.17\,i}\\\
0.00195\,e^{-1.39\,i}&0.00841\,e^{-1.41\,i}&0.0227\,e^{-1.20\,i}\\\
0.00659\,e^{-2.64\,i}&0.0245\,e^{-2.89\,i}&0.230\,e^{0.45\,i}\\\
\end{pmatrix}$ (76)
$\displaystyle\begin{pmatrix}(Y_{126})_{u_{R}d_{R}}&(Y_{126})_{u_{R}s_{R}}&(Y_{126})_{u_{R}b_{R}}\\\
(Y_{126})_{c_{R}d_{R}}&(Y_{126})_{c_{R}s_{R}}&(Y_{126})_{c_{R}b_{R}}\\\
(Y_{126})_{t_{R}d_{R}}&(Y_{126})_{t_{R}s_{R}}&(Y_{126})_{t_{R}b_{R}}\\\
\end{pmatrix}$ $\displaystyle=\begin{pmatrix}2.78\cdot
10^{-6}\,e^{-2.03\,i}&1.21\cdot 10^{-6}\,e^{0.99\,i}&0.00171\,e^{2.14\,i}\\\
0.00233\,e^{0.98\,i}&0.0100\,e^{0.97\,i}&0.236\,e^{1.11\,i}\\\
0.00686\,e^{0.49\,i}&0.0241\,e^{0.25\,i}&0.251\,e^{-0.41\,i}\\\ \end{pmatrix}$
(77)
$\displaystyle\begin{pmatrix}(Y_{120})_{u_{R}d_{R}}&(Y_{120})_{u_{R}s_{R}}&(Y_{120})_{u_{R}b_{R}}\\\
(Y_{120})_{c_{R}d_{R}}&(Y_{120})_{c_{R}s_{R}}&(Y_{120})_{c_{R}b_{R}}\\\
(Y_{120})_{t_{R}d_{R}}&(Y_{120})_{t_{R}s_{R}}&(Y_{120})_{t_{R}b_{R}}\\\
\end{pmatrix}$
$\displaystyle=\begin{pmatrix}0.000396\,e^{0.01\,i}&0.00172&0.000353\,e^{1.61\,i}\\\
0.00173\,e^{-3.13\,i}&0.000397\,e^{0.02\,i}&0.000145\,e^{-0.07\,i}\\\
0.000272\,e^{-2.82\,i}&0.000169\,e^{1.45\,i}&6.78\cdot
10^{-6}\,e^{-3.11\,i}\\\ \end{pmatrix}$ (78)
$\displaystyle\begin{pmatrix}(Y_{10})_{u_{L}d_{L}}&(Y_{10})_{u_{L}s_{L}}&(Y_{10})_{u_{L}b_{L}}\\\
(Y_{10})_{c_{L}d_{L}}&(Y_{10})_{c_{L}s_{L}}&(Y_{10})_{c_{L}b_{L}}\\\
(Y_{10})_{t_{L}d_{L}}&(Y_{10})_{t_{L}s_{L}}&(Y_{10})_{t_{L}b_{L}}\\\
\end{pmatrix}$ $\displaystyle=\begin{pmatrix}4.70\cdot
10^{-6}\,e^{-1.04\,i}&1.17\cdot 10^{-6}\,e^{-1.52\,i}&0.00164\,e^{0.58\,i}\\\
0.00195\,e^{-2.53\,i}&0.00840\,e^{0.61\,i}&0.0227\,e^{-0.45\,i}\\\
0.00493\,e^{1.94\,i}&0.0250\,e^{-0.85\,i}&0.230\,e^{1.20\,i}\\\ \end{pmatrix}$
(79)
$\displaystyle\begin{pmatrix}(Y_{126})_{u_{L}d_{L}}&(Y_{126})_{u_{L}s_{L}}&(Y_{126})_{u_{L}b_{L}}\\\
(Y_{126})_{c_{L}d_{L}}&(Y_{126})_{c_{L}s_{L}}&(Y_{126})_{c_{L}b_{L}}\\\
(Y_{126})_{t_{L}d_{L}}&(Y_{126})_{t_{L}s_{L}}&(Y_{126})_{t_{L}b_{L}}\\\
\end{pmatrix}$ $\displaystyle=\begin{pmatrix}4.23\cdot
10^{-6}\,e^{0.65\,i}&1.35\cdot 10^{-6}\,e^{0.28\,i}&0.00171\,e^{2.89\,i}\\\
0.00233\,e^{-0.15\,i}&0.0100\,e^{2.99\,i}&0.0236\,e^{1.86\,i}\\\
0.00517\,e^{-1.17\,i}&0.0246\,e^{2.30\,i}&0.251\,e^{0.345\,i}\\\
\end{pmatrix}$ (80)
$\displaystyle\begin{pmatrix}(Y_{120})_{u_{L}d_{L}}&(Y_{120})_{u_{L}s_{L}}&(Y_{120})_{u_{L}b_{L}}\\\
(Y_{120})_{c_{L}d_{L}}&(Y_{120})_{c_{L}s_{L}}&(Y_{120})_{c_{L}b_{L}}\\\
(Y_{120})_{t_{L}d_{L}}&(Y_{120})_{t_{L}s_{L}}&(Y_{120})_{t_{L}b_{L}}\\\
\end{pmatrix}$
$\displaystyle=\begin{pmatrix}0.000396\,e^{-1.12\,i}&0.00172\,e^{2.01\,i}&0.000354\,e^{2.36\,i}\\\
0.00173\,e^{-1.12\,i}&0.000401\,e^{-1.14\,i}&0.000143\,e^{0.69\,i}\\\
0.000284\,e^{-0.58\,i}&0.000148\,e^{-2.01\,i}&7.33\cdot
10^{-6}\,e^{-2.36\,i}\\\ \end{pmatrix}$ (81)
$\displaystyle\begin{pmatrix}(Y_{10})_{u_{L}u_{L}}&(Y_{10})_{u_{L}c_{L}}&(Y_{10})_{u_{L}t_{L}}\\\
&(Y_{10})_{c_{L}c_{L}}&(Y_{10})_{c_{L}t_{L}}\\\ &&(Y_{10})_{t_{L}t_{L}}\\\
\end{pmatrix}$ $\displaystyle=\begin{pmatrix}2.36\cdot
10^{-6}\,e^{0.97\,i}&0.0000700\,e^{0.59\,i}&0.00164\,e^{0.58\,i}\\\
&0.00914\,e^{0.52\,i}&0.0225\,e^{-0.46\,i}\\\ &&0.231\,e^{1.20\,i}\\\
\end{pmatrix}$ (82)
$\displaystyle\begin{pmatrix}(Y_{126})_{u_{L}u_{L}}&(Y_{126})_{u_{L}c_{L}}&(Y_{126})_{u_{L}t_{L}}\\\
&(Y_{126})_{c_{L}c_{L}}&(Y_{126})_{c_{L}t_{L}}\\\ &&(Y_{126})_{t_{L}t_{L}}\\\
\end{pmatrix}$ $\displaystyle=\begin{pmatrix}2.52\cdot
10^{-6}\,e^{-1.77\,i}&0.0000729\,e^{2.89\,i}&0.00171\,e^{2.89\,i}\\\
&0.0108\,e^{2.90\,i}&0.0234\,e^{1.84\,i}\\\ &&0.251\,e^{0.34\,i}\\\
\end{pmatrix}$ (83)
$\displaystyle\begin{pmatrix}(Y_{120})_{u_{L}u_{L}}&(Y_{120})_{u_{L}c_{L}}&(Y_{120})_{u_{L}t_{L}}\\\
&(Y_{120})_{c_{L}c_{L}}&(Y_{120})_{c_{L}t_{L}}\\\ &&(Y_{120})_{t_{L}t_{L}}\\\
\end{pmatrix}$
$\displaystyle=\begin{pmatrix}0&0.00177\,e^{2.02\,i}&0.000283\,e^{2.45\,i}\\\
&0&0.000149\,e^{0.70\,i}\\\ &&0\\\ \end{pmatrix}$ (84) $\displaystyle
r_{1}=0.794,\ \ \ r_{2}=0.960\,e^{0.84\,i},\ \ \ r_{e}=1.00\,e^{1.48\,i},\ \ \
r_{D}=0.488\,e^{-2.93\,i}.$ (85)

In Eqs. (82)-(84), we do not display some off-diagonal components because in
this flavor basis, $Y_{10},Y_{126}$ are symmetric and $Y_{120}$ is
antisymmetric.

## Appendix C: Prediction on Neutrino Parameters

The result of the analysis of Section 5.1, shown in Appendix B, gives the
following prediction on the Dirac CP phase of the neutrino mixing matrix,
$\delta_{\rm CP}$, the sum of the neutrino mass, $\sum_{i=1}^{3}m_{i}$, and
the effective neutrino mass for neutrinoless double $\beta$ decay, $|m_{ee}|$:

$\displaystyle\delta_{\rm CP}$ $\displaystyle=-1.42~{}{\rm rad},$ (86)
$\displaystyle\sum_{i=1}^{3}m_{i}$ $\displaystyle=0.0631~{}{\rm eV},$ (87)
$\displaystyle|m_{ee}|$ $\displaystyle=0.000237~{}{\rm eV}.$ (88)

We caution that the prediction on $\delta_{\rm CP}$ is not stable in the
analysis, namely, results of the analysis of Section 5.1 with slightly higher
values of Eq. (54) yield significantly different predictions on $\delta_{\rm
CP}$. Therefore, Eq. (86) should not be fully trusted.

## References

  * [1] H. Georgi, “The State of the Art—Gauge Theories,” AIP Conf. Proc. 23, 575 (1975).
  * [2] H. Fritzsch and P. Minkowski, “Unified Interactions of Leptons and Hadrons,” Annals Phys. 93, 193 (1975).
  * [3] M. Gell-Mann, P. Ramond and R. Slansky, “Complex Spinors and Unified Theories,” Conf. Proc. C 790927, 315-321 (1979) [arXiv:1306.4669 [hep-th]].
  * [4] T. Yanagida, “Horizontal gauge symmetry and masses of neutrinos,” Conf. Proc. C 7902131, 95-99 (1979) KEK-79-18-95.
  * [5] T. Yanagida, “Horizontal Symmetry and Mass of the Top Quark,” Phys. Rev. D 20, 2986 (1979)
  * [6] R. N. Mohapatra and G. Senjanovic, “Neutrino Mass and Spontaneous Parity Violation,” Phys. Rev. Lett. 44, 912 (1980).
  * [7] S. Weinberg, “Supersymmetry at Ordinary Energies. 1. Masses and Conservation Laws,” Phys. Rev. D 26, 287 (1982).
  * [8] N. Sakai and T. Yanagida, “Proton Decay in a Class of Supersymmetric Grand Unified Models,” Nucl. Phys. B 197, 533 (1982).
  * [9] T. Goto and T. Nihei, “Effect of RRRR dimension five operator on the proton decay in the minimal SU(5) SUGRA GUT model,” Phys. Rev. D 59, 115009 (1999) [hep-ph/9808255].
  * [10] K. Abe et al. [Super-Kamiokande], “Search for proton decay via $p\to\nu K^{+}$ using 260 kiloton·year data of Super-Kamiokande,” Phys. Rev. D 90, no.7, 072005 (2014) [arXiv:1408.1195 [hep-ex]].
  * [11] K. Matsuda, Y. Koide and T. Fukuyama, “Can the SO(10) model with two Higgs doublets reproduce the observed fermion masses?,” Phys. Rev. D 64, 053015 (2001) [hep-ph/0010026].
  * [12] K. Matsuda, Y. Koide, T. Fukuyama and H. Nishiura, “How far can the SO(10) two Higgs model describe the observed neutrino masses and mixings?,” Phys. Rev. D 65, 033008 (2002) Erratum: [Phys. Rev. D 65, 079904 (2002)] [hep-ph/0108202].
  * [13] T. Fukuyama and N. Okada, “Neutrino oscillation data versus minimal supersymmetric SO(10) model,” JHEP 0211, 011 (2002) [hep-ph/0205066].
  * [14] B. Bajc, G. Senjanovic and F. Vissani, “b - tau unification and large atmospheric mixing: A Case for noncanonical seesaw,” Phys. Rev. Lett. 90, 051802 (2003) [hep-ph/0210207].
  * [15] H. S. Goh, R. N. Mohapatra and S. P. Ng, “Minimal SUSY SO(10), b tau unification and large neutrino mixings,” Phys. Lett. B 570, 215 (2003) [hep-ph/0303055].
  * [16] C. S. Aulakh, B. Bajc, A. Melfo, G. Senjanovic and F. Vissani, “The Minimal supersymmetric grand unified theory,” Phys. Lett. B 588, 196 (2004) [hep-ph/0306242].
  * [17] H. S. Goh, R. N. Mohapatra and S. P. Ng, “Minimal SUSY SO(10) model and predictions for neutrino mixings and leptonic CP violation,” Phys. Rev. D 68, 115008 (2003) [hep-ph/0308197].
  * [18] B. Dutta, Y. Mimura and R. N. Mohapatra, “CKM CP violation in a minimal SO(10) model for neutrinos and its implications,” Phys. Rev. D 69, 115014 (2004) [hep-ph/0402113].
  * [19] B. Bajc, G. Senjanovic and F. Vissani, “Probing the nature of the seesaw in renormalizable SO(10),” Phys. Rev. D 70, 093002 (2004) [hep-ph/0402140].
  * [20] S. Bertolini, M. Frigerio and M. Malinsky, “Fermion masses in SUSY SO(10) with type II seesaw: A Non-minimal predictive scenario,” Phys. Rev. D 70, 095002 (2004) [arXiv:hep-ph/0406117 [hep-ph]].
  * [21] W. M. Yang and Z. G. Wang, “Fermion masses and flavor mixing in a supersymmetric SO(10) model,” Nucl. Phys. B 707, 87-99 (2005) [arXiv:hep-ph/0406221 [hep-ph]].
  * [22] B. Dutta, Y. Mimura and R. N. Mohapatra, “Neutrino masses and mixings in a predictive SO(10) model with CKM CP violation,” Phys. Lett. B 603, 35-45 (2004) [arXiv:hep-ph/0406262 [hep-ph]].
  * [23] B. Dutta, Y. Mimura and R. N. Mohapatra, “Suppressing proton decay in the minimal SO(10) model,” Phys. Rev. Lett. 94, 091804 (2005) [arXiv:hep-ph/0412105 [hep-ph]].
  * [24] K. S. Babu and C. Macesanu, “Neutrino masses and mixings in a minimal SO(10) model,” Phys. Rev. D 72, 115003 (2005) [hep-ph/0505200].
  * [25] B. Dutta, Y. Mimura and R. N. Mohapatra, “Neutrino mixing predictions of a minimal SO(10) model with suppressed proton decay,” Phys. Rev. D 72, 075009 (2005) [arXiv:hep-ph/0507319 [hep-ph]].
  * [26] S. Bertolini, T. Schwetz and M. Malinsky, “Fermion masses and mixings in SO(10) models and the neutrino challenge to SUSY GUTs,” Phys. Rev. D 73, 115012 (2006) [hep-ph/0605006].
  * [27] A. S. Joshipura and K. M. Patel, “Fermion Masses in SO(10) Models,” Phys. Rev. D 83, 095002 (2011) [arXiv:1102.5148 [hep-ph]].
  * [28] A. Dueck and W. Rodejohann, “Fits to SO(10) Grand Unified Models,” JHEP 1309, 024 (2013) [arXiv:1306.4468 [hep-ph]].
  * [29] T. Fukuyama, K. Ichikawa and Y. Mimura, “Revisiting fermion mass and mixing fits in the minimal SUSY $SO(10)$ GUT,” Phys. Rev. D 94, no. 7, 075018 (2016) [arXiv:1508.07078 [hep-ph]].
  * [30] T. Fukuyama, K. Ichikawa and Y. Mimura, “Relation between proton decay and PMNS phase in the minimal SUSY $SO(10)$ GUT,” Phys. Lett. B 764, 114 (2017) [arXiv:1609.08640 [hep-ph]].
  * [31] T. Fukuyama, N. Okada and H. M. Tran, “Sparticle spectroscopy of the minimal SO(10) model,” Phys. Lett. B 767, 295 (2017) [arXiv:1611.08341 [hep-ph]].
  * [32] K. S. Babu, B. Bajc and S. Saad, “Yukawa Sector of Minimal SO(10) Unification,” JHEP 02, 136 (2017) [arXiv:1612.04329 [hep-ph]].
  * [33] K. S. Babu, B. Bajc and S. Saad, “Resurrecting Minimal Yukawa Sector of SUSY SO(10),” JHEP 1810, 135 (2018) [arXiv:1805.10631 [hep-ph]].
  * [34] T. Deppisch, S. Schacht and M. Spinrath, “Confronting SUSY SO(10) with updated Lattice and Neutrino Data,” JHEP 1901, 005 (2019) [arXiv:1811.02895 [hep-ph]].
  * [35] T. Fukuyama, N. Okada and H. M. Tran, “Alternative renormalizable $SO$(10) GUTs and data fitting,” Nucl. Phys. B 954, 114992 (2020) [arXiv:1907.02948 [hep-ph]].
  * [36] N. Haba, Y. Mimura and T. Yamada, “Enhanced $\Gamma(p\to K^{0}\mu^{+})/\Gamma(p\to K^{+}\bar{\nu}_{\mu})$ as a signature of minimal renormalizable SUSY $SO$ (10) GUT,” PTEP 2020, no.9, 093B01 (2020) [arXiv:2002.11413 [hep-ph]].
  * [37] N. Haba, Y. Mimura and T. Yamada, “Renormalizable $SO(10)$ grand unified theory with suppressed dimension-5 proton decays,” PTEP 2021, no.2, 023B01 (2021) [arXiv:2008.05362 [hep-ph]].
  * [38] H. S. Goh, R. N. Mohapatra and S. Nasri, “SO(10) symmetry breaking and type II seesaw,” Phys. Rev. D 70, 075022 (2004) [arXiv:hep-ph/0408139 [hep-ph]].
  * [39] T. Fukuyama, A. Ilakovac, T. Kikuchi, S. Meljanac and N. Okada, “General formulation for proton decay rate in minimal supersymmetric SO(10) GUT,” Eur. Phys. J. C 42, 191 (2005) [hep-ph/0401213].
  * [40] C. S. Aulakh and A. Girdhar, “SO(10) MSGUT: Spectra, couplings and threshold effects,” Nucl. Phys. B 711, 275 (2005) [hep-ph/0405074].
  * [41] T. Fukuyama, A. Ilakovac, T. Kikuchi, S. Meljanac and N. Okada, “SO(10) group theory for the unified model building,” J. Math. Phys. 46, 033505 (2005) [hep-ph/0405300].
  * [42] T. Fukuyama, A. Ilakovac, T. Kikuchi, S. Meljanac and N. Okada, “Higgs masses in the minimal SUSY SO(10) GUT,” Phys. Rev. D 72, 051701 (2005) [hep-ph/0412348].
  * [43] B. Bajc, A. Melfo, G. Senjanovic and F. Vissani, “The Minimal supersymmetric grand unified theory. 1. Symmetry breaking and the particle spectrum,” Phys. Rev. D 70, 035007 (2004) [hep-ph/0402122].
  * [44] B. Bajc, A. Melfo, G. Senjanovic and F. Vissani, “Fermion mass relations in a supersymmetric SO(10) theory,” Phys. Lett. B 634, 272 (2006) [hep-ph/0511352].
  * [45] R. L. Workman et al. [Particle Data Group], “Review of Particle Physics,” PTEP 2022, 083C01 (2022)
  * [46] A. Bazavov et al. [MILC Collaboration], “MILC results for light pseudoscalars,” PoS CD 09, 007 (2009) [arXiv:0910.2966 [hep-ph]].
  * [47] S. Durr et al., “Lattice QCD at the physical point: light quark masses,” Phys. Lett. B 701, 265 (2011) [arXiv:1011.2403 [hep-lat]].
  * [48] S. Durr et al., “Lattice QCD at the physical point: Simulation and analysis details,” JHEP 1108, 148 (2011) [arXiv:1011.2711 [hep-lat]].
  * [49] C. McNeile, C. T. H. Davies, E. Follana, K. Hornbostel and G. P. Lepage, “High-Precision c and b Masses, and QCD Coupling from Current-Current Correlators in Lattice and Continuum QCD,” Phys. Rev. D 82, 034512 (2010) [arXiv:1004.4285 [hep-lat]].
  * [50] T. Blum et al. [RBC and UKQCD Collaborations], “Domain wall QCD with physical quark masses,” Phys. Rev. D 93, no. 7, 074505 (2016) [arXiv:1411.7017 [hep-lat]].
  * [51] A. Bazavov et al., “Staggered chiral perturbation theory in the two-flavor case and SU(2) analysis of the MILC data,” PoS LATTICE 2010, 083 (2010) [arXiv:1011.1792 [hep-lat]].
  * [52] S. Aoki et al., “Review of lattice results concerning low-energy particle physics,” Eur. Phys. J. C 77, no. 2, 112 (2017) [arXiv:1607.00299 [hep-lat]].
  * [53] K. G. Chetyrkin, J. H. Kuhn, A. Maier, P. Maierhofer, P. Marquard, M. Steinhauser and C. Sturm, Phys. Rev. D 80, 074010 (2009) doi:10.1103/PhysRevD.80.074010 [arXiv:0907.2110 [hep-ph]].
  * [54] A. M. Sirunyan et al. [CMS], “Measurement of $\mathrm{t\bar{t}}$ normalised multi-differential cross sections in pp collisions at $\sqrt{s}=13$ TeV, and simultaneous determination of the strong coupling strength, top quark pole mass, and parton distribution functions,” Eur. Phys. J. C 80, no.7, 658 (2020) [arXiv:1904.05237 [hep-ex]].
  * [55] J. Charles et al. [CKMfitter Group], “CP violation and the CKM matrix: Assessing the impact of the asymmetric $B$ factories,” Eur. Phys. J. C 41, no. 1, 1 (2005) [hep-ph/0406184], updated results and plots available at: http://ckmfitter.in2p3.fr
  * [56] B. A. Kniehl, A. F. Pikelner and O. L. Veretin, “mr: a C++ library for the matching and running of the Standard Model parameters,” Comput. Phys. Commun. 206, 84 (2016) [arXiv:1601.08143 [hep-ph]].
  * [57] F. Jegerlehner, M. Y. Kalmykov and O. Veretin, “MS versus pole masses of gauge bosons: Electroweak bosonic two loop corrections,” Nucl. Phys. B 641, 285 (2002) [hep-ph/0105304]; F. Jegerlehner, M. Y. Kalmykov and O. Veretin, “MS-bar versus pole masses of gauge bosons. 2. Two loop electroweak fermion corrections,” Nucl. Phys. B 658, 49 (2003) [hep-ph/0212319].
  * [58] F. Jegerlehner and M. Y. Kalmykov, “O(alpha alpha(s)) correction to the pole mass of the t quark within the standard model,” Nucl. Phys. B 676, 365 (2004) [hep-ph/0308216]; F. Jegerlehner and M. Y. Kalmykov, “O(alpha alpha(s)) relation between pole- and MS-bar mass of the t quark,” Acta Phys. Polon. B 34, 5335 (2003) [hep-ph/0310361].
  * [59] F. Bezrukov, M. Y. Kalmykov, B. A. Kniehl and M. Shaposhnikov, “Higgs Boson Mass and New Physics,” JHEP 1210, 140 (2012) [arXiv:1205.2893 [hep-ph]].
  * [60] P. Marquard, A. V. Smirnov, V. A. Smirnov and M. Steinhauser, “Quark Mass Relations to Four-Loop Order in Perturbative QCD,” Phys. Rev. Lett. 114, no. 14, 142002 (2015) [arXiv:1502.01030 [hep-ph]].
  * [61] B. A. Kniehl, A. F. Pikelner and O. L. Veretin, “Two-loop electroweak threshold corrections in the Standard Model,” Nucl. Phys. B 896, 19 (2015) [arXiv:1503.02138 [hep-ph]].
  * [62] M. E. Machacek and M. T. Vaughn, “Two Loop Renormalization Group Equations in a General Quantum Field Theory. 1. Wave Function Renormalization,” Nucl. Phys. B 222, 83-103 (1983)
  * [63] M. E. Machacek and M. T. Vaughn, “Two Loop Renormalization Group Equations in a General Quantum Field Theory. 2. Yukawa Couplings,” Nucl. Phys. B 236, 221-232 (1984)
  * [64] M. E. Machacek and M. T. Vaughn, “Two Loop Renormalization Group Equations in a General Quantum Field Theory. 3. Scalar Quartic Couplings,” Nucl. Phys. B 249, 70-92 (1985)
  * [65] I. Esteban, M. C. Gonzalez-Garcia, M. Maltoni, T. Schwetz and A. Zhou, “The fate of hints: updated global analysis of three-flavor neutrino oscillations,” JHEP 09, 178 (2020) [arXiv:2007.14792 [hep-ph]].
  * [66] NuFIT 5.1 (2021), www.nu-fit.org.
  * [67] S. Borsanyi, S. Durr, Z. Fodor, S. Krieg, A. Schafer, E. E. Scholz and K. K. Szabo, “SU(2) chiral perturbation theory low-energy constants from 2+1 flavor staggered lattice simulations,” Phys. Rev. D 88, 014513 (2013) [arXiv:1205.0788 [hep-lat]].
  * [68] Y. Aoki, T. Izubuchi, E. Shintani and A. Soni, “Improved lattice computation of proton decay matrix elements,” Phys. Rev. D 96, no.1, 014506 (2017) [arXiv:1705.01338 [hep-lat]].
  * [69] J. Hisano, H. Murayama and T. Yanagida, “Nucleon decay in the minimal supersymmetric SU(5) grand unification,” Nucl. Phys. B 402, 46-84 (1993) [arXiv:hep-ph/9207279 [hep-ph]].
  * [70] R. Matsumoto et al. [Super-Kamiokande], “Search for proton decay via $p\rightarrow\mu^{+}K^{0}$ in 0.37 megaton-years exposure of Super-Kamiokande,” Phys. Rev. D 106, no.7, 072003 (2022) [arXiv:2208.13188 [hep-ex]].
  * [71] K. Kobayashi et al. [Super-Kamiokande], “Search for nucleon decay via modes favored by supersymmetric grand unification models in Super-Kamiokande-I,” Phys. Rev. D 72, 052007 (2005) [arXiv:hep-ex/0502026 [hep-ex]].
  * [72] K. Abe et al. [Hyper-Kamiokande], “Hyper-Kamiokande Design Report,” [arXiv:1805.04163 [physics.ins-det]].