# MOLUCINATE: A Generative Model for Molecules in 3D Space

Michael Arcidiacono
<EMAIL_ADDRESS>&David Ryan Koes
Comp. & Systems Biology
University of Pittsburgh
Pittsburgh, PA 15213
<EMAIL_ADDRESS>

###### Abstract

Recent advances in machine learning have enabled generative models for both
optimization and _de novo_ generation of drug candidates with desired
properties. Previous generative models have focused on producing SMILES
strings or 2D molecular graphs, while attempts at producing molecules in 3D
have focused on reinforcement learning (RL), distance matrices, and pure atom
density grids. Here we present MOLUCINATE (MOLecUlar ConvolutIoNal generATive
modEl), a novel architecture that simultaneously generates topological and 3D
atom position information. We demonstrate the utility of this method by using
it to optimize molecules for desired radius of gyration. In the future, this
model can be used for more useful optimization such as binding affinity for a
protein target.

##  1 Introduction

Computational drug discovery aims to generate molecules with optimized
activity against protein targets. Given the success of machine learning (ML)
techniques in image and text processing, there has been interest in applying
similar techniques to _de novo_ drug design. Recent successes at generation of
SMILES strings [31, 7, 14, 9, 18] and graphs [3, 32, 27, 12] are promising,
but most do not generate 3D conformer information. Since a molecule’s binding
affinity against a protein target is dependant on its 3D conformation within
the binding site, it is desirable to generate this information as well. [30,
21, 11, 6].

Current work in this area poses the problem of 3D molecular generation as a
reinforcement learning (RL) problem [5, 26, 16, 19], which show promise for
_de novo_ molecule generation. However, RL-based methods do not share certain
advantages with generative methods that produce a latent space of molecules,
allowing interpolation between molecules and optimization of properties
directly via gradient descent. Among non-RL-based methods, there has been some
work on generating distance matrices [4, 8, 20], but this research is
preliminary and so far restricted to very small molecules and constitutional
isomers. Additionally, Samanta et al. [25] generate 3D coordinates directly
using graph convolutions, but this approach fails to encode inductive biases
about 3D geometry and thus generates unrealistic coordinates. Lastly, Ragoza
et al. [21] generate atomic density grids, which has the advantage of being
the same format as the input to convolutional networks predicting binding
affinity [30, 21, 11, 6], but requires an extra overhead for decoding the
molecule topology from the grid. In this work, we build upon this
representation to also incorporate information about the molecular graph. By
simultaneously generating the molecular graph and its 3D structure, we can
create a latent space of molecules that enables easy optimization of 3D
properties while maintaining topological similarity to the original molecule.
In the future, this model can be used for tasks such as _de novo_ drug
discovery and optimization of candidate molecules for potency against a
protein target.

In this work, we use a novel representation of 3D molecules that use per-atom
voxel grids to encode atom position information. By combining spatial and
graph convolutions into a novel architecture, we create a variational auto-
encoder (VAE) [13] to generate a latent space of 3D molecules. Additionally,
we demonstrate the usefulness of this auto-encoder by using it to optimize
molecules for radius of gyration (Rg). This is an inherently 3D property of a
molecule and can be readily and unambiguously computed for any molecule. In
the future, we hope to use this architecture to optimize molecules for more
interesting metrics (such as binding affinity against protein targets) and
generate molecules conditional on a protein binding pocket.

##  2 Methods

###  2.1 Representation of Molecules

In order to represent molecules in 3D, we store each molecule as a graph
$\mathcal{G}$ with a list of nodes $\mathcal{A}$ and edges $\mathcal{E}$. The
atoms are ordered according to the canonical SMILES representation (as
returned by RDKit [1]). Each atom $a_{i}\in\mathcal{A}$ contains information
about its type
$t_{i}\in\\{\text{C},\text{N},\text{O},\text{P},\text{S},\text{F},\text{Cl},\text{Br},\text{I},\text{END}\\}$
and valence $v_{i}\in\mathbb{N}^{3}$. The atom valences tensor specifies the
number of total bonds of each type (single, double triple) that the atom has
to other (heavy) atoms. For instance, a carbon in a benzene ring would have
its valence encoded as $[1,1,0]$. A final, virtual, atom of type END is also
appended to the graph.

To store 3D information, each atom has a position tensor with shape $(G,G,G)$,
where $G$ is the grid size. For this application, an 8 Å grid with 0.5 Å
resolution was used, so $G=16$. We used two separate density encodings for the
input and the output of the VAE. For the input molecules, the density function
is taken from Ragoza et al. [21] and is based on 3D exponential decay defined
by the Van Der Waals radius. For the output, we one-hot encode the atom
positions: we set the voxel that contains the atom center to 1 and set all
other voxels to zero. While the one-hot representation is too sparse for the
encoder’s 3D convolutions [15], it is useful for the output because it enables
us to softmax the output “densities” over all space, allowing us to interpret
the output as a 3D probability distribution.

For simplicity, we store bond types as per-atom information. Each atom $a_{i}$
has a bond type list
$b_{ij}\in\\{\text{SINGLE},\text{DOUBLE},\text{TRIPLE},\text{NO\\_BOND}\\}$
defined as the list of bond types to previous atoms $a_{j}$, where $j<i$. Each
atom can have a maximum of 4 bonds to previous atoms and lists with less than
4 elements are padded with the extra bond type NO_BOND. Note that, since
double and triple bonds are encoded as individual “bonds” with different
types, the 4 bond maximum still allows relevant hypervalent moieties such as
phosphate groups.

###  2.2 Combining Graph and Spatial Convolutions

To encode and decode the atom position grids, graph and spatial convolutions
were combined. For layer $l$ in the network, for each atom $i$, we define a
voxel hidden state $h_{i}^{(l)}$ with shape $(C_{l},G_{l},G_{l},G_{l})$, where
$C_{l}$ is the number of channels (the input atom positions have $C_{0}$ = 1)
and $G_{l}$ is the grid size for that layer. To perform the graph-spatial
convolution, the summed channels of each atom’s neighbors are concatenated
with the atom’s channels, and then a 3x3x3 spatial convolution is performed.

###  2.3 Bond prediction

To predict bonded atoms, we take inspiration from the attention mechanism. For
each atom $a_{i}$, we separately predict 4 bonds to atoms before in the range
$[0,i)$. Once the VAE has decoded a hidden state $h_{i}$, for each bond type
$b\in b_{i}$ we use a linear layer (followed by a leaky ReLU) to compute keys
$k_{i}^{b}$. We compute values $v_{i}^{b}$ in the same way. The score for atom
$a_{i}$ bonded to atom $a_{j<i}$ is simply $k_{i}^{b\top}v_{j}^{b}$. When
decoding the final molecule, a bond is created between $a_{i}$ and the top
scoring atom $a_{j}$ if $b\neq\text{NO\\_BOND}$.

###  2.4 VAE Architecture

Figure 1: Model architecture.

The overall auto-encoder architecture is shown in Figure 1. To encode a
molecule into the latent space, we first encode the atom density grids into
single per-atom vectors by alternating spatial-graph convolutional layers and
2x2x2 max pooling. This hidden state is then concatenated with the hidden
state obtained by running graph convolutions on the remaining atom features
(atom type, valences, and bonds types). After concatenation, further graph
convolutions are used to produce the final per-atom hidden state. To turn this
into a single latent code, a gated recurrent unit (GRU) [2] is used. The
latent space size was 512.

To decode, we generate new atoms autoregressively. To generate atom $a_{i}$,
we encode atoms $a_{0},...,a_{i-1}$ via an encoder with a similar architecture
to the one described above – however, all bonds to atoms $i$ or greater are
masked out of the graph convolutions to prevent cheating. This encoded
representation is concatenated with the latent code and a GRU is run to
produce a new hidden representation. This is then fed into a multi-headed
attention module [29] with 8 heads. To run graph convolutions, we need bonding
information, so the bond predictor is immediately run from this hidden state.
Using the predicted bonds, we run graph convolutions to produce the atom
types, atom valences, and bond types. We additionally use spatial-graph
convolutions to produce the predicted atom positions. To generate new
molecules, we iteratively run the decoder to produce atoms one by one until an
atom is reached with type END.

Additionally, we desire to predict Rg from the latent code. This is
accomplished with a simple two-layer multi-layer perceptron (MLP). This
property predictor is trained concurrently with the auto-encoder.

###  2.5 Training

The VAE was trained separately on the QM9 [24, 22] and ZINC 15 datasets [28,
10]. The datasets were filtered down to molecules with 16 or fewer heavy atoms
that fit into an 8 Å grid (60,888 files for ZINC, 121,836 for QM9). The QM9
dataset was divided according to a 80:10:10 train:validation:test split, while
ZINC was divided with a 90:10 train:validation split.111Due to an error early
in training, a proper test set for ZINC was not used. No such mistake was made
for QM9. During training, molecules were loaded, kekulized, rotated randomly,
and then fed into the VAE. The model was trained to simultaneously reconstruct
the input and predict Rg. To do this, we utilized seven loss functions: the
cross-entropy for the atom types, atom valences, atom positions, bond types,
and bonded atoms, mean-squared-error (MSE) for Rg, and Kullback–Leibler (KL)
divergence of the latent code to a multivariate normal distribution. The loss
weights were, respectively, 1.0, 1.0, 0.6, 2.0, 2.0, 1.0, and 0.1. The VAE was
optimized using AdamW [17] with a learning rate of 0.001 and batch size of 64
for 168 epochs.

##  3 Results

The model was evaluated according to its ability to reconstruct, generate, and
optimize molecules. To test reconstruction accuracy, molecules were sampled
from the test dataset, encoded, and then decoded. The topological accuracy is
the fraction of molecules whose decoded molecular graphs were exactly the same
as the input molecules. Of the molecules that were correctly decoded, the RMSD
of the conformers was also measured. On the QM9 test set, our model achieved a
reconstruction accuracy of 94.7% and the mean RMSD was 0.15 Å. On the ZINC
validation set, the Reconstruction accuracy was 50.6% and the mean RMSD was
0.62 Å.

To test generation, 16,000 latent points were randomly sampled from a
multivariate normal distribution. The topological validity was defined to be
the fraction of generated molecules that did not throw an exception when using
RDKit’s `SanitizeMol`. Additionally, the stability of the generated conformers
was tested by running Unified Force Field (UFF) optimization [23]. The
geometric validity was the fraction of topologically valid molecules that did
not throw an exception when running UFF optimization. The RMSD between the
(aligned) optimized conformer and the generated conformer was also measured
for each geometrically valid molecule. Lastly, the distribution of computed
properties of the generated molecules was compared to the distribution of the
filtered original dataset. Of the molecules generated from the QM9 model,
84.0% were topologically valid. Of those, 75.0% were geometrically valid. Of
the geometrically valid molecules, the mean UFF RMSD was 1.23 Å. For the ZINC
model, the topological validity was 60.3%, the geometric validity was 75.0%,
and the UFF RMSD was 1.05 Å.

To test optimization, molecules were chosen from the ZINC validation set and
encoded into latent vectors. Rg was predicted using the prediction network on
the latent vector, and the derivative of the predicted radius with respect to
the latent vector was computed. Gradient descent was then run on the latent
vector for 500 iterations (learning rate 0.2) to minimize the radius. The
results of Rg minimization are shown in Figure 2. After optimizing 1,000
molecules, 75.0% of the resulting molecules were valid, 99.5% of the valid
molecules had reduced Rg, and the average difference in Rg was -0.53 Å.
Qualitatively, the results are consistent with the algorithm “shrinking” the
molecules while preserving overall shape and features.

Figure 2: Results of minimizing Rg of molecules sampled from the ZINC
validation set. The top row shows the original molecules, and the bottom row
shows the molecules after optimization. The optimized molecules are shown
after UFF relaxation; the raw output of the model is overlaid transparently.

##  4 Conclusions

By using a novel graph representation of molecules that incorporates voxelized
atom position information, we have created a generative model that allows
encoding, decoding, interpolation, and optimization of both the geometric and
topological properties of molecules. Our model produces topologically valid
molecules 60.3% of the time on the ZINC validation set, and the UFF RMSD of
1.0 Å demonstrates that the molecules produced are often close to low energy
conformations. However, the fact that the UFF optimization routine failed to
converge for a quarter of valid molecules implies that the actual mean RMSD to
a low-energy structure might be larger. More tweaking of the UFF optimization
parameters needs to be done so that it does not fail so often.

We have also demonstrated that this model can optimize molecules for
characteristics defined by the molecule’s 3D geometry – here, Rg. While this
is admittedly a toy problem, the optimization serves as a useful proof of
concept. In the future, we hope to use this model to optimize molecules for
more useful characteristics such as binding affinity for protein targets.

Perhaps the biggest current limitation of the model was the small size of
molecules used: the model was only trained and tested on molecules with 16 or
fewer heavy atoms that fit into an 8 Å cube. These parameters were chosen for
ease of iteration; since the model run time scales as $O(ag^{3})$, where $a$
is the number of atoms and $g$ is the grid size, supporting larger molecules
takes more time to train. Now that we have narrowed in on an architecture that
works for these small molecules, we hope to scale up the model to support
larger and more useful molecules.

The full source code for training and evaluating this model is available at
https://github.com/mixarcid/molucinate.

##  5 Acknowledgements

The authors thank Dr. Alexander Tropsha, Matthew Ragoza, and Jack Lynch for
their insightful comments while discussing this work. Additionally, this work
is supported by R01GM108340 from the National Institute of General Medical
Sciences.

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Neutrino Mass in Non-Supersymmetric $SO(10)$ GUT

Naoyuki Habaa, Yasuhiro Shimizua 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

We study a prediction on neutrino observables in a non-supersymmetric
renormalizable $SO(10)$ GUT model that contains a 10 complex scalar field and
a 126 scalar field whose Yukawa couplings with 16 matter fields provide the
quark and charged lepton Yukawa couplings, neutrino Dirac Yukawa coupling and
Majorana mass for the singlet neutrinos. The $SO(10)$ breaking is achieved in
two steps by a ${\cal O}(10^{15})$ GeV VEV of a 54 real scalar field and a
${\cal O}(10^{14})$ GeV VEV of the 126 field. First, we analyze the gauge
coupling unification conditions and determine the VEV of the 126 field. Next,
we constrain the Yukawa couplings of the ${\bf 10}$ and ${\bf 126}$ fields at
the scale of the 126 field’s VEV from experimental data on quark and charged
lepton masses and quark flavor mixings. Then we express the active neutrino
mass with the above Yukawa couplings and the 126 field’s VEV based on the
Type-1 seesaw mechanism, and fit neutrino oscillation data, thereby deriving a
prediction on poorly or not measured neutrino observables. What distinguishes
our work from previous studies is that we do not assign Peccei-Quinn charges
on visible sector fields so that the 10 scalar field and its complex conjugate
both have Yukawa couplings with 16 matter fields. From the fitting of neutrino
oscillation data, we find that not only the normal neutrino mass hierarchy,
but also the inverted hierarchy can be realized. We also reveal that in the
normal hierarchy case, the Dirac CP phase of the neutrino mixing matrix
$\delta_{CP}$ is likely in the ranges of $-2.4<\delta_{\rm CP}<-1.2$ and
$1.2<\delta_{\rm CP}<2.4$, and not in the region with $\delta_{\rm
CP}\sim\pi$, and that in the normal hierarchy case, $\theta_{23}$ is likely in
the upper octant and in the range of $0.50\lesssim\sin^{2}\theta_{23}\lesssim
0.55$.

##  1 Introduction

The $SO(10)$ grand unified theory (GUT) [1, 2] is a viable candidate for
physics beyond the Standard Model (SM), because it accounts for the origin of
the hypercharge quantization, and it is automatically equipped with the seesaw
mechanism that naturally explains the tiny neutrino mass [3, 4, 5, 6]. There
are four classes of $SO(10)$ GUT models which are either supersymmetric or
non-supersymmetric and either renormalizable or non-renormalizable. In
supersymmetric models, the gauge coupling unification is achieved without
intermediate scale, whereas in non-supersymmetric models, one or more
intermediate scales are necessary for the successful unification. In
renormalizable models [7, 8], one introduces ${\bf 10}$ and ${\bf
126}+{\bf\overline{126}}$ (and optionally ${\bf 120}$) representation fields
from which the electroweak-symmetry-breaking Higgs field originates, and the
renormalizable couplings of ${\bf 10}$ and ${\bf\overline{126}}$ with ${\bf
16}$ matter fields give rise to realistic SM Yukawa couplings and neutrino
Dirac Yukawa coupling. Additionally, the renormalizable coupling of
${\bf\overline{126}}$ and its vacuum expectation value (VEV) generate Majorana
mass for the singlet neutrinos. In non-renormalizable models, one introduces
${\bf 10}$ and ${\bf 16}+{\bf\overline{16}}$ fields (let us denote the latter
by ${\bf 16}_{H}+{\bf\overline{16}}_{H}$ to avoid confusion), and the
renormalizable coupling of ${\bf 10}$ with ${\bf 16}$ matter fields and the
non-renormalizable couplings of two ${\bf\overline{16}}_{H}$’s with ${\bf 16}$
matter fields, combined with the VEV of ${\bf\overline{16}}_{H}$, generate
realistic SM Yukawa couplings, neutrino Dirac Yukawa coupling and Majorana
mass for the singlet neutrinos. The renormalizable models are attractive
because the flavor structures of the neutrino Dirac Yukawa coupling and
Majorana mass term can be constrained from experimental data on quark and
charged lepton masses and quark flavor mixings, so that one can make a
restrictive prediction on the neutrino mass and mixings. However, the
supersymmetric renormalizable models confront a serious trouble that the
$SO(10)$ gauge coupling becomes non-perturbative near the unification scale
because the Dynkin index of ${\bf 126}$ representation is large and both
scalar and fermionic components contribute to the renormalization group (RG)
evolution of the gauge coupling. For the above reasons, the non-supersymmetric
renormalizable models are worth for scrutiny.

In this paper, we study a non-supersymmetric renormalizable $SO(10)$ GUT model
with emphasis on its prediction on the neutrino mass and mixings.
Specifically, we consider a non-supersymmetric $SO(10)$ GUT model containing a
${\bf 10}$ complex scalar field and a ${\bf 126}$ scalar field that couple
with 16 matter fields. The GUT breaking is achieved by a high-scale (${\cal
O}(10^{15})$ GeV) VEV of a ${\bf 54}$ real scalar field and an intermediate-
scale (${\cal O}(10^{14})$ GeV) VEV of the ${\bf 126}$ field [9]. Our analysis
proceeds as follows: First, we analyze the gauge coupling unification
conditions and evaluate the intermediate-scale VEV. Next, we constrain the
Yukawa couplings of the ${\bf 10}$ and ${\bf 126}^{*}$ fields at the
intermediate scale from experimental data on the quark and charged lepton
masses and quark flavor mixings. Then we express the neutrino Dirac Yukawa
coupling and Majorana mass for the singlet neutrinos with these Yukawa
couplings and the intermediate-scale VEV and calculate the neutrino mass
matrix based on the Type-1 seesaw model. Finally, we fit experimental data on
the neutrino mixing angles and mass differences, and make a prediction on
poorly or not measured neutrino observables.

Previously, non-supersymmetric renormalizable $SO(10)$ GUT models and their
implication on neutrino physics have been studied in a number of papers
[9]-[29]. Our work differs from them in that we consider a more general model
where the ${\bf 10}$ complex scalar field and its complex conjugate both have
Yukawa couplings with 16 matter fields, as Peccei-Quinn charges [30] are not
assigned to these fields. As a result, the fitting of neutrino data becomes
easier and we obtain an interesting finding that the $SO(10)$ GUT model can be
consistent with not only the normal hierarchy but also the inverted hierarchy
of the neutrino mass 111 There is a recent report [29] that a model with a
${\bf 10}$ and a ${\bf 120}$ real scalar fields and a ${\bf 126}$ complex
scalar field can also fit the inverted neutrino mass hierarchy. . Another
feature of our study is that we include information on the VEV of the 126
field in the fitting of neutrino data, which allows us to determine the
portions of the SM Higgs field in the ${\bf 126}$ field ($c_{3},c_{4}$ defined
in Eq. (8)).

This paper is organized as follows: In Section 2, we describe the non-
supersymmetric renormalizable $SO(10)$ GUT model we consider. In Section 3, we
analyze the gauge coupling unification conditions and evaluate the
intermediate-scale VEV of the ${\bf 126}$ scalar field. In Section 4, we
constrain the Yukawa couplings of the ${\bf 10}$ and ${\bf 126}$ scalar fields
at the intermediate scale from experimental data on the quark and charged
lepton masses and quark flavor mixings, express the neutrino mass matrix with
these Yukawa couplings and the intermediate-scale VEV, and fit neutrino
oscillation data to derive a prediction on neutrino observables. In Section 5,
we inspect the validity of approximations made in Sections 3,4. Section 6
summarizes the paper.

##  2 Non-supersymmetric Renormalizable $SO(10)$ GUT Model

The model is a non-supersymmetric $SO(10)$ gauge theory with the following
field content: Three generations of left-handed Weyl spinors in 16
representation, denoted by ${\bf 16}^{i}$ with $i$ the flavor index; a complex
scalar field in 10, denoted by ${\bf 10}_{H}$; a complex scalar field in 126,
denoted by ${\bf 126}_{H}$; a real scalar field in 54, denoted by ${\bf
54}_{H}$. The Yukawa couplings are given by

$\displaystyle-{\cal L}_{\rm Yukawa}=(Y_{10})_{ij}\ {\bf 16}^{i}\ {\bf
10}_{H}\ {\bf 16}^{j}+(Z_{10})_{ij}\ {\bf 16}^{i}\ {\bf 10}_{H}^{*}\ {\bf
16}^{j}+(Y_{126})_{ij}\ {\bf 16}^{i}\ {\bf 126}_{H}^{*}\ {\bf 16}^{j}+{\rm
H.c.}$ (1)

The gauge symmetry breaking proceeds as follows: The component of ${\bf
54}_{H}$ with charge $({\bf 1},{\bf 1},{\bf 1})$ in the $SU(4)\times
SU(2)_{L}\times SU(2)_{R}$ subgroup develops a VEV and breaks $SO(10)$. We
write the $SO(10)$-breaking scale as $\mu=\mu_{\rm GUT}$. The effective theory
below scale $\mu=\mu_{\rm GUT}$ is a Pati-Salam model with $SU(4)\times
SU(2)_{L}\times SU(2)_{R}$ gauge group [31]. We assume that the $({\bf 1},{\bf
2},{\bf 2})$ component of ${\bf 10}_{H}$ and the $({\bf\overline{10}},{\bf
1},{\bf 3})+({\bf 10},{\bf 3},{\bf 1})+({\bf 15},{\bf 2},{\bf 2})$ components
of ${\bf 126}_{H}$ have mass much smaller than $\mu_{\rm GUT}$ and remain in
the effective Pati-Salam model. The $({\bf\overline{10}},{\bf 1},{\bf 3})$ and
$({\bf 10},{\bf 3},{\bf 1})$ components have the same mass as a consequence of
$D$-parity [32, 33]. We write the fields of these components as $({\bf 1},{\bf
2},{\bf 2})_{H},({\bf\overline{10}},{\bf 1},{\bf 3})_{H},({\bf 10},{\bf
3},{\bf 1})_{H},({\bf 15},{\bf 2},{\bf 2})_{H}$, respectively, and write the
fields of the $({\bf 4},{\bf 2},{\bf 1})+({\bf\overline{4}},{\bf 1},{\bf 2})$
components of ${\bf 16}^{i}$ fermions as $({\bf 4},{\bf 2},{\bf
1})^{i}+({\bf\overline{4}},{\bf 1},{\bf 2})^{i}$. The effective Pati-Salam
model contains the following Yukawa couplings:

$\displaystyle-{\cal L}_{\rm Yukawa}$ $\displaystyle\supset(Y_{1})_{ij}\ ({\bf
4},{\bf 2},{\bf 1})^{i}({\bf 1},{\bf 2},{\bf 2})_{H}({\bf\overline{4}},{\bf
1},{\bf 2})^{j}$ $\displaystyle+(Z_{1})_{ij}\ ({\bf 4},{\bf 2},{\bf
1})^{i}({\bf 1},{\bf 2},{\bf 2})_{H}^{*}({\bf\overline{4}},{\bf 1},{\bf
2})^{j}$ $\displaystyle+(Y_{15})_{ij}\ ({\bf 4},{\bf 2},{\bf 1})^{i}({\bf
15},{\bf 2},{\bf 2})_{H}^{*}({\bf\overline{4}},{\bf 1},{\bf 2})^{j}$
$\displaystyle+\frac{1}{2}(Y_{N})_{ij}\ ({\bf\overline{4}},{\bf 1},{\bf
2})^{i}({\bf\overline{10}},{\bf 1},{\bf 3})_{H}^{*}({\bf\overline{4}},{\bf
1},{\bf 2})^{j}+\frac{1}{2}(Y_{N})_{ij}\ ({\bf 4},{\bf 2},{\bf 1})^{i}({\bf
10},{\bf 3},{\bf 1})_{H}^{*}({\bf 4},{\bf 2},{\bf 1})^{j}$ $\displaystyle+{\rm
H.c.},$ (2)

where the Yukawa couplings satisfy the following matching conditions at scale
$\mu=\mu_{\rm GUT}$ at tree level [36]:

$\displaystyle Y_{1}$ $\displaystyle=-2\sqrt{2}\,Y_{10},$ (3) $\displaystyle
Z_{1}$ $\displaystyle=-2\sqrt{2}\,Z_{10},$ (4) $\displaystyle Y_{15}$
$\displaystyle=8\sqrt{2}\,Y_{126},$ (5) $\displaystyle Y_{N}$
$\displaystyle=8\,Y_{126}.$ (6)

Next, the component of the $({\bf\overline{10}},{\bf 1},{\bf 3})_{H}$ field
with charge $({\bf 1},{\bf 1},0)$ in the $SU(3)_{C}\times SU(2)_{L}\times
U(1)_{Y}$ subgroup develops a VEV, $v_{\rm PS}$, as

$\displaystyle\langle({\bf\overline{10}},{\bf 1},{\bf 3})_{H}\rangle=v_{\rm
PS}$ (7)

and breaks $SU(4)\times SU(2)_{R}$. We write the $SU(4)\times
SU(2)_{R}$-breaking scale as $\mu=\mu_{\rm PS}$. The effective theory below
scale $\mu=\mu_{\rm PS}$ is the SM with $SU(3)_{C}\times SU(2)_{L}\times
U(1)_{Y}$ gauge group. The pair of $({\bf 1},{\bf 2},\pm\frac{1}{2})$
components in the $({\bf 1},{\bf 2},{\bf 2})_{H}$ field, denoted by
$H_{u},H_{d}$, and the pair of $({\bf 1},{\bf 2},\pm\frac{1}{2})$ components
in the $({\bf 15},{\bf 2},{\bf 2})_{H}$ field, denoted by $\Phi_{u},\Phi_{d}$,
gain a mass matrix and the $H_{u},\epsilon
H_{d}^{*},\Phi_{u},\epsilon\Phi_{d}^{*}$ fields mix with each other
($\epsilon$ denotes the antisymmetric tensor in $SU(2)_{L}$ space). This mass
matrix is assumed to have one negative eigenvalue at the electroweak scale and
three positive eigenvalues at ${\cal O}(v_{\rm PS}^{2})$. The eigenstate
belonging to the negative eigenvalue is identified with the SM Higgs field,
denoted by $H$. We express the $H$ component of each field as

$\displaystyle H_{u}=c_{1}\,H+...,$ $\displaystyle\epsilon
H_{d}^{*}=c_{2}\,H+....,$ $\displaystyle\Phi_{u}=c_{3}\,H+...,$
$\displaystyle\epsilon\Phi_{d}^{*}=c_{4}\,H+....,$ (8)

where $c_{1},c_{2},c_{3},c_{4}$ are numbers satisfying
$|c_{1}|^{2}+|c_{2}|^{2}+|c_{3}|^{2}+|c_{4}|^{2}=1$, and “…” is an
abbreviation for other mass eigenstates. The $({\bf 4},{\bf 2},{\bf 1})^{i}$
and $({\bf\overline{4}},{\bf 1},{\bf 2})^{i}$ fermions are decomposed into the
isospin-doublet quarks, isospin-singlet up-type quarks, isospin-singlet down-
type quarks, isospin-doublet leptons, isospin-singlet charged leptons and
singlet neutrinos, denoted by $q^{i}$, $u^{c\,i}$, $d^{c\,i}$, $\ell^{i}$,
$e^{c\,i}$, $\nu^{c\,i}$, respectively. This effective theory contains the
following Yukawa couplings:

$\displaystyle-{\cal L}_{\rm Yukawa}\supset(Y_{u})_{ij}\ q^{i}\epsilon
Hu^{c\,j}+(Y_{d})_{ij}\ q^{i}H^{*}d^{c\,j}+(Y_{e})_{ij}\
\ell^{i}H^{*}e^{c\,j}+(Y_{D})_{ij}\ \ell^{i}\epsilon H\nu^{c\,j}+{\rm H.c.},$
(9)

where the Yukawa couplings satisfy the following matching conditions at scale
$\mu=\mu_{\rm PS}$ at tree level:

$\displaystyle Y_{u}$
$\displaystyle=c_{1}\,Y_{1}-c_{2}\,Z_{1}-\frac{1}{2\sqrt{3}}c_{4}\,Y_{15},$
(10) $\displaystyle Y_{d}$
$\displaystyle=-c_{2}^{*}\,Y_{1}+c_{1}^{*}\,Z_{1}+\frac{1}{2\sqrt{3}}c_{3}^{*}\,Y_{15},$
(11) $\displaystyle Y_{e}$
$\displaystyle=-c_{2}^{*}\,Y_{1}+c_{1}^{*}\,Z_{1}-\frac{\sqrt{3}}{2}c_{3}^{*}\,Y_{15},$
(12) $\displaystyle Y_{D}$
$\displaystyle=c_{1}\,Y_{1}-c_{2}\,Z_{1}+\frac{\sqrt{3}}{2}c_{4}\,Y_{15}.$
(13)

The singlet neutrinos gain Majorana mass term below,

$\displaystyle-{\cal L}_{\rm Yukawa}\supset\frac{1}{2\sqrt{2}}v_{\rm
PS}(Y_{N})_{ij}\,\nu^{c\,i}\nu^{c\,j}+{\rm H.c.},$ (14)

and are integrated out at scale $\mu=\mu_{\rm PS}$. Then the Weinberg operator
is derived as

$\displaystyle-{\cal L}_{\rm eff}=\frac{1}{2}(C_{\nu})_{ij}\,\ell^{i}\epsilon
H\,\ell^{j}\epsilon H+{\rm H.c.},$ (15)

where $C_{\nu}$ satisfies at scale $\mu=\mu_{\rm PS}$ at tree level,

$\displaystyle C_{\nu}$ $\displaystyle=-\frac{\sqrt{2}}{v_{\rm
PS}}Y_{D}Y_{N}^{-1}Y_{D}^{T}.$ (16)

We comment that in the present model, the $H_{u},\epsilon H_{d}^{*}$ fields
and the $\Phi_{u},\epsilon\Phi_{d}^{*}$ fields are allowed to have mixing
terms because quartic terms in the $SO(10)$ gauge theory below,

$\displaystyle\lambda\,{\bf 126}_{H}^{*}{\bf 126}_{H}^{2}{\bf
10}_{H}+\lambda^{\prime}\,{\bf 126}_{H}{\bf 126}_{H}^{*\,2}{\bf 10}_{H}+{\rm
H.c.},$ (17)

give rise to quartic terms in the effective Pati-Salam model below,

$\displaystyle\tilde{\lambda}\,({\bf\overline{10}},{\bf 1},{\bf
3})_{H}^{*}({\bf\overline{10}},{\bf 1},{\bf 3})_{H}({\bf 15},{\bf 2},{\bf
2})_{H}({\bf 1},{\bf 2},{\bf
2})_{H}+\tilde{\lambda}^{\prime}\,({\bf\overline{10}},{\bf 1},{\bf
3})_{H}^{*}({\bf\overline{10}},{\bf 1},{\bf 3})_{H}({\bf 15},{\bf 2},{\bf
2})_{H}^{*}({\bf 1},{\bf 2},{\bf 2})_{H}+{\rm H.c.}$ (18)

When the $({\bf 1},{\bf 1},0)$ component of the $({\bf\overline{10}},{\bf
1},{\bf 3})_{H}$ field develops the VEV $v_{\rm PS}$, Eq. (18) yields mixing
terms for the $H_{u},\epsilon H_{d}^{*}$ fields and the
$\Phi_{u},\epsilon\Phi_{d}^{*}$ fields. Remarkably, these mixing terms are
${\cal O}(v_{\rm PS}^{2})$ and of the same order as the mass terms of the
$({\bf 1},{\bf 2},{\bf 2})_{H}$ and $({\bf 15},{\bf 2},{\bf 2})_{H}$ fields.
As a result, large mixings of $H_{u},\epsilon H_{d}^{*}$ and
$\Phi_{u},\epsilon\Phi_{d}^{*}$ can be realized.

We comment on the strong CP problem and dark matter in the present model.
Unlike previous models of non-supersymmetric $SO(10)$ GUT, we do not assign
$U(1)$ Peccei-Quinn charges [30] to the ${\bf 16}^{i}$ matter fields and the
${\bf 10}_{H},{\bf 126}_{H}$ scalar fields. Nevertheless, we can implement the
Peccei-Quinn mechanism by introducing new ${\bf 16}$ and ${\bf\overline{16}}$
Weyl fermions and ${\bf 1}$ complex scalar and assigning $U(1)$ Peccei-Quinn
charge $+1$ to ${\bf 16},{\bf\overline{16}}$ and $-2$ to ${\bf 1}$. Then a
KSVZ axion [37, 38] emerges and can solve the strong CP problem. It can also
be a dark matter candidate.

##  3 Gauge Coupling Unification

In this section and the next section, we adopt the following experimental
values of the gauge coupling constants, quark and charged lepton masses and
quark flavor mixings: The QCD and QED gauge coupling constants in 5-quark-
flavor QCD$\times$QED theory are fixed as $\alpha_{s}^{(5)}(M_{Z})=0.1181$ and
$\alpha^{(5)}(M_{Z})=1/127.95$. The lepton pole masses and $W$, $Z$, Higgs
boson pole masses are taken from Particle Data Group [39]. We use the results
of lattice calculations of the individual up and down quark masses, the
strange quark mass, the charm quark mass and the bottom quark mass in
$\overline{{\rm MS}}$ scheme reviewed in Ref. [40], which read $m_{u}(2~{}{\rm
GeV})=2.14(8)~{}{\rm MeV}$, $m_{d}(2~{}{\rm GeV})=4.70(5)~{}{\rm MeV}$ [41,
42], $m_{s}(2~{}{\rm GeV})=93.40(57)~{}{\rm MeV}$ [41, 43, 44, 45],
$m_{c}(3~{}{\rm GeV})=0.988(11)~{}{\rm GeV}$ [41, 43, 45, 46, 47],
$m_{b}(m_{b})=4.203(11)~{}{\rm GeV}$ [41, 45, 48, 49, 50, 51]. We use the top
quark pole mass measured by CMS in Ref. [52], which reads $M_{t}=170.5(8)$
GeV. We calculate the CKM mixing angles and CP phase from the Wolfenstein
parameters in Ref. [53]. The above data are translated into the values of the
quark and lepton Yukawa coupling matrices and the gauge coupling constants at
scale $\mu=M_{Z}$ in $\overline{{\rm MS}}$ scheme with the help of the code
[54] based on Refs. [55]-[59].

We analyze the gauge coupling unification conditions and evaluate the
$SU(4)\times SU(2)_{R}$-breaking VEV $v_{\rm PS}$. To this end, we solve the
two-loop RG equations [60]-[62] of SM, and match the theory with the
$SU(4)\times SU(2)_{L}\times SU(2)_{R}$ gauge theory containing three
generations of Weyl fermions $({\bf 4},{\bf 2},{\bf
1})^{i},({\bf\overline{4}},{\bf 1},{\bf 2})^{i}$ and complex scalars $({\bf
1},{\bf 2},{\bf 2})_{H},({\bf\overline{10}},{\bf 1},{\bf 3})_{H},({\bf
10},{\bf 3},{\bf 1})_{H},({\bf 15},{\bf 2},{\bf 2})_{H}$ at scale
$\mu=\mu_{\rm PS}$. Then we calculate the two-loop RG equations of the
$SU(4)\times SU(2)_{L}\times SU(2)_{R}$ gauge theory, and match the theory
with the $SO(10)$ gauge theory at scale $\mu=\mu_{\rm GUT}$. From the above
matching conditions, we evaluate $v_{\rm PS}$. Additionally, we evaluate the
mass of the $SO(10)$ gauge boson that gains mass along the breaking of
$SO(10)$ to $SU(4)\times SU(2)_{L}\times SU(2)_{R}$.

We make two approximations: First, we approximate that the scalar particles
decoupled at scale $\mu=\mu_{\rm PS}$ have a common mass $M_{H_{\rm PS}}$.
This has little impact on the evaluation of $v_{\rm PS}$ because the power of
$M_{H_{\rm PS}}$ in the equation determining $v_{\rm PS}$ Eq. (26) is
relatively small. Second, when solving the two-loop RG equations of the gauge
couplings of the $SU(4)\times SU(2)_{L}\times SU(2)_{R}$ gauge theory, we omit
two-loop contributions involving the Yukawa couplings. Later in Section 5 we
will check that this approximation does not affect the result.

Given the approximation on the scalar particle masses, the matching conditions
around scale $\mu\sim\mu_{\rm PS}$ in $\overline{{\rm MS}}$ scheme are given
by

$\displaystyle\frac{1}{g_{2R}^{2}(\mu)}-\frac{2}{48\pi^{2}}$
$\displaystyle=\frac{5}{3}\left(\frac{1}{g_{1}^{2}(\mu)}-\frac{1}{8\pi^{2}}\frac{28}{5}\ln\frac{M_{G({\bf
3},{\bf
1},\frac{2}{3})}}{\mu}-\frac{1}{8\pi^{2}}\frac{21}{5}\ln\frac{M_{G({\bf
1},{\bf 1},1)}}{\mu}+\frac{1}{8\pi^{2}}\frac{67}{6}\ln\frac{M_{H_{\rm
PS}}}{\mu}\right)$
$\displaystyle-\frac{2}{3}\left(\frac{1}{g_{3}^{2}(\mu)}-\frac{1}{8\pi^{2}}\frac{7}{2}\ln\frac{M_{G({\bf
3},{\bf
1},\frac{2}{3})}}{\mu}+\frac{1}{8\pi^{2}}\frac{67}{6}\ln\frac{M_{H_{\rm
PS}}}{\mu}-\frac{3}{48\pi^{2}}\right),$ (19)
$\displaystyle\frac{1}{g_{2L}^{2}(\mu)}-\frac{2}{48\pi^{2}}$
$\displaystyle=\frac{1}{g_{2}^{2}(\mu)}+\frac{1}{8\pi^{2}}\frac{71}{6}\ln\frac{M_{H_{\rm
PS}}}{\mu}-\frac{2}{48\pi^{2}},$ (20)
$\displaystyle\frac{1}{g_{4}^{2}(\mu)}-\frac{4}{48\pi^{2}}$
$\displaystyle=\frac{1}{g_{3}^{2}(\mu)}-\frac{1}{8\pi^{2}}\frac{7}{2}\ln\frac{M_{G({\bf
3},{\bf
1},\frac{2}{3})}}{\mu}+\frac{1}{8\pi^{2}}\frac{67}{6}\ln\frac{M_{H_{\rm
PS}}}{\mu}-\frac{3}{48\pi^{2}},$ (21)

and those around scale $\mu\sim\mu_{\rm GUT}$ in $\overline{{\rm MS}}$ scheme
are given by

$\displaystyle\frac{1}{g_{10}^{2}(\mu)}-\frac{8}{48\pi^{2}}$
$\displaystyle=\frac{1}{g_{2R}^{2}(\mu)}-\frac{1}{8\pi^{2}}21\,\ln\frac{M_{G({\bf
6},{\bf 2},{\bf 2})}}{\mu}+\frac{1}{8\pi^{2}}\ln\frac{M_{H({\bf 1},{\bf
3},{\bf 3})}}{\mu}-\frac{2}{48\pi^{2}},$ (22)
$\displaystyle\frac{1}{g_{10}^{2}(\mu)}-\frac{8}{48\pi^{2}}$
$\displaystyle=\frac{1}{g_{2L}^{2}(\mu)}-\frac{1}{8\pi^{2}}21\,\ln\frac{M_{G({\bf
6},{\bf 2},{\bf 2})}}{\mu}+\frac{1}{8\pi^{2}}\ln\frac{M_{H({\bf 1},{\bf
3},{\bf 3})}}{\mu}-\frac{2}{48\pi^{2}},$ (23)
$\displaystyle\frac{1}{g_{10}^{2}(\mu)}-\frac{8}{48\pi^{2}}$
$\displaystyle=\frac{1}{g_{4}^{2}(\mu)}-\frac{1}{8\pi^{2}}14\,\ln\frac{M_{G({\bf
6},{\bf 2},{\bf 2})}}{\mu}+\frac{1}{8\pi^{2}}\frac{1}{3}\ln\frac{M_{H({\bf
6},{\bf 1},{\bf 1})}^{2}M_{H({\bf 20}^{\prime},{\bf 1},{\bf
1})}^{4}}{\mu^{6}}-\frac{4}{48\pi^{2}},$ (24)

where $g_{3},g_{2},g_{1}$ denote the gauge couplings of the $SU(3)_{C}\times
SU(2)_{L}\times U(1)_{Y}$ gauge theory ($g_{1}$ is in the GUT normalization),
$g_{4},g_{2L},g_{2R}$ denote those of the $SU(4)\times SU(2)_{L}\times
SU(2)_{R}$ gauge theory, and $g_{10}$ denotes that of the $SO(10)$ gauge
theory. $M_{G({\bf 3},{\bf 1},\frac{2}{3})}$, $M_{G({\bf 1},{\bf 1},1)}$
denote the masses of the gauge bosons that become massive along the
$SU(4)\times SU(2)_{R}$ breaking (subscripts display the charges in
$SU(3)_{C}\times SU(2)_{L}\times U(1)_{Y}$), and $M_{G({\bf 6},{\bf 2},{\bf
2})}$ denotes the mass of the gauge boson that becomes massive along the
$SO(10)$ breaking into $SU(4)\times SU(2)_{L}\times SU(2)_{R}$. $M_{H_{\rm
PS}}$ denotes the common mass of the scalar particles decoupled at scale
$\mu=\mu_{\rm PS}$, and $M_{H({\bf 6},{\bf 1},{\bf 1})}$, $M_{H({\bf
20}^{\prime},{\bf 1},{\bf 1})}$, $M_{H({\bf 1},{\bf 3},{\bf 3})}$ denote the
masses of the scalar particles decoupled at scale $\mu=\mu_{\rm GUT}$
(subscripts display the charges in $SU(4)\times SU(2)_{L}\times SU(2)_{R}$).
There are two complex scalar particles with the same charge $({\bf 6},{\bf
1},{\bf 1})$, and $M_{H({\bf 6},{\bf 1},{\bf 1})}$ should be regarded as the
geometric mean of their masses. The scalar particles with charge $({\bf
20}^{\prime},{\bf 1},{\bf 1})$ and $({\bf 1},{\bf 3},{\bf 3})$ are real.

From Eqs. (22),(23) and the particle content of the $SU(4)\times
SU(2)_{L}\times SU(2)_{R}$ gauge theory, we see that $g_{2R}=g_{2L}$ holds at
any scale, even if the Yukawa couplings are not neglected when solving the RG
equations of the gauge couplings of the $SU(4)\times SU(2)_{L}\times
SU(2)_{R}$ gauge theory. This is in accord with the fact that $D$-parity is
unbroken when the VEV of ${\bf 54}_{H}$ breaks $SO(10)$. From Eqs. (19),(20)
and the fact that $g_{2R}=g_{2L}$, we obtain the one-loop relation below,

$\displaystyle\frac{M_{G({\bf 3},{\bf 1},\frac{2}{3})}^{21}M_{G({\bf 1},{\bf
1},1)}^{21}}{\mu^{42}}\frac{M_{H_{\rm
PS}}^{2}}{\mu^{2}}=\exp\left[8\pi^{2}\left(\frac{5}{g_{1}^{2}(\mu)}-\frac{3}{g_{2}^{2}(\mu)}-\frac{2}{g_{3}^{2}(\mu)}\right)+2\right].$
(25)

We solve the two-loop RG equations of SM and insert the result into Eq. (25),
thereby obtaining

$\displaystyle M_{G({\bf 3},{\bf 1},\frac{2}{3})}^{21}M_{G({\bf 1},{\bf
1},1)}^{21}M_{H_{\rm PS}}^{2}=e^{2}(10^{13.70}\,{\rm GeV})^{44}.$ (26)

From the above relation, we evaluate $v_{\rm PS}$. We note $M_{G({\bf 3},{\bf
1},\frac{2}{3})}^{2}=g_{4}^{2}\,v_{\rm PS}^{2}$, $M_{G({\bf 1},{\bf
1},1)}^{2}=g_{2R}^{2}\,v_{\rm PS}^{2}$. The value of $M_{H_{\rm PS}}$ has
little impact on the evaluation of $v_{\rm PS}$ because of its comparably
small power of 2. If $M_{H_{\rm PS}}$ lies in a natural range of $0.3\,v_{\rm
PS}>M_{H_{\rm PS}}>0.03\,v_{\rm PS}$, we get

$\displaystyle v_{\rm PS}=10^{14.0}\,{\rm GeV}.$ (27)

From Eqs. (23),(24) and the fact that $g_{2R}=g_{2L}$, we obtain the one-loop
relation below,

$\displaystyle\frac{\mu}{M_{H({\bf 1},{\bf 3},{\bf 3})}}\frac{M_{H({\bf
6},{\bf 1},{\bf 1})}^{2/3}M_{H({\bf 20}^{\prime},{\bf 1},{\bf
1})}^{4/3}}{\mu^{2}}\frac{M_{G({\bf 6},{\bf 2},{\bf
2})}^{7}}{\mu^{7}}=\exp\left[8\pi^{2}\left(\frac{1}{g_{2L}^{2}(\mu)}-\frac{1}{g_{4}^{2}(\mu)}\right)+\frac{1}{3}\right].$
(28)

We solve the two-loop RG equations of the $SU(4)\times SU(2)_{L}\times
SU(2)_{R}$ gauge theory and insert the result into Eq. (28), thereby obtaining

$\displaystyle\frac{M_{H({\bf 6},{\bf 1},{\bf 1})}^{2/3}M_{H({\bf
20}^{\prime},{\bf 1},{\bf 1})}^{4/3}}{M_{H({\bf 1},{\bf 3},{\bf 3})}}M_{G({\bf
6},{\bf 2},{\bf 2})}^{7}=e^{1/3}(10^{15.04}~{}{\rm GeV})^{8}.$ (29)

If we assume a mild hierarchy among the scalar particle masses as

$\displaystyle M_{H({\bf 6},{\bf 1},{\bf 1})}=M_{H({\bf 20}^{\prime},{\bf
1},{\bf 1})}\simeq 10^{13.5}~{}{\rm GeV},$ (30) $\displaystyle M_{H({\bf
1},{\bf 3},{\bf 3})}\simeq 10^{16.5}~{}{\rm GeV},$ (31)

then we get $M_{G({\bf 6},{\bf 2},{\bf 2})}\simeq 6\times 10^{15}$ GeV and the
current bound on the $p\to e^{+}\pi^{0}$ partial lifetime as well as those of
other nucleon decay modes are satisfied.

##  4 Fitting of Neutrino Data

We constrain the Yukawa couplings of the $SU(4)\times SU(2)_{L}\times
SU(2)_{R}$ gauge theory $Y_{1},Z_{1},Y_{15},Y_{N}$ at scale $\mu=\mu_{\rm
PS}$, from experimental data on the quark and charged lepton masses and quark
flavor mixings. Then we express the neutrino mass matrix with
$Y_{1},Z_{1},Y_{15},Y_{N}$ and $v_{\rm PS}$ in Eq. (27) based on the Type-1
seesaw mechanism. Finally, we fit experimental data on the neutrino mixing
angles and mass differences with $Y_{1},Z_{1},Y_{15},Y_{N}$ under the above
constraints.

First, we calculate the up-type quark, down-type quark and charged lepton
Yukawa couplings in SM at scale $\mu=\mu_{\rm PS}$ by solving the SM two-loop
RG equations. We take $\mu_{\rm PS}=10^{13.7}\,{\rm GeV}$, in accordance with
Eq. (26). The result is presented in Table 1 in the form of the singular
values of the Yukawa coupling matrices and the parameters of the CKM matrix at
scale $\mu=\mu_{\rm PS}$.

Table 1: The singular values of the Yukawa coupling matrices and the CKM mixing angles and CP phase in SM at scale $\mu=\mu_{\rm PS}=10^{13.7}$ GeV. Also shown are the errors of the quark Yukawa couplings, propagated from the experimental errors of the corresponding masses, and maximal errors of the CKM parameters, obtained by assuming maximal correlation of experimental errors of the Wolfenstein parameters. | Value
---|---
$y_{u}$ | 2.98(11)$\times 10^{-6}$
$y_{c}$ | 0.001519(17)
$y_{t}$ | 0.4458(42)
$y_{d}$ | 6.729(72)$\times 10^{-6}$
$y_{s}$ | 0.00013369(82)
$y_{b}$ | 0.006402(20)
$y_{e}$ | 2.732$\times 10^{-6}$
$y_{\mu}$ | 0.0005767
$y_{\tau}$ | 0.009803
$\cos\theta_{13}^{\rm ckm}\sin\theta_{12}^{\rm ckm}$ | 0.22503(24)
$\cos\theta_{13}^{\rm ckm}\sin\theta_{23}^{\rm ckm}$ | 0.04576(77)
$\sin\theta_{13}^{\rm ckm}$ | 0.00403(22)
$\delta_{\rm km}$ (rad) | 1.148(33)

Due to $D$-parity, $Y_{1},Z_{1},Y_{15}$ at scale $\mu=\mu_{\rm PS}$ have
symmetric flavor indices. Therefore, in the flavor basis where the isospin-
doublet down-type quarks have a diagonal Yukawa coupling, the up-type quark,
down-type quark and charged lepton Yukawa coupling matrices
$Y_{u},Y_{d},Y_{e}$ at scale $\mu=\mu_{\rm PS}$ can be written as

$\displaystyle Y_{u}$ $\displaystyle=V_{\rm
CKM}^{T}\begin{pmatrix}y_{u}&0&0\\\ 0&y_{c}\,e^{2i\,d_{2}}&0\\\
0&0&y_{t}\,e^{2i\,d_{3}}\\\ \end{pmatrix}V_{\rm CKM},$ (32) $\displaystyle
Y_{d}$ $\displaystyle=\begin{pmatrix}y_{d}&0&0\\\ 0&y_{s}&0\\\ 0&0&y_{b}\\\
\end{pmatrix},$ (33) $\displaystyle Y_{e}$
$\displaystyle=U_{e}^{T}\begin{pmatrix}y_{e}&0&0\\\ 0&y_{\mu}&0\\\
0&0&y_{\tau}\\\ \end{pmatrix}U_{e},$ (34)

where $y_{u},y_{c},y_{t},y_{d},y_{s},y_{b},y_{e},y_{\mu},y_{\tau}$ are given
in Table 1, $V_{\rm CKM}$ is the CKM matrix whose parameters are given in
Table 1, $d_{2},d_{3}$ are unknown phases and $U_{e}$ is an unknown unitary
matrix. $d_{2},d_{3},U_{e}$ are not constrained experimentally. Using Eqs.
(10)-(12), we can write $Y_{1},Z_{1},Y_{15}$ at scale $\mu=\mu_{\rm PS}$ as

$\displaystyle Y_{1}$
$\displaystyle=\frac{1}{4c_{3}^{*}(|c_{1}|^{2}-|c_{2}|^{2})}\left\\{4c_{1}^{*}c_{3}^{*}Y_{u}+(c_{1}^{*}c_{4}+3c_{2}c_{3}^{*})Y_{d}+(-c_{1}^{*}c_{4}+c_{2}c_{3}^{*})Y_{e}\right\\},$
(35) $\displaystyle Z_{1}$
$\displaystyle=\frac{1}{4c_{3}^{*}(|c_{1}|^{2}-|c_{2}|^{2})}\left\\{4c_{2}^{*}c_{3}^{*}Y_{u}+(c_{2}^{*}c_{4}+3c_{1}c_{3}^{*})Y_{d}+(-c_{2}^{*}c_{4}+c_{1}c_{3}^{*})Y_{e}\right\\},$
(36) $\displaystyle Y_{15}$
$\displaystyle=\frac{\sqrt{3}}{2}\frac{1}{c_{3}^{*}}(Y_{d}-Y_{e})$ (37)

with $Y_{u},Y_{d},Y_{e}$ given by Eqs. (32)-(34). $Y_{N}$ is related to
$Y_{d}-Y_{e}$ in the following way: Eqs. (5),(6) give that
$Y_{N}=Y_{15}/\sqrt{2}$ at scale $\mu=\mu_{\rm GUT}$. Then $Y_{N}$ and
$Y_{15}$ evolve from $\mu=\mu_{\rm GUT}$ to lower energy scales through
different RG equations in the $SU(4)\times SU(2)_{L}\times SU(2)_{R}$ gauge
theory. At scale $\mu=\mu_{\rm PS}$, $Y_{15}$ is proportional to $Y_{d}-Y_{e}$
as Eq. (37). Hence, to relate $Y_{N}$ to $Y_{d}-Y_{e}$, we have to solve the
RG equations of $Y_{N}$ and $Y_{15}$ in the $SU(4)\times SU(2)_{L}\times
SU(2)_{R}$ gauge theory. Unfortunately, this is not possible because the RG
equations depend on the Yukawa couplings $Y_{1},Z_{1},Y_{15}$ that are
undetermined before the fitting analysis is finished. Therefore, we
approximate $Y_{N}=Y_{15}/\sqrt{2}$ at scale $\mu=\mu_{\rm PS}$. Later in
Section 5 we will assess the impact of this approximation after
$Y_{1},Z_{1},Y_{15}$ are determined. Given the above approximation, we can
express the coefficient of the Weinberg operator at scale $\mu=\mu_{\rm PS}$
using Eqs. (35)-(37) and Eqs. (13),(16) as

$\displaystyle
C_{\nu}=\frac{2\sqrt{2}}{\sqrt{3}}c_{3}^{*}\frac{\sqrt{2}}{v_{\rm
PS}}\left(Y_{u}+\frac{c_{4}}{c_{3}^{*}}(Y_{d}-Y_{e})\right)(Y_{d}-Y_{e})^{-1}\left(Y_{u}+\frac{c_{4}}{c_{3}^{*}}(Y_{d}-Y_{e})\right)\
\ \ \ \ {\rm at}\ \mu=\mu_{\rm PS}.$ (38)

The fitting analysis is performed as follows: First, we evaluate Eq. (38) with
the central values in Table 1 and the estimate of $v_{\rm PS}=10^{14.0}$ GeV
in Eq. (27). At this stage, phases $d_{2},d_{3}$, unitary matrix $U_{e}$ and
complex numbers $c_{3},c_{4}$ are free parameters except that the latter two
satisfy

$\displaystyle|c_{3}|^{2}+|c_{4}|^{2}<1.$ (39)

Next, we solve the 1-loop RG equation for the Wilson coefficient of the
Weinberg operator $C_{\nu}$ from scale $\mu=\mu_{\rm PS}$ to $\mu=M_{Z}$, and
evaluate the neutrino mass matrix as

$\displaystyle M_{\nu}=\frac{v^{2}}{2}C_{\nu}(M_{Z})$ (40)

with $v=246$ GeV. From $M_{\nu}$ above, we derive the three neutrino mixing
angles and the two neutrino mass differences. Finally, we fit the neutrino
oscillation data in NuFIT5.1 (with SK atmospheric data) [63, 64] with the free
parameters $d_{2},d_{3},U_{e},c_{3},c_{4}$. We consider both the normal
hierarchy and the inverted hierarchy of the neutrino mass. We perform the
fitting repeatedly and collect multiple fitting results in which two mixing
angles $\sin^{2}\theta_{12},\sin^{2}\theta_{13}$ and the ratio of the neutrino
mass differences $\Delta m^{2}_{21}/|\Delta m^{2}_{31}|$ are within the
$2\sigma$ ranges and mixing angle $\sin^{2}\theta_{23}$ is within the
$3\sigma$ range of the NuFIT5.1 data.

We plot the fitting results on the plane of $|c_{3}|$ versus $|c_{4}|$ in Fig.
1. Recall that $c_{3},c_{4}$ quantify the portions of $({\bf 1},{\bf
2},\pm\frac{1}{2})$ components of $({\bf 15},{\bf 2},{\bf 2})_{H}$ field in
the SM Higgs field as defined in Eq. (8). The left panel is for the case of
the normal neutrino mass hierarchy and the right panel is for the case of the
inverted hierarchy.

Figure 1: Fitting results on the plane of $|c_{3}|$ versus $|c_{4}|$, where
$c_{3},c_{4}$ quantify the portions of $({\bf 1},{\bf 2},\pm\frac{1}{2})$
components of $({\bf 15},{\bf 2},{\bf 2})_{H}$ field in the SM Higgs field as
defined in Eq. (8). The left panel is for the case of the normal neutrino mass
hierarchy and the right panel is for the case of the inverted hierarchy.

We see from Fig. 1 that $|c_{3}|$ is ${\cal O}(0.01)$ in both normal and
inverted hierarchy cases. $|c_{4}|$ is ${\cal O}(10)$ times larger than
$|c_{3}|$ in the normal hierarchy case, while it is on the same order or
smaller than $|c_{3}|$ in the inverted hierarchy case. This implies that from
the point of view of naturalness of the mass matrix of $({\bf 1},{\bf
2},\pm\frac{1}{2})$ components of $({\bf 15},{\bf 2},{\bf 2})_{H}$ field, the
inverted hierarchy is favored because $|c_{3}|$ and $|c_{4}|$ can be on the
same order. Also, Fig. 1 and Eq. (38) indicate that the normal hierarchy is
realized when the neutrino Dirac Yukawa coupling is dominated by the term
proportional to $Y_{d}-Y_{e}$, whereas the inverted hierarchy is realized when
the term proportional to $Y_{u}$ is dominant or comparable to that
proportional to $Y_{d}-Y_{e}$.

We examine the prediction of the model on neutrino observables, by plotting
the fitting results on the planes of neutrino mixing angle
$\sin^{2}\theta_{23}$ versus the Dirac CP phase of the neutrino mixing matrix
$\delta_{\rm CP}$, the effective neutrino mass for neutrinoless double beta
decay $|m_{ee}|$, and the neutrino mass sum $\sum_{i=1}^{3}m_{i}$ in Fig. 2.
The left-side panels are for the case of the normal neutrino mass hierarchy
and the right-side panels are for the case of the inverted hierarchy. The
reason that we focus on $\sin^{2}\theta_{23}$ is that it still has large
uncertainty and further improvement of its measurement is anticipated.

Figure 2:  Panels in the first row: Fitting results on the plane of neutrino
mixing angle $\sin^{2}\theta_{23}$ versus the Dirac CP phase of the neutrino
mixing matrix $\delta_{\rm CP}$. Panels in the second row: Those on the plane
of $\sin^{2}\theta_{23}$ versus the effective neutrino mass for neutrinoless
double beta decay $|m_{ee}|$. Panels in the third row: Those on the plane of
$\sin^{2}\theta_{23}$ versus the neutrino mass sum $\sum_{i=1}^{3}m_{i}$. The
left-side panels are for the case of the normal neutrino mass hierarchy and
the right-side panels are for the case of the inverted hierarchy.

An interesting finding in Fig. 2 is that in the normal hierarchy case, the
Dirac CP phase $\delta_{\rm CP}$ is mostly in the ranges of $-2.4<\delta_{\rm
CP}<-1.2$ and $1.2<\delta_{\rm CP}<2.4$, and it is unlikely that $\delta_{\rm
CP}\sim\pi$. This is in clear contrast with the inverted hierarchy case, where
$\delta_{\rm CP}$ is equally distributed in the whole range. Also, in the
normal hierarchy case, mixing angle $\theta_{23}$ is likely in the upper
octant and in a narrow range of $0.50\lesssim\sin^{2}\theta_{23}\lesssim
0.55$. The predictions on the effective neutrino mass for neutrinoless double
beta decay $|m_{ee}|$ and the neutrino mass sum $\sum_{i=1}^{3}m_{i}$ are less
interesting, since these predictions simply correspond to the situation with
small lightest neutrino mass ($m_{1}\ll\Delta m_{21}^{2}$ in the normal
hierarchy case and $m_{3}\ll\Delta m_{21}^{2}$ in the inverted hierarchy
case).

We further study the correlations between the predictions, by plotting the
fitting results on the planes of $\delta_{\rm CP}$ versus $|m_{ee}|$;
$\delta_{\rm CP}$ versus $\sum_{i=1}^{3}m_{i}$; $|m_{ee}|$ versus
$\sum_{i=1}^{3}m_{i}$ in Fig. 3 in Appendix. Unfortunately, we do not observe
clear correlations among the predictions on $\delta_{\rm CP}$, $|m_{ee}|$ and
$\sum_{i=1}^{3}m_{i}$.

##  5 Validity of the Approximations

We have made two approximations: First, in Section 3, we have neglected the
contribution of the Yukawa couplings to the two-loop RG equations of the gauge
couplings in the $SU(4)\times SU(2)_{L}\times SU(2)_{R}$ gauge theory. Second,
in Section 4, we have neglected the RG evolutions of Yukawa couplings
$Y_{15},Y_{N}$ in the $SU(4)\times SU(2)_{L}\times SU(2)_{R}$ gauge theory
from scale $\mu=\mu_{\rm GUT}$ to $\mu=\mu_{\rm PS}$.

Now we inspect the validity of the above approximations. For this purpose, we
solve the full RG equations of the $SU(4)\times SU(2)_{L}\times SU(2)_{R}$
gauge theory including the Yukawa couplings $Y_{1},Z_{1},Y_{15},Y_{N}$. Here
the initial conditions of $Y_{1},Z_{1},Y_{15}$ at scale $\mu=\mu_{\rm PS}$ are
given by Eqs. (35)-(37),(32)-(34) with the fitting results of Section 4
inserted into $d_{2},d_{3},U_{e},c_{3},c_{4}$. Parameters $c_{1},c_{2}$ have
not been determined in the analysis of Section 4 and so we take
$c_{1}=\sqrt{1-|c_{3}|^{2}-|c_{4}|^{2}}$ and $c_{2}=0$ to minimize the
magnitudes of $Y_{1},Z_{1}$. We approximate $Y_{N}=Y_{15}/\sqrt{2}$ at scale
$\mu=\mu_{\rm PS}$ (which should hold exactly at scale $\mu=\mu_{\rm GUT}$,
not at $\mu=\mu_{\rm PS}$) and study the relation between $Y_{N}$ and
$Y_{15}/\sqrt{2}$ at scale $\mu=\mu_{\rm GUT}$ under the above approximation.

By solving the full RG equations of the $SU(4)\times SU(2)_{L}\times
SU(2)_{R}$ gauge theory, we find that the estimate of $10^{15.04}$ GeV in Eq.
(29) is valid even if the two-loop contributions of the Yukawa couplings are
included in the gauge coupling running. We also find that the components of
$Y_{15}/\sqrt{2}$ and $Y_{N}$ at scale $\mu=\mu_{\rm GUT}$ differ by at most
0.1%, which implies that the approximation of taking $Y_{N}=Y_{15}/\sqrt{2}$
at scale $\mu=\mu_{\rm PS}$ (instead of at scale $\mu=\mu_{\rm GUT}$) has
negligible impact compared to errors of the NuFIT5.1 data used in the fitting
analysis.

##  6 Summary

We have studied a prediction on neutrino observables in a non-supersymmetric
renormalizable $SO(10)$ GUT model that contains a 10 complex scalar field and
a 126 scalar field. The 10 field and its complex conjugate and the complex
conjugate of the 126 field have Yukawa couplings with the 16 matter fields
$Y_{10},Z_{10},Y_{126}$, which give rise to the SM Yukawa couplings and
neutrino Dirac Yukawa coupling. The $SO(10)$ is broken into $SU(4)\times
SU(2)_{L}\times SU(2)_{R}$ by the VEV of a 54 real scalar field, and it is
broken into the SM gauge groups by the VEV of the 126 field. The latter VEV
and $Y_{126}$ generate Majorana mass for the singlet neutrinos. For the above
model, we have determined the VEV of the 126 field from the gauge coupling
unification conditions. We have constrained the Yukawa couplings of the ${\bf
10}$ and ${\bf 126}$ fields at the scale of the 126 VEV from experimental data
on the quark and charged lepton masses and quark flavor mixings, expressed the
neutrino mass matrix with the above Yukawa couplings and the 126 VEV based on
the Type-1 seesaw mechanism, fit the neutrino oscillation data, and derived a
prediction on neutrino observables.

We have found that both the normal hierarchy and the inverted hierarchy of the
neutrino mass can be fit and that the inverted hierarchy is favored from the
point of view of naturalness of the mass matrix of $({\bf 1},{\bf
2},\pm\frac{1}{2})$ components of $({\bf 15},{\bf 2},{\bf 2})_{H}$ field.
Also, in the normal hierarchy case, the Dirac CP phase of the neutrino mixing
matrix $\delta_{CP}$ is predicted to be likely in the ranges of
$-2.4<\delta_{\rm CP}<-1.2$ and $1.2<\delta_{\rm CP}<2.4$, and not in the
range of $\delta_{\rm CP}\sim\pi$. In the normal hierarchy case, mixing angle
$\theta_{23}$ is predicted to be likely in the upper octant and in a narrow
range of $0.50\lesssim\sin^{2}\theta_{23}\lesssim 0.55$.

## Acknowledgement

This work is partially supported by Scientific Grants by the Ministry of
Education, Culture, Sports, Science and Technology of Japan Nos. 21H000761
(NH) and No. 19K147101 (TY).

## Appendix

To study the correlations between the predictions on neutrino observables, we
plot the fitting results of Section 4 on the planes of $\delta_{\rm CP}$
versus $|m_{ee}|$; $\delta_{\rm CP}$ versus $\sum_{i=1}^{3}m_{i}$; $|m_{ee}|$
versus $\sum_{i=1}^{3}m_{i}$ in Fig. 3.

Figure 3:  Panels in the first row: Fitting results on the plane of
$\delta_{\rm CP}$ versus $|m_{ee}|$. Panels in the second row: Those on the
plane of $\delta_{\rm CP}$ versus $\sum_{i=1}^{3}m_{i}$. Panels in the third
row: Those on the plane of $|m_{ee}|$ versus $\sum_{i=1}^{3}m_{i}$. The left-
side panels are for the case of the normal neutrino mass hierarchy, and the
right-side panels are for the case of the inverted hierarchy.

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