aainstitutetext: Department of Physics, Oklahoma State University, Stillwater,
OK 74078, USAbbinstitutetext: Maryland Center for Fundamental Physics &
Department of Physics, University of Maryland, College Park, Maryland 20742,
USAccinstitutetext: Department of Physics, University of Virginia,
Charlottesville, Virginia 22904-4714, USA

# Predictive Dirac Neutrino Spectrum with Strong CP Solution
in $\boldsymbol{SU(5)_{L}\times SU(5)_{R}}$ Unification

K.S. Babu<EMAIL_ADDRESS>b Rabindra N. Mohapatra<EMAIL_ADDRESS>c Anil
Thapa<EMAIL_ADDRESS>
###### Abstract

We develop a grand unified theory of matter and forces based on the gauge
symmetry $SU(5)_{L}\times SU(5)_{R}$ with parity interchanging the two factor
groups. Our main motivation for such a construction is to realize a minimal
GUT embedding of left-right symmetric models that provide a parity solution to
the strong CP problem without the axion. We show how the gauge couplings unify
with an intermediate gauge symmetry $SU(3)_{cL}\times SU(2)_{2L}\times
U(1)_{L}\times SU(5)_{R}$, and establish its consistency with proton decay
constraints. The model correctly reproduces the observed fermion masses and
mixings and leads to naturally light Dirac neutrinos with their Yukawa
couplings suppressed by a factor $M_{I}/M_{G}$, the ratio of the intermediate
scale to the GUT scale. We call this mechanism type II-Dirac seesaw.
Furthermore, the model predicts $\delta_{CP}=\pm(130.4\pm 1.2)^{\circ}$ and
$m_{\nu_{1}}=(4.8-8.4)$ meV for the Dirac CP phase and the lightest neutrino
mass. We demonstrate how the model solves the strong CP problem via parity
symmetry.

##  1 Introduction

Grand unification of forces and matter Pati:1974yy ; Georgi:1974sy ;
Georgi:1974yf  is an appealing framework for beyond the standard model
physics, as it provides an understanding of the disparate forces of nature,
and also provides a connection between apparently diverse building blocks of
matter such as the quarks and leptons. It further promises to elucidate the
nature of the neutrino, explains its tiny mass, and addresses the origin of
matter-antimatter asymmetry in the universe. Concrete realizations of this
idea predict new phenomena such as the decay of the proton which can be used
to test this approach and has rightly spurred dedicated decades-long
experimental efforts to discover such decays. Grand unification (GUT) has thus
become a widely accepted paradigm for physics beyond the standard model for
nearly half a century.

The most common frameworks for realizing GUTs are based on simple gauge
symmetry groups such as $SU(5)$, $SO(10)$ and $E_{6}$. While these models are
phenomenologically appealing and quite successful, there has been no direct
experimental support for any of them so far. It is important, therefore, to
pursue alternative unification ideas that are testable. In this paper, we
develop and analyze in detail one such alternative based on the gauge group
$SU(5)_{L}\times SU(5)_{R}$ with parity symmetry that interchanges the two
$SU(5)$ factors Davidson:1987mi ; Cho:1993jb ; Mohapatra:1996fu ; Lee:2016wiy
; Emmanuel-Costa:2011hwa ; Tavartkiladze:2016imo ; Lonsdale:2014wwa , so that
there is a single unified force. The main motivation for developing such a
theory is to provide a minimal GUT embedding of left-right symmetric models
which solve the strong CP problem via parity symmetry without the axion. A
successful left-right symmetric model of this type was proposed by two of us
some time ago Babu:1989rb  which has been followed up in several recent papers
Hall:2018let ; Hall:2019qwx ; Dunsky:2020dhn ; Harigaya:2022wzt ; Carrasco-
Martinez:2023nit ; Craig:2020bnv ; deVries:2021pzl ; Dcruz:2022rjg ;
Babu:2023srr . The key idea behind this solution to the strong CP problem is
that the instanton-induced strong CP violating Lagrangian term
$\overline{\theta}G\widetilde{G}$ is parity violating and therefore a theory
that obeys parity symmetry could lead to vanishing $\overline{\theta}$
Beg:1978mt ; Mohapatra:1978fy ; Mohapatra:1995xd ; Kuchimanchi:1995rp ;
Mohapatra:1997su ; Babu:2001se ; Kuchimanchi:2023imj . Parity is of course
broken in nature, which would induce nonzero $\overline{\theta}$, but in the
model of Ref. Babu:1989rb  this occurs only at the two-loop level. The
$SU(5)_{L}\times SU(5)_{R}$ model with parity symmetry thus has the potential
to solve the strong CP problem without the axion, in the framework of a GUT.

Unified theories based on product groups accompanied by a discrete symmetry,
such as $SU(3)_{c}\times SU(3)_{L}\times SU(3)_{R}$ trinification with a
$Z_{3}$ symmetry Rujula:1984 ; Babu:1985gi , $SU(3)_{q}\times
SU(3)_{\ell}\times SU(3)_{L}\times SU(3)_{R}$ quartification with a $Z_{4}$
symmetry Babu:2003nw , as well as $SO(10)_{1}\times SO(10)_{2}\times
SO(10)_{3}$ Babu:2007mb  and $SU(5)_{1}\times SU(5)_{2}\times SU(5)_{3}$
FernandezNavarro:2023hrf  with a $Z_{3}$ symmetry, where each gauge group acts
on a separate family in the last two cases, have been developed to varying
degrees of detail in the literature. The $SU(5)_{L}\times SU(5)_{R}$ model
developed here is similar in spirit to these models, but with the main
motivation being parity solution to the strong CP problem.

Achieving the desired symmetry breaking pattern consistent with gauge coupling
unification, along with realistic fermion mass generation within
$SU(5)_{L}\times SU(5)_{R}$ grand unification turns out to be somewhat
challenging. One reason for this is that the GUT scale value of the weak
mixing angle in this framework is ${\rm
sin}^{2}~{}\theta_{W}(M_{G})=\frac{3}{16}$, as opposed to this value being
${\rm sin}^{2}~{}\theta_{W}(M_{G})=\frac{3}{8}$ in conventional GUTs such as
$SU(5)$. This mixing angle should therefore increase in running down in
energy. Consistent unification can be achieved, as we show here, with an
intermediate symmetry group $SU(3)_{cL}\times SU(2)_{2L}\times U(1)_{L}\times
SU(5)_{R}$. This is the simplest scenario that we have found for symmetry
breaking. With an unbroken $SU(5)_{R}$ surviving down to the intermediate
scale, one should worry about proton decay mediated by the
($X_{R}^{\mu},\,Y_{R}^{\mu})$ gauge bosons of this group. We have found a
natural flavor structure that emerges within the framework which suppresses
proton decay mediated by these gauge bosons completely, as these couplings
always involve a heavy field.

The fermion mass matrices of the model have a general structure that resembles
the universal seesaw mechanism Davidson:1987mh ; Berezhiani:1985in  wherein
the usual fermions acquire their masses by mixing with vector-like fermions
present in the theory. However, such a seesaw cannot be universal with the
assumed intermediate symmetry, since in that case there would be rapid proton
decay. In the absence of certain scalars that are needed for universal seesaw,
we find that proton decay is naturally suppressed.

The model developed here is very predictive in the neutrino sector. As we
shall see, neutrinos are naturally light Dirac fermions in this framework,
with their Yukawa couplings suppressed by an overall factor of $M_{I}/M_{G}$,
the ratio of the intermediate scale to the GUT scale. We call such a
suppression mechanism type-II Dirac seesaw due to its similarity with type-II
seesaw in Majorana neutrino models. The model predicts normal ordering of
neutrino masses. The quark-lepton connection present in the unified theory
enables us to predict the Dirac CP phase appearing in neutrino oscillations to
be $\delta_{CP}=(130.4\pm 1.2)^{\circ}$ or $(229.6\pm 1.2)^{\circ}$, and the
lightest neutrino mass to be $m_{\nu_{1}}=(4.8-8.4)$ meV. These predictions
will provide tests of the model. Naturally, there will be no neutrinoless
double beta decay within this model.

The main results of the paper can be summarized as follows. (i) we have
presented a successful embedding of the parity solution to strong CP problem
in an $SU(5)_{L}\times SU(5)_{R}$ GUT framework and showed its consistency
with gauge coupling unification; (ii) neutrinos are naturally light Dirac
fermions within the model with quantitative predictions for two: the CP phase
and the lightest neutrino mass; (iii) there is no issue with rapid proton
decay even with the intermediate symmetry containing $SU(5)_{R}$ owing to an
emergent flavor structure; and (iv) we have shown the vanishing of
$\overline{\theta}$ through one-loop diagrams, including various contributions
from the GUT symmetry breaking sector. The two-loop induced
$\overline{\theta}$ is compatible with neutron electric dipole moment (EDM)
limits, with a mild fine-tuning of parameters at the few percent level.
Neutron EDM should be within reach of forthcoming experiments for the model to
be not finely tuned.

The paper is organized as follows. We describe the salient features of the
model including the symmetry breaking sector and its motivations in Sec. 2.
The mechanism of fermion mass generation is discussed in Sec. 3. In Sec. 4, we
discuss the unification of gauge couplings where we also determine the scales
and the various couplings at different scales. In Sec. 5 we show how the
neutrino and the down-type quark mass matrices are connected leading to
predictions for the leptonic CP phases and the lightest neutrino mass. Sec. 6
is devoted to a discussion of proton decay in the model where we show the
consistency of the model. In Sec. 7, we evaluate all the one-loop corrections
to the strong CP parameter $\overline{\theta}$ and show that they are all
vanishing. Here we also estimate the leading two-loop contributions and
discuss its implications for neutron electric dipole moment. In Sec. 8 we
conclude. In several appendices we present various technical details,
including decomposition under subgroups, threshold corrections, and two-loop
renormalization group equations for the Yukawa coupling matrices that are used
in our analysis of gauge coupling unification, fermion mass fitting and
$\overline{\theta}$ estimation.

##  2 Model

The unified theory that we develop in this paper is based on the gauge group
$SU(5)_{L}\times SU(5)_{R}$ with parity symmetry which interchanges the two
$SU(5)$ factors. This theory provides a natural embedding of the left-right
symmetric theory of Ref. Babu:1989rb  which provides a parity solution to the
strong CP problem. We start with a brief review of the model of Ref.
Babu:1989rb . It is based on the gauge group $SU(3)_{c}\times SU(2)_{L}\times
SU(2)_{R}\times U(1)_{B-L}$ with the assignment of the three families of
fermions as follows:

$\displaystyle Q_{L}\ (3,2,1,1/3)$ $\displaystyle=\begin{pmatrix}u_{L}\\\
d_{L}\\\ \end{pmatrix},\hskip 14.22636ptQ_{R}\
(3,1,2,1/3)=\begin{pmatrix}u_{R}\\\ d_{R}\\\ \end{pmatrix},$
$\displaystyle\ell_{L}\ (1,2,1,-1)$ $\displaystyle=\begin{pmatrix}\nu_{L}\\\
e_{L}\\\ \end{pmatrix},\hskip 14.22636pt\ell_{R}\
(1,1,2,-1)=\begin{pmatrix}\nu_{R}\\\ \ell_{R}\\\ \end{pmatrix}.$ (1)

This is supplemented by three families of vector-like quarks and leptons
denoted as

$U_{L,R}(3,1,1,4/3),~{}~{}~{}D_{L,R}(3,1,1,-2/3),~{}~{}~{}E_{L,R}(1,1,1,-2)~{}.$
(2)

Parity symmetry ($P$) can be defined within the model, under which
$f_{L}\leftrightarrow f_{R}$ where $f$ stands for any of the fermions, along
with $W_{L}^{\mu}\leftrightarrow W_{R}^{\mu}$ for the $SU(2)_{L,R}$ gauge
bosons. Note the absence of an electrically neutral vector-like lepton, which
is crucial for realizing Dirac neutrinos in the framework Babu:1988yq ;
Babu:2022ikf . The Higgs sector of the model is very simple, consisting of a
pair of parity symmetric doublets, $\\{\chi_{L}(1,2,1,1)+\chi_{R}(1,1,2,1)\\}$
with the transformation $\chi_{L}\leftrightarrow\chi_{R}$ under $P$.

We now proceed to embed this model into a grand unified $SU(5)_{L}\times
SU(5)_{R}$ framework. To achieve this, we note that under the $SU(5)_{L}\times
SU(5)_{R}$ gauge symmetry, all left-handed fermions of Eq. (1) neatly fit into
${\bf 10}+{\bf\overline{5}}$ of $SU(5)_{L}$, while all right-handed fermions
fit into ${\bf 10}+{\bf\overline{5}}$ of $SU(5)_{R}$. The vector-like fermions
of Eq. (2) complete these multiplets without needing any other fermions. Thus
the full fermion content of this left-right symmetric model fit into two basic
anomaly-free chiral representations of $SU(5)_{L}\times SU(5)_{R}$. Their
grouping under $SU(5)_{L}\times SU(5)_{R}$ is given by

$\displaystyle\psi_{L,R}=\left[\begin{array}[]{c}D_{1}^{c}\\\ D_{2}^{c}\\\
D_{3}^{c}\\\ e\\\ -\nu\end{array}\right]_{L,R}\,,\hskip
28.45274pt\chi_{L,R}=\frac{1}{\sqrt{2}}\left[\begin{array}[]{ccccc}0&U_{3}^{c}&-U_{2}^{c}&u_{1}&d_{1}\\\
-U_{3}^{c}&0&U_{1}^{c}&u_{2}&d_{2}\\\ U_{2}^{c}&-U_{1}^{c}&0&u_{3}&d_{3}\\\
-u_{1}&-u_{2}&-u_{3}&0&E^{c}\\\
-d_{1}&-d_{2}&-d_{3}&-E^{c}&0\end{array}\right]_{L,R}\,.$ (13)

We denote these fields transforming under $SU(5)_{L}\times SU(5)_{R}$ as
{$\psi_{L}(\overline{\bf 5},1)+\psi_{R}(1,\overline{\bf 5})\\}$, and
$\\{\chi_{L}({\bf 10},1)+\chi_{R}(1,{\bf 10})\\}$. There are three copies of
them corresponding to the three generations. As in the left-right symmetric
model, right-handed neutrinos, $\nu_{R}$, are naturally present in the
$\psi_{R}(1,\overline{\bf 5})$ multiplet. Under parity, the fields transform
as $\psi_{L}\leftrightarrow\psi_{R}$, $\chi_{L}\leftrightarrow\chi_{R}$ along
with $V^{\mu}_{L}\leftrightarrow V^{\mu}_{R}$, where $V^{\mu}_{L,R}$ denote
the gauge bosons of $SU(5)_{L,R}$ symmetry. Parity symmetry would imply that
there is a single gauge coupling in the theory, identified as
$\alpha_{5L}=\alpha_{5R}=\alpha_{G}$. One of the main goals of this paper is
to construct a realistic symmetry breaking chain that admits gauge coupling
unification consistent with proton decay constraints, while reproducing the
fermion masses correctly and preserving the parity solution to the strong CP
problem as in the left-right symmetric model of Ref. Babu:1989rb .

To break $SU(5)_{L}\times SU(5)_{R}$ spontaneously all the way down to
$SU(3)_{c}\times U(1)_{\rm em}$ and to generate realistic fermion masses and
mixings, we choose the following Higgs multiplets:111While the choice of
$\\{({\bf 24},1)+(1,{\bf 24})\\}$ instead of $\\{({\bf 75},1)+(1,{\bf 75})\\}$
is more economical, it will not go well with the strong CP solution, since
terms in the Higgs potential of the type $({\bf 24},1)H^{\dagger}_{R}\Phi
H_{L}$ and $({\bf 24},1)\eta^{\dagger}\Phi\Phi$ with complex coefficients
would be allowed, in spite of parity symmetry. These complex couplings will
spoil the strong CP solution via parity symmetry, owing to scalar–pseudoscalar
mixings that they generate. In presence of such mixings, the loop induced
$\overline{\theta}$ would be typically too large. It is possible to forbid
such terms with an additional discrete symmetry, but not with parity alone.

$\displaystyle\\{\Sigma_{L}({\bf 75},1)+\Sigma_{R}(1,{\bf 75})\\},\hskip
8.53581pt\\{H_{L}({\bf 5},1)+H_{R}(1,{\bf 5})\\},\hskip
8.53581pt\Phi(\overline{\bf 5},{\bf 5}),\hskip 8.53581pt\eta(\overline{\bf
15},{\bf 15})\,.$ (14)

Here the field $\Sigma_{L}$ is used to break the $SU(5)_{L}$ symmetry down to
$SU(3)_{cL}\times SU(2)_{L}\times U(1)_{L}$ and $\Sigma_{R}$ appears as its
parity partner. The $H_{L,R}$ fields and the $\Phi$ field are used to generate
fermion masses and to break the surviving symmetry down to $SU(3)_{c}\times
U(1)_{\rm em}$. We discuss the details of fermion mass generation in Sec. 3,
with the mass matrices for quarks and leptons given in Eq. (29). The simplest
symmetry breaking chain that we have found, consistent with realistic fermion
masses and proton decay constraints, assumes a single intermediate scale,
denoted as $M_{I}$, and proceeds as follows:

$\displaystyle SU(5)_{L}\times SU(5)_{R}$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\downarrow
M_{G}\sim\langle\Sigma_{L}\rangle$ $\displaystyle SU(3)_{cL}\times
SU(2)_{L}\times U(1)_{L}\times SU(5)_{R}$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\downarrow
M_{I}\sim\langle\Phi\rangle,\langle H_{R}\rangle$ $\displaystyle
SU(3)_{c}\times SU(2)_{L}\times U(1)_{Y}$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\downarrow
M_{W}\sim\langle H_{L}\rangle$ $\displaystyle SU(3)_{c}\times
U(1)_{\mathrm{em}}$ (15)

Here the energy scales $\langle\Phi\rangle$ and $\langle H_{R}\rangle$ could
in principle be different, but with the simplified assumption of having a
single intermediate scale, they are identified. A hierarchy of VEVs is
necessary with $M_{W}\ll M_{I}\ll M_{G}$ so as to be consistent with
phenomenology, viz., proton decay constraints as well as collider constraint
on the mass of new gauge bosons. This is achieved in the model, as we discuss
towards the end of this section.

The $\eta(\overline{\bf 15},{\bf 15})$ field of Eq. (14) plays no role in the
symmetry breaking, which can be arranged by choosing its mass term
$M^{2}_{\eta}>0$, which we shall assume. Its purpose is to enable gauge
coupling unification. In order to generate the correct value of
$\sin^{2}\theta_{W}(M_{Z})$, which must increase in running down in energy, it
is beneficial to have new matter/scalar multiplets at scales below $M_{G}$
that transform nontrivially under $SU(2)_{L}$, but carry very small $U(1)_{Y}$
charges. The fragment $(\overline{\bf 3},{\bf 2},-1/6,{\bf 15})$ under
$[SU(3)_{cL}\times SU(2)_{L}\times U(1)_{L}]\times SU(5)_{R}$ from the
$\eta(\overline{\bf 15},{\bf 15})$ field has all the desired properties, and
can lead to successful unification, consistent with low energy phenomenology,
if its mass is assumed to be at or around $M_{I}$. Evolution of gauge
couplings is discussed in detail in Sec. 4 where we have shown in Fig. 2
successful unification when this scalar fragment $(\overline{\bf 3},{\bf
2},-1/6,{\bf 15})$ has a mass equal to $M_{I}$. This is a simpler way of
achieving gauge coupling unification in $SU(5)_{L}\times SU(5)_{R}$ theories
compared to other attempts Davidson:1987mi ; Cho:1993jb ; Mohapatra:1996fu ;
Lee:2016wiy ; Emmanuel-Costa:2011hwa .

A scalar field belonging to $(\overline{\bf 10},{\bf 10})$ could be considered
to help with gauge coupling unification instead of the $\eta(\overline{\bf
15},{\bf 15})$ field, (and has been introduced in Ref. Davidson:1987mi ;
Cho:1993jb ; Lee:2016wiy ; Emmanuel-Costa:2011hwa ), which contains a fragment
$(\overline{\bf 3},{\bf 2},-1/6,{\bf 10})$ under $[SU(3)_{cL}\times
SU(2)_{L}\times U(1)_{L}]\times SU(5)_{R}$ with the right properties to
increase $\sin^{2}\theta_{W}$ while running down in energies. However, we find
this choice to have three problems in the present context: (i) It would allow
the $(2,2)$ blocks of the up-type quark and charged lepton mass matrices
${\cal M}_{u}$ and ${\cal M}_{\ell}$ (see Eq. (29)) to be nonzero, which would
result in rapid proton decay mediated by the $SU(5)_{R}$ gauge bosons having
masses of order $M_{I}$; (ii) it would lead to complex couplings in the scalar
potential of the type $(\overline{\bf 10},{\bf 10})^{*}\Phi\Phi\Sigma_{L}$ and
$\Phi(\overline{\bf 10},{\bf 10})(\overline{\bf 10},{\bf 10})\Sigma_{L}$,
potentially spoiling the parity solution to the strong CP problem; and (iii)
it would lead to $g_{5R}(M_{I})$ becoming non-perturbative while running from
$M_{G}$ to $M_{I}$, making calculations unreliable. This is the rationale
behind adopting the $\eta(\overline{\bf 15},{\bf 15})$ multiplet instead of
the $(\overline{\bf 10},{\bf 10})$ scalar field.

Under left-right parity symmetry the scalar multiplets transform as

$\displaystyle H_{L}\leftrightarrow H_{R}\,,\
~{}\Phi\leftrightarrow\Phi^{\dagger}\,,\
~{}\eta\leftrightarrow\eta^{\dagger}\,,\
~{}\Sigma_{L}\leftrightarrow\Sigma_{R}\,.$ (16)

We have constructed the full scalar potential with these fields and cross-
checked it against the software package Sym2Int Fonseca:2017lem . We find that
there are three non-trivial couplings allowed by the gauge symmetry that can
be complex:

$\displaystyle V\supset$ $\displaystyle\ \mu_{1}\ H_{R}^{\dagger}\Phi
H_{L}+\mu_{2}\ \eta^{\dagger}\Phi\Phi+\lambda\
H_{L}^{\dagger}\eta^{\dagger}\Phi H_{R}+h.c..$ (17)

However, with the imposition of parity symmetry, see Eq. (16), all these
couplings become real. Thus, the full Higgs potential of the model is CP
invariant, which would admit a vacuum structure that preserves CP (for some
range of parameters of the potential). In this case the scalar and
pseudoscalar fields would remain unmixed, which is significant for the theory
to provide parity-based solution to the strong CP problem, especially in the
vanishing of the $\overline{\theta}$ parameter at the one-loop level. This is
discussed in more detail in Sec. 7.

The real vacuum expectation values (VEVs) of these Higgs fields are denoted as

$\displaystyle\langle H_{L,R}\rangle$ $\displaystyle=\kappa_{L,R}\
(0,0,0,0,1)^{T}\,,\hskip 14.22636pt\langle\Phi\rangle=v_{\Phi}\ {\rm
diag(1,1,1,0,0)}\,$ (18) $\displaystyle\hskip
42.67912pt\langle(\Sigma_{L})^{ab}_{cd}\rangle=\langle\Sigma_{L}\rangle({\bf
1_{SM}})^{ab}_{cd}\,$ (19)

where the ${\bf 1_{SM}}$ singlet from the ${\bf 75}$ is the (un-normalized)
combination given by

$\displaystyle({\bf 1_{SM}})^{ab}_{cd}$
$\displaystyle=\Sigma_{L}\left({~{}}^{14}_{14}+{~{}}^{15}_{15}+{~{}}^{24}_{24}+{~{}}^{25}_{25}+{~{}}^{34}_{34}+{~{}}^{35}_{35}-{~{}}^{12}_{12}-{~{}}^{13}_{13}-{~{}}^{23}_{23}-3.\,{~{}}^{45}_{45}\right)\,.$
(20)

Here $(\Sigma_{L})^{ab}_{cd}=-(\Sigma_{L})^{ba}_{cd}=-(\Sigma_{L})^{ab}_{dc}$
and $(\Sigma_{L})^{ab}_{ad}=0$ with $(a,b,..=1...5)$ Abud:1984ni ;
Hubsch:1984pg . Note that the SM singlet from the ${\bf 75}$, which may be
obtained from the product ${\bf 10}\times{\bf\overline{10}}={\bf 1}+{\bf
24}+{\bf 75}$ after removing the ${\bf 24}$ and ${\bf 1}$ fragments, is the
combination $(q\overline{q}-U^{c}\overline{U^{c}}-3.\,E^{c}\overline{E^{c}})$,
in the notation of Eq. (13).222The SM singlets from the 1 and 24 are
respectively $(q\overline{q}+U^{c}\overline{U^{c}}+E^{c}\overline{E^{c}})$ and
$(q\overline{q}-4.\,U^{c}\overline{U^{c}}+6.\,E^{c}\overline{E^{c}})$. This is
the combination appearing in Eq. (20).333 This index structure is in agreement
with the result given in Ref. Abud:1984ni , after making the interchange
$\\{4\leftrightarrow 1,\,3\leftrightarrow 5\\}$. The VEV for the $\Sigma_{R}$
field can be written in an analogous fashion by the replacement $L\to R$ in
Eqs. (19)-(20).

In the symmetry breaking scheme displayed in Eq. (15), parity is spontaneously
broken at a scale $\langle\Sigma_{L}\rangle\sim M_{G}$. It can be shown that
starting with a $P$-invariant potential, a minimum where
$\langle\Sigma_{R}\rangle=0$ and $\langle\Sigma_{L}\rangle\neq 0$ can be
realized, for a certain range of parameters of the potential Senjanovic:1975rk
. Since at the scale $M_{G}$, only the $\Sigma_{L,R}$ fields acquire VEVs, we
can write the relevant potential as

$V(\langle\Sigma_{L,R}\rangle)=-\mu^{2}\left(\langle{\Sigma_{L}}\rangle^{2}+\langle{\Sigma_{R}}\rangle^{2}\right)+\lambda_{1}\left(\langle{\Sigma_{L}}\rangle^{2}+\langle\Sigma_{R}\rangle^{2}\right)^{2}+\lambda_{2}\left(\langle{\Sigma_{L}}\rangle^{2}-\langle{\Sigma_{R}}\rangle^{2}\right)^{2}~{}.$
(21)

Here $\lambda_{1,2}$ are combinations of various quartic couplings involving
the $\Sigma_{L,R}$ fields that appear in the potential. For the choice
$\lambda_{1}>0$ and $\lambda_{2}<0$, and $\mu^{2}>0$, the minimum of the
potential occurs at $\\{\langle\Sigma_{L}\rangle\neq
0,\,\langle\Sigma_{R}\rangle=0\\}$, or at $\\{\langle\Sigma_{R}\rangle\neq
0,\,\langle\Sigma_{L}\rangle=0\\}$ Senjanovic:1975rk , among which we choose
the former solution without any loss of generality. For subsequent symmetry
breaking at the intermediate scale $M_{I}$, the cross couplings of $H_{L}$ and
$H_{R}$ fields with the $\Sigma_{L,R}$ fields would generate unequal masses
for the Higgs doublets in $H_{L}^{d}$ and $H_{R}^{d}$. Such a setup allows for
the realization of the hierarchy $\langle H_{L}^{d}\rangle\ll\langle
H_{R}^{d}\rangle\sim\langle\Phi\rangle\neq 0$. We shall adopt this chain and
mechanism of symmetry breaking in our analysis. While we have constructed the
full Higgs potential of the model, we shall only make use of the reality of
the couplings that admits real-valued VEVs, see Eq. (17), and the mechanism of
spontaneous parity breaking which leads to a hierarchical VEV structure as
discussed here.

##  3 Fermion Mass Generation

In this section we develop the scheme for fermion mass generation. This
includes a natural mechanism for small Dirac neutrino masses, which we term as
type-II Dirac seesaw. We also show here the predictivity of the model in the
neutrino sector, while delegating a detailed numerical analysis to Sec. 5.

###  3.1 Yukawa Lagrangian

The most general gauge-invariant Yukawa interactions of the model, that are
also invariant under parity symmetry, are given by the Lagrangian

$\displaystyle-{\cal L}_{\rm Yuk}$ $\displaystyle=$
$\displaystyle\frac{(Y_{u}^{*})_{ij}}{4}\epsilon_{\alpha\beta\gamma\delta\rho}\left\\{\chi_{Li}^{\alpha\beta}\chi_{Lj}^{\gamma\delta}H_{L}^{\rho}+\chi_{Ri}^{\alpha\beta}\chi_{Rj}^{\gamma\delta}H_{R}^{\rho}\right\\}+\sqrt{2}\,(Y_{\ell}^{*})_{ij}\left\\{\psi_{Li\alpha}\chi_{Lj}^{\alpha\beta}H_{L\beta}^{*}+\psi_{Ri\alpha}\chi_{Rj}^{\alpha\beta}H_{R\beta}^{*}\right\\}$
(22) $\displaystyle+$ $\displaystyle(Y_{D}^{*})_{ij}\
\overline{\psi}_{Li}^{\alpha}\,\Phi^{\beta}_{\alpha}\,\psi_{Rj\beta}+h.c.$

Here $(i,\,j)$ are family indices, while $(\alpha,\,\beta...)$ are
$SU(5)_{L,R}$ indices. We have $(Y_{u}^{*})_{ij}=(Y_{u}^{*})_{ji}$ by
$SU(5)_{L,R}$ symmetry, and $(Y_{D}^{*})_{ij}=(Y_{D})_{ji}$, owing to parity.
After spontaneous symmetry breaking, with the VEVs as shown in Eq. (18), the
mass matrices induced for the up-type quarks, charged leptons and down-type
quarks can be written, in the notation $-{\cal L}_{\rm
mass}\supset\overline{f_{L}}M_{f}f_{R}$, as

$\displaystyle\mathcal{M}_{u}=\left(\begin{array}[]{cc}0&Y_{u}\kappa_{L}\\\
Y_{u}^{\dagger}\kappa_{R}&0\end{array}\right),~{}~{}~{}~{}\mathcal{M}_{\ell}=\left(\begin{array}[]{cc}0&Y_{\ell}\kappa_{L}\\\
Y_{\ell}^{\dagger}\kappa_{R}&0\end{array}\right),~{}~{}~{}~{}\mathcal{M}_{d}=\left(\begin{array}[]{cc}0&Y_{\ell}^{T}\kappa_{L}\\\
Y_{\ell}^{*}\kappa_{R}&Y_{D}v_{\Phi}\end{array}\right)\,.$ (29)

Here the $3\times 3$ sub-matrices obey $Y_{u}=Y_{u}^{T}$ and
$Y_{D}=Y_{D}^{\dagger}$, while $Y_{\ell}$ is a general complex matrix. While
these mass matrices resemble those appearing in the context of universal
seesaw left-right symmetric models of Ref. Davidson:1987mh ; Berezhiani:1983hm
; Babu:1989rb , there is a significant difference in that the $(2,2)$ blocks
of ${\cal M}_{u}$ and ${\cal M}_{\ell}$ are zero here. This departure from
universal seesaw is crucial for the model to be compatible with proton decay
constraints, with the assumed $SU(5)_{R}$ intermediate symmetry. Nonzero
entries in the $(2,2)$ blocks of ${\cal M}_{u}$ and ${\cal M}_{\ell}$ will
lead to significant mixing between $u$ and $U$ quarks as well as between $e$
and $E$ leptons. This in turn would result in rapid proton decay mediated by
the $(X_{R}^{\mu},\,Y_{R}^{\mu})$ gauge bosons of $SU(5)_{R}$. With these
blocks being zero, as in Eq. (29), the $(X_{R}^{\mu},\,Y_{R}^{\mu})$ gauge
bosons will have baryon and lepton number violating couplings only between
light fermions and heavy vector-like fermions, preventing rapid proton decay.
This issue is further discussed in Sec. 6.

###  3.2 Type-II Dirac seesaw for small neutrino masses

Neutrinos turn out to be naturally light Dirac particles within the model.
Although baryon $(B$) and lepton numbers ($L$) are both broken by the gauge
interactions of $SU(5)_{L}\times SU(5)_{R}$, $(B-L)$ symmetry is left
unbroken, which prevents any renormalizable couplings that would allow
Majorana masses for the neutrinos. There is in fact a natural mechanism for
the Dirac neutrinos Yukawa couplings to be extremely small: They are
proportional to the ratio $(M_{I}/M_{G})$, the scales associated with
intermediate and GUT symmetry breaking. The effective Dirac neutrino masses
and Yukawa couplings are of the form

Figure 1: Diagram generating small Dirac neutrino masses via type-II Dirac
seesaw. Here $\Phi_{5}^{5}$ is the neutral member of a Higgs doublet that has
GUT scale mass, which acquires a small induced VEV after $SU(2)_{L,R}$
symmetry breaking triggered by the VEVs $\langle H^{0}_{L,R}\rangle$.

${\cal L}_{\nu-{\rm mass}}^{\rm Dirac}=\frac{\overline{\nu}_{L}\nu_{R}\langle
H_{L}^{0}\rangle\langle H_{R}^{0}\rangle}{M_{G}}\hskip
8.53581pt\Rightarrow\hskip 8.53581ptY_{\nu}^{\rm
Dirac}\sim\frac{M_{I}}{M_{G}}\approx 10^{-7}~{}.$ (30)

Here we have inserted the numerical value of the ratio $(M_{I}/M_{G})$
obtained from gauge coupling unification, see Eq. (61) of Sec. 4, as a
reference value. In addition to this suppression, there are flavor-dependent
suppression factors in $Y_{\nu}^{\rm Dirac}$, leading to consistent neutrino
oscillation phenomenology.

The effective interactions shown in Eq. (30) arise as follows. In presence of
the cubic scalar coupling $\mu_{1}H_{R}^{\dagger}\Phi H_{L}$, as given in Eq.
(17), the neutral member ($\Phi_{5}^{5}$) of the Higgs doublet
$\Phi_{d}(1,2,1/2)$ (the quantum numbers here refer to SM gauge symmetry)
contained in $\Phi(\overline{\bf 5},{\bf 5})$ field will acquire an induced
VEV, $\langle\Phi^{5}_{5}\rangle=v_{\nu}$, proportional to the product of VEVs
$\kappa_{L}\kappa_{R}$ that break $SU(2)_{L}\times SU(2)_{R}$ symmetry, as
shown in Fig. 1 Lee:2016wiy . The third term of the Yukawa Lagrangian of Eq.
(22) will in turn lead to neutrino masses:

$v_{\nu}\simeq\frac{\mu_{1}\kappa_{L}\kappa_{R}}{M^{2}_{\Phi_{d}}(1,2,1/2)}\,\hskip
8.53581pt\Rightarrow\hskip 14.22636ptM_{\nu}=Y_{D}^{*}\,v_{\nu}\,.$ (31)

Note here that the mass of $\Phi_{d}(1,2,1/2)$ scalar is at the GUT scale,
consistent with the extended survival hypothesis Dimopoulos:1984ha  that we
adopt, while $\kappa_{R}=M_{I}$. The original survival hypothesis of Georgi
Georgi:1979md  applies to fermions in theories with multiple scales and
assumes that only those fermions survive down to lower energy that cannot
acquire masses consistent with symmetries. The extended survival hypothesis
generalizes this idea to the scalar sector and assumes that only those scalar
fields that are needed for subsequent symmetry breaking survive below the GUT
scale Dimopoulos:1984ha ; Mohapatra:1982aq . We interpret the extended version
of the hypothesis somewhat more broadly to also includes realism in
phenomenology, with a fragment of $\eta(\overline{\bf 15},{\bf 15})$ not
involved in symmetry breaking surviving below $M_{G}$ to help with gauge
coupling unification. The mass parameter $\mu_{1}$ can be as large as $M_{G}$,
but it could be smaller as well. Taking $\mu_{1}\sim M_{G}$, one would obtain
$Y_{\nu}^{\rm Dira}\sim(M_{I}/M_{G})\sim 10^{-7}$, which is the largest
allowed value for these couplings. This mechanism is somewhat analogous to the
familiar type-II seesaw mechanism for Majorana neutrinos, where a Higgs
triplet acquires an induced VEV, and leads to an effective $d=5$ operator for
neutrino masses once the triplet field is integrated out. We therefore call
this type-II Dirac seesaw Berbig:2022hsm ; Bonilla:2017ekt . This is in
contrast to type-I Dirac seesaw where local symmetries in the right handed
neutrino sector leads directly to operators of the type
$(\ell_{L}H_{L})(\ell_{R}H_{R})/M$ Berezhiani:1995yi ; Gu:2006dc .

###  3.3 Reparametrization and predictive neutrino spectrum

Without loss of generality one can work in a basis where the Yukawa coupling
matrices $Y_{D}$ and $Y_{u}$ of Eq. (29) are chosen to be real and diagonal.
This is possible by independent rotations on the three families of $\\{({\bf
10},1)+(1,{\bf 10})\\}$ and $\\{({\bf\overline{5}},1)+(1,\overline{\bf 5})\\}$
fermion fields. In this basis, the $3\times 3$ up-type quark mass matrix and
the Dirac neutrino mass matrix take the form:

$\hat{M}_{u}\equiv\hat{Y}_{u}\,\kappa_{L}={\rm
diag.}(m_{u},\,m_{c},\,m_{t}),\hskip 14.22636pt\hat{M}_{\nu}\equiv
Y_{D}^{*}\,v_{\nu}={\rm diag.}(m_{\nu_{1}}\,,m_{\nu_{2}}\,,m_{\nu_{3}})\,.$
(32)

The $3\times 3$ charged lepton mass matrix $M_{\ell}$ will have a general form
which can be written as

$M_{\ell}\equiv Y_{\ell}\,\kappa_{L}=\hat{U}^{\dagger}_{\rm
PMNS}\hat{M}_{\ell}V_{R}^{T}\,,\hskip 14.22636pt{\rm where}\
\hat{M}_{\ell}={\rm diag.}(m_{e},\,m_{\mu},\,m_{\tau})\,.$ (33)

Here $\hat{U}_{\rm PMNS}=PU_{\rm PMNS}Q$, where $U_{\rm PMNS}$ is the charged
lepton mixing matrix written in the canonical form with a single CP-violating
phase $\delta_{CP}$, while $P$ and $Q$ are diagonal phase matrices that are
unobservable in weak interactions. (Note that there are no Majorana phases
associated with charged current weak interactions in this model, since
neutrinos are Dirac fermions.) The matrix $V_{R}$ in Eq. (33) is an arbitrary
unitary matrix.

We redefine the down-type quarks $(d,D)$ and the charged leptons $(e,E)$ to go
from the original basis in which the mass matrices of Eq. (29) are written, to
a new primed basis given by

$\displaystyle d_{L}=V_{R}\,P^{*}d^{\prime}_{L},\hskip
8.53581ptd_{R}=V_{R}\,P^{*}d^{\prime}_{R},\hskip 8.53581ptD_{L}=Q\,U_{\rm
PMNS}^{T}\,D^{\prime}_{L},\hskip 8.53581ptD_{R}=Q\,U_{\rm
PMNS}^{T}\,D^{\prime}_{R}$ $\displaystyle e_{L}=Q^{*}U_{\rm
PMNS}^{\dagger}e_{L}^{\prime},\hskip 8.53581pte_{R}=Q^{*}\,U_{\rm
PMNS}^{\dagger}e_{R}^{\prime},\hskip
8.53581pt\nu_{L}=Q^{*}\nu_{L}^{\prime},\hskip
8.53581pt\nu_{R}=Q^{*}\,\nu_{R}^{\prime}$ $\displaystyle
E_{L}=V_{R}^{*}\,PE_{L}^{\prime},\hskip
14.22636ptE_{R}=V_{R}^{*}PE_{R}^{\prime}\,.$ (34)

These primes states are not quite the mass eigenstates; see further
redefinitions given below in Eq. (41) that achieve this. The fermion mass
matrices of Eq. (29) at the GUT scale in the new primed basis read as

$\displaystyle{\cal
M}_{u}=\left(\begin{matrix}0&\hat{M}_{u}\cr\hat{M}_{u}\frac{\kappa_{R}}{\kappa_{L}}&0\end{matrix}\right),~{}~{}~{}~{}{\cal
M}_{\ell}=\left(\begin{matrix}0&\hat{M}_{\ell}\\\
\hat{M}_{\ell}\frac{\kappa_{R}}{\kappa_{L}}&0\end{matrix}\right),$ (35)
$\displaystyle{\cal M}_{d}=\begin{pmatrix}0&\hat{M}_{\ell}\\\
\hat{M}_{\ell}\frac{\kappa_{R}}{\kappa_{L}}&~{}~{}\frac{v_{\phi}}{v_{\nu}}U_{\rm
PMNS}^{*}\hat{M}_{\nu}U_{\rm PMNS}^{T}\end{pmatrix}~{}.$ (36)

The charged current interactions of the $W_{L}^{\pm\mu}$ gauge bosons with the
quarks and leptons in the new basis of Eq. (34) read as

${\cal L}_{\rm
cc}=\frac{g_{2L}}{\sqrt{2}}\left\\{\left(\bar{u}_{L}\gamma_{\mu}V_{R}P^{*}d^{\prime}_{L}\right)W^{\mu+}_{L}+\left(\bar{e}^{\prime}_{L}U_{\rm
PMNS}\ \gamma_{\mu}\nu^{\prime}_{L}\right)W^{\mu-}_{L}\right\\}+h.c.$ (37)

We see that the $(e^{\prime},\,\nu^{\prime})$ fields correspond to physical
leptons with their interactions written in the canonical form. In the quark
sector, while the up-type quark mass matrix is diagonal, ${\cal M}_{d}$ of Eq.
(36) needs to be diagonalized to get to the physical basis. This can be done
by a bi-unitary transformation:

$\xi_{L}^{\dagger}{\cal M}_{d}\,\xi_{R}={\rm
diag.}(m_{d},\,m_{s},\,m_{b},\,m_{D_{1}},\,m_{D_{2}},\,m_{D_{3}})$ (38)

where $\xi_{L,R}$ are $6\times 6$ unitary matrices which can be parametrized
in $3\times 3$ block form as

$\displaystyle\xi_{L,R}=\left(\begin{matrix}\xi^{11}&\xi^{12}\cr\xi^{21}&\xi^{22}\end{matrix}\right)_{L,R}~{}.$
(39)

Denoting the light down-type mass eigenstates collectively as $d^{0}$ and the
heavy ones as $D^{0}$, we have

$\displaystyle d^{\prime}_{L}$ $\displaystyle=$
$\displaystyle\xi^{11}_{L}\,d_{L}^{0}+\xi_{L}^{12}\,D_{L}^{0}$ $\displaystyle
D_{L}^{\prime}$ $\displaystyle=$
$\displaystyle\xi_{L}^{21}\,d_{L}^{0}+\xi_{L}^{22}\,D_{L}^{0}~{}$ (40)

along with analogous relations for $(d^{\prime}_{R},\,D^{\prime}_{R})$
obtained with the replacement $L\rightarrow R$. The charged current quark
interactions in this basis will involve the matrix $V_{R}P^{*}\xi_{L}^{11}$,
which should be identified as the CKM mixing matrix, $\hat{V}_{\rm CKM}$. To
bring this (essentially) unitary matrix444The $3\times 3$ matrix
$\xi_{L}^{11}$, being a sub-block of the $6\times 6$ unitary matrix $\xi_{L}$,
is not unitary in general. However, departure from unitarity in $\xi_{L}$ is
extremely small, of order $(m_{d_{i}}/m_{D_{i}})\leq 10^{-10}$. to the
canonical form with a single phase, we write $\hat{V}_{\rm
CKM}=P^{\prime}V_{\rm CKM}Q^{\prime}$, where $V_{\rm CKM}$ has a single phase,
and where $P^{\prime},\,Q^{\prime}$ are diagonal phase matrices. We further
redefine the fields so that the charged current quark interactions have the
canonical form with a single phase in $V_{\rm CKM}$ while also enuring that
the mass eigenvalues remain real:

$\displaystyle u_{L}=P^{\prime}\hat{u}_{L},\hskip
8.53581ptU_{R}=P^{\prime}\hat{U}_{R}$ $\displaystyle
d_{L}^{0}=Q^{\prime*}\hat{d}_{L},\hskip
8.53581ptd_{R}^{0}=Q^{\prime*}\hat{d}_{R}$ (41)

These hatted fields $(\hat{u},\,\hat{d})$ are the mass eigenstates, and their
charged current weak interactions involve the canonical CKM matrix. These
unitary transformations are relevant for proton decay discussions, which will
be addressed in Sec. 6.

The CKM matrix, which is given by

$V_{\rm CKM}=P^{\prime*}\,V_{R}\,P^{*}\,\xi_{L}^{11}\,Q^{\prime*}~{},$ (42)

is unconstrained in this scenario, as it contains the unspecified unitary
matrix $V_{R}$, along with certain unspecified diagonal phase matrices.
Nevertheless, the mass matrices of Eqs. (35)-(36) are constrained, as they
have less parameters than observables. To see this, notice that ${\cal M}_{d}$
in Eq. (36) contains only a single unknown parameter, the VEV ratio factor
$(\kappa_{R}/\kappa_{L})(v_{\nu}/v_{\Phi})$, if we assume complete knowledge
of the neutrino masses and mixings. With this single parameter all three light
quark masses $(m_{d},\,m_{s},\,m_{b})$ should be fitted, which leads to two
quantitative predictions. In practice, in the fit that we carry out in Sec. 5,
we use the known quark masses to predict two of the currently unknown
parameters in neutrino oscillations, viz., $\delta_{CP}$ and $m_{\nu_{1}}$,
the Dirac CP phase and the lightest neutrino mass. It may be noted that in
spite of two VEV ratios $(\kappa_{R}/\kappa_{L})$ and $(v_{\Phi}/v_{\nu})$
that appear in the mass matrix ${\cal M}_{d}$ of Eq. (36), only a single
combination appears in the light down-type quark mass matrix. This can be seen
by considering a general block matrix of the form

$\displaystyle{\cal M}=\begin{pmatrix}0&m\\\ M^{\prime}&M\end{pmatrix}\,$ (43)

which obeys the hierarchy $m\ll M^{\prime}\sim M$. Integrating out the heavy
states will lead to the light-sector mass matrix ${\cal M}_{\rm
Light}=m\left[\mathbb{1}-(\mathbb{1}+x\ x^{\dagger})^{-1}\right]^{1/2}$, or
equivalently

$\displaystyle{\cal M}_{\rm Light}{\cal M}_{\rm
Light}^{\dagger}=m\left[\mathbb{1}-(\mathbb{1}+x\
x^{\dagger})^{-1}\right]m^{\dagger}\,,\hskip 14.22636pt\ {\rm where}\ \quad
x=M^{-1}M^{\prime}\,.$ (44)

In the limit of $x\ll 1$, this matrix reproduces the usual seesaw formula, but
Eq. (44) is valid even when $x\sim 1$ Babu:1995uu . It becomes clear then that
only a single parameter, the ratio of $(\kappa_{R}/\kappa_{L})$ and
$(v_{\Phi}/v_{\nu})$, will appear in the light down-type quark mass matrix. In
Sec. 5 where we carry out numerical fits to fermion masses and mixings we have
used the full $6\times 6$ matrix given in Eq. (36), but we have cross-checked
with the analytic expression of Eq. (44) for consistency. We defer a
discussion of fermion fits to Sec. 5, since that requires the values of the
various Yukawa and gauge couplings at the GUT scale and at the intermediate
scale, to which we now turn.

##  4 Gauge Coupling Unification

Under parity symmetry the gauge bosons of the two $SU(5)_{L,R}$ groups
transform as $V_{L}^{\mu}\leftrightarrow V_{R}^{\mu}$, which requires equality
of the two coupling, $\alpha_{5L}=\alpha_{5R}\equiv\alpha_{G}$. This, in turn,
implies that the three gauge couplings of the SM obey, at the unification
scale $M_{G}$, the relations

$2\ \alpha_{3}=\alpha_{2}=\frac{13}{3}\ \alpha_{Y}=\alpha_{G}\,.$ (45)

The factor $2$ in front of $\alpha_{3}$ arises because the $SU(3)_{c}$
symmetry is embedded as a diagonal subgroup of $SU(5)_{L}\times SU(5)_{R}$,
unlike $SU(2)_{L}$ which is entirely inside $SU(5)_{L}$. The factor $13/3$ in
front of $\alpha_{Y}$ arises since hypercharge is contained in both $SU(5)$
factors, with the relations

$(B-L)=Y_{L}+Y_{R},\hskip 17.07164pt\frac{Y}{2}=T_{3R}+\frac{(B-L)}{2}~{}.$
(46)

where the $(B-L)$ charges are those shown in Eqs. (1)-(2), and $T_{3R}$ is the
third component of right-handed isospin. The GUT-normalized hypercharge is
therefore $\sqrt{3/13}\,(Y/2)$, leading to the factor shown in Eq.
(45).555$\sum_{i}(Y_{i}/2)^{2}=26/3$, where the sum goes over a family of
fermions, including vector-like fermions shown in Eqs. (1)-(2). With the
normalization factor in ($Y/2)$ of $\sqrt{3/13}$ included, this factor becomes
2, which is the same normalization as for the generators of $SU(2)_{L}$ – a
family contains four $SU(2)_{L}$ doublets. This yields
$\sin^{2}\theta_{W}(M_{G})=3/16$ at the unification scale, distinct from its
value of $3/8$ in conventional GUTs such as $SU(5)$. Here $\sin^{2}\theta_{W}$
should increase while running down to the weak scale, as opposed to an
decreasing value in conventional GUTs. As discussed in Sec. 2, this can be
achieved in a phenomenologically consistent manner by lowering the mass of the
scalar sub-multiplet $({\bf\bar{3}},{\bf 2},-1/6,{\bf
15})\subset({\bf\overline{15},15})$ (under $SU(3)_{cL}\times SU(2)_{L}\times
U(1)_{L}\times SU(5)_{R}\subset SU(5)_{L}\times SU(5)_{R}$) below the GUT
scale. Above its mass, in going to higher momenta, $\alpha_{2L}$ will decrease
faster than $\alpha_{Y}$, leading to a smaller value of
$\sin^{2}\theta_{W}(M_{G})$, while being consistent with phenomenology.666In
the absence of the $({\bf\bar{3}},{\bf 2},-1/6,{\bf 15})$ sub-multiplet at the
intermediate scale we found that $\alpha_{5R}$ grows in running down from
$M_{G}$ to $M_{I}$ and becomes non-perturbative before reaching $M_{I}$.
Inclusion of the $\eta({\bf\overline{15},15})$ preserves the strong CP
solution via parity, since it induces no complex couplings in the Higgs
potential, nor does it have any Yukawa interactions.

Figure 2: Evolution of the gauge coupling leading to unification in the model
with the breaking chain given in Eq. (15). We have included full three-loop
RGE effects in obtaining this result. See text for a description of the
benchmark point used here.

We have carried out a detailed analysis of the evolution of gauge couplings to
see the prospects for unification of all gauge couplings with the symmetry
breaking chain shown in Eq. (15) having a single intermediate scale $M_{I}$.
As shown in Eq. (15), above $M_{I}$, the gauge symmetry is $SU(3)_{cL}\times
SU(2)_{L}\times U(1)_{L}\times SU(5)_{R}$, while it is the Standard Model
below $M_{I}$. The two-loop renormalization group equations (RGE) for the
gauge coupling evolution can be written in the form

$\displaystyle
16\pi^{2}\frac{dg_{i}}{dt}=g_{i}^{3}b_{i}+\frac{g_{i}^{3}}{16\pi^{2}}\left[\sum_{j}b_{ij}g_{j}^{2}-\sum_{k}C_{ik}{\rm
Tr}\left(Y_{k}^{\dagger}Y_{k}\right)\right].$ (47)

Here $b_{i}$ are the one-loop beta-function ($\beta$) coefficients, while
$b_{ij}$ and $C_{ik}$ are the two-loop coefficients arising from gauge and
Yukawa interactions respectively. For $m_{t}<\mu<M_{I}$, the indices take the
values $i,j=(1_{Y},2_{L},3_{c})$ and $k=(u,d,\ell)$. For $m_{I}<\mu<M_{G}$, we
replace $\\{b_{i},b_{ij},C_{ik}\\}$ by
$\\{b^{\prime}_{i},b^{\prime}_{ij},C^{\prime}_{ik}\\}$ where
$i,j=(1_{L},2_{L},3_{cL},5_{R})$ and $k=(u_{L},u_{R},d_{L},d_{R},\ell_{L},D)$.
The Yukawa coupling matrices in this momentum range are defined through Eq.
(144) of Appendix A.4.

In the momentum range $m_{t}\leq\mu\leq M_{I}$ we evolve the normalized gauge
couplings $\alpha_{1Y}=(13/3)\alpha_{Y}$, as well as $\alpha_{2L}$ and
$\alpha_{3cL}$. In the range $M_{I}\leq\mu\leq M_{G}$ we run the normalized
couplings $(\alpha_{1L},\,\alpha_{2L},\,\alpha_{3L}$) where
$\alpha_{1L}=(5/3)\alpha_{L}$ with the charges under $U(1)_{L}$ of the various
fields given in Eq. (136) of Appendix. A.1. The factor (5/3) is the familiar
$SU(5)$ normalization factor, but now associated with $U(1)_{L}$ rather than
$U(1)_{Y}$. The unification conditions at $\mu=M_{G}$ are then

$\alpha_{1L}^{-1}(M_{G})=\alpha_{2L}^{-1}(M_{G})=\alpha_{3cL}^{-1}(M_{G})=\alpha_{5R}^{-1}(M_{G})\equiv\alpha_{G}^{-1}~{}.$
(48)

At $M_{I}$ the following boundary conditions hold:

$\displaystyle\alpha_{1Y}^{-1}\left(M_{I}\right)$
$\displaystyle=\frac{8}{13}\alpha_{5R}^{-1}\left(M_{I}\right)+\frac{5}{13}\alpha_{1L}^{-1}\left(M_{I}\right)\,,$
(49) $\displaystyle\alpha_{3c}^{-1}\left(M_{I}\right)$
$\displaystyle=\alpha_{3cL}^{-1}\left(M_{I}\right)+\alpha_{5R}^{-1}\left(M_{I}\right)\,.$
(50)

The entire set of fermion fields will survive down to the intermediate scale
$M_{I}$, since their masses arise only after the intermediate symmetry
breaking. These fields, and the minimal set of scalar fields at $M_{I}$ needed
for symmetry breaking, fermion mass generation, and to achieve gauge coupling
unification, under the gauge group $[SU(3)_{cL}\times SU(2)_{L}\times
U(1)_{L}]\times SU(5)_{R}$, are:

$\displaystyle{\rm Fermions:}$ $\displaystyle Q_{L}({\bf 3},{\bf
2},1/6,1)+\chi_{R}(1,1,0,{\bf 10})+L_{L}(1,{\bf 2},-1/2,1)+D_{R}({\bf
3},1,-1/3,1)$ (51) $\displaystyle+~{}U_{R}({\bf
3},1,2/3,1)+E_{R}(1,1,-1,1)+\psi_{R}(1,1,0,{\bf\overline{5}})$
$\displaystyle{\rm Scalars:}$ $\displaystyle H_{R}(1,1,0,{\bf
5})+\Phi_{D}({\bf 3},1,-\frac{1}{3},{\bf\overline{5}})+H_{L}^{d}(1,{\bf
2},\frac{1}{2},1)+\eta_{K}({\bf\bar{3}},{\bf 2},-1/6,{\bf 15})$ (52)

The scalar fields $(H_{R},\,\Phi_{D},\,H_{L}^{d})$ of Eq. (52) are required
for symmetry breaking and fermion mass generation, while the scalar field
$\eta_{K}({\bf\bar{3}},{\bf 2},-1/6,{\bf 15})$ is used to achieve gauge
coupling unification. This choice of scalar spectrum is consistent with the
extended survival hypothesis Dimopoulos:1984ha ; Mohapatra:1982aq  that we
have adopted.

The one-loop and two-loop $\beta$-function coefficients below $M_{I}$ are
those of the SM (with the normalization $\alpha_{1Y}=(13/3)\alpha_{Y}$) and
are given by

$\displaystyle\left(b_{1Y},b_{2L},b_{3c}\right)=\left(\frac{41}{26},-\frac{19}{6},-7\right),$
(53) $\displaystyle b_{ij}=\begin{pmatrix}199/338&27/26&44/13\\\
9/26&35/6&12\\\
11/26&9/2&-26\end{pmatrix},~{}~{}~{}~{}~{}~{}~{}C_{ik}=\begin{pmatrix}17/26&5/26&15/26\\\
3/2&3/2&1/2\\\ 2&2&0\end{pmatrix}~{}.$ (54)

The gauge symmetry above $M_{I}$ is enhanced to $[SU(3)_{cL}\times
SU(2)_{L}\times U(1)_{L}]\times SU(5)_{R}$. With the fermions and scalar
fields shown in Eqs. (51)-(52) contributing to the RGE above $M_{I}$, the beta
function coefficients are:

$\displaystyle\left(b_{1L}^{\prime},b_{2L}^{\prime},b_{3cL}^{\prime},b_{5R}^{\prime}\right)=\left(\frac{74}{15},\frac{13}{3},-\frac{7}{6},-\frac{20}{3}\right)\,,$
(55) $\displaystyle b^{\prime}_{ij}=\begin{pmatrix}326/75&36/5&332/15&216/5\\\
12/5&310/3&132&504\\\ 83/30&99/2&307/3&360\\\
9/5&63&120&6338/15\end{pmatrix},$ $\displaystyle
C^{\prime}_{ik}=\begin{pmatrix}17/10&0&1/2&0&3/2&1\\\ 3/2&0&3/2&0&1/2&0\\\
2&0&2&0&0&5/2\\\ 0&9/2&0&5&0&3/2\end{pmatrix}\,.$ (56)

These $\beta$-functions coefficient are obtained using PyR@TE package
Sartore:2020gou  and cross-checked against known results.

Before exploring gauge coupling unification numerically, it is worthwhile to
analyze the analytic solutions to the one-loop RGE, which are given in Eq.
(138) in Appendix A.3. There we have also included the calculable threshold
effects from the vector-like fermion sector, which arise owing to the
relations $M_{U_{1}}:M_{U_{2}}:M_{U_{3}}=m_{u}:m_{c}:m_{t}$,
$M_{E_{1}}:M_{E_{2}}:M_{E_{3}}=m_{e}:m_{\mu}:m_{\tau}$ and a less trivial
relation for the vector-like down-quark mass ratios. If we ignore these
threshold effects from vector-like fermions for simplicity we would obtain the
following expression for $\sin^{2}\theta_{W}(m_{t})$ to one-loop accuracy:

$\displaystyle\sin^{2}\theta_{W}(m_{t})=\frac{3}{16}\left[1+\frac{\alpha}{6\pi}\left\\{13(b_{1Y}-b_{2L})\log\frac{m_{t}}{M_{I}}+(5b^{\prime}_{1L}-13b^{\prime}_{2L}+8b^{\prime}_{5R})\log\frac{M_{I}}{M_{G}}\right\\}\right]\,.$
(57)

Substituting the one-loop $\beta$-function coefficients from Eqs. (53) and
(55), we obtain

$\sin^{2}\theta_{W}(m_{t})=\frac{3}{16}\left[1+\frac{\alpha}{6\pi}\left\\{-\frac{185}{3}\log\frac{M_{I}}{m_{t}}+(46+39)\log\frac{M_{G}}{M_{I}}\right\\}\right]\,.$
(58)

It is clear from the above equation that without the intermediate scale
symmetry, which may be realized by setting $M_{I}\to M_{G}$,
$\sin^{2}\theta_{W}(m_{t})$ will be smaller than its value at $M_{G}$, which
is inconsistent. The factor 39 multiplying log$(M_{G}/M_{I})$ in Eq. (58)
arises from the $\eta_{K}({\bf\bar{3}},{\bf 2},-1/6,{\bf 15})$ scalar
fragment, which helps in realizing the right value of
$\sin^{2}\theta_{W}(m_{t})\simeq 0.23$, consistent with other phenomenological
constraints. Without this contribution, $M_{I}$ would be too low to be
consistent with LHC limits on vector-like quarks, see Eq. (143) and
discussions in Appendix A.3.

We have carried out the full two-loop RGE analysis numerically, which improves
the one-loop results. The two-loop coefficient of $g_{5R}$ is relatively
large, $b^{\prime}_{5_{R}5_{R}}=6338/15$, and has an opposite sign compared to
the one loop coefficient of $b^{\prime}_{5_{R}}=-20/3$. Since
$\alpha_{5R}(M_{I})\sim 0.1$, the two-loop terms correct the one-loop result
significantly. We also investigated the three-loop beta-function coefficient
of the $g_{5R}^{7}$ term, and found it to be Poole:2019kcm ; Sartore:2020gou
$b^{\prime\prime}_{5_{R}5_{R}5_{R}}=2035937/108$, which shows slow
convergence. For better accuracy we therefore used the full three-loop beta
funcations for our analysis. The following input values at
$m_{t}(m_{t})=162.8$ GeV were used for the gauge couplings.

$\displaystyle\alpha_{1}(m_{t})=0.01024\,,\quad\alpha_{2}(m_{t})=0.0340\,,\quad\alpha_{3}(m_{t})=0.1094\,.$
(59)

We first run these gauge couplings using Eq. (47) from $\mu=m_{t}$ to to the
intermediate scale $M_{I}$, which is to be determined self-consistently. We
have also included the two-loop threshold effects from the vector-like
fermions by evaluating RGE at different vector-like fermion mass scales. These
masses are obtained self-consistently from the fermion mass fits. For the up-
type vector-like quarks and charged leptons we use the ratios
$M_{U_{1}}:M_{U_{2}}:M_{U_{3}}=m_{u}:m_{c}:m_{t}$, and
$M_{E_{1}}:M_{E_{2}}:M_{E_{3}}=m_{e}:m_{\mu}:m_{\tau}$, and set
$M_{U_{3}}=M_{I}$, along with $M_{E_{3}}=(m_{\tau}/m_{t})M_{I}$. The running
masses of the light quarks and leptons are listed at a scale $\mu=M_{I}$ in
Table 1, which are used to determine the masses of the vector-like fermions.
For the down-type vector-like quarks we use the values obtained from our
fermions mass fits given for our benchmark point in Eq. (72). We ignore all
GUT scale threshold effects. We also ignore such threshold effects from scalar
fields at $M_{I}$. Furthermore, the scalar field $\eta_{K}({\bf\bar{3}},{\bf
2},-1/6,{\bf 15})$ is kept at $M_{I}$ for this numerical scan. Since this
field is not involved in symmetry breaking, keeping its mass at $M_{I}$ is not
necessary, but is assumed for simplicity. We have also explored two other
scenarios where its mass is kept away from $M_{I}$, which will be discussed
later in this section. Above $M_{I}$, the gauge couplings
$\\{\alpha_{1L},\alpha_{2L},\alpha_{3cL},\alpha_{5R}\\}$ were run all the way
to the unification scale $M_{G}$ identified from the matching conditions given
in Eq. (50). This numerical procedure involved scanning the space of three
unknown parameters $\\{\alpha_{5R},M_{I},M_{G}\\}$ numerically such that all
four gauge couplings are unified at a single value at $\mu=M_{G}$. We found
successful unification when the gauge couplings at the intermediate scale take
values given by

$\displaystyle\alpha_{1L}(M_{I})=0.0367,\hskip
8.53581pt\alpha_{2L}(M_{I})=0.0255,\hskip
8.53581pt\alpha_{3L}(M_{I})=0.049,\hskip 8.53581pt\alpha_{5R}(M_{I})=0.106\,.$
(60)

The intermediate scale ($M_{I}$), unification scale ($M_{G}$) and the unified
gauge coupling ($\alpha_{G}$) were found for the benchmark point to be

$M_{I}=9.02\times 10^{10}\ \mathrm{GeV},\quad M_{G}=8.0\times 10^{17}\
\mathrm{GeV},\quad\alpha_{G}^{-1}=13.18~{}.$ (61)

The evolution of the gauge couplings in the various momentum regimes that lead
to successful unification is depicted in Fig. 2 where we have used the full
three-loop RGE. We note that without the inclusion of the three-loop effects,
the values of $M_{I}$ and $M_{G}$ would be $M_{I}=3.38\times 10^{10}$ GeV,
$M_{G}=3.09\times 10^{17}$ GeV and the gauge coupling $\alpha_{5R}$ would be
$\alpha_{5R}(M_{I})=0.122$. The slow convergence of the loop-expansion can be
understood partly because $\alpha_{5R}(M_{I})=0.122$ is not so small, and
partly because the actual expansion parameter is not $\alpha/(4\pi)$, but
rather $N\alpha/(4\pi)$ where $N$ is an effective number of degrees of
freedom, which is also not so small in the present scenario.

Note that the unification scale shown in Eq. (61) is below the Planck scale,
and the unified gauge coupling $\alpha_{G}$ has a perturbative value. These
aspects are important for the consistency of the theory and the reliability of
calculations. The value of $M_{G}$ is also quite compatible with proton decay
limits. In Sec. 6 we shall establish the consistency of the framework with
proton decay mediated by the $SU(5)_{R}$ gauge bosons which have masses of
order $M_{I}\simeq 10^{11}$ GeV.

(a) (b)

Figure 3: Gauge coupling evolution leading to unification with the mass of
$\eta_{K}({\bf\bar{3}},{\bf 2},-1/6,{\bf 15})\subset({\bf\overline{15},15})$
above the intermediate scale. Left (Right) panel corresponds to keeping its
mass at 28 (270) times the intermediate scale $M_{I}$.

The mass of the scalar field $\eta_{K}$ $({\bf\bar{3}},{\bf 2},-1/6,{\bf
15})\subset({\bf\overline{15},15})$ is not required to be at $M_{I}$, as it
has no role in symmetry breaking. It is therefore of interest to see how the
unification picture changes if its mass is taken to be different from $M_{I}$.
If its mass is below $M_{I}$, we see no significant difference in the
unification shown in Fig. 2. However, if its mass is above $M_{I}$, the scale
of unification can be somewhat lowered. This is depicted in Fig. 3 for two
benchmark scenarios. The figure on the left (right) panel corresponds to
keeping the mass $\eta_{K}$ $({\bf\bar{3}},{\bf 2},-1/6)$ at $28\times M_{I}$
($270\times M_{I}$). The intermediate scale $M_{I}$, the unification scale
$M_{G}$, and the various gauge couplings for these two scenarios of Fig. 3 (a)
and (b) are found to be:

$\displaystyle(a)\ m_{\eta_{K}}$ $\displaystyle=28\ M_{I}:$
$\displaystyle\alpha_{1L}(M_{I})=0.0280,\hskip
8.53581pt\alpha_{2L}(M_{I})=0.0264,\hskip
8.53581pt\alpha_{3L}(M_{I})=0.0472,\hskip
8.53581pt\alpha_{5R}(M_{I})=0.191\,,$ $\displaystyle M_{I}=4.08\times 10^{9}\
\mathrm{GeV},\quad M_{G}=1.65\times 10^{17}\
\mathrm{GeV},\quad\alpha_{G}^{-1}=21.4\,.$ (62) $\displaystyle(b)\
m_{\eta_{K}}$ $\displaystyle=270\ M_{I}:$
$\displaystyle\alpha_{1L}(M_{I})=0.0252,\hskip
8.53581pt\alpha_{2L}(M_{I})=0.0270,\hskip
8.53581pt\alpha_{3L}(M_{I})=0.0468,\hskip 8.53581pt\alpha_{5R}(M_{I})=0.30\,,$
$\displaystyle M_{I}=9.14\times 10^{8}\ \mathrm{GeV},\quad M_{G}=7.20\times
10^{16}\ \mathrm{GeV},\quad\alpha_{G}^{-1}=25.10\,.$ (63)

It is important to note that further increasing the mass of $\eta_{K}$ will
decrease the GUT scale. However, such an increase would also result in the
increase of the gauge coupling $\alpha_{5R}$, making it non-perturbative.

We conclude this section by noting that successful unification of gauge
couplings has been achieved with perturbative values of all gauge couplings
and the unification scale lying below the Planck scale, in the range
$M_{G}=(7\times 10^{16}-8\times 10^{17})$ GeV.

##  5 Fermion Mass Fitting

In this section, we show that the model can successfully reproduce all fermion
masses and mixings. In fact, the model is quite predictive in the neutrino
sector, which can serve as a test of the model. Having determined the gauge
couplings at various momentum scales, we are now ready to analyze these
predictions quantitatively. As discussed in Sec. 3, a successful fit to all
fermion masses would require finding acceptable values of the light down-type
quark masses ($m_{d},\,m_{s},\,m_{b})$ from the eigenvalues of the $6\times 6$
matrix ${\cal M}_{d}$ of Eq. (36), while being consistent with neutrino
oscillation data. In practice, we use the down-type quark masses as inputs and
predict the currently unknown parameters in the neutrino sector, viz.,
($\delta_{CP}$, $m_{\nu_{1}}$) – the CP-violating oscillation parameter and
the lightest neutrino mass.

Figure 4: $1\sigma$ (blue) and $2\sigma$ (orange) normal hierarchy prediction
for $\delta_{CP}$ and the lightest neutrino mass $m_{\nu_{1}}$ with best fit
for $\delta_{CP}=(130.4\pm 1.2)^{\circ}$ or $(229.6\pm 1.2)^{\circ}$. We use
$3\sigma$ uncertainties for the rest of the oscillation parameters.

Neutrino Dirac masses arise from the Yukawa couplings $Y_{D}$ and the induced
VEV $v_{\nu}$, given in Eq. (31). To obtain fits to the down-type quark masses
arising from ${\cal M}_{d}$ of Eq. (36), we use the neutrino oscillation data
(within $3\sigma$ uncertainties Esteban:2020cvm ) as input and find the
eigenvalues of $\mathcal{M}_{d}^{\dagger}\mathcal{M}_{d}$ from which we derive
the light mass eigenvalues $(m_{d},m_{s},m_{b})$. The eigenvectors of ${\cal
M}_{d}$ of Eq. (36) do not play any role here, since the CKM matrix is
arbitrary containing an unknown unitary matrix $V_{R}$ (see Eq. (42)) which
may be adjusted freely.

We find it convenient to perform the fermion mass fits at the intermediate
scale $M_{I}$, rather than at the unification scale $M_{G}$, where Eq. (36) is
valid. This is because we don’t have simple expressions for the eigenvalues of
${\cal M}_{d}$ that can be applied at $M_{G}$. The form of ${\cal M}_{d}$ will
be modified at $M_{I}$ due to the renormalization group evolution of its
various elements. We express ${\cal M}_{d}(M_{I})$ as

${\cal
M}_{d}(M_{I})=\left(\begin{array}[]{cc}0&m_{\ell}(m_{t})\frac{\eta_{\ell}\eta^{\prime}_{\ell_{L}}}{\eta^{\prime}_{d_{L}}}\\\
m_{\ell}(m_{t})\frac{\kappa_{R}}{\kappa_{L}}\frac{\eta_{\ell}\eta^{\prime}_{\ell_{L}}}{\eta^{\prime}_{d_{R}}}&~{}~{}~{}\frac{v_{\phi}\eta_{\nu}\eta_{U}^{2}}{v_{\nu}\eta^{\prime}_{D}}U_{\rm
PMNS}^{*}\hat{M}_{\nu}U_{\rm PMNS}^{T}\end{array}\right)\,.$ (64)

Here we have introduced various scaling factors $\eta_{i}$ and
$\eta^{\prime}_{\alpha}$. The $\eta_{i}$ factors correspond to the running of
the Standard Model Yukawa couplings from $\mu=m_{t}$ to $\mu=M_{I}$:

$\eta_{u_{i},d_{i},e_{i},\nu_{i}}=\frac{Y_{u_{i},d_{i},e_{i},\nu_{i}}(M_{I})}{Y_{u_{i},d_{i},e_{i},\nu_{i}}(m_{t})}~{}.$
(65)

These factors are the same as the ratios of the running masses at the two
energy scales. Since the effect of the lighter quark and lepton Yukawa
couplings on the RG evolution is negligible, we have $\eta_{u}=\eta_{c}$,
$\eta_{e}=\eta_{\mu}=\eta_{\tau}\equiv\eta_{\ell}$, and $\eta_{d}=\eta_{s}$.
The factor $\eta_{U}$ is the RGE factor for the running of $U_{\rm PMNS}$,
which is in principle flavor dependent, but all these factors are essentially
one, with the largest deviation from one being proportional to
$Y_{\tau}^{2}\sim 10^{-4}$. It is also worth noting that the $\eta$ factors
appearing in the (2,2) block of Eq. (64) may be absorbed into $v_{\Phi}$, and
therefore are not really needed for the fit. The $\eta^{\prime}_{\alpha}$
factors are the running factors of the various Yukawa couplings of Eq. (144)
between $M_{I}$ and $M_{G}$ defined as

$\eta_{\alpha}^{\prime}=\frac{Y_{\alpha}(M_{G})}{Y_{\alpha}(M_{I})}~{}~{}~{}~{}{\rm
for}~{}\alpha=(u_{L},\,u_{R},\,d_{L},\,d_{R},\,D)~{}.$ (66)

Here again there is a flavor dependence, but the lighter fermions of the same
type have the same RGE factor.

The $\eta_{i}$ factors are obtained by solving the SM renormalization group
equations to two-loop accuracy. We choose the benchmark values of
$M_{I}=9.02\times 10^{10}$ GeV, corresponding to the unification picture shown
in Fig. 2. We include the threshold effects from the vector-like fermions
which have masses below $M_{I}$ (except for $U_{3}$ which has a mass
$M_{U_{3}}=M_{I}$ for this fit). The $\eta_{i}$ values are found to be

$\displaystyle\\{\eta_{u,c},\ \eta_{t},\ \eta_{d,s},\ \eta_{b},\
\eta_{\ell}\\}=\\{0.491,0.537,0.50,0.458,1.00\\}\,,$ (67)

where $\eta_{\ell}\equiv\eta_{e,\mu,\tau}$.

To obtain the $\eta^{\prime}_{\alpha}$ factors to go from $M_{I}$ to $M_{G}$,
we numerically solve the two-loop beta functions for the Yukawa couplings
$(Y_{uL},Y_{uR},Y_{dL},Y_{\ell L},Y_{D})$. These matrices relevant for
$M_{I}\leq\mu\leq M_{G}$ are defined in Eq. (144) in Appendix A.4 along with
their boundary conditions at $M_{G}$ given in Eq. (152). The full set of two-
loop RGE is presented in Appendix A.4. Corresponding to the unification
picture of Fig. 2 and the values of the gauge couplings given in Eq. (60) and
the scales given in Eq. (61) we find the $\eta_{\alpha}^{\prime}$ factors to
be

$\displaystyle\\{\eta^{\prime}_{u_{L},c_{L}},\ \eta^{\prime}_{t_{L}},\
\eta^{\prime}_{u_{R},c_{R}},\ \eta^{\prime}_{t_{R}},\
\eta^{\prime}_{d_{L},s_{L}}\\}=\\{0.468,0.48,0.0951,0.134,0.468\\},$
$\displaystyle\\{\eta^{\prime}_{b_{L}},\ \eta^{\prime}_{\ell_{L}},\
\eta^{\prime}_{d_{R},s_{R}},\ \eta^{\prime}_{b_{R}},\
\eta^{\prime}_{D}\\}=\\{0.475,0.805,0.150,0.092,0.30\\}\,.$ (68)

These values are obtained by setting the boundary conditions $Y_{\ell}=Y_{\ell
L}$ and $Y_{u}=Y_{uL}$ at $M_{I}$ and then randomly choosing the rest of the
Yukawa couplings and accepting only those that match the boundary conditions
at the GUT scale, $Y_{uL}=Y_{uR}$ and $Y_{dL}=Y_{dR}=Y_{\ell}^{T}$, see Eq.
(152). It turns out that due to the smallness of all Yukawa couplings, except
for $Y_{t}$, in this running no off-diagonal entries in $Y_{u_{L}}$,
$Y_{u_{R}}$ and $Y_{\ell_{L}}$ are induced by the RGE flow. This makes it
relatively easy to select the random Yukawa couplings that match the GUT scale
boundary conditions.

We choose as input the quark and lepton masses at a scale $\mu=m_{t}$ the
values tabulated in Ref. Huang:2020hdv . These values are summarized in Table
1. Here we also list the running masses at $\mu=M_{I}$, using the $\eta_{i}$
factors given in Eq. (67). The goal for the fermion mass fit is then to
reproduce correctly the masses of the down-type quarks arising from Eq. (64)
with the charged lepton masses at $M_{I}$ taken as input. For this analysis we
set $M_{I}=9.02\times 10^{10}$ GeV, see Eq. (61), and vary the known neutrino
oscillation parameters in their 3-sigma ranges. We then numerically scan the
parameters of Eq. (64) to obtain fits to the down-type quarks masses. Correct
masses are reproduced, but only for a narrow range of the currently unknown
parameters $m_{\nu_{1}}$ and $\delta_{CP}$. The 1-sigma and 2-sigma allowed
regions of these parameters are shown in the $(m_{\nu_{1}}-\delta_{CP})$ plane
in Fig. 4. At the 2-sigma level this analysis shows that the neutrino
parameters should lie in the range

$\displaystyle\delta_{CP}$ $\displaystyle=$ $\displaystyle(130.4\pm
1.2)^{\circ}~{}{\rm or}~{}(229.6\pm 1.2)^{\circ}$ $\displaystyle m_{\nu_{1}}$
$\displaystyle=$ $\displaystyle(4.8-8.4)~{}{\rm meV}~{}.$ (69)

The two solutions for $\delta_{CP}$, which differ by a change of its sign,
lead to the same eigenvalues of ${\cal M}_{d}^{\dagger}{\cal M}_{d}$, and
cannot be distinguished by the down-type quark masses.

$\mu/{\rm GeV}$ |  $m_{t}$ Huang:2020hdv  | $M_{I}$
---|---|---
$m_{d}/{\rm MeV}$ | $2.56\pm 0.18$ | $1.28\pm 0.090$
$m_{s}/{\rm MeV}$ | $50.9\pm 4.41$ | $25.45\pm 2.205$
$m_{b}/{\rm GeV}$ | $2.702\pm 0.025$ | $1.237\pm 0.0114$
$m_{u}/{\rm MeV}$ | $1.18\pm 0.20$ | $0.579\pm 0.098$
$m_{c}/{\rm GeV}$ | $0.594\pm 0.017$ | $0.2916\pm 0.0083$
$m_{t}/{\rm GeV}$ | $161.98\pm 0.75$ | $86.98\pm 0.40$
$m_{e}/{\rm MeV}$ | $0.48583\pm 0.00045$ | $0.48568\pm 0.00045$
$m_{\mu}/{\rm GeV}$ | $0.102347\pm 0.000019$ | $0.10231\pm 0.000019$
$m_{\tau}/{\rm GeV}$ | $1.73850\pm 0.00014$ | $1.73815\pm 0.00014$
Table 1: Running masses of the quarks and charged lepton at $\mu=m_{t}$ and at
$\mu=M_{I}$. These are obtained with $M_{I}=9.02\times 10^{10}$ GeV, and by
including the threshold corrections arising from vector-like fermions
$(U,D,E)$ which have masses below $M_{I}$.

We present a benchmark fit for the masses at the intermediate scale by setting
the parameter
$v_{\phi}\eta_{\nu}\eta_{u}^{2}/(v_{\nu}\eta^{\prime}_{D})=1.0\times
10^{7}M_{I}$:

$\displaystyle{\cal M}_{d}(M_{I})=$
$\displaystyle\begin{pmatrix}0&0&0&-0.008&0&0\\\ 0&0&0&0&0.17&0\\\
0&0&0&0&0&-2.95\\\ -7.53\cdot 10^{5}&0&0&1.19\cdot 10^{4}&4.42\cdot
10^{6}\,e^{1.06\,i}&9.60\cdot 10^{6}\,e^{2.76\,i}\\\ 0&1.58\cdot
10^{8}&0&\bullet&2.75\cdot 10^{7}&2.03\cdot 10^{7}\,e^{-0.044\,i}\\\
0&0&-4.38\cdot 10^{9}&\bullet&\bullet&2.26\cdot 10^{7}\\\ \end{pmatrix}$ (70)

Here the $``\bullet"$ represents the complex conjugated entry such that the
lower $(2,2)$ block of Eq. (70) is hermitian. The matrix ${\cal M}_{d}$ can be
block-diagonalized with a biunitary transformation given in Eq. (38). While
the $d_{L}-D_{L}$ mixing angles, parametrized by the off-diagonal entries of
$\xi_{L}$ are extremely small, of order $m_{d_{i}}/M_{D_{i}}$, this is not the
case for $d_{R}-D_{R}$ mixing, which can be as large as ${\cal O}(1)$. As a
result, the seesaw approximation is not very good when applied to Eq. (70).
Numerical diagonalization of Eq. (70) yields the light and heavy eigenvalues
to be

$\displaystyle\\{m_{d},m_{s},m_{b}\\}(M_{I})=\\{8.06\times 10^{-4}\ {\rm
GeV},\ 2.83\times 10^{-2}\ {\rm GeV},\ 1.24\ {\rm GeV}\\},\ $ (71)
$\displaystyle\\{m_{D_{1}},m_{D_{2}},m_{D_{3}}\\}(M_{I})=\\{1.05\times 10^{7}\
{\rm GeV},\ 1.62\times 10^{8}\ {\rm GeV},\ 4.38\times 10^{9}\ {\rm GeV}\\}.$
(72)

This fit corresponds to $m_{d}\ (2\,{\rm GeV})=2.92\ {\rm MeV}$, $m_{s}(2\
{\rm GeV})=0.102$ GeV, and $m_{b}(m_{b})=4.16$ GeV. These values are in good
agreement with observations, although the value of $m_{d}$ is on the lower
side. However, this is consistent with one lattice evaluation which finds
$m_{d}(2\ {\rm GeV})=3.68\pm 0.29\pm 0.10$ of Ref. Aoki:2012st .

The model only admits normal ordering of neutrino masses. The predictions of
the model for the CP violating phase and the lightest neutrino mass are shown
in Fig. 4 where the allowed region is the shaded one. For values that lie
outside of the shaded region we find that $m_{d}(2\,{\rm GeV})$ becomes way
smaller than acceptable. For instance, if $\delta_{CP}=200^{\circ}$, which is
outside the shaded region, the maximum value of the down-quark mass is
$m_{d}\,(2\,{\rm GeV})\simeq 24.9$ keV, well below the experimental limit.
Similarly, for $m_{\rm Lightest}=1$ meV, which is outside the shaded region,
the maximum down-quark mass is found to be $m_{d}\,(2\,{\rm GeV})\simeq 23.5$
keV.

With the neutrino fit, we can evaluate the VEV $v_{\Phi}$ and from there the
value of the cubic scalar coupling $\mu_{1}$ (see Eq. (31)). For this
benchmark fit we find $v_{\nu}=3.3\times 10^{-7}$ GeV, leading to
$\mu_{1}=1.36\times 10^{16}$ GeV. This shows self-consistency of the analysis
and confirms the naturalness of the small Dirac neutrino masses within the
framework.

##  6 Proton Decay

In this section, we show the consistency of the model with proton decay
constraints. Since the gauge bosons of $SU(5)_{R}$ have masses of order
$M_{I}\sim 10^{11}$ GeV, it is imperative to establish that they do not
mediate rapid proton decay. The $H_{R}(1,5)$ Higgs field, which also has a
mass of order $M_{I}$, contains a color-triplet scalar that could potentially
mediate proton decay which should be consistent with experimental limits.
Finally, the $(X_{L}^{\mu},\,Y_{L}^{\mu})$ gauge bosons of $SU(5)_{L}$ would
mediate proton decay for which we estimate the lifetime.

The couplings of $X_{R}^{\mu}$ and $Y_{R}^{\mu}$ gauge bosons of $SU(5)_{R}$
to fermions are given, in the original basis, as Langacker:1980js

$\displaystyle{\cal L}_{X_{R}^{\mu},Y_{R}^{\mu}}$
$\displaystyle=\frac{g_{5R}}{\sqrt{2}}\left[\epsilon_{\alpha\beta\gamma}{(U^{T\gamma}_{L}})C\gamma_{\mu}u_{R}^{\beta}X_{R}^{\mu\alpha}+(E_{L}^{T})C\gamma_{\mu}d_{R}^{\alpha}X^{\mu*}_{R\alpha}\right.+(D_{L}^{T\alpha})C\gamma_{\mu}e_{R}X^{\mu*}_{R\alpha}$
$\displaystyle~{}~{}~{}+\epsilon_{\alpha\beta\gamma}(U_{L}^{T\gamma})C\gamma_{\mu}d_{R}^{\beta}Y_{R}^{\mu\alpha}-\left.(D_{L}^{T\alpha})C\gamma_{\mu}\nu_{R}Y^{\mu*}_{R\alpha}-(E_{L}^{T})C\gamma_{\mu}u_{R}^{\alpha}Y^{\mu*}_{R\alpha}\right]+h.c.$
(73)

It turns out that these interactions do not lead to proton decay, owing to the
structure of the mass matrices of the model, as given in Eq. (29). These
matrices imply that in order to identify the light up-type quark and charged
lepton fields as ($u_{R},\,e_{R})$, one must interchange the fields as
$u_{R}\leftrightarrow U_{R}$ and $e_{R}\leftrightarrow E_{R}$. Furthermore,
the fields $U_{L}$ and $E_{L}$ are heavy with no mixing with the light $u_{L}$
and $e_{L}$ fields. In the down-type quark sector there exists $d_{R}-D_{R}$
mixing, which could be of order one, as well as $d_{L}-D_{L}$ mixing which is
suppressed by factors of the type $(Y_{d_{i}}/Y_{D_{i}})(M_{W}/M_{I})$, which
turns out to be less than $10^{-10}$ for the first two families of quarks
relevant for proton decay. It follows then that in Eq. (73) all terms involve
at least one heavy field, except for the second last term with the $\nu_{R}$
field, with a suppressed coupling when converted to light $d_{L}$ quark field.
But this single term does not lead to proton decay since it conserves baryon
and lepton numbers, as can be seen by assigning $B$ and $L$ numbers of 1/3 and
1 to the $Y_{R}^{\mu}$ gauge boson. We thus conclude that the proposed form of
the mass matrices, given in Eq. ((29)), is consistent with proton lifetime
limits, even when the $SU(5)_{R}$ gauge bosons have masses of order $M_{I}\sim
10^{11}$ GeV.

The $B$-violating interactions of $(X_{L}^{\mu},\,Y_{L}^{\mu})$ gauge bosons
of $SU(5)_{L}$, which have masses of order $M_{G}$, are given in the original
basis of fermion fields by

$\displaystyle{\cal L}_{X_{L}^{\mu},Y_{L}^{\mu}}$
$\displaystyle=\frac{g_{5}}{\sqrt{2}}\left[\epsilon_{\alpha\beta\gamma}{(U^{T\gamma}_{R}})C\gamma_{\mu}u_{L}^{\beta}X_{L}^{\mu\alpha}+(E_{R}^{T})C\gamma_{\mu}d_{L}^{\alpha}X^{\mu*}_{L\alpha}\right.+(D_{R}^{T\alpha})C\gamma_{\mu}e_{L}X^{\mu*}_{L\alpha}$
$\displaystyle~{}~{}~{}+\epsilon_{\alpha\beta\gamma}(U_{R}^{T\gamma})C\gamma_{\mu}d_{L}^{\beta}Y_{L}^{\mu\alpha}-\left.(D_{R}^{T\alpha})C\gamma_{\mu}\nu_{L}Y^{\mu*}_{L\alpha}-(E_{R}^{T})\gamma_{\mu}u_{L}^{\alpha}Y^{\mu*}_{L\alpha}\right]+h.c.$
(74)

We now write down this Lagrangian in terms of the physical mass eigenstates of
quark and lepton fields. We adopt the convention where the CKM matrix in the
quark sector and the PMNS matrix in the lepton sector have their canonical
forms each with a single phase. The relevant transformation is given in Eq.
(34) and Eq. (41), along with the interchanges $u_{R}\leftrightarrow U_{R}$
and $e_{R}\leftrightarrow E_{R}$ so that the light fields are denoted as
$(u,d,e)$. The transformed Lagrangian is then given by (with the hats and
superscript 0 dropped from the fields):

$\displaystyle{\cal L}_{X_{L}^{\mu},Y_{L}^{\mu}}$
$\displaystyle=\frac{g_{5}}{\sqrt{2}}\left[\epsilon_{\alpha\beta\gamma}{(u^{T\gamma}_{R}})C\gamma_{\mu}(P^{\prime})^{2}\,u_{L}^{\beta}X_{L}^{\mu\alpha}+(e_{R}^{T})C\gamma_{\mu}\xi_{L}^{11}Q^{\prime*}d_{L}^{\alpha}X^{\mu*}_{L\alpha}\right.$
$\displaystyle~{}~{}~{}+(d_{R}^{T\alpha})Q^{\prime*}(\xi_{R}^{21})^{T}C\gamma_{\mu}e_{L}X^{\mu*}_{L\alpha}+\epsilon_{\alpha\beta\gamma}(u_{R}^{T\gamma})C\gamma_{\mu}(P^{\prime})^{2}V_{\rm
CKM}d_{L}^{\beta}Y_{L}^{\mu\alpha}$
$\displaystyle~{}~{}~{}-\left.(d_{R}^{T\alpha})Q^{\prime*}(\xi_{R}^{21})^{T}U_{\rm
PMNS}C\gamma_{\mu}\nu_{L}Y^{\mu*}_{L\alpha}-(e_{R}^{T})C\gamma_{\mu}(\xi_{L}^{11}Q^{\prime*}V_{\rm
CKM}^{\dagger})u_{L}^{\alpha}Y^{\mu*}_{L\alpha}\right]+h.c.$ (75)

It is straightforward to derive the baryon number violating effective four-
fermion operator by integrating out the $(X_{L}^{\mu},\,Y_{L}^{\mu})$ gauge
bosons. It may appear that Eq. (75) differs significantly, owing to the
appearance of the matrices $(\xi_{R}^{21})^{T}$ from its analogues in standard
$SU(5)$ GUT, but this is not the case. The numerical structure of the matrices
$|\xi_{L}^{11}|$ and $|\xi_{R}^{21}|$ corresponding to the benchmark fit for
fermion masses, Eq. (70), are found to be:

$\displaystyle|\xi_{L}^{11}|=\left(\begin{array}[]{ccc}0.9967&0.0808&0.00029\\\
0.0802&0.989&0.1239\\\ 0.00974&0.1235&0.9922\\\ \end{array}\right)\,,\hskip
14.22636pt|\xi_{R}^{21}|=\left(\begin{array}[]{ccc}0.9997&0.00545&0.00298\\\
0.00193&0.1948&0.8686\\\ 0&0&0.4370\\\ \end{array}\right)\,.$ (82)

This structure implies that the light $d_{R}$ field is contained almost
entirely in the original $D_{R}$ field. For the leading decay mode of the
proton, $p\rightarrow e^{+}\pi^{0}$, is a function of $(\xi_{R}^{21})_{11}$,
which is close to unity. Thus the model prediction for the lifetime of the
proton is very similar to that in standard $SU(5)$, except that in the present
case it is much longer, owing to the higher unification scale $M_{G}=(7\times
10^{16}-8\times 10^{17})$ GeV. We estimate the corresponding lifetime of the
proton to be of order $(10^{38}-10^{42})$ yrs, which is well beyond the reach
of forthcoming experiments JUNO, HyperKamiokande and DUNE (for reviews on
current status and future prospects see Refs. Babu:2013jba ; Dev:2022jbf .) We
note, however, that if the GUT scale is lowered in some variations of the
present model by a factor of 10, the lifetime of the proton will be shortened
by a factor of $10^{4}$, in which case the forthcoming experiments will be
sensitive to its decay.

Color-triplet scalars arising from $H_{R}(1,{\bf 5})$ and $H_{L}(5,1)$,
denoted as $H_{R,L}^{c}$, can mediate proton decay. Since the mass of
$H_{R}^{c}$ is at $M_{I}$, one should verify that this field does not mediate
rapid proton decay. We shall see that with the flavor structure of the mass
matrices given in Eq. (29), $H_{R}^{c}$ field having a mass of order $M_{I}$
is indeed safe from proton decay constraints. The interactions of the
$H_{R}^{c}$ field with quarks and leptons are given by the Lagrangian (in the
original basis)

$\displaystyle{\cal L}_{\rm Yuk}^{H_{R}^{c}}$ $\displaystyle\supset$
$\displaystyle(Y_{u})_{ij}^{*}\left[\frac{1}{2}(u_{Ri}d_{Rj}-d_{Ri}u_{Rj})H_{R}^{c}+U_{Li}E_{Lj}(H_{R}^{c})^{*}\right]$
(83) $\displaystyle+$
$\displaystyle(Y_{\ell}^{\dagger})_{ij}\left[(\nu_{Ri}d_{Rj}-e_{Ri}u_{Rj})(H_{R}^{c})^{*}+D_{Li}U_{Lj}H_{R}^{c}\right]+h.c.$

Analogous interactions of the $H_{L}^{c}$ field are obtained by the
interchange $R\rightarrow L$ in Eq. (83), with identical Yukawa couplings
owing to parity symmetry. Now, with the fermion mass matrix structure of Eq.
(29), one must make the interchanges {$u_{R}\leftrightarrow
U_{R},e_{R}\leftrightarrow E_{R}\\}$ so that the light states are labeled as
$(u,\,e)$. Furthermore, the $(U_{L},\,E_{L})$ fields are entirely in the heavy
states. These features imply that the $H_{R}^{c}$ color-triplet scalar has
couplings which involve at least one heavy state, except for the term
$\nu_{R}d_{R}H_{R}^{c}$, which has terms with only light fermion fields.
However, with this single term, there is no baryon number violation, as can be
seen by assigning $H_{R}^{c}$ a $B$ charge of $-1/3$. The field $H_{R}^{c}$
behaves as a leptoquark in this case, not mediating proton decay.

Let us comment briefly about proton decay mediated by higher dimensional
operators suppressed by the Planck scale. One might worry, with the
$SU(5)_{R}$ surviving down to $M_{I}\sim 10^{11}$ GeV, that the gauge bosons
could mediate proton decay in presence of such Planck suppressed operators.
One such operator is the $\bar{\chi}_{R}\chi_{L}\Phi\Phi/M_{\rm Pl}$. Such a
term, if present, would induce a bare mass of order $M_{I}^{2}/M_{P}=10$ GeV
in the (2,2) block of ${\cal M}_{u}$ of (29). Such a mass term would induce
$u_{U}$ mixing, and thus lead to proton decay mediated by the
$(X_{R}^{\mu},\,Y_{R}^{\mu})$ gauge bosons. Keeping in mind that the mass
$U_{1}$ is $10^{-5}M_{I}=10^{5}$ GeV, the $u_{R}-U_{R}$ mixing angle arising
from here is about $10^{-4}$. The $u_{L}-U_{L}$ mixing is estimated to be much
smaller, of order $(m_{u}\times 10\ {\rm GeV})/(10^{5}\ {\rm
GeV})^{2}=10^{-12}$. These operators do not induce mass terms in the (2,2)
block of ${\cal M}_{\ell}$ of Eq. (29). Consequently, the leading proton decay
diagram from Eq. (73) would arise by combining the fourth and fifth term of
this equation. The amplitude of this diagram has a suppression factor of
$10^{-22}$, arising from the $u_{L}-U_{L}$ mixing suppression of $10^{-12}$
and the $d_{L}-D_{L}$ mixing suppression of order $10^{-10}$. The lifetime of
the proton arising from these diagrams is of order $10^{58}$ yrs, which is
well within limits.

##  7 Strong CP Solution via Parity

Parity solution to the strong CP problem Beg:1978mt ; Mohapatra:1978fy ;
Mohapatra:1995xd ; Kuchimanchi:1995rp ; Mohapatra:1997su ; Babu:2001se  is an
alternative to the popular axion solution Peccei:1977hh ; Weinberg:1977ma ;
Wilczek:1977pj . The parity solution has an advantage over the axion solution
in that unlike the currently viable invisible axion models Kim:1979if ;
Shifman:1979if ; Dine:1981rt ; Zhitnitsky:1980tq , which are subject to
destabilization by quantum gravitational effects which violate all global
symmetries Kamionkowski:1992mf ; Barr:1992qq ; Holman:1992us , these models
are stable as long as the parity breaking scale is less than about 100 TeV
Berezhiani:1992pq . The latter property makes these low scale parity-symmetric
models experimentally testable.

A realistic UV-complete implementation of the parity solution to the strong CP
problem was provided in Ref. Babu:1989rb . The fermion content of that left-
right symmetric model is the same as in the present $SU(5)_{L}\times
SU(5)_{R}$ model. There is therefore a good chance that the GUT embedding of
the model can also solve the strong CP problem via parity symmetry. Indeed, we
find this to be the case, as will be detailed in this section.

###  7.1 $\overline{\theta}$ at tree-level

The strong interactions admit a CP violating $\theta$-parameter in the
Lagrangian denoted as

${\cal L}^{\theta}_{\rm QCD}=\frac{\theta_{\rm
QCD}}{32\pi^{2}}G_{\mu\nu}^{a}\tilde{G}^{a}_{\mu\nu}~{}$ (84)

where $G_{\mu\nu}^{a}$ stand for the gluon fields and where
$\tilde{G}^{a}_{\mu\nu}=\frac{1}{2}\epsilon_{\mu\nu\alpha\beta}\,G^{a\,\alpha\beta}$.
The parameter $\theta_{\rm QCD}$ itself is unphysical, as its value can be
altered by chiral rotations on the quark fields. A physical observable remain
invariant and is given by

$\overline{\theta}=\theta_{\rm QCD}+{\rm arg}\left\\{{\rm
det}\left(M_{q}\right)\right\\}$ (85)

where $M_{q}$ is the quark mass matrix. $\theta_{\rm QCD}$ is odd under
parity, and therefore in a parity-symmetric theory it would vanish. The quark
mass matrix is in general not parity symmetric, since its entries arise from
parity breaking VEVs. However, if the determinant of the quark mass matrix is
real in a parity-symmetric theory, then $\overline{\theta}=0$ at tree-level.
Quantum corrections would in general induce nonzero but finite value for
$\overline{\theta}$, but this may be within the experimentally allowed range
of $\overline{\theta}\leq 1.19\times 10^{-10}$, arising from neutron EDM
limits Dragos:2019oxn ; ParticleDataGroup:2022pth .

In the $SU(5)_{L}\times SU(5)_{R}$ with parity, the gauge fields transform
under $P$ as

$\displaystyle G_{L\mu}^{a}(t,x)\rightarrow G_{R\mu}^{a}(t,-x)\times s(\mu)$
$\displaystyle G_{R\mu}^{a}(t,x)\rightarrow G_{L\mu}^{a}(t,-x)\times s(\mu)$
(86) $\text{where}~{}~{}s(\mu)=\begin{cases}+1&\text{for}~{}~{}\mu=0\\\
-1&\text{for}~{}~{}\mu=i=1,2,3\end{cases}$ (87)

Here $G_{L\mu}^{a}$ are the $SU(5)_{L}$ gauge bosons, while $G_{R\mu}^{a}$
stand for their $SU(5)_{R}$ counterparts. Consistent with this transformation,
a certain $\theta^{\prime}$ term is allowed in the $SU(5)_{L}\times SU(5)_{R}$
theory, which is given by

${\cal
L}_{\theta}^{\prime}=\frac{\theta^{\prime}}{32\pi^{2}}\left(G_{L\mu\nu}^{a}\tilde{G}^{a}_{L\mu\nu}-G_{R\mu\nu}^{a}\tilde{G}_{R\mu\nu}^{a}\right)~{}.$
(88)

$SU(3)_{c}$ is embedded as a diagonal subgroup of $SU(3)_{cL}\times
SU(3)_{cR}$ in the full theory. The axigluons of the broken $SU(3)$ are
$G_{A\mu}^{a}=(G_{L\mu}^{a}+G_{R\mu}^{a})/\sqrt{2}$, while the massless QCD
gluons are $G_{\mu}^{a}=(G_{L\mu}^{a}-G_{R\mu}^{a})/\sqrt{2}$. The terms from
Eq. (88) involve only the axigluons, and therefore $\theta_{\rm QCD}$
involving the usual gluon fields is zero.

With the quark mass matrices given in Eq. (29), det[$M_{q}]={\rm det}[{\cal
M}_{u}{\cal M}_{d}$] is real in the present GUT framework, implying that
$\overline{\theta}=0$ at the tree-level. We now proceed to show that all one-
loop corrections to $\overline{\theta}$ are also vanishing, owing to the
structure of the theory.

###  7.2 Vanishing of one-loop $\overline{\theta}$ contributions

Here we show that all one-loop diagrams which could potentially generate
nonzero contributions to $\overline{\theta}$ are vanishing within the model.
We follow the procedure adopted in Ref. Babu:1989rb  to perform this
calculation, which we find very convenient. First of all, we evaluate the one-
loop diagrams at the GUT scale, so that the fermion mass matrices have the
form given in Eq. (29). We work in the original basis of fermion fields,
before any field rotations are performed. For ease of writing we shall use the
notation $Y_{d}$ which is to be identified as $Y_{\ell}^{T}$ for the Yukawa
coupling matrices appearing in ${\cal M}_{d}$ of Eq. (29).

The one-loop contribution to $\overline{\theta}$ are computed from corrections
to the quark mass matrices ${\cal M}_{u}$ and ${\cal M}_{d}$ of Eq. (29).
These corrections arise via the exchange of scalar fields as well as gauge
fields present in the theory. We summarize these interactions here. The Yukawa
interactions of scalar fields with fermions are contained in the left-right
symmetric Higgs doublet couplings that appear in ${\cal M}_{u}$ and ${\cal
M}_{d}$ of Eq. (29), the color-triplet couplings given in Eq. (83) and their
left-handed counterparts, and the Yukawa couplings of the
$\Phi(\overline{5},5)$ field given in Eq. (22), which can be expanded to the
form

$\displaystyle-{\cal L}_{\rm Yuk}$ $\displaystyle\supset$
$\displaystyle(Y_{D}^{*})_{ij}\left[\left(\overline{D}_{Rj\beta}D_{Li}^{\alpha}\right)(\phi_{DD}^{o})^{\beta}_{\alpha}+\left(\overline{D}_{Rj\alpha}D_{Li}^{\alpha}\right)\phi_{DD}^{s}+\left(D_{Ri}^{T}C\nu_{Rj}\right)\phi_{D\nu}\right.$
(89) $\displaystyle+$
$\displaystyle\left.\left(D_{Ri}^{T}Ce_{Rj}\right)\phi_{De}+\left(\overline{L}_{Li}D^{c}_{Rj}\right)\phi_{LD}+\left(\overline{L}_{Li}\nu_{Rj}\right)\phi_{L\nu}+\left(\overline{L}_{Li}e_{Rj}\right)\phi_{Le}\right]~{}.$

Here the notation used for the various scalar fields is self-explanatory, and
their quantum numbers under the SM gauge symmetry are listed in Eq. (137) of
Appendix A.2. Other scalar fields, such as the $\Sigma_{L}(75,1)$ and
$\eta(\overline{15},15)$ used in the model play no role in this calculation,
since they have no Yukawa couplings to the fermions.

The gauge bosons of the full $SU(5)_{L}\times SU(5)_{R}$ theory have various
interactions involving fermions. In particular, the
$(X_{R}^{\mu},\,Y_{R}^{\mu})$ gauge interactions are shown in Eq. (73), while
the $(X_{L}^{\mu},\,Y_{L}^{\mu})$ interactions are given in Eq. (74).
Additionally, the gauge interactions of the axi-gluon fields $G_{A}^{\mu}$ and
the axi-$Z^{\prime}$ field $Z_{A}^{\mu}$, which have masses of order $M_{I}$,
are given by

$\displaystyle{\cal
L}^{G^{\mu}_{A},Z^{\mu}_{A}}=-\sqrt{\frac{3}{10}}g_{5}(Y_{L}-Y_{R})\left(\bar{f}_{L,R}^{i}\gamma_{\mu}f_{L,R}^{i}\right)Z_{A}^{\mu}\
-\frac{g_{5}}{\sqrt{2}}(T^{a}G_{A}^{\mu
a})_{\alpha\beta}\left(\overline{f}_{L\alpha}^{i}\gamma_{\mu}f_{L\beta}^{i}-\overline{f}_{R\alpha}^{i}\gamma_{\mu}f_{R\beta}^{i}\right).$
(90)

Here $Y_{L}$ and $Y_{R}$ are the $U(1)_{L}$ and $U(1)_{R}$ charges of various
fields, listed in Eq. (136) of Appendix A.1 (The $Y_{R}$ charges are identical
to $Y_{L}$ charges, but when referred to fields transforming under
$SU(5)_{R}$). In addition, the theory has $W_{R}^{\pm}$ and $Z_{R}^{0}$ gauge
bosons, as well as the SM gauge bosons. We shall collectively denote the
($G_{A}^{\mu},\,G^{\mu},\,Z^{\mu},A^{\mu},Z_{A}^{\mu}$) as $V^{\mu}$. It is
noteworthy that all these $V^{\mu}$ fields have flavor diagonal couplings to
fermions, unlike the $(X^{\mu}_{L,R},\,Y^{\mu}_{L,R})$ gauge bosons. It is
also worth noting that the $W_{L,R}^{\pm}$ gauge bosons, having couplings only
to fermions of one specific chirality, will not participate in the quark mass
matrix corrections, when evaluated in the Feynman gauge, which we adopt.

Having identified all interactions that can correct the quark mass matrices,
we proceed to evaluate the relevant one-loop diagrams. We adopt standard
perturbation theory techniques to evaluate $\overline{\theta}$, following the
procedure outlined in Ref. Babu:1989rb .777More recently, Fock-Schwinger
method has been applied to evaluate the loop contributions to $\bar{\theta}$
in Ref. Hisano:2023izx . We work in the flavor basis, where Eq. (29) is
written. The mass matrices for ${\cal M}_{u}$ and ${\cal M}_{d}$ are treated
as part of the interaction Lagrangian. All the one-loop diagrams that can
potentially contribute to $\overline{\theta}$ in this basis are shown in Fig.
5 for ${\cal M}_{d}$ and in Fig. 6 for ${\cal M}_{u}$. The crosses in the
internal fermion lines in these diagrams represent chirality flipping
interactions. Since we work in the flavor basis, there could be multiple such
chirality flips which should be summed. This can be easily done, which leads
to the full tree-level propagator with all possible mass insertion, in the
down-type quark sector, given by

$\overline{f}_{R}\left(\mathcal{M}_{d}^{\dagger}\frac{k^{2}}{k^{2}-\mathcal{M}_{d}\mathcal{M}_{d}^{\dagger}}\right)f_{L}$
(91)

where $f_{R,L}\equiv(d_{R,L},\,D_{R,L})^{T}$ are 6-component column vectors.
Analogous expressions hold for ${\cal M}_{u}$ and ${\cal M}_{\ell}$ as well.

We can write down the loop-corrected quark mass matrix as

$\displaystyle M_{q}~{}=M^{(0)}_{q}+\delta M_{q}=~{}M^{(0)}_{q}(1+C)$ (92)

where $M^{(0)}_{q}$ is the tree-level quark mass matrix for $q=u,d$ given by
Eq. (29) and $C=C_{1}+C_{2}+...$ is the loop contribution with subscripts
denoting 1-loop, 2-loop, etc. Since Det($M^{(0)}_{q}$) is real for $q=u,d$,
one can write $\overline{\theta}$ as

$\displaystyle\overline{\theta}~{}=~{}{\rm Im~{}Tr}C_{1}+~{}{\rm
Im~{}Tr}(C_{2}-\frac{1}{2}C_{1}^{2})+...$ (93)

We denote the one-loop correction to the quark mass matrix as

$\displaystyle\delta M_{q}=\left(\begin{array}[]{cc}\delta M^{q}_{LL}&\delta
M^{q}_{LH}\\\ \delta M^{q}_{HL}&\delta M^{q}_{HH}\end{array}\right)\,,$ (96)

where $H,L$ stand for the heavy and light sector respectively. The induced
$\overline{\theta}$ from $q=d$ sector is then given by

$\displaystyle\overline{\theta}~{}=~{}{\rm
ImTr}\left[-\frac{1}{\kappa_{L}\kappa_{R}}\delta
M^{d}_{LL}(Y_{d}^{\dagger})^{-1}M_{D}Y_{d}^{-1}\right.\left.+\frac{1}{\kappa_{L}}\delta
M_{LH}^{d}Y_{d}^{-1}+\frac{1}{\kappa_{R}}\delta
M_{HL}^{d}(Y_{d}^{\dagger})^{-1}\right].$ (97)

Here we have defined $M_{D}=Y_{D}v_{\Phi}$, which is the (2,2) block of ${\cal
M}_{d}$. The contribution for for $q=u$ can be obtained similarly by replacing
$Y_{d}\to Y_{u}$, but without the first term in Eq. (97) since $M_{U}=0$. Note
that $\delta M_{HH}^{q}$ does not contribute to $\overline{\theta}$ at one-
loop order. As a result we do not include diagrams where both external legs
are the heavy $D$-quarks or $U$-quarks in Fig. 5 and in Fig. 6. It is also
worth noting that the $\overline{\theta}$ corrections arising from $\delta
M_{LL}^{u}$ will be automatically zero since $M_{U}=0$, even though the
correction to the mass matrix itself is nonzero. We have included such
diagrams in Fig. 6 (a) and (h), but their contribution to $\overline{\theta}$
vanishes.

The propagators relevant to the evaluation of Figs. 5 and 6 can be obtained
from the full tree-level propagator of Eq. (91) by appropriate projection
operators. It is helpful to define the inverse of the propagator matrix as

$\left(\mathcal{M}_{d}\mathcal{M}_{d}^{\dagger}-k^{2}\right)^{-1}\equiv\left(\begin{array}[]{ll}x(k^{2})&~{}~{}y(k^{2})\\\
y^{\dagger}(k^{2})&~{}~{}z(k^{2})\end{array}\right)~{}.$ (98)

Here $x=x^{\dagger}$, $z=z^{\dagger}$ and $y$ are $3\times 3$ block matrices,
which obey the following relations from matrix multiplication:

$\displaystyle\left(\kappa_{R}^{2}Y_{d}^{\dagger}Y_{d}+M_{D}M_{D}^{\dagger}-k^{2}\right)y^{\dagger}=-\kappa_{L}M_{D}Y_{d}^{\dagger}x,$
(99)
$\displaystyle\kappa_{L}Y_{d}Y_{d}^{\dagger}X+Y_{d}M_{D}^{\dagger}y^{\dagger}=\frac{1}{\kappa_{L}}\left(I+k^{2}x\right),$
(100) $\displaystyle y=-\kappa_{L}HY_{d}M_{D}^{\dagger}z,~{}~{}{\rm
where}~{}~{}H=H^{\dagger}=\left(\kappa_{L}^{2}Y_{d}Y_{d}^{\dagger}-k^{2}\right)^{-1}~{},$
(101)
$\displaystyle\kappa_{L}y^{\dagger}Y_{d}M_{D}^{\dagger}+z(M_{D}M_{D}^{\dagger}+\kappa_{R}^{2}Y_{d}^{\dagger}Y_{d}-k^{2})=I~{}.$
(102)

Although we have denoted $M_{D}^{\dagger}$ to be different from $M_{D}$ for
generality, in our case owing to parity symmetry we have
$M_{D}=M_{D}^{\dagger}$, which we shall adopt. The tree-level interactions
corresponding to the crosses in Fig. (5) can now be read off from the
effective Lagrangian given by

$\displaystyle-\mathcal{L}_{\mathrm{eff}}^{\mathrm{tree}}=$
$\displaystyle\bar{D}_{R}\left[\frac{k^{4}}{\kappa_{L}}Y_{d}^{-1}y(k^{2})\right]D_{L}+\bar{d}_{R}\left[k^{2}Y_{d}\,\kappa_{R}\
\,z(k^{2})\right]D_{L}$
$\displaystyle+\bar{D}_{R}\left[\frac{k^{2}}{\kappa_{L}}Y_{d}^{-1}\left[I+k^{2}\
x(k^{2})\right]\right]d_{L}+\bar{d}_{R}\left[k^{2}Y_{d}\,\kappa_{R}\,y^{\dagger}(k^{2})\right]d_{L}+\text{
h.c. }$ (103)

Similar expressions are valid for the up-type quark sector as well as the
charged leptons sector, except that since $M_{U}=M_{E}=0$ for these sectors,
$y(k^{2})=0$, and consequently the analogs of the first and last terms of Eq.
(103) will be vanishing.

We are now ready to evaluate the contributions of each graph in Fig. 5 and
Fig. 6 to $\overline{\theta}$. We begin with Fig. 5 (a). The correction to the
down quark mass matrix from here is given by

$\displaystyle\delta M_{LL}^{d}=$
$\displaystyle\int\frac{d^{4}k}{(2\pi)^{4}}Y_{d}\frac{1}{\kappa_{L}}(Y_{d})^{-1}\frac{k^{2}y\left(k^{2}\right)Y_{d}^{\dagger}\lambda\,\kappa_{L}\kappa_{R}}{\left[(p-k)^{2}-M_{H_{L}^{0}}^{2}\right]\left[(p-k)^{2}-M_{H_{R}^{0}}^{2}\right]}~{}.$
(104)

Here $\lambda$ is a certain quartic coupling in the Higgs potential. It is
important to recall that all couplings in the Higgs potential are real-valued
in the model. While fields with identical quantum numbers under
$SU(3)_{c}\times U(1)_{\rm em}$ will mix, there is no mixing between scalar
fields and pseudo-scalar fields owing to the reality of the Higgs potential
couplings. Using Eq. (97) one can write the $\overline{\theta}$ contribution
as

$\displaystyle\overline{\theta}=-\operatorname{Im}\operatorname{Tr}\left(\frac{\lambda}{\kappa_{L}}\int\frac{d^{4}k}{(2\pi)^{4}}\frac{k^{2}y\left(k^{2}\right)M_{D}(Y_{d})^{-1}}{\left[(p-k)^{2}-M_{H_{L}^{0}}^{2}\right]\left[(p-k)^{2}-M_{H_{R}^{0}}^{2}\right]}\right)$
(105)

The trace can be performed before doing the momentum integral, and we find

$\displaystyle\operatorname{Tr}\left[y(k^{2})M_{D}(Y_{d})^{-1}\right]=-\kappa_{L}\operatorname{Tr}\left[\left(Y_{d}^{\dagger}Y_{d}\kappa_{L}^{2}-k^{2}\right)^{-1}M_{D}^{\dagger}\
z(k^{2})M_{D}\right]~{}.$ (106)

The matrix that is traced over here is a product of two hermitian matrices,
and thus its trace is real. Consequently the contribution to
$\overline{\theta}$ from Fig. (5) (a) is zero.

For the other diagrams we follow the same technique and summarize our results
here. Fig. 5 (b) contains two diagrams. While the diagram with $d_{R}$
incoming and $D_{L}$ outgoing contributes to $\delta M_{HL}^{d}$, it is the
hermitian conjugate of the diagram where $d_{L}$ is incoming and $D_{R}$
outgoing that contributes to $\delta M_{LH}^{d}$ of Eq. (96). This remark
applies to other diagrams as well. We summarize the flavor structure of these
diagrams and show that each contribution to $\overline{\theta}$ is vanishing.

  * •

5-(b): Fig. 5 (b) has the following flavor structure:

$\displaystyle d_{R}-D_{L}\ (\delta M_{HL}^{d}):{\rm Im\ Tr}\
[Y_{d}^{\dagger}Y_{d}\ z(k^{2})Y_{d}^{\dagger}(Y_{d}^{\dagger})^{-1}]=0$ (107)
$\displaystyle D_{R}-d_{L}\ (\delta M_{LH}^{d}):{\rm Im\ Tr}\
[Y_{d}Y_{d}^{-1}\ [I+k^{2}x(k^{2})]Y_{d}Y_{d}^{-1}]=0$ (108)

The first trace is over the product of two hermitian matrices, while the
second one is over a hermitian matrix, both of which are real, leading to zero
contribution to $\overline{\theta}$.

  * •

5-(c):

$\displaystyle d_{R}-D_{L}\ (\delta M_{HL}^{d}):{\rm Im\ Tr}\
[Y_{d}^{\dagger}Y_{u}\ z(k^{2})Y_{u}^{\dagger}(Y_{d}^{\dagger})^{-1}]=0$ (109)
$\displaystyle D_{R}-d_{L}\ (\delta M_{LH}^{d}):{\rm Im\ Tr}\
[Y_{u}Y_{u}^{-1}\ [I+k^{2}x(k^{2})]Y_{d}Y_{d}^{-1}]=0$ (110)

  * •

5-(d):

$\displaystyle d_{R}-D_{L}\ (\delta M_{HL}^{d}):{\rm Im\ Tr}\
[Y_{d}^{\dagger}\ z^{T}(k^{2})Y_{u}^{T}Y_{u}^{*}(Y_{d}^{\dagger})^{-1}]=0$
(111) $\displaystyle D_{R}-d_{L}\ (\delta M_{LH}^{d}):{\rm Im\ Tr}\
[Y_{u}^{T}[I+k^{2}x(k^{2})](Y_{u}^{T})^{-1}\ Y_{d}Y_{d}^{-1}]=0$ (112)

  * •

5-(e):

$\displaystyle d_{R}-D_{L}\ (\delta M_{HL}^{d}):{\rm Im\ Tr}\
[(\kappa_{L}^{2}Y_{d}^{\dagger}Y_{d}-k^{2})^{-1}M_{D}^{\dagger}z(k^{2})Y_{D}]=0$
(113) $\displaystyle D_{R}-d_{L}\ (\delta M_{LH}^{d}):{\rm Im\ Tr}\
[(Y_{d}^{\dagger}Y_{d}\kappa_{L}^{2}-k^{2})^{-1}M_{D}^{\dagger}z(k^{2})Y_{D}]=0$
(114)

  * •

5-(f):

$\displaystyle d_{R}-D_{L}\ (\delta M_{HL}^{d}):{\rm Im\ Tr}\
[z^{\dagger}(k^{2})Y_{d}^{\dagger}(Y_{d}^{\dagger})^{-1}]=0$ (115)
$\displaystyle D_{R}-d_{L}\ (\delta M_{LH}^{d}):{\rm Im\ Tr}\
[(I+k^{2}x(k^{2}))^{\dagger}(Y_{d}Y_{d}^{\dagger})^{-1}]=0$ (116)

  * •

5-(g):

$\displaystyle d_{R}-D_{L}\ (\delta M_{HL}^{d}):{\rm Im\ Tr}\
[z^{\dagger}(k^{2})Y_{d}^{\dagger}(Y_{d}^{\dagger})^{-1}]=0$ (117)
$\displaystyle D_{R}-d_{L}\ (\delta M_{LH}^{d}):{\rm Im\ Tr}\
[(I+k^{2}x(k^{2}))^{\dagger}(Y_{d}Y_{d}^{\dagger})^{-1}]=0$ (118)

  * •

5-(h):

$\displaystyle d_{R}-d_{L}\ (\delta M_{HL}^{d}):{\rm Im\ Tr}\
[(Y_{d}^{\dagger}Y_{d}\kappa_{L}^{2}-k^{2})^{-1}M_{D}^{\dagger}z(k^{2})M_{D}]=0$
(119)

Now we turn to the diagrams correcting the up-type quark mass matrix shown in
Fig. 6and summarize our results here. As noted earlier, even though the
contributions from Fig. 6 (a) and (h) to $\delta M_{LL}^{u}$ are nonzero, they
do not contribute to $\overline{\theta}$ owing to the condition $M_{U}=0$, see
Eq. (97).

  * •

6-(b):

$\displaystyle u_{R}-U_{L}\ (\delta M_{HL}^{u}):{\rm Im\ Tr}\
[Y_{u}^{\dagger}Y_{d}z(k^{2})Y_{d}^{\dagger}(Y_{u}^{\dagger})^{-1}]=0$ (120)
$\displaystyle U_{R}-u_{L}(\delta M_{LH}^{u}):{\rm Im\ Tr}\
[Y_{d}Y_{d}^{-1}[I+k^{2}x(k^{2})]Y_{u}Y_{u}^{-1}]=0$ (121)

  * •

6-(c):

$\displaystyle u_{R}-U_{L}\ (\delta M_{HL}^{u}):{\rm Im\ Tr}\
[Y_{d}^{*}z^{T}(k^{2})Y_{d}^{T}Y_{u}^{\dagger}(Y_{u}^{\dagger})^{-1}]=0$ (122)
$\displaystyle U_{R}-u_{L}(\delta M_{LH}^{u}):{\rm Im\ Tr}\
[Y_{u}[I+k^{2}x(k^{2})]^{T}(Y_{d}^{T})^{-1}Y_{d}^{T}Y_{u}^{-1}]=0$ (123)

  * •

6-(d):

$\displaystyle u_{R}-U_{L}\ (\delta M_{HL}^{u}):{\rm Im\ Tr}\
[Y_{u}^{\dagger}z^{T}(k^{2})Y_{d}^{\dagger}Y_{d}(Y_{u}^{\dagger})^{-1}]=0$
(124) $\displaystyle U_{R}-u_{L}(\delta M_{LH}^{u}):{\rm Im\ Tr}\
[Y_{d}[I+k^{2}x(k^{2})]^{T}Y_{d}^{-1}Y_{u}Y_{u}^{-1}]=0$ (125)

  * •

6-(e):

$\displaystyle u_{R}-U_{L}\ (\delta M_{HL}^{u}):{\rm Im\ Tr}\
[z^{\dagger}(k^{2})Y_{u}^{\dagger}(Y_{u}^{\dagger})^{-1}]=0$ (126)
$\displaystyle U_{R}-u_{L}(\delta M_{LH}^{u}):{\rm Im\ Tr}\
[(I+k^{2}x(k^{2}))^{\dagger}(Y_{u}Y_{u}^{\dagger})^{-1}]=0$ (127)

  * •

6-(f):

$\displaystyle u_{R}-U_{L}\ (\delta M_{HL}^{u}):{\rm Im\ Tr}\
[Y_{u}^{*}z^{*}(k^{2})(Y_{u}^{\dagger})^{-1}]=0$ (128) $\displaystyle
U_{R}-u_{L}(\delta M_{LH}^{u}):{\rm Im\ Tr}\
[(Y_{u}^{*})^{-1}(I+k^{2}x(k^{2}))^{*}Y_{u}^{-1}]=0$ (129)

  * •

6-(g):

$\displaystyle u_{R}-U_{L}\ (\delta M_{HL}^{u}):{\rm Im\ Tr}\
[Y_{u}^{\dagger}Y_{u}z(k^{2})Y_{u}^{\dagger}(Y_{u}^{\dagger})^{-1}]=0$ (130)
$\displaystyle U_{R}-u_{L}(\delta M_{LH}^{u}):{\rm Im\ Tr}\
[Y_{u}Y_{u}^{-1}(I+k^{2}x(k^{2}))Y_{u}Y_{u}^{-1}]=0$ (131)

(a) (b) (c)

(d) (e) (f)

(g) (h)

Figure 5: Diagrams leading to one-loop radiative corrections to the down-type
quark mass matrix. Here $V^{\mu}$ stands collectively for the gauge bosons
($G_{A}^{\mu},\,G^{\mu},\,Z^{\mu},A^{\mu},Z_{A}^{\mu}$) which all have flavor-
consering interactions.

(a) (b) (c)

(d) (e) (f)

(g) (h)

Figure 6: Diagrams leading to one-loop radiative corrections to the up-type
quark mass matrix.

This proves that there is no induced $\overline{\theta}$ in the model at the
one-loop level.

###  7.3 Renormalization group evolution of ${\overline{\theta}}$ and neutron
EDM

We have seen that the one-loop induced $\overline{\theta}$ at the scale
$M_{G}$ is vanishing. There is the possibility that extrapolation of the
Yukawa couplings by the renormalization group equations (RGE) from the GUT
scale to the weak scale could generate a nonzero $\bar{\theta}$. To study this
question, we analyze the RG equations for the Yukawa matrices of the model
relevant for the momentum range $M_{I}\leq\mu\leq M_{G}$ given in Appendix
A.4. First we note that the evolution of $\overline{\theta}$ will involve
determinant of the matrix $Y_{uL}Y^{\dagger}_{uR}$ and those with $u$ replaced
by $d$. From the RG equations given in Appendix A.4, we see that the one loop
expression for $\frac{d}{dt}{(Y_{uL}Y^{\dagger}_{uR}})$ is a hermitian matrix,
and therefore does not generate $\bar{\theta}$ if the initial $\bar{\theta}$
is zero.

To show this in more detail, we take infinitesimal extrapolations of the
$\bar{\theta}$ parameters in steps of $dt$. These are given by
$\frac{d}{dt}(Y_{uL}Y^{\dagger}_{uR})dt$. We see that the induced value of
$\bar{\theta}$ depends on how the integrand behaves. If the integrand is real,
no $\bar{\theta}$ is induced in this infinitesimal interval. Successive
iterations can lead to finite shifts in $t$. The induced $\bar{\theta}$ via
RGE from the up-quark sector can be written as

$\displaystyle\delta(\bar{\theta})={\rm
ImTr(\frac{d}{dt}(Y_{uL}Y^{\dagger}_{uR}})(Y_{uL}Y^{\dagger}_{uL})^{-1})~{}.$
(132)

Using the one-loop expressions for the RGE from the Appendix A.4 and setting
$Y_{uL}=Y_{uR}$ as the initial value, we see that the expression within the
bracket is hermitian and therefore induced $\delta(\bar{\theta})=0$. They
would therefore keep the GUT scale $\bar{\theta}$ value unchanged to one loop.
Several of the two-loop corrections can also similarly be seen to give zero
contributions. However at the two-loop level there are nonzero contributions
to $\overline{\theta}$. In particular, the 8th term of Eq. (156) in the RGE
for $Y_{u_{R}}$ generates a nonzero $\overline{\theta}$ which can be estimated
to be

$\overline{\theta}\approx\frac{8}{(16\pi^{2})^{2}}{\rm Im}{\rm
Tr}\left(Y_{dR}^{T}Y_{d_{R}}^{*}Y_{u_{R}}Y_{d_{R}}^{\dagger}Y_{d_{R}}(Y_{u_{L}}Y_{u_{L}}^{\dagger})^{-1}\right)\rm{ln}\left(\frac{M_{H_{L}^{c}}}{M_{H_{R}^{c}}}\right)~{}.$
(133)

This term can be seen to be originating from the two-loop diagram shown in
Fig. 7. Note that this diagram is log-divergent. There is an analogous diagram
where the color-triplet field $H_{R}^{c}$ is replaced by $H_{L}^{c}$ and the
quark helicities are flipped. Since in the computation of the RGE beta
functions, it was assumed that $H_{L}^{c}$ has a mass of order $M_{G}$, while
$H_{R}^{c}$ is at $M_{I}$, only the diagram of Fig. 7 contributes below
$M_{G}$. Above the mass scale of $H_{L}^{c}$, the combined contributions to
$\overline{\theta}$ from $H_{R}^{c}$ and $H_{L}^{c}$ would nearly vanish,
since this is the parity symmetric limit.

To estimate the induced $\overline{\theta}$ from Eq. (133), we set the GUT-
scale values of the Yukawa coupling matrices, namely,
$Y_{d_{R}}=Y_{d_{L}}=Y_{\ell}^{T}$ and $Y_{u_{R}}=Y_{u_{L}}=Y_{u}$. Then we
use the transformed basis where the fermion fit was given, with the mass
matrices given as in Eq. (36). We can estimate $\overline{\theta}$ to be

$\overline{\theta}\approx\frac{8}{(4\pi)^{4}}{\rm Im}{\rm Tr}\left[Q^{*}U_{\rm
PMND}^{\dagger}\,\hat{Y}_{\ell}^{2}U_{\rm PMNS}Q^{2}\,\hat{Y}_{u}\,U_{\rm
PMNS}^{T}\hat{Y}_{\ell}^{2}\,U_{\rm
PMNS}^{*}(\hat{Y}_{u})^{-1}\right]\rm{ln}\left(\frac{M_{H_{L}^{c}}}{M_{H_{R}^{c}}}\right)~{}.$
(134)

Here all the parameters are known, except for the two phases in the diagonal
matrix $Q={\rm diag}.(e^{i\alpha_{1}},\,e^{i\alpha_{2}},\,1)$. These two
phases are unobservable in low energy experiments, except through its
contributions to $\overline{\theta}$ and thus to neutron EDM.

Figure 7: Dominant two-loop radiative correction contributing to
$\bar{\theta}$.

In Fig. 8 we have presented the induced value of $\overline{\theta}$ arising
from this dominant two-loop diagram, as a function of one of the phase
parameters, $\alpha_{2}$. We have fixed the phase $\alpha_{1}=0.128$ modulo
integer multiples of $\pi/2$. This is the preferred value of this angle to be
consistent with neutron EDM limits. We have also shown the correlations with
neutron EDM as well as its current limit and future sensitivity. In the left
panel of Fig. 8, we kept the mass of $H_{L}^{c}$ equal to $M_{G}$ while
$H_{R}^{c}$ mass is $M_{I}$. In the right panel, $H_{L}^{c}$ mass is kept at 3
times $H_{R}^{c}$ mass. Such a lowering of the $H_{L}^{c}$ mass to a scale of
order $10^{12}$ GeV is compatible with proton decay constraints, and has very
little effect on gauge coupling unification.

A mild fine-tuning of parameters of the model is needed for the induced
$\overline{\theta}$ to be within allowed range from neutron EDM. This can be
seen by expanding the imaginary part of the trace appearing in Eq. (134).
Making an expansion in small quark masses, we find the leading term in the
ImTr[] to be

${\rm Im}{\rm Tr}[~{}]\simeq Y_{b}^{4}\frac{Y_{t}}{Y_{u}}{\rm
Im}\left[e^{-2i\alpha_{1}}(U_{31}^{*}U_{33})^{2}\right]~{}.$ (135)

For typical value of the phase $\alpha_{1}$, one would get
$\overline{\theta}\sim 10^{-9}$ corresponding to the right panel of Fig. 8,
and $\overline{\theta}\sim 10^{-7}$ for the left panel. The best fit to
fermion masses provides the PMNS matrix element to be
$U_{31}=0.483e^{-0.127i}$, and the choice of $\alpha_{1}$ nearly cancels this
phase. The tuning needed is at the level of $10^{-3}$ for the left panel,
while it is only a few percent for the right panel. One could turn this
observation around and state that neutron EDM should not be too much smaller
than the current experimental limit within the model; otherwise the model will
be more finely tuned. It should be noted that other contributions to neutron
EDM arising from heavy particle exchange are highly suppressed since the new
particles have masses of order $M_{I}\simeq 10^{11}$ GeV.

Figure 8: Prediction of the model for neutron edm $d_{n}$ as a function phase
$\alpha_{2}$ marginalized over the phase $\alpha_{1}$. The allowed values
consistent with the current EDM limit for the phase $\alpha_{1}$ are
$0.128+n\pi/2$ $(n=0,1,2,3)$. Here $Q\equiv{\rm
diag}.(e^{i\alpha_{1}},e^{i\alpha_{2}},1)$. Gray shaded region is the
exclusion from the current limit Dragos:2019oxn ; ParticleDataGroup:2022pth
and black dashed line represent the future sensitivity reach of ${\cal
O}(10^{-28})$ e-cm Alarcon:2022ero .

If parity symmetry is broken at low energy scales, Planck suppressed operators
will not destabilize the parity solution to the strong CP problem
Berezhiani:1995yi . In the GUT embedding one necessarily has a high scale, and
there are operators such as
$\Sigma_{L}V^{\mu\nu}_{L}\widetilde{V}_{L,\mu\nu}/M_{P}$ contributing to
$\overline{\theta}$ at the level of $10^{-3}$, which would need to be tuned.
In comparison to the tuning needed in popular axion model, which is at the
level of $10^{-44}$ Kamionkowski:1992mf ; Barr:1992qq ; Holman:1992us , here
the needed turning is at the level of $10^{-7}$.

##  8 Conclusions

We have developed in this paper a grand unified framework based on the gauge
symmetry $SU(5)_{L}\times SU(5)_{R}$ that embeds a class of left-right
symmetric models which solves the strong CP problem by parity symmetry. Of the
many possible symmetry breaking chains, we have found one that allows
successful gauge coupling unification with a single intermediate scale with
$M_{I}\sim 10^{11}$ GeV. The intermediate scale gauge symmetry is
$SU(3)_{cL}\times SU(2)_{L}\times U(1)_{L}\times SU(5)_{R}$. We have further
shown that the gauge bosons of $SU(5)_{R}$ that survive down to $M_{I}$ do not
cause any problem with rapid proton decay. This is achieved with an
interesting flavor structure for the fermion mass matrices which disallows
interactions of the $(X_{R}^{\mu},\,Y_{R}^{\mu})$ gauge bosons of $SU(5)_{R}$
with light fermion fields alone. This flavor structure also explains the
observed masses and mixing in the quark and lepton sectors.

A novel feature of the model is that neutrinos are naturally light Dirac
fermions. Their Yukawa couplings are suppressed by a factor $M_{I}/M_{G}\sim
10^{-7}$, since the Higgs doublet that induces its mass has a GUT scale mass
and acquires only an induced VEV, $v_{\nu}\sim v\,(M_{I}/M_{G})$, where $v\sim
174$ GeV. We call this mechanism of obtaining suppressed Dirac mass as the
type-II Dirac seesaw. Furthermore, the model makes several predictions in the
neutrino sector, which can serve as its tests. The CP violating phase in
neutrino oscillations is found to lie in the range $\delta_{CP}=\pm(130.4\pm
1.2)^{\circ}$. The lightest neutrino mass is constrained to lie in the range
$m_{\nu_{1}}=(4.8-8.4)$ meV. The model predicts normal ordering of neutrino
masses, and obviously neutrinoless double beta decay is forbidden within the
model.

We have shown that the full GUT embedding of the left-right symmetric model
preserves the strong CP solution via parity symmetry. This is possible even in
presence of a large number of new particles at the GUT scale as well as at the
intermediate scale. We have shown by explicit computations that the one-loop
induced $\overline{\theta}$ is zero. Nonzero $\overline{\theta}$ is induced at
the two-loop level, which we have estimated and shown to be compatible with
neutron EDM limits. A mild fine-tuning of parameters of the model is needed,
at the level of a few percent, for consistency with neutron EDM limits. This
observation could be turned around to state that neutron EDM is likely within
reach of the forthcoming round of experiments, or else the model will be fine-
tuned.

## Acknowledgement

The work of KSB was supported in part by the United States Department of
Energy Grant No. DE-SC0016013. The work of AT was supported in part by the
National Science Foundation under Grant PHY-2210428. KSB and AT wish to
acknowledge the Center for Theoretical Underground Physics and Related Areas
(CETUP*) and the Institute for Underground Science at SURF, where part of this
work was done, for hospitality and for providing a stimulating environment.

##  Appendix A Appendix

###  A.1 Decomposition of fields under intermediate symmetry

Here we list the decomposition of various fields under the intermediate
symmetry $SU(3)_{cL}\times SU(2)_{L}\times U(1)_{L}\times SU(5)_{R}$. This
list is helpful in deciding which fragments of the Higgs fields survive down
to $M_{I}$.

$\displaystyle({\bf 5},\overline{\bf 5})$ $\displaystyle=({\bf
3},1,-1/3,{\bf\overline{5}})+(1,{\bf 2},1/2,{\bf\overline{5}})$
$\displaystyle({\bf 75},1)$ $\displaystyle=({\bf\overline{3}},1,-5/3,1)+({\bf
6},{\bf 2},-5/6,1)+({\bf 3},{\bf 2},-5/6,1)+({\bf 8},{\bf 3},0,1)$
$\displaystyle~{}~{}+({\bf 8},1,0,1)+(1,1,0,1)+({\bf 6},{\bf 2},5/6,1)+({\bf
3},{\bf 2},5/6,1)+({\bf 3},1,5/3,1)$ $\displaystyle({\bf 5},1)$
$\displaystyle=({\bf 3},1,-1/3,1)+(1,{\bf 2},1/2,1)$ $\displaystyle({\bf
10},1)$ $\displaystyle=({\bf\overline{3}},1,-2/3,1)+({\bf 3},{\bf
2},1/6,1)+(1,1,1,1)$ $\displaystyle({\bf 15},{\bf\overline{15}})$
$\displaystyle=({\bf 3},{\bf 2},1/6,{\bf\overline{15}})+(1,{\bf
3},1,{\bf\overline{15}})+({\bf\overline{6}},1,-2/3,{\bf\overline{15}})$ (136)

###  A.2 Decomposition of scalar fields under SM gauge group

We define the following fields under SM gauge group $SU(3)_{c}\times
SU(2)_{L}\times U(1)_{Y}$:

$\displaystyle H_{L}({\bf 5},1)$ $\displaystyle=H_{L}^{c}({\bf
3},1,-1/3)+H_{L}^{d}(1,{\bf 2},1/2)$ $\displaystyle H_{R}(1,{\bf 5})$
$\displaystyle=H_{R}^{c}({\bf 3},1,-1/3)+H_{R}^{+}(1,1,1)+H_{R}^{0}(1,1,0)$
$\displaystyle\Phi({\bf\overline{5}},{\bf 5})$
$\displaystyle=\phi_{DD}^{o}({\bf
8},1,0)+\Phi_{DD}^{s}(1,1,0)+\Phi_{D\nu}({\bf\overline{3}},1,1/3)+\Phi_{De}({\bf\overline{3}},1,4/3)$
$\displaystyle~{}~{}~{}+\Phi_{LD}({\bf 3},{\bf 2},-5/6)+\Phi_{L\nu}(1,{\bf
2},-1/2)+\Phi_{Le}(1,{\bf 2},1/2)\,.$ (137)

###  A.3 Analytical solution of one loop RGE to gauge couplings including
threshold corrections from vector-like fermions

The solutions to the one-loop RGE, $\alpha_{i}(\mu)$, can be written down as
functions of momentum starting from the scale $M_{G}$ where the couplings are
unified. At the scale $M_{I}$, we impose the boundary conditions given in Eq.
(50). For the three gauge couplings at $\mu=m_{t}$ we obtain

$\displaystyle\alpha_{1Y}^{-1}(m_{t})$ $\displaystyle=$
$\displaystyle\alpha_{G}^{-1}-\left(\frac{5}{13}b_{1}^{\prime}+\frac{8}{13}b_{5}^{\prime}\right)\frac{1}{2\pi}{\rm
log}\left(\frac{M_{I}}{M_{G}}\right)-\frac{b_{1}}{2\pi}{\rm
log}\left(\frac{m_{t}}{M_{I}}\right)-\Delta_{\alpha_{1Y}}$
$\displaystyle\alpha_{2L}^{-1}(M_{t})$ $\displaystyle=$
$\displaystyle\alpha_{G}^{-1}-\frac{b_{2}^{\prime}}{2\pi}{\rm
log}\left(\frac{M_{I}}{M_{G}}\right)-\frac{b_{2}}{2\pi}{\rm
log}\left(\frac{m_{t}}{M_{I}}\right)-\Delta_{\alpha_{2L}}$
$\displaystyle\alpha_{3c}^{-1}(Mm_{t})$ $\displaystyle=$ $\displaystyle
2\alpha_{G}^{-1}-\frac{b_{3}^{\prime}+b_{5}^{\prime}}{2\pi}{\rm
log}\left(\frac{M_{I}}{M_{G}}\right)-\frac{b_{3}}{2\pi}{\rm
log}\left(\frac{m_{t}}{M_{I}}\right)-\Delta_{\alpha_{3c}}~{}.$ (138)

Here $\Delta_{\alpha_{i}}$ are threshold corrections at $M_{I}$ arising from
the spread in the masses of the $U,\,D,\,E$ fermions. These masses are not all
degenerate, as they obey the relations

$M_{U_{1}}:M_{U_{2}}:M_{U_{3}}=m_{u}:m_{c}:m_{t},~{}~{}~{}M_{E_{1}}:M_{E_{2}}:M_{E_{3}}=m_{e}:m_{\mu}:m_{\tau}$
(139)

which needs to be incorporated via $\Delta_{\alpha_{i}}$ in Eq. (138). In
addition, the masses of the $D_{i}$ quarks are also hierarchical, as shown in
Eq. (72) from our numerical fit to fermion masses. The threshold corrections
$\Delta_{\alpha_{i}}$ from these spreads in fermion masses at $\mu=M_{I}$ are
given by

$\Delta_{\alpha_{i}}=\sum_{F=U,D,E}\left[\frac{\Delta b_{i}^{F}}{2\pi}{\rm
log}\left(\frac{M_{F_{1}}}{M_{F_{2}}}\right)+\frac{2\Delta
b_{i}^{F}}{2\pi}{\rm
log}\left(\frac{M_{F_{2}}}{M_{F_{3}}}\right)+\frac{3\Delta
b_{i}^{F}}{2\pi}{\rm log}\left(\frac{M_{F_{3}}}{M_{I}}\right)\right]~{}.$
(140)

The contributions to $\Delta b_{i}^{F}$ from the vector-like fermions
$U,\,D,\,E$ are:

$\displaystyle(\Delta b_{1}^{U},\,\Delta b_{1}^{D},\,\Delta b_{1}^{E})$
$\displaystyle=$
$\displaystyle\left(\frac{16}{39},\,\frac{4}{9},\,\frac{4}{13}\right)$
$\displaystyle(\Delta b_{2}^{U},\,\Delta b_{2}^{D},\,\Delta b_{2}^{E})$
$\displaystyle=$ $\displaystyle(0,\,0,\,0)$ $\displaystyle(\Delta
b_{3}^{U},\,\Delta b_{3}^{D},\,\Delta b_{3}^{E})$ $\displaystyle=$
$\displaystyle\left(\frac{2}{3},\,\frac{2}{3},\,0\right)~{}.$ (141)

When the threshold corrections are ignored, one would obtain from Eq. (138)
the result given in Eq. (57) for the weak mixing angle
$\sin^{2}\theta_{W}(m_{t})$.

Now, let us consider the one-loop solutions to the RGE including the threshold
effects shown in Eq.(138). If we use the fermion and scalar spectrum at
$M_{I}$ shown in Eqs. (51)-(52), but remove the $\eta_{K}$ scalar, the one-
loop $\beta$-function coefficients will be

$(b_{1Y},b_{2L},b_{3c})=\left(\frac{41}{26},-\frac{19}{6},-7\right),~{}~{}(b^{\prime}_{1L},b^{\prime}_{2L},b^{\prime}_{3cL},b^{\prime}_{5R})=\left(\frac{133}{30},-\frac{19}{6},-\frac{37}{6},-\frac{41}{3}\right)~{}.$
(142)

Taking $M_{U_{3}}=M_{I}$, $M_{E_{3}}=(m_{\tau}/m_{t})M_{I}$, with the
relations given in Eq. (139), and with $M_{D_{i}}$ values given in Eq. (72),
one obtains for $M_{I}$, $M_{G}$ and $\alpha_{G}^{-1}$ the following values:

$M_{I}=6.3\times 10^{5}~{}{\rm GeV},~{}~{}~{}M_{G}=3.3\times 10^{15}~{}{\rm
GeV},~{}~{}~{}\alpha_{G}^{-1}=45.0~{}.$ (143)

This choice is however inconsistent with LHC data CMS:2016edj , since
$M_{U_{3}}=6.3\times 10^{5}~{}{\rm GeV}$ would imply that $M_{U_{1}}\sim 4$
GeV. The value of $M_{I}$ should be raised to $M_{I}\geq 10^{8}$ GeV for
consistency with LHC limit of $M_{U_{1}}\geq 1$ TeV. This is indeed what is
achieved with the inclusion of the $\eta_{K}$ scalar with a mass around
$M_{I}$. We should also note that without the $\eta_{K}$ field at $M_{I}$, the
$SU(5)_{R}$ gauge coupling $\alpha_{5R}$ tends to take non-perturbative values
at $\mu=M_{I}$, although this issue may be ameliorated by including a few
percent threshold effect at the GUT scale that lowers $\alpha_{5R}(M_{G})$
from its unified value.

###  A.4 Renormalization group equations for Yukawa couplings for
$M_{I}\leq\mu\leq M_{G}$

Here we give the full two-loop beta functions for the Yukawa couplings of the
model in the momentum range $M_{I}\leq\mu\leq M_{G}$. These were generated
with PyR@TE package Sartore:2020gou  and cross-checked with other known
results. We have used these two-loop RGEs for the numerical results obtained
for gauge coupling unification in Sec. 4, as well as for determining the
running factors $\eta_{i}^{\prime}$ precisely for the fermion mass parameters
of Eq. (68) of Sec. 5.

The most general renormalizable Yukawa interaction involving the fermion and
scalar fields listed in Eqs. (51)-(52), valid for the momentum range
$M_{I}\leq\mu\leq M_{G}$ when the intermediate scale gauge symmetry is
$SU(3)_{cL}\times SU(2)_{L}\times U(1)_{L}\times SU(5)_{R}$, can be written
down as

$\displaystyle\autobreak-\mathcal{L}_{Y}=\par
Y_{uL}{\overline{Q}_{L}}\widetilde{H}_{L}^{d}{U_{R}}\par+\frac{1}{4}Y_{uR}^{\dagger}{\chi_{R}}{H_{R}}{\chi_{R}}\par+Y_{dL}{\overline{Q}_{L}}{H}_{L}^{d}{D_{R}}\par+Y_{lL}{\overline{L}_{L}}{H}_{L}^{d}{E_{R}}\par+\sqrt{2}Y_{dR}^{\dagger}\psi_{R}{H_{R}^{\dagger}}{\chi_{R}}\par+Y_{D}{D_{R}}{\Phi}_{D}^{\dagger}{\psi_{R}}\par+\text{h.c.}$
(144)

Here the $SU(5)_{R}$ contractions are not shown explicitly, but these are
identical to those given in Eq. (22). The resulting quark and lepton mass
matrices take the form:

$\displaystyle\mathcal{M}_{u}=\left(\begin{array}[]{cc}0&Y_{uL}^{T}\kappa_{L}\\\
Y_{uR}^{*}\kappa_{R}&0\end{array}\right),~{}~{}\mathcal{M}_{\ell}=\left(\begin{array}[]{cc}0&Y_{\ell
L}^{T}\kappa_{L}\\\
Y_{dR}^{*}\kappa_{R}&0\end{array}\right),~{}~{}\mathcal{M}_{d}=\left(\begin{array}[]{cc}0&Y_{dL}\kappa_{L}\\\
Y_{dR}^{\dagger}\kappa_{R}&Y_{D}^{*}v_{\Phi}\end{array}\right)\,.$ (151)

We have the relation $Y_{uR}=Y_{uR}^{T}$ in this momentum regime owing to
$SU(5)_{R}$ symmetry. At $\mu=M_{G}$, these matrices obey the following
boundary conditions (see Eq. (29)):

$Y_{uL}=Y_{uR}=Y_{u},~{}~{}~{}Y_{dR}=Y_{dL}=Y_{\ell L}^{T}=Y_{\ell}^{T}~{}.$
(152)

Following the definition of Ref. Sartore:2020gou , the RGEs for the Yukawa
couplings canbe written as

$\beta\left(X\right)\equiv\mu\frac{dX}{d\mu}\equiv\cdot\frac{1}{\left(4\pi\right)^{2}}\beta^{(1)}(X)+\cdot\frac{1}{\left(4\pi\right)^{4}}\beta^{(2)}(X)\,,$

where $\beta^{(1)}(X)$ and $\beta^{(2)}(X)$ are the one-loop and two-loop beta
functions. These beta functions, valid in the momentum range $M_{I}\leq\mu\leq
M_{G}$ are given by

$\displaystyle\autobreak\beta^{(1)}(Y_{uL})=\par+\frac{3}{2}Y_{uL}Y_{uL}^{\dagger}Y_{uL}\par-\frac{3}{2}Y_{dL}Y_{dL}^{\dagger}Y_{uL}\par+3\mathrm{Tr}\left(Y_{uL}^{\dagger}Y_{uL}\right)Y_{uL}\par+3\mathrm{Tr}\left(Y_{dL}^{\dagger}Y_{dL}\right)Y_{uL}\par+\mathrm{Tr}\left(Y_{lL}^{\dagger}Y_{lL}\right)Y_{uL}\par-\frac{17}{20}g_{1L}^{2}Y_{uL}\par-\frac{9}{4}g_{2L}^{2}Y_{uL}\par-8g_{3L}^{2}Y_{uL}\,.$
(153)
$\displaystyle\autobreak\beta^{(2)}(Y_{uL})=\par+\frac{3}{2}Y_{uL}Y_{uL}^{\dagger}Y_{uL}Y_{uL}^{\dagger}Y_{uL}\par-\frac{1}{4}Y_{uL}Y_{uL}^{\dagger}Y_{dL}Y_{dL}^{\dagger}Y_{uL}\par-
Y_{dL}Y_{dL}^{\dagger}Y_{uL}Y_{uL}^{\dagger}Y_{uL}\par+\frac{11}{4}Y_{dL}Y_{dL}^{\dagger}Y_{dL}Y_{dL}^{\dagger}Y_{uL}\par+\frac{35}{8}Y_{dL}Y_{D}^{*}Y_{D}^{\mathrm{T}}Y_{dL}^{\dagger}Y_{uL}\par-\frac{27}{4}\mathrm{Tr}\left(Y_{uL}^{\dagger}Y_{uL}Y_{uL}^{\dagger}Y_{uL}\right)Y_{uL}\par-\frac{27}{4}\mathrm{Tr}\left(Y_{uL}^{\dagger}Y_{uL}\right)Y_{uL}Y_{uL}^{\dagger}Y_{uL}\par+\frac{15}{4}\mathrm{Tr}\left(Y_{uL}^{\dagger}Y_{uL}\right)Y_{dL}Y_{dL}^{\dagger}Y_{uL}\par+\frac{3}{2}\mathrm{Tr}\left(Y_{uL}^{\dagger}Y_{dL}Y_{dL}^{\dagger}Y_{uL}\right)Y_{uL}\par-\frac{27}{4}\mathrm{Tr}\left(Y_{dL}^{\dagger}Y_{dL}\right)Y_{uL}Y_{uL}^{\dagger}Y_{uL}\par-\frac{27}{4}\mathrm{Tr}\left(Y_{dL}^{\dagger}Y_{dL}Y_{dL}^{\dagger}Y_{dL}\right)Y_{uL}\par+\frac{15}{4}\mathrm{Tr}\left(Y_{dL}^{\dagger}Y_{dL}\right)Y_{dL}Y_{dL}^{\dagger}Y_{uL}\par-\frac{45}{4}\mathrm{Tr}\left(Y_{dL}^{\dagger}Y_{dL}Y_{D}^{*}Y_{D}^{\mathrm{T}}\right)Y_{uL}\par-\frac{9}{4}\mathrm{Tr}\left(Y_{lL}^{\dagger}Y_{lL}\right)Y_{uL}Y_{uL}^{\dagger}Y_{uL}\par+\frac{5}{4}\mathrm{Tr}\left(Y_{lL}^{\dagger}Y_{lL}\right)Y_{dL}Y_{dL}^{\dagger}Y_{uL}\par-\frac{9}{4}\mathrm{Tr}\left(Y_{lL}^{\dagger}Y_{lL}Y_{lL}^{\dagger}Y_{lL}\right)Y_{uL}\par+\frac{223}{80}g_{1L}^{2}Y_{uL}Y_{uL}^{\dagger}Y_{uL}\par+\frac{135}{16}g_{2L}^{2}Y_{uL}Y_{uL}^{\dagger}Y_{uL}\par+16g_{3L}^{2}Y_{uL}Y_{uL}^{\dagger}Y_{uL}\par-\frac{43}{80}g_{1L}^{2}Y_{dL}Y_{dL}^{\dagger}Y_{uL}\par+\frac{9}{16}g_{2L}^{2}Y_{dL}Y_{dL}^{\dagger}Y_{uL}\par-16g_{3L}^{2}Y_{dL}Y_{dL}^{\dagger}Y_{uL}\par+\frac{17}{8}g_{1L}^{2}\mathrm{Tr}\left(Y_{uL}^{\dagger}Y_{uL}\right)Y_{uL}\par+\frac{45}{8}g_{2L}^{2}\mathrm{Tr}\left(Y_{uL}^{\dagger}Y_{uL}\right)Y_{uL}\par+20g_{3L}^{2}\mathrm{Tr}\left(Y_{uL}^{\dagger}Y_{uL}\right)Y_{uL}\par+\frac{5}{8}g_{1L}^{2}\mathrm{Tr}\left(Y_{dL}^{\dagger}Y_{dL}\right)Y_{uL}\par+\frac{45}{8}g_{2L}^{2}\mathrm{Tr}\left(Y_{dL}^{\dagger}Y_{dL}\right)Y_{uL}\par+20g_{3L}^{2}\mathrm{Tr}\left(Y_{dL}^{\dagger}Y_{dL}\right)Y_{uL}\par+\frac{15}{8}g_{1L}^{2}\mathrm{Tr}\left(Y_{lL}^{\dagger}Y_{lL}\right)Y_{uL}\par+\frac{15}{8}g_{2L}^{2}\mathrm{Tr}\left(Y_{lL}^{\dagger}Y_{lL}\right)Y_{uL}\par+\frac{5561}{1800}g_{1L}^{4}Y_{uL}\par-\frac{9}{20}g_{1L}^{2}g_{2L}^{2}Y_{uL}\par+\frac{19}{15}g_{1L}^{2}g_{3L}^{2}Y_{uL}\par+\frac{67}{4}g_{2L}^{4}Y_{uL}\par+9g_{2L}^{2}g_{3L}^{2}Y_{uL}\par-\frac{202}{9}g_{3L}^{4}Y_{uL}\,.$
(154)
$\displaystyle\autobreak\beta^{(1)}(Y_{uR})=\par+3Y_{uR}Y_{uR}^{*}Y_{uR}\par-3Y_{uR}Y_{dR}^{\dagger}Y_{dR}\par-3Y_{dR}^{\mathrm{T}}Y_{dR}^{*}Y_{uR}\par+3\mathrm{Tr}\left(Y_{uR}^{*}Y_{uR}\right)Y_{uR}\par+4\mathrm{Tr}\left(Y_{dR}^{\dagger}Y_{dR}\right)Y_{uR}\par-\frac{108}{5}g_{5R}^{2}Y_{uR}\,.$
(155)
$\displaystyle\autobreak\beta^{(2)}(Y_{uR})=\par-\frac{33}{4}Y_{uR}Y_{uR}^{*}Y_{uR}Y_{uR}^{*}Y_{uR}\par-6Y_{uR}Y_{uR}^{*}Y_{uR}Y_{dR}^{\dagger}Y_{dR}\par-\frac{3}{4}Y_{uR}Y_{uR}^{*}Y_{dR}^{\mathrm{T}}Y_{dR}^{*}Y_{uR}\par-\frac{3}{4}Y_{uR}Y_{dR}^{\dagger}Y_{dR}Y_{uR}^{*}Y_{uR}\par+11Y_{uR}Y_{dR}^{\dagger}Y_{dR}Y_{dR}^{\dagger}Y_{dR}\par+\frac{21}{4}Y_{uR}Y_{dR}^{\dagger}Y_{D}^{\dagger}Y_{D}Y_{dR}\par-6Y_{dR}^{\mathrm{T}}Y_{dR}^{*}Y_{uR}Y_{uR}^{*}Y_{uR}\par-8Y_{dR}^{\mathrm{T}}Y_{dR}^{*}Y_{uR}Y_{dR}^{\dagger}Y_{dR}\par+11Y_{dR}^{\mathrm{T}}Y_{dR}^{*}Y_{dR}^{\mathrm{T}}Y_{dR}^{*}Y_{uR}\par+\frac{21}{4}Y_{dR}^{\mathrm{T}}Y_{D}^{\mathrm{T}}Y_{D}^{*}Y_{dR}^{*}Y_{uR}\par-\frac{27}{2}\mathrm{Tr}\left(Y_{uR}^{*}Y_{uR}Y_{uR}^{*}Y_{uR}\right)Y_{uR}\par-\frac{27}{2}\mathrm{Tr}\left(Y_{uR}^{*}Y_{uR}\right)Y_{uR}Y_{uR}^{*}Y_{uR}\par+\frac{15}{2}\mathrm{Tr}\left(Y_{uR}^{*}Y_{uR}\right)Y_{uR}Y_{dR}^{\dagger}Y_{dR}\par+6\mathrm{Tr}\left(Y_{uR}^{*}Y_{uR}Y_{dR}^{\dagger}Y_{dR}\right)Y_{uR}\par+\frac{15}{2}\mathrm{Tr}\left(Y_{uR}^{*}Y_{uR}\right)Y_{dR}^{\mathrm{T}}Y_{dR}^{*}Y_{uR}\par-18\mathrm{Tr}\left(Y_{dR}^{\dagger}Y_{dR}\right)Y_{uR}Y_{uR}^{*}Y_{uR}\par+10\mathrm{Tr}\left(Y_{dR}^{\dagger}Y_{dR}\right)Y_{uR}Y_{dR}^{\dagger}Y_{dR}\par-18\mathrm{Tr}\left(Y_{dR}^{\dagger}Y_{dR}Y_{dR}^{\dagger}Y_{dR}\right)Y_{uR}\par+10\mathrm{Tr}\left(Y_{dR}^{\dagger}Y_{dR}\right)Y_{dR}^{\mathrm{T}}Y_{dR}^{*}Y_{uR}\par-9\mathrm{Tr}\left(Y_{dR}^{\dagger}Y_{D}^{\dagger}Y_{D}Y_{dR}\right)Y_{uR}\par+\frac{396}{5}g_{5R}^{2}Y_{uR}Y_{uR}^{*}Y_{uR}\par-\frac{261}{5}g_{5R}^{2}Y_{uR}Y_{dR}^{\dagger}Y_{dR}\par-\frac{261}{5}g_{5R}^{2}Y_{dR}^{\mathrm{T}}Y_{dR}^{*}Y_{uR}\par+54g_{5R}^{2}\mathrm{Tr}\left(Y_{uR}^{*}Y_{uR}\right)Y_{uR}\par+60g_{5R}^{2}\mathrm{Tr}\left(Y_{dR}^{\dagger}Y_{dR}\right)Y_{uR}\par-\frac{2427}{25}g_{5R}^{4}Y_{uR}\,.$
(156)
$\displaystyle\autobreak\beta^{(1)}(Y_{dL})=\par-\frac{3}{2}Y_{uL}Y_{uL}^{\dagger}Y_{dL}\par+\frac{3}{2}Y_{dL}Y_{dL}^{\dagger}Y_{dL}\par+\frac{5}{2}Y_{dL}Y_{D}^{*}Y_{D}^{\mathrm{T}}\par+3\mathrm{Tr}\left(Y_{uL}^{\dagger}Y_{uL}\right)Y_{dL}\par+3\mathrm{Tr}\left(Y_{dL}^{\dagger}Y_{dL}\right)Y_{dL}\par+\mathrm{Tr}\left(Y_{lL}^{\dagger}Y_{lL}\right)Y_{dL}\par-\frac{1}{4}g_{1L}^{2}Y_{dL}\par-\frac{9}{4}g_{2L}^{2}Y_{dL}\par-8g_{3L}^{2}Y_{dL}\,.$
(157)
$\displaystyle\autobreak\beta^{(2)}(Y_{dL})=\par+\frac{11}{4}Y_{uL}Y_{uL}^{\dagger}Y_{uL}Y_{uL}^{\dagger}Y_{dL}\par-
Y_{uL}Y_{uL}^{\dagger}Y_{dL}Y_{dL}^{\dagger}Y_{dL}\par-\frac{1}{4}Y_{dL}Y_{dL}^{\dagger}Y_{uL}Y_{uL}^{\dagger}Y_{dL}\par+\frac{3}{2}Y_{dL}Y_{dL}^{\dagger}Y_{dL}Y_{dL}^{\dagger}Y_{dL}\par-\frac{5}{2}Y_{dL}Y_{D}^{*}Y_{dR}^{*}Y_{dR}^{\mathrm{T}}Y_{D}^{\mathrm{T}}\par-\frac{5}{8}Y_{dL}Y_{D}^{*}Y_{D}^{\mathrm{T}}Y_{dL}^{\dagger}Y_{dL}\par-\frac{15}{8}Y_{dL}Y_{D}^{*}Y_{D}^{\mathrm{T}}Y_{D}^{*}Y_{D}^{\mathrm{T}}\par-\frac{27}{4}\mathrm{Tr}\left(Y_{uL}^{\dagger}Y_{uL}Y_{uL}^{\dagger}Y_{uL}\right)Y_{dL}\par+\frac{15}{4}\mathrm{Tr}\left(Y_{uL}^{\dagger}Y_{uL}\right)Y_{uL}Y_{uL}^{\dagger}Y_{dL}\par-\frac{27}{4}\mathrm{Tr}\left(Y_{uL}^{\dagger}Y_{uL}\right)Y_{dL}Y_{dL}^{\dagger}Y_{dL}\par+\frac{3}{2}\mathrm{Tr}\left(Y_{uL}^{\dagger}Y_{dL}Y_{dL}^{\dagger}Y_{uL}\right)Y_{dL}\par+\frac{15}{4}\mathrm{Tr}\left(Y_{dL}^{\dagger}Y_{dL}\right)Y_{uL}Y_{uL}^{\dagger}Y_{dL}\par-\frac{27}{4}\mathrm{Tr}\left(Y_{dL}^{\dagger}Y_{dL}Y_{dL}^{\dagger}Y_{dL}\right)Y_{dL}\par-\frac{27}{4}\mathrm{Tr}\left(Y_{dL}^{\dagger}Y_{dL}\right)Y_{dL}Y_{dL}^{\dagger}Y_{dL}\par-\frac{45}{4}\mathrm{Tr}\left(Y_{dL}^{\dagger}Y_{dL}Y_{D}^{*}Y_{D}^{\mathrm{T}}\right)Y_{dL}\par+\frac{5}{4}\mathrm{Tr}\left(Y_{lL}^{\dagger}Y_{lL}\right)Y_{uL}Y_{uL}^{\dagger}Y_{dL}\par-\frac{9}{4}\mathrm{Tr}\left(Y_{lL}^{\dagger}Y_{lL}\right)Y_{dL}Y_{dL}^{\dagger}Y_{dL}\par-\frac{9}{4}\mathrm{Tr}\left(Y_{lL}^{\dagger}Y_{lL}Y_{lL}^{\dagger}Y_{lL}\right)Y_{dL}\par-\frac{15}{4}\mathrm{Tr}\left(Y_{D}^{\dagger}Y_{D}\right)Y_{dL}Y_{D}^{*}Y_{D}^{\mathrm{T}}\par-\frac{79}{80}g_{1L}^{2}Y_{uL}Y_{uL}^{\dagger}Y_{dL}\par+\frac{9}{16}g_{2L}^{2}Y_{uL}Y_{uL}^{\dagger}Y_{dL}\par-16g_{3L}^{2}Y_{uL}Y_{uL}^{\dagger}Y_{dL}\par+\frac{187}{80}g_{1L}^{2}Y_{dL}Y_{dL}^{\dagger}Y_{dL}\par+\frac{135}{16}g_{2L}^{2}Y_{dL}Y_{dL}^{\dagger}Y_{dL}\par+16g_{3L}^{2}Y_{dL}Y_{dL}^{\dagger}Y_{dL}\par+\frac{11}{12}g_{1L}^{2}Y_{dL}Y_{D}^{*}Y_{D}^{\mathrm{T}}\par+\frac{55}{3}g_{3L}^{2}Y_{dL}Y_{D}^{*}Y_{D}^{\mathrm{T}}\par+51g_{5R}^{2}Y_{dL}Y_{D}^{*}Y_{D}^{\mathrm{T}}\par+\frac{17}{8}g_{1L}^{2}\mathrm{Tr}\left(Y_{uL}^{\dagger}Y_{uL}\right)Y_{dL}\par+\frac{45}{8}g_{2L}^{2}\mathrm{Tr}\left(Y_{uL}^{\dagger}Y_{uL}\right)Y_{dL}\par+20g_{3L}^{2}\mathrm{Tr}\left(Y_{uL}^{\dagger}Y_{uL}\right)Y_{dL}\par+\frac{5}{8}g_{1L}^{2}\mathrm{Tr}\left(Y_{dL}^{\dagger}Y_{dL}\right)Y_{dL}\par+\frac{45}{8}g_{2L}^{2}\mathrm{Tr}\left(Y_{dL}^{\dagger}Y_{dL}\right)Y_{dL}\par+20g_{3L}^{2}\mathrm{Tr}\left(Y_{dL}^{\dagger}Y_{dL}\right)Y_{dL}\par+\frac{15}{8}g_{1L}^{2}\mathrm{Tr}\left(Y_{lL}^{\dagger}Y_{lL}\right)Y_{dL}\par+\frac{15}{8}g_{2L}^{2}\mathrm{Tr}\left(Y_{lL}^{\dagger}Y_{lL}\right)Y_{dL}\par-\frac{31}{1800}g_{1L}^{4}Y_{dL}\par-\frac{27}{20}g_{1L}^{2}g_{2L}^{2}Y_{dL}\par+\frac{31}{15}g_{1L}^{2}g_{3L}^{2}Y_{dL}\par+\frac{67}{4}g_{2L}^{4}Y_{dL}\par+9g_{2L}^{2}g_{3L}^{2}Y_{dL}\par-\frac{202}{9}g_{3L}^{4}Y_{dL}\,.$
(158)
$\displaystyle\autobreak\beta^{(1)}(Y_{lL})=\par+\frac{3}{2}Y_{lL}Y_{lL}^{\dagger}Y_{lL}\par+3\mathrm{Tr}\left(Y_{uL}^{\dagger}Y_{uL}\right)Y_{lL}\par+3\mathrm{Tr}\left(Y_{dL}^{\dagger}Y_{dL}\right)Y_{lL}\par+\mathrm{Tr}\left(Y_{lL}^{\dagger}Y_{lL}\right)Y_{lL}\par-\frac{9}{4}g_{1L}^{2}Y_{lL}\par-\frac{9}{4}g_{2L}^{2}Y_{lL}\,.$
(159)
$\displaystyle\autobreak\beta^{(2)}(Y_{lL})=\par+\frac{3}{2}Y_{lL}Y_{lL}^{\dagger}Y_{lL}Y_{lL}^{\dagger}Y_{lL}\par-\frac{27}{4}\mathrm{Tr}\left(Y_{uL}^{\dagger}Y_{uL}Y_{uL}^{\dagger}Y_{uL}\right)Y_{lL}\par-\frac{27}{4}\mathrm{Tr}\left(Y_{uL}^{\dagger}Y_{uL}\right)Y_{lL}Y_{lL}^{\dagger}Y_{lL}\par+\frac{3}{2}\mathrm{Tr}\left(Y_{uL}^{\dagger}Y_{dL}Y_{dL}^{\dagger}Y_{uL}\right)Y_{lL}\par-\frac{27}{4}\mathrm{Tr}\left(Y_{dL}^{\dagger}Y_{dL}Y_{dL}^{\dagger}Y_{dL}\right)Y_{lL}\par-\frac{27}{4}\mathrm{Tr}\left(Y_{dL}^{\dagger}Y_{dL}\right)Y_{lL}Y_{lL}^{\dagger}Y_{lL}\par-\frac{45}{4}\mathrm{Tr}\left(Y_{dL}^{\dagger}Y_{dL}Y_{D}^{*}Y_{D}^{\mathrm{T}}\right)Y_{lL}\par-\frac{9}{4}\mathrm{Tr}\left(Y_{lL}^{\dagger}Y_{lL}Y_{lL}^{\dagger}Y_{lL}\right)Y_{lL}\par-\frac{9}{4}\mathrm{Tr}\left(Y_{lL}^{\dagger}Y_{lL}\right)Y_{lL}Y_{lL}^{\dagger}Y_{lL}\par+\frac{387}{80}g_{1L}^{2}Y_{lL}Y_{lL}^{\dagger}Y_{lL}\par+\frac{135}{16}g_{2L}^{2}Y_{lL}Y_{lL}^{\dagger}Y_{lL}\par+\frac{17}{8}g_{1L}^{2}\mathrm{Tr}\left(Y_{uL}^{\dagger}Y_{uL}\right)Y_{lL}\par+\frac{45}{8}g_{2L}^{2}\mathrm{Tr}\left(Y_{uL}^{\dagger}Y_{uL}\right)Y_{lL}\par+20g_{3L}^{2}\mathrm{Tr}\left(Y_{uL}^{\dagger}Y_{uL}\right)Y_{lL}\par+\frac{5}{8}g_{1L}^{2}\mathrm{Tr}\left(Y_{dL}^{\dagger}Y_{dL}\right)Y_{lL}\par+\frac{45}{8}g_{2L}^{2}\mathrm{Tr}\left(Y_{dL}^{\dagger}Y_{dL}\right)Y_{lL}\par+20g_{3L}^{2}\mathrm{Tr}\left(Y_{dL}^{\dagger}Y_{dL}\right)Y_{lL}\par+\frac{15}{8}g_{1L}^{2}\mathrm{Tr}\left(Y_{lL}^{\dagger}Y_{lL}\right)Y_{lL}\par+\frac{15}{8}g_{2L}^{2}\mathrm{Tr}\left(Y_{lL}^{\dagger}Y_{lL}\right)Y_{lL}\par+\frac{2021}{200}g_{1L}^{4}Y_{lL}\par+\frac{27}{20}g_{1L}^{2}g_{2L}^{2}Y_{lL}\par+\frac{67}{4}g_{2L}^{4}Y_{lL}\,.$
(160)
$\displaystyle\autobreak\beta^{(1)}(Y_{dR})=\par-\frac{9}{2}Y_{dR}Y_{uR}^{*}Y_{uR}\par+3Y_{dR}Y_{dR}^{\dagger}Y_{dR}\par+\frac{3}{2}Y_{D}^{\dagger}Y_{D}Y_{dR}\par+3\mathrm{Tr}\left(Y_{uR}^{*}Y_{uR}\right)Y_{dR}\par+4\mathrm{Tr}\left(Y_{dR}^{\dagger}Y_{dR}\right)Y_{dR}\par-18g_{5R}^{2}Y_{dR}\,.$
(161)