aainstitutetext: University of Split, Faculty of Electrical Engineering,
Mechanical Engineering and Naval Architecture in Split, Ruđera Boškovića 32,
HR-21000 Split, Croatiabbinstitutetext: J. Stefan Institute, Jamova 39, P. O.
Box 3000, SI-1001 Ljubljana, Sloveniaccinstitutetext: University of Zagreb,
Faculty of Science, Department of Physics, Bijenička cesta 32, HR-10000
Zagreb, Croatiaddinstitutetext: University of Sarajevo, Faculty of Mechanical
Engineering, Vilsonovo šetalište 9, BA-71000 Sarajevo, Bosnia and
Herzegovinaeeinstitutetext: Department of Physics, University of Ljubljana,
Jadranska 19, SI-1000 Ljubljana, Sloveniaffinstitutetext: Department of
Physics, University of Basel, Klingelbergstrasse 82, CH-4056 Basel,
Switzerland

# Gauge and Scalar Boson Mediated Proton Decay in a Predictive SU(5) GUT Model

Ilja Doršner c,d Emina Džaferović-Mašić b,e Svjetlana Fajfer f and Shaikh Saad
<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

We assess proton decay signatures in the simplest viable $SU(5)$ model with
regard to constraints on parameters governing the Standard Model fermion mass
spectrum. Experimental signals for all eight two-body proton decay processes
result from exchange of two gauge bosons, a single scalar leptoquark, or their
combination. Consequently, it enables us to delve into an in-depth anatomy of
proton decay modes and anticipate future signatures. Our findings dictate that
observing a proton decay into $p\to\pi^{0}e^{+}$ indicates gauge boson
mediation, with the potential for observation of $p\to\eta^{0}e^{+}$ mode.
Alternatively, if decay is through $p\to K^{+}\overline{\nu}$ process, it is
mediated by a scalar leptoquark, possibly allowing the observation of
$p\to\pi^{0}\mu^{+}$. Detection of both $p\to\pi^{0}e^{+}$ and $p\to
K^{+}\overline{\nu}$ could enhance $p\to\pi^{0}\mu^{+}$ through constructive
interference. The model predicts inaccessibility of
$p\to\pi^{+}\overline{\nu}$, $p\to\eta^{0}\mu^{+}$, $p\to K^{0}e^{+}$, and
$p\to K^{0}\mu^{+}$, regardless of the dominant mediation type, in the coming
decades. In summary, through a comprehensive analysis of proton decay signals,
gauge coupling unification, and fermion masses and mixing, we precisely
constrain the parameter space of the $SU(5)$ model in question.

##  1 Introduction

Grand Unified Theories Pati:1973rp ; Pati:1974yy ; Georgi:1974sy ;
Georgi:1974yf ; Georgi:1974my ; Fritzsch:1974nn  (GUTs) are theoretical
frameworks that seek to unify three fundamental forces of the Standard Model
(SM) of elementary particle physics — the electromagnetic, weak, and strong
interactions — into a single one. Within these frameworks, the fundamental
constituents of matter, namely quarks and leptons, are also partially or
completely unified. One of the hallmark predictions of GUTs is thus that
protons are not absolutely stable. Proton decay is, therefore, a crucial
signature of GUTs that offers a potential avenue to test these theories, even
though the relevant energy scales are far beyond the direct reach with current
or any foreseeable experimental capabilities.

The discovery of proton decay would be a groundbreaking shift in the field of
particle physics and an important development in our understanding of the
universe. The search for proton decay has consequently motivated development
of humongous detectors of unprecedented sensitivity, such as Super-Kamiokande
and its successor, Hyper-Kamiokande (Hyper-K). It is worthwhile to point out
that other forthcoming large-volume neutrino experiments, specifically JUNO
and DUNE, are expected to probe proton decay lifetimes with an efficacy
nearing that of the maximum achievable Hyper-K reach.

Hyper-K is the next-generation neutrino observatory and is scheduled to start
observation in 2027. Remarkably, due to its enhanced sensitivity and a larger
detection volume, Hyper-K will significantly improve the chances of observing
proton decay with partial lifetimes as high as $10^{35}$ years within a decade
of operation. Owing to these advancements on the experimental side, it is
timely to conduct an in-depth evaluation of the potential for observing proton
decay in predictive and simple unified theories, which is the prime objective
of the current manuscript.

The Georgi-Glashow model Georgi:1974sy  played a pivotal role in the early
development of GUTs. However, this model, consisting of only an adjoint Higgs
to break the GUT symmetry and a fundamental Higgs to break the electroweak
symmetry, is not compatible with experimental data due to its simplicity. More
precisely, within this model (i) gauge couplings do not unify, (ii) one cannot
accommodate neutrino masses, and (iii) there is a degeneracy between the down-
type quark and charged lepton masses that contradicts the data.

It is, therefore, essential to extend the Georgi-Glashow model to overcome the
limitations listed above. While making such an extension, it is desirable to
keep the model as simple as possible and, more importantly, predictive. In
light of this, one of the simplest and realistic GUTs, based on $SU(5)$ gauge
symmetry, was recently proposed in Ref. Dorsner:2019vgf . This minimal
realization utilizes only the first five lowest-lying representations of
dimensions 5, 10, 15, 24, and 35. It has a limited number of model parameters
that makes it a highly predictive setup. Intriguingly, apart from two
superheavy gauge bosons, only a single scalar leptoquark participates in
proton decay mediation. This specific feature, coupled with the intricate
interplay of model parameters necessary to replicate the experimentally
observed masses and mixings of charged fermions and neutrinos, yields
predictions regarding both the unification scale and the rates of various
proton decay modes. The aim of this manuscript is to thoroughly examine the
relationship between these decay modes and to arrive at robust predictions
poised for potential observation in the forthcoming Hyper-K experiment.

To concisely highlight our findings, we present the following summary. Our
study infers that the decay of a proton into $p\to\pi^{0}e^{+}$ indicates
gauge boson mediation, which also opens a possibility of observing the
$p\to\eta^{0}e^{+}$ decay. Conversely, a proton decay that follows the $p\to
K^{+}\overline{\nu}$ route suggests an exchange of a scalar leptoquark,
raising a potential to detect the $p\to\pi^{0}\mu^{+}$ decay. The concurrent
observation of $p\to\pi^{0}e^{+}$ and $p\to K^{+}\overline{\nu}$ decays might
amplify the chances of seeing the $p\to\pi^{0}\mu^{+}$ decay due to an
interference effect. Our model forecasts that decay modes
$p\to\pi^{+}\overline{\nu}$, $p\to\eta^{0}\mu^{+}$, $p\to K^{0}e^{+}$, and
$p\to K^{0}\mu^{+}$ will remain out of reach in the coming decades, regardless
of the dominant decay mechanism.

Our manuscript is organized as follows. We present main features of the model,
such as the particle content interactions and symmetry breaking intricacies,
in Sec. 2. Since The SM fermion mass generation relies on three different
mechanisms for its viability, we discuss relevant subtleties of these
mechanisms in Sec. 3. The proton decay predictions, for both the gauge boson
and scalar leptoquark mediations, are discussed at length in Sec. 4.
Predictions of our model, with regard to gauge coupling unification and proton
decay signatures for all eight two-body decay processes and both types of
mediations, are presented in Sec. 5. We briefly conclude in Sec. 6.

##  2 Model

The model under consideration contains scalars ($5_{H}$, $24_{H}$, and
$35_{H}$), fermions ($\overline{5}_{Fi}$, $10_{Fi}$, and
$15_{F}+\overline{15}_{F}$), and gauge fields ($24_{G}$), where $i=1,2,3$. The
full scalar and fermionic content of this $SU(5)$ model, including its
decomposition under the SM gauge group, is summarized in Table I.

SCALARS | FERMIONS
---|---
$SU(5)$ | Standard Model | $(b_{3},b_{2},b_{1})$ | $SU(5)$ | Standard Model | $(b_{3},b_{2},b_{1})$
| $\Lambda_{1}\left(1,2,+\frac{1}{2}\right)$ | $\left(0,\frac{1}{6},\frac{1}{10}\right)$ |  | $L_{i}\left(1,2,-\frac{1}{2}\right)$ | $\left(0,1,\frac{3}{5}\right)$
$\Lambda=5_{H}$ | $\Lambda_{3}\left(3,1,-\frac{1}{3}\right)$ | $\left(\frac{1}{6},0,\frac{1}{15}\right)$ | $F_{i}={\overline{5}_{F}}_{i}$ | $d_{i}^{c}\left(\overline{3},1,+\frac{1}{3}\right)$ | $\left(1,0,\frac{2}{5}\right)$
| $\phi_{0}\left(1,1,0\right)$ | $\left(0,0,0\right)$ |  | $Q_{i}\left(3,2,+\frac{1}{6}\right)$ | $\left(2,3,\frac{1}{5}\right)$
| $\phi_{1}\left(1,3,0\right)$ | $\left(0,\frac{1}{3},0\right)$ | $T_{i}={10_{F}}_{i}$ | $u_{i}^{c}\left(\overline{3},1,-\frac{2}{3}\right)$ | $\left(1,0,\frac{8}{5}\right)$
$\phi=24_{H}$ | $\phi_{3}\left(3,2,-\frac{5}{6}\right)$ | $\left(\frac{1}{6},\frac{1}{4},\frac{5}{12}\right)$ |  | $e_{i}^{c}\left(1,1,+1\right)$ | $\left(0,0,\frac{6}{5}\right)$
| $\phi_{\overline{3}}\left(\overline{3},2,+\frac{5}{6}\right)$ | $\left(\frac{1}{6},\frac{1}{4},\frac{5}{12}\right)$ |  | $\Sigma_{1}(1,3,+1)$ | $\left(0,\frac{4}{3},\frac{6}{5}\right)$
| $\phi_{8}\left(8,1,0\right)$ | $\left(\frac{1}{2},0,0\right)$ | $\Sigma=15_{F}$ | $\Sigma_{3}\left(3,2,+\frac{1}{6}\right)$ | $\left(\frac{2}{3},1,\frac{1}{15}\right)$
| $\Phi_{1}\left(1,4,-\frac{3}{2}\right)$ | $\left(0,\frac{5}{3},\frac{9}{5}\right)$ |  | $\Sigma_{6}\left(6,1,-\frac{2}{3}\right)$ | $\left(\frac{5}{3},0,\frac{16}{15}\right)$
| $\Phi_{3}\left(\overline{3},3,-\frac{2}{3}\right)$ | $\left(\frac{1}{2},2,\frac{4}{5}\right)$ |  | $\overline{\Sigma}_{1}\left(1,3,-1\right)$ | $\left(0,\frac{4}{3},\frac{6}{5}\right)$
$\Phi=35_{H}$ | $\Phi_{6}\left(\overline{6},2,+\frac{1}{6}\right)$ | $\left(\frac{5}{3},1,\frac{1}{15}\right)$ | $\overline{\Sigma}=\overline{15}_{F}$ | $\overline{\Sigma}_{3}\left(\overline{3},2,-\frac{1}{6}\right)$ | $\left(\frac{2}{3},1,\frac{1}{15}\right)$
| $\Phi_{10}\left(\overline{10},1,+1\right)$ | $\left(\frac{5}{2},0,2\right)$ |  | $\overline{\Sigma}_{6}\left(\overline{6},1,+\frac{2}{3}\right)$ | $\left(\frac{5}{3},0,\frac{16}{15}\right)$
Table I: Particle content of the model, its decomposition under the SM gauge
group, and associated $\beta$-function coefficients. $i=1,2,3$ represents a
generation index.

The full scalar potential reads

$\displaystyle\mathcal{L}_{V}=$
$\displaystyle-\mu^{2}_{\Lambda}\left(\Lambda^{*}_{\alpha}\Lambda^{\alpha}\right)+\lambda^{\Lambda}_{0}\left(\Lambda^{*}_{\alpha}\Lambda^{\alpha}\right)^{2}+\mu_{1}\Lambda^{*}_{\alpha}\Lambda^{\beta}\phi^{\alpha}_{\beta}+\lambda^{\Lambda}_{1}\left(\Lambda^{*}_{\alpha}\Lambda^{\alpha}\right)\left(\phi^{\beta}_{\gamma}\phi_{\beta}^{\gamma}\right)+\lambda^{\Lambda}_{2}\Lambda^{*}_{\alpha}\Lambda^{\beta}\phi_{\beta}^{\gamma}\phi_{\gamma}^{\alpha}$
$\displaystyle-\mu^{2}_{\phi}\left(\phi^{\beta}_{\gamma}\phi_{\beta}^{\gamma}\right)+\mu_{2}\phi^{\alpha}_{\beta}\phi^{\beta}_{\gamma}\phi^{\gamma}_{\alpha}+\lambda^{\phi}_{0}\left(\phi^{\beta}_{\gamma}\phi_{\beta}^{\gamma}\right)^{2}+\lambda^{\phi}_{1}\phi^{\alpha}_{\beta}\phi^{\beta}_{\gamma}\phi^{\gamma}_{\delta}\phi^{\delta}_{\alpha}+\mu^{2}_{\Phi}\left(\Phi^{*\alpha\beta\gamma}\Phi_{\alpha\beta\gamma}\right)$
$\displaystyle+\lambda^{\Phi}_{0}\left(\Phi^{*\alpha\beta\gamma}\Phi_{\alpha\beta\gamma}\right)^{2}+\lambda^{\Phi}_{1}\Phi^{*\alpha\beta\gamma}\Phi_{\alpha\beta\delta}\Phi^{*\delta\rho\sigma}\Phi_{\rho\sigma\gamma}+\lambda_{0}\left(\Phi^{*\alpha\beta\gamma}\Phi_{\alpha\beta\gamma}\right)\left(\phi^{\delta}_{\rho}\phi_{\delta}^{\rho}\right)$
$\displaystyle+\lambda_{0}^{\prime}\left(\Phi^{*\alpha\beta\gamma}\Phi_{\alpha\beta\gamma}\right)\left(\Lambda^{*}_{\rho}\Lambda^{\rho}\right)+\lambda_{0}^{\prime\prime}\Phi^{*\alpha\beta\gamma}\Phi_{\beta\gamma\delta}\Lambda^{\delta}\Lambda^{*}_{\alpha}+\mu_{3}\Phi^{*\alpha\beta\gamma}\Phi_{\beta\gamma\delta}\phi^{\delta}_{\alpha}$
$\displaystyle+\lambda_{1}\Phi^{*\alpha\beta\gamma}\Phi_{\alpha\delta\rho}\phi^{\delta}_{\beta}\phi^{\rho}_{\gamma}+\lambda_{2}\Phi^{*\alpha\beta\rho}\Phi_{\alpha\beta\delta}\phi^{\gamma}_{\rho}\phi^{\delta}_{\gamma}+\left\\{\lambda^{\prime}\Lambda^{\alpha}\Lambda^{\beta}\Lambda^{\gamma}\Phi_{\alpha\beta\gamma}+\mathrm{h.c.}\right\\},$
(2.1)

where, again, $\Lambda=5_{H}$, $\phi=24_{H}$, $\Phi=35_{H}$, and
$\alpha,\beta,\gamma,\delta,\sigma,\rho=1,2,3,4,5$ are used to denote
associated $SU(5)$ indices.

The complete Yukawa part of the Lagrangian reads

$\displaystyle\mathcal{L}_{Y}=$
$\displaystyle\left\\{Y^{u}_{ij}T^{\alpha\beta}_{i}T^{\gamma\delta}_{j}\Lambda^{\rho}\epsilon_{\alpha\beta\gamma\delta\rho}+Y^{d}_{ij}T^{\alpha\beta}_{i}F_{\alpha
j}\Lambda^{\ast}_{\beta}+Y^{a}_{i}\Sigma^{\alpha\beta}F_{\alpha
i}\Lambda^{\ast}_{\beta}+Y^{b}_{i}\overline{\Sigma}_{\beta\gamma}F_{\alpha
i}\Phi^{*\alpha\beta\gamma}\right.$
$\displaystyle\left.+Y^{c}_{i}T^{\alpha\beta}_{i}\overline{\Sigma}_{\beta\gamma}\phi^{\gamma}_{\alpha}+\mathrm{h.c.}\right\\}+M_{\Sigma}\overline{\Sigma}_{\alpha\beta}\Sigma^{\alpha\beta}+y\overline{\Sigma}_{\alpha\beta}\Sigma^{\beta\gamma}\phi_{\gamma}^{\alpha},$
(2.2)

where we specify contractions in both the $SU(5)$ and flavor spaces. $Y^{u}$
and $Y^{d}$ are, in general, arbitrary $3\times 3$ complex Yukawa coupling
matrices, $Y^{a}$, $Y^{b}$, and $Y^{c}$ are complex Yukawa coupling vectors of
length 3, whereas $y$ is just a real number. The relevant matrix elements are
denoted with $Y^{u}_{ij}$, $Y^{d}_{ij}$, $Y^{a}_{i}$, $Y^{b}_{i}$,
$Y^{c}_{i}$, and $y$, where, again, $i,j=1,2,3$ represent generation indices.

We note that it is possible, without loss of generality, to use unitary field
rotations of representations $\overline{5}_{Fi}$ and ${10_{F}}_{j}$ to go into
a specific basis where $Y^{d}$ is a diagonal matrix with real and positive
entries. We will use this basis for subsequent discussion and our numerical
analysis. Also, all physical interactions at the SM level that feature $Y^{u}$
are always proportional to linear combinations $Y^{u}_{ij}+Y^{u}_{ji}$ that
are manifestly symmetric in the flavor space. One can thus either redefine
$Y^{u}$ to be a symmetric matrix and keep track of factors of two associated
with that redefinition or continue to treat $Y^{u}$ as an arbitrary $3\times
3$ complex matrix. We opt for the latter approach.

The $SU(5)$ symmetry of the model is broken directly down to the SM gauge
group $SU(3)\times SU(2)\times U(1)$ when $\phi_{0}\in 24_{H}$ acquires a
specific vacuum expectation value (VEV). The SM is subsequently broken at the
electroweak scale by a VEV of $\Lambda_{1}\in 5_{H}$ down to $SU(3)\times
U(1)_{\mathrm{em}}$. This two-step process can be schematically represented as

$\displaystyle SU(5)\xrightarrow[]{\langle 24_{H}\rangle}SU(3)\times
SU(2)\times U(1)\xrightarrow[]{\langle 5_{H}\rangle}SU(3)\times
U(1)_{\mathrm{em}}\;.$ (2.3)

The relevant VEVs, for our study, are

$\displaystyle\langle
24_{H}\rangle=v_{24}\mathrm{diag}\left(-1,-1,-1,3/2,3/2\right),$ (2.4)
$\displaystyle\langle 5_{H}\rangle=(0\quad 0\quad 0\quad 0\quad
v_{5}/\sqrt{2})^{T},$ (2.5)

where $v_{5}\approx 246$ GeV reproduces the masses of the SM gauge boson
fields $W_{\mu}^{\pm}$ and $Z_{\mu}$. The masses of the superheavy gauge
bosons $X_{\mu}^{\pm 4/3}\in 24_{G}$ and $Y_{\mu}^{\pm 1/3}\in 24_{G}$, with
the VEV of $24_{H}$ given via Eq. (2.4), turn out to be

$\displaystyle M_{X}=M_{Y}=\sqrt{\frac{25}{8}}g_{\mathrm{GUT}}v_{24},$ (2.6)

where $g_{\mathrm{GUT}}$ is a gauge coupling constant at the unification
scale.

One especially convenient feature of $SU(5)$ is that the gauge coupling
unification scale $M_{\mathrm{GUT}}$ can be identified, for all practical
purposes, with the mass of proton decay mediating gauge bosons $X_{\mu}^{\pm
4/3}$ and $Y_{\mu}^{\pm 1/3}$. We can thus set $M_{X}=M_{Y}\equiv
M_{\mathrm{GUT}}$ when we discuss proton decay signatures via gauge mediation.
Also, our model has only one scalar leptoquark $\Lambda_{3}\in 5_{H}$ of mass
$M_{\Lambda_{3}}$ that mediates proton decay.

Note that our discussion of the symmetry breaking procedure does not include
possible VEV of an electrically neutral component of $\phi_{1}\in 24_{H}$
that, if present, contributes to masses of $W_{\mu}^{\pm}$ gauge boson fields
Dorsner:2007fy . We also omit a VEV of an electrically neutral component of
$\Phi_{1}\in 35_{H}$ in our symmetry breaking discussion. These VEVs are
expected to be much smaller than the electroweak scale and can thus be safely
neglected from our symmetry breaking discussion.

If $SU(5)$ symmetry is exact, all the SM multiplets within a given $SU(5)$
representation are degenerate in mass. This, of course, is no longer true
after we spontaneously break it. We accordingly find two particular relations
between the masses of the SM multiplets residing in $35_{H}$ and
$15_{F}+\overline{15}_{F}$ that are especially relevant for the gauge coupling
unification study. Namely, an analysis of the potential in Eq. (2.1), after
the first stage of symmetry breaking, stipulates that

$M^{2}_{\Phi_{10}}=M^{2}_{\Phi_{1}}-3M^{2}_{\Phi_{3}}+3M^{2}_{\Phi_{6}},$
(2.7)

whereas the last two terms in Eq. (2.2) yield

$M_{\Sigma_{6}}=2M_{\Sigma_{3}}-M_{\Sigma_{1}}.$ (2.8)

Our one-loop level gauge coupling unification analysis reveals that fields
$\Phi_{10}$ and $\Phi_{1}$ need to be degenerate in mass and heavy, i.e., of
the order of $10^{11}$ GeV, whereas $\Phi_{3}$ and $\Phi_{6}$ also need to be
degenerate in mass but light, i.e., of the order of $10$ TeV, if
$M_{\mathrm{GUT}}$ is to be at, or exceed, existing experimental limits on
partial proton decay lifetimes. Moreover, under the assumption that parameters
$M_{\Sigma_{6}}$, $M_{\Sigma_{3}}$, and $M_{\Sigma_{1}}$ are all positive, the
masses of $\Sigma_{6}$, $\Sigma_{3}$, and $\Sigma_{1}$ fields are degenerate
and, as such, are not constrained by the gauge coupling unification
requirement, at all. It turns out, however, that phenomenological viability of
neutrino masses requires $M_{\Sigma_{1}}$ to be of the order of
$M_{\Phi_{1}}$, as we demonstrate later on.

We implement one-loop level gauge coupling unification analysis in order to
find the largest possible value of unification scale $M_{\mathrm{GUT}}$ and
associated value of $\alpha_{\mathrm{GUT}}=g^{2}_{\mathrm{GUT}}/(4\pi)$ for
fixed values of $M_{\Phi_{1}}$ and $M_{\Sigma_{1}}$. Again, since
$M_{\Phi_{1}}\sim M_{\Phi_{10}}$ and $M_{\Sigma_{1}}\sim M_{\Sigma_{3}}\sim
M_{\Sigma_{6}}$, the only remaining parameters we need to vary within our
unification analysis are the masses of $\Phi_{3},\Phi_{6}\in 35_{H}$,
$\phi_{1},\phi_{8}\in 24_{H}$, and $\Lambda_{3}\in 5_{H}$, while the relevant
SM input parameters are $M_{Z}=91.1876$ GeV, $\alpha_{S}(M_{Z})=0.1193\pm
0.0016$, $\alpha^{-1}(M_{Z})=127.906\pm 0.019$, and
$\sin^{2}\theta_{W}(M_{Z})=0.23126\pm 0.00005$ Agashe:2014kda . Since we also
require $M_{\Lambda_{3}}$ to be at, or above, $3\times 10^{11}$ GeV due to the
scalar mediated proton decay requirements Dorsner:2012uz , the only masses one
actually needs to vary in our approach are $M_{\Phi_{3}}$, $M_{\Phi_{6}}$,
$M_{\phi_{1}}$, and $M_{\phi_{8}}$.

The procedure we perform is as follows. First we specify a lower limit on the
masses of the new physics states to establish a connection between the most
accessible scale of new physics and $M_{\mathrm{GUT}}$. This lower limit is
simply denoted with $M=\min(M_{J})$, where
$J(=\Phi_{1},\Phi_{3},\Phi_{6},\Phi_{10},\Sigma_{1},\Sigma_{3},\Sigma_{6},\phi_{1},\phi_{8},\Lambda_{3}$)
is set at $1$ TeV and $10$ TeV. For example, the $M\geq 1$ TeV scenario simply
means that the new physics states $J$ cannot reside below $1$ TeV scale.
Again, this lower limit, in practice, only affects masses of $\Phi_{3}$,
$\Phi_{6}$, $\phi_{1}$, and $\phi_{8}$ as all other fields need to be much
more massive.

We, next, produce a discrete set of values for $M_{\Phi_{1}}$ and
$M_{\Sigma_{1}}$, where we take
$\log_{10}(M_{\Phi_{1}}/1\,\mathrm{GeV})\in[10.2,12.9]$ and
$\log_{10}(M_{\Sigma_{1}}/1\,\mathrm{GeV})\in[8.4,14.2]$ with the lattice
spacing of 0.1 in these $\log$ units in both directions within
$M_{\Phi_{1}}$-$M_{\Sigma_{1}}$ plane. The particular choice of ranges for
$M_{\Phi_{1}}$ and $M_{\Sigma_{1}}$ is dictated by a need to generate high
enough $M_{\mathrm{GUT}}$ to avoid rapid proton decay and to simultaneously
obtain viable neutrino mass scale. For each of these
$(M_{\Phi_{1}},M_{\Sigma_{1}})$ points we freely vary remaining $M_{J}$ masses
until we find maximal possible $M_{\mathrm{GUT}}$. Note that our procedure, at
every point in $M_{\Phi_{1}}$-$M_{\Sigma_{1}}$ plane, generates
$\alpha_{\mathrm{GUT}}$, $M_{\mathrm{GUT}}$, and associated mass spectrum. We
also account for conditions given with Eqs. (2.7) and (2.8) when generating
$M_{\mathrm{GUT}}$, where we discard all those points with
$M_{\mathrm{GUT}}\leq 6\times 10^{15}$ GeV due to gauge mediated proton decay
requirements to save on computing time in subsequent steps. We show in Fig. 2
contours of constant value of $\alpha_{\mathrm{GUT}}$ (blue dot-dashed lines)
and $M_{\mathrm{GUT}}/(10^{15}\,\mathrm{GeV})$ (red solid lines) that we
obtain with this procedure for $M\geq 1$ TeV and $M\geq 10$ TeV scenarios.

One thing to note is that the value of $M_{\mathrm{GUT}}$ is independent of
$M_{\Sigma_{1}}$ due to degeneracy of fields in $15_{F}+\overline{15}_{F}$, in
agreement with our previous discussion. Also, to produce $M_{\mathrm{GUT}}$
values, as given in Fig. 2, we need to have four multiplets to be exactly at
$M$ scale. These multiplets are $\Phi_{3},\Phi_{6}\in 35_{H}$ and
$\phi_{1},\phi_{8}\in 24_{H}$. If the mass of any of these four multiplets is
simply shifted up, the unification scale $M_{\mathrm{GUT}}$ would go down.
This means that we are discussing the most conservative scenario with regard
to the proton decay signatures via gauge boson exchange. Also, one can treat
scale $M$, for all practical purposes, as a geometric mean of masses
$M_{\Phi_{3}}$, $M_{\Phi_{6}}$, $M_{\phi_{1}}$, and $M_{\phi_{8}}$ when one
studies impact of its change on $M_{\mathrm{GUT}}$.

The second step of the numerical analysis is to run the masses and mixing
parameters of the SM charged fermions from $M_{Z}$ to $M_{\mathrm{GUT}}$ using
the factual new physics mass spectrum associated with every unification point
that was obtained in the previous step. The charged fermion mass
renormalization group running is performed at the one-loop level Arason:1991ic
. We separate the gauge coupling unification study from the running of the SM
charged fermion parameters since the latter provides feedback to the former
only at the two-loop level, whereas the former impacts the latter already at
the one-loop level. A summary of experimentally measured observables with the
associated $1\,\sigma$ uncertainties of both the charged and neutral fermion
sectors at low-energy scale is provided in Table II.

5cm$m(M_{Z})$ (GeV) | Fit input |  $\theta^{\rm{CKM},\rm{PMNS}}_{ij}$ & $\delta^{\rm{CKM}}$ & $\Delta m^{2}_{ij}$ (eV2) | Fit input
---|---|---|---
$m_{u}/10^{-3}$ | $1.158\pm 0.392$ | $\sin\theta^{\rm{CKM}}_{12}$ | $0.2254\pm 0.00072$
$m_{c}$ | $0.627\pm 0.019$ | $\sin\theta^{\rm{CKM}}_{23}/10^{-2}$ | $4.207\pm 0.064$
$m_{t}$ | $171.675\pm 1.506$ | $\sin\theta^{\rm{CKM}}_{13}/10^{-3}$ | $3.640\pm 0.130$
$m_{d}/10^{-3}$ | $2.864\pm 0.286$ | $\delta^{\rm{CKM}}$ | $1.208\pm 0.054$
$m_{s}/10^{-3}$ | $54.407\pm 2.873$ | $\Delta m^{2}_{21}/10^{-5}$ |  7.425$\pm$0.205
$m_{b}$ | $2.854\pm 0.026$ | $\Delta m^{2}_{3\ell}/10^{-3}$ |  2.515$\pm$0.028
$m_{e}/10^{-3}$ | $0.486576$ | $\sin^{2}\theta^{\rm{PMNS}}_{12}/10^{-1}$ |  3.045$\pm$0.125
$m_{\mu}$ | $0.102719$ | $\sin^{2}\theta^{\rm{PMNS}}_{23}$ |  0.554$\pm$0.021
$m_{\tau}$ | $1.74618$ | $\sin^{2}\theta^{\rm{PMNS}}_{13}/10^{-2}$ | $2.224\pm 0.065$

Table II: Experimental measurements related to charged fermions
Antusch:2013jca  and neutrinos for normal ordering Esteban:2020cvm  with 1
$\sigma$ uncertainties.

Due to a minimal impact of the running of neutrino observables from $M_{Z}$ to
$M_{\mathrm{GUT}}$, we opt to always use corresponding low-energy scale values
in our numerical analysis.

##  3 Fermion Masses

The SM fermion mass generation, in our model, relies on three different
mechanisms for its viability. We discuss relevant subtleties of these
mechanisms in what follows.

###  3.1 Charged fermion sector

We first revisit the charged fermion mass generation without inclusion of the
vector-like fermion corrections for clarity of exposition. Recall, $i$-th
generation of the SM fermions, in the Georgi-Glashow $SU(5)$, is entirely
embedded in $\overline{5}_{Fi}$ and $10_{Fi}$ Georgi:1974sy . For example, the
first generation is

$\displaystyle\overline{5}_{F1}=\begin{pmatrix}d^{C}_{1}\\\ d^{C}_{2}\\\
d^{C}_{3}\\\ e\\\
-\nu_{e}\end{pmatrix},\;\;\;10_{F1}=\frac{1}{\sqrt{2}}\begin{pmatrix}0&u^{C}_{3}&-u^{C}_{2}&u_{1}&d_{1}\\\
-u^{C}_{3}&0&u^{C}_{1}&u_{2}&d_{2}\\\ u^{C}_{2}&-u^{C}_{1}&0&u_{3}&d_{3}\\\
-u_{1}&-u_{2}&-u_{3}&0&e^{C}\\\ -d_{1}&-d_{2}&-d_{3}&-e^{C}&0\end{pmatrix},$
(3.9)

where all the fields are left-handed.

Once the electroweak symmetry is broken via VEV of Eq. (2.5), the first two
terms in Eq. (2.2) generate the following mass matrices for the up-type
quarks, charged leptons, and down-type quarks:

$\displaystyle M_{U}$
$\displaystyle=\sqrt{2}v_{5}\left(Y^{u}+Y^{u\,T}\right),$ (3.10)
$\displaystyle M_{E}$ $\displaystyle=\frac{v_{5}}{2}Y^{d\,T},$ (3.11)
$\displaystyle M_{D}$ $\displaystyle=\frac{v_{5}}{2}Y^{d}.$ (3.12)

We again note that we work in a basis where $Y^{d}$ is a real diagonal matrix.
Clearly, Eqs. (3.11) and (3.12) predict that $m_{e}=m_{d}$, $m_{\mu}=m_{s}$,
and $m_{\tau}=m_{b}$, where these mass relations hold at the unification scale
$M_{\mathrm{GUT}}$.

To generate experimentally observed mismatch between the masses of the down-
type quarks and charged leptons we clearly need to introduce corrections to
either $M_{E}$ or $M_{D}$, or both. Required corrections, in our model, stem
from interactions of the SM charged fermions in $\overline{5}_{Fi}$ and
${10_{F}}_{j}$ with the vector-like fermions in $15_{F}+\overline{15}_{F}$
Shafi:1999rm ; Shafi:1999ft ; Oshimo:2009ia ; Dorsner:2019vgf ;
Dorsner:2021qwg ; Antusch:2023jok ; Antusch:2023mqe . Namely, once both the
$SU(5)$ and electroweak symmetries are broken, fermion submultiplets with the
same transformation properties under the $SU(3)\times U(1)_{\mathrm{em}}$
gauge group that reside in $\overline{5}_{Fi}$, ${10_{F}}_{j}$, $15_{F}$, and
$\overline{15}_{F}$ will mix. The mixing terms of interest arise from the
third, fourth, and fifth contractions of Eq. (2.2) and read

$\displaystyle\mathcal{L}_{Y}\supset$
$\displaystyle-Y^{a}_{i}\;\left(\Sigma^{0}\nu_{i}\frac{h^{0}}{\sqrt{2}}+\Sigma^{e^{C}}e^{-}_{i}\frac{v_{5}}{\sqrt{2}}+\Sigma^{d}d^{C}_{i}\frac{v_{5}}{\sqrt{2}}\right)$
$\displaystyle-Y^{b}_{i}\;\overline{\Sigma}^{0}\nu_{i}\frac{\Phi^{0}_{\mathrm{Re}}}{\sqrt{2}}-\frac{5v_{24}}{4}Y^{c}_{i}\;\left(d_{i}\overline{\Sigma}^{d}+u_{i}\overline{\Sigma}^{u}\right),$
(3.13)

where we introduce the electric charge eigenstate fields
$\Sigma^{0},\Sigma^{e^{C}}\in\Sigma_{1}(1,3,1)$ and
$\Sigma^{u},\Sigma^{d}\in\Sigma_{3}(3,2,1/6)$. Moreover,
$\Phi^{0}_{\mathrm{Re}}$ denotes the real part of $\Phi_{1}\in 35_{H}$,
whereas $h^{0}\in\Lambda_{1}\in 5_{H}$ denotes what will primarily be the SM
Higgs boson. We stress that $\Phi^{0}_{\mathrm{Re}}$ and $h^{0}$, strictly
speaking, are not exact mass eigenstates as these two states mix, as we
discuss later. The masses of associated linear combinations of
$\Phi^{0}_{\mathrm{Re}}$ and $h^{0}$ will subsequently be denoted with
$M_{\Phi_{1}}$ and $M_{h}$, where $M_{\Phi_{1}}\gg M_{h}$. Also, since
$M_{\Sigma_{1}}$ needs to be of the order of $M_{\Phi_{1}}$ that is heavy and
$M_{\Sigma_{1}}\approx M_{\Sigma_{3}}\approx M_{\Sigma_{6}}$, we can safely
neglect any mass splitting between fermions $\Sigma^{0}$ and $\Sigma^{e^{C}}$
or $\Sigma^{u}$ and $\Sigma^{d}$ that is induced through the mixing with the
SM fermions.

Finally, Eqs. (3.10), (3.11), (3.12) and (3.13), when put together, yield
$4\times 4$ mass matrices for the charged fermions that read

$\displaystyle\mathcal{L}_{Y}\supset$
$\displaystyle\begin{pmatrix}u_{i}&\Sigma^{u}\end{pmatrix}\begin{pmatrix}\begin{array}[]{c|c}\sqrt{2}v_{5}(Y^{u}_{ij}+Y^{u}_{ji})&-\frac{5v_{24}}{4}Y^{c}_{i}\\\
\hline\cr 0&M_{\Sigma_{3}}\end{array}\end{pmatrix}\begin{pmatrix}u^{C}_{j}\\\
\overline{\Sigma}^{u}\end{pmatrix}$
$\displaystyle+\begin{pmatrix}d_{i}&\Sigma^{d}\end{pmatrix}\begin{pmatrix}\begin{array}[]{c|c}\frac{1}{2}v_{5}Y^{d}_{ij}&-\frac{5v_{24}}{4}Y^{c}_{i}\\\
\hline\cr\frac{1}{\sqrt{2}}v_{5}Y^{a}_{j}&M_{\Sigma_{3}}\end{array}\end{pmatrix}\begin{pmatrix}d^{C}_{j}\\\
\overline{\Sigma}^{d}\end{pmatrix}$
$\displaystyle+\begin{pmatrix}e_{i}&\overline{\Sigma}^{e^{C}}\end{pmatrix}\begin{pmatrix}\begin{array}[]{c|c}\frac{1}{2}v_{5}Y^{d}_{ji}&\frac{1}{\sqrt{2}}v_{5}Y^{a}_{i}\\\
\hline\cr 0&M_{\Sigma_{1}}\end{array}\end{pmatrix}\begin{pmatrix}e^{C}_{j}\\\
\Sigma^{e^{C}}\end{pmatrix}\,.$ (3.14)

Again, we are interested in a scenario where $M_{\Sigma_{i}}\gg v_{5}$. In
this particular limit, the $3\times 3$ mass matrices of the charged fermions
of the SM take the form

$\displaystyle
M_{U}=\left(\mathbb{I}+\epsilon^{2}\;Y^{c}{Y^{c}}^{\dagger}\right)^{-\frac{1}{2}}\;\sqrt{2}v_{5}(Y^{u}+Y^{u\,T}),$
(3.15) $\displaystyle M_{E}=\frac{1}{2}v_{5}Y^{d\,T},$ (3.16) $\displaystyle
M_{D}=\left(\mathbb{I}+\epsilon^{2}\;Y^{c}{Y^{c}}^{\dagger}\right)^{-\frac{1}{2}}\left(M_{E}^{T}+\frac{1}{\sqrt{2}}v_{5}\;\epsilon\;Y^{c}Y^{a}\right),$
(3.17)

where we define

$\displaystyle\epsilon=\frac{5}{4}\frac{v_{24}}{M_{\Sigma_{3}}}.$ (3.18)

Our numerical analysis reveals that
$\mathbb{I}+\epsilon^{2}\;Y^{c}{Y^{c}}^{\dagger}\approx\mathbb{I}$ to a great
accuracy. This guarantees that $M_{U}=M_{U}^{T}$ and makes transparent that a
mismatch between the down-type quark and charged lepton masses originates
solely from a single rank-one matrix with elements proportional to the product
$Y^{c}_{i}Y^{a}_{j}$. Also, a nice feature of this set-up is that the values
of charged fermion masses directly determine the entries of diagonal matrix
$Y^{d}$ via Eq. (3.16).

###  3.2 Neutrino sector

Neutrino masses receive contributions at both the tree- and one-loop levels
Babu:2009aq ; Bambhaniya:2013yca . However, in the viable parameter space of
our model, the tree-level contribution to the neutrino masses can be
completely neglected when compared to the latter one. We, nevertheless, quote
it for completeness. It reads

$(M_{N})^{\mathrm{tree}}_{ij}=-\frac{\lambda^{\prime}v^{4}_{5}}{4M_{\Sigma_{1}}M^{2}_{\Phi_{1}}}\left(Y^{a}_{i}Y^{b}_{j}+Y^{b}_{i}Y^{a}_{j}\right).$
(3.19)

Eq. (3.19) implies that the need to have rather heavy $\Phi_{1}$, i.e.,
$M_{\Phi_{1}}\sim 10^{11}$ GeV, in order to unify gauge coupling constants of
the SM at high enough scale $M_{\mathrm{GUT}}$, does not allow for large
enough neutrino mass scale even for electroweak scale $M_{\Sigma_{1}}$.

The leading contribution to neutrino masses is of the one-loop level topology
and is shown in Fig. 1. (For other radiative neutrino mass models within the
$SU(5)$ setup, see, for example, Refs. Wolfenstein:1980sf ; Barbieri:1981yw ;
Perez:2016qbo ; Kumericki:2017sfc ; Saad:2019vjo ; Klein:2019jgb .)

Figure 1: The Feynman diagrams of the leading order contribution towards
Majorana neutrino masses at the $SU(5)$ (left panel) and the SM (right panel)
levels.

A necessary ingredient to generate non-zero masses for the neutrinos, as can
be seen from Fig. 1, is the very last term in Eq. (2.1) that leads to a mixing
between $h^{0}\in 5_{H}$ and a field $\Phi^{0}_{\mathrm{Re}}\in 35_{H}$. This
term reads

$\displaystyle\mathcal{L}_{V}\supset\lambda^{\prime}\;\Lambda^{\alpha}\Lambda^{\beta}\Lambda^{\gamma}\Phi_{\alpha\beta\gamma}+\mathrm{h.c.}\supset
2\times\lambda^{\prime}\;\frac{v_{5}^{2}}{4}h^{0}\Phi^{0}_{\mathrm{Re}},$
(3.20)

where we explicitly show the $SU(5)$ origin of the relevant SM interaction.

From the above term, the mixing angle $\theta$ between $h^{0}$ and
$\Phi^{0}_{\mathrm{Re}}$ is given by

$\displaystyle\sin
2\theta=\frac{\lambda^{\prime}v^{2}_{5}}{M^{2}_{\Phi_{1}}-M^{2}_{h}}\;,$
(3.21)

where, again, $M_{\Phi_{1}}$ and $M_{h}$ denote physical masses of linear
combinations of $h^{0}\in\Lambda_{1}\in 5_{H}$ and
$\Phi^{0}_{\mathrm{Re}}\in\Phi_{1}\in 35_{H}$. Here, $M_{h}$ is the mass of
the SM Higgs boson. If we properly take into account all the mixing and loop
factors, we derive the following neutrino mass formula:

$\displaystyle(M_{N})^{\mathrm{loop}}_{ij}$
$\displaystyle=\frac{\lambda^{\prime}v_{5}^{2}}{64\pi^{2}}(Y^{a}_{i}Y^{b}_{j}+Y^{b}_{i}Y^{a}_{j})\frac{M_{\Sigma_{1}}}{M^{2}_{\Phi_{1}}-M^{2}_{h}}\bigg{\\{}\frac{M^{2}_{\Phi_{1}}\ln\frac{M^{2}_{\Sigma_{1}}}{M^{2}_{\Phi_{1}}}}{M^{2}_{\Sigma_{1}}-M^{2}_{\Phi_{1}}}-\frac{M^{2}_{h}\ln\frac{M^{2}_{\Sigma_{1}}}{M^{2}_{h}}}{M^{2}_{\Sigma_{1}}-M^{2}_{h}}\bigg{\\}}.$
(3.22)

In the limit when $M_{\Phi_{1}},M_{\Sigma_{1}}\gg M_{h}$, we find

$(M_{N})^{\mathrm{loop}}_{ij}=m_{0}(Y^{a}_{i}Y^{b}_{j}+Y^{b}_{i}Y^{a}_{j}),$
(3.23)

where we introduce dimensionful parameter $m_{0}$ that sets the neutrino mass
scale. It reads

$m_{0}=\frac{\lambda^{\prime}v_{5}^{2}}{64\pi^{2}}\frac{M_{\Sigma_{1}}}{M^{2}_{\Sigma_{1}}-M^{2}_{\Phi_{1}}}\ln\frac{M^{2}_{\Sigma_{1}}}{M^{2}_{\Phi_{1}}}.$
(3.24)

The mass dependence of Eq. (3.24) clearly demonstrates that it is possible to
have satisfactorily large value of $m_{0}$ for $M_{\Phi_{1}}\sim 10^{11}$ GeV
as long as $M_{\Sigma_{1}}\gg M_{h}$.

We note that both the tree- and one-loop level contributions towards neutrino
mass matrix are proportional to $Y^{a}_{i}Y^{b}_{j}+Y^{b}_{i}Y^{a}_{j}$
combination. This insures that the lightest of the neutrinos is a massless
particle, which is consistent with the current neutrino oscillation data. We
can thus write

$(M_{N})_{ij}=m_{0}\left(Y^{a}_{i}Y^{b}_{j}+Y^{b}_{i}Y^{a}_{j}\right)=(N\;\mathrm{diag}(0,m_{2},m_{3})\;N^{T})_{ij}\,,$
(3.25)

where $N$ is a unitary matrix and $m_{2}$ and $m_{3}$ are neutrino mass
eigenstates. $N$ can actually be written as

$N=\mathrm{diag}(e^{i\gamma_{1}},e^{i\gamma_{2}},e^{i\gamma_{3}})V_{\mathrm{PMNS}}^{*},$
(3.26)

where $V_{\mathrm{PMNS}}$ is the Pontecorvo-Maki-Nakagawa-Sakata (PMNS)
unitary mixing matrix with three mixing angles, two Majorana phases, and one
CP violating Dirac phase.

We can use results of Refs. Cordero-Carrion:2018xre ; Cordero-Carrion:2019qtu
to obtain entries of $Y^{a}$ and $Y^{b}$ from Eq. (3.25) as a function of $N$,
$m_{2}$, and $m_{3}$. Namely, we find, for the normal neutrino mass ordering
scenario, the following expressions for ${Y^{a}}^{T}$ and ${Y^{b}}^{T}$:

$Y^{a\,T}=\frac{1}{\rho\sqrt{2}}\begin{pmatrix}i\;r_{2}\;N_{12}+r_{3}\;N_{13}\\\
i\;r_{2}\;N_{22}+r_{3}\;N_{23}\\\
i\;r_{2}\;N_{32}+r_{3}\;N_{33}\end{pmatrix},\;\;Y^{b\,T}=\frac{\rho}{\sqrt{2}}\begin{pmatrix}-i\;r_{2}\;N_{12}+r_{3}\;N_{13}\\\
-i\;r_{2}\;N_{22}+r_{3}\;N_{23}\\\
-i\;r_{2}\;N_{32}+r_{3}\;N_{33}\end{pmatrix},$ (3.27)

where $r_{2}=\sqrt{m_{2}/m_{0}}$, $r_{3}=\sqrt{m_{3}/m_{0}}$, and $\rho$ is an
unknown parameter that accounts for the fact that the matrix elements of
$Y^{a}$ and $Y^{b}$ are always featured as products $Y^{a}_{i}Y^{b}_{j}$ in
Eq. (3.25). Since $V_{\mathrm{PMNS}}$ has three phases, i.e., one CP violating
Dirac phase and two Majorana phases, unitary matrix $N$ has, all in all, six
phases. Hence in Eq. (3.27) we have six arbitrary phases. This is consistent
with the process of trading six real parameters and six phases from $Y^{a}$
and $Y^{b}$ for three PMNS angles, three neutrino masses, and six phases
during the inversion procedure.

We can use experimental input from neutrino sector to directly constrain
viable parameter space of the model. Namely, since the neutrino mass scale
parameter $m_{0}$ depends solely on $M_{\Phi_{1}}$, $M_{\Sigma_{1}}$, and
$\lambda^{\prime}$, we can establish a part of parameter space in
$M_{\Phi_{1}}$-$M_{\Sigma_{1}}$ plane where $m_{0}$ is simply too small to
accommodate experimentally observed neutrino mass-square differences with
perturbative couplings. Namely, if we demand that the dimensionless couplings
do not exceed value of one, we can exclude part of parameter space with
requirement that $m_{0}\leq m_{3}/2$. This naive approach excludes parameter
space to the right of a green curve labeled with $1$ in both panels of Fig. 2,
where we set $|\lambda^{\prime}|=1$. To be precise, green curves in both
panels of Fig. 2 are contours of constant $m_{0}$ in units of $\sqrt{\Delta
m^{2}_{31}}/2$. For example, green curve labeled $10$ can be interpreted in
two different ways. If, for example, we set $|\lambda^{\prime}|=1$, the Yukawa
coupling product $\max(|Y^{a}_{i}Y^{b}_{j}|)$, along that curve, needs to be
$10^{-1}$. Or, if we demand that the Yukawa couplings do not exceed value of
one, the value of $|\lambda^{\prime}|$ is $10^{-1}$ along green curve labeled
with 10.

We can do a more accurate analysis with regard to perturbativity of the Yukawa
couplings in the neutrino sector. Namely, since $Y^{a}$ and $Y^{b}$ entries,
as given via Eq. (3.27), are completely determined via six unknown phases and
$\lambda^{\prime}$, we can vary these phases and demand that both
$\max(|Y^{a}_{i}|)\leq 1$ and $\max(|Y^{b}_{i}|)\leq 1$, where $i=1,2,3$, to
insure perturbativity for fixed values of $M_{\Phi_{1}}$, $M_{\Sigma_{1}}$,
and $|\lambda^{\prime}|=1$. If we do that, we find that the parameter space to
the right of the outermost dashed black curve in both panels of Fig. 2 is
ruled out for any choice of aforementioned six phases. In other words, it is
impossible to have perturbative Yukawa couplings in the neutrino sector for
any point in $M_{\Phi_{1}}$-$M_{\Sigma_{1}}$ plane to the right of outermost
dashed black curve. Similarly, for any choice of these six phases we always
find perturbative solution to the left of the innermost dashed black curve.
Clearly, green curve labeled with $1$ is a good approximation of a more
accurate numerical study. The region between two dashed black curves
represents a region where one can have satisfactory numerical fit of neutrino
masses with perturbative Yukawa couplings but only for very specific choices
of six phases that enter $Y^{a}$ and $Y^{b}$ via Eq. (3.27).

###  3.3 Fermion mass fit

The $3\times 3$ mass matrices $M_{U}$, $M_{E}$, $M_{D}$, and $M_{N}$ of the SM
fermions, as given in Eqs. (3.15), (3.16), (3.17), and (3.25), respectively,
can be diagonalized via

$\displaystyle M_{U}=U_{L}M_{U}^{\mathrm{diag}}U_{R}^{\dagger}\,,$ (3.28)
$\displaystyle M_{D}=D_{L}M_{D}^{\mathrm{diag}}D_{R}^{\dagger}\,,$ (3.29)
$\displaystyle M_{E}=E_{L}M_{E}^{\mathrm{diag}}E_{R}^{\dagger}\,,$ (3.30)
$\displaystyle M_{N}=NM_{N}^{\mathrm{diag}}N^{T}\,,$ (3.31)

where $U_{L}$, $U_{R}$, $D_{L}$, $D_{R}$, $E_{L}$, $E_{R}$, and $N$ are
unitary matrices that implement transition from flavor to mass-eigenstate
basis.

Our numerical study yields $D_{L}$, $D_{R}$, and $N$, as we discuss in detail
later on, whereas the features of the model stipulate that

$\displaystyle U_{L}$
$\displaystyle=D_{L}\mathrm{diag}(1,e^{i\eta_{1}},e^{i\eta_{2}})V_{\mathrm{CKM}}^{T}\,\mathrm{diag}(e^{i\kappa_{1}},e^{i\kappa_{2}},e^{i\kappa_{3}})\equiv
D_{L}D(\eta)V_{\mathrm{CKM}}^{T}D(\kappa)$ (3.32) $\displaystyle U_{R}$
$\displaystyle=U_{L}^{*}\,\mathrm{diag}(e^{i\xi_{1}},e^{i\xi_{2}},e^{i\xi_{3}})\equiv
U_{L}^{*}D(\xi),$ (3.33) $\displaystyle E_{L}$ $\displaystyle=\mathbb{I}\,,$
(3.34) $\displaystyle E_{R}$ $\displaystyle=\mathbb{I}\,,$ (3.35)

where $V_{\mathrm{CKM}}$ is the Cabibbo-Kobayashi-Maskawa (CKM) mixing matrix,
which contains one CP violating Dirac phase $\delta^{\mathrm{CKM}}$. We
introduce in Eqs. (3.32) and (3.33) convenient notation for diagonal phase
matrices $D(\eta)$, $D(\kappa)$, and $D(\xi)$. Note, also, that it is a
symmetric nature of $M_{U}$ matrix that relates $U_{L}$ and $U_{R}$, as given
in Eq. (3.33).

To performe numerical fit, we use charged lepton masses at $M_{\mathrm{GUT}}$
as an input to determine the diagonal matrix $Y^{d}$ via
$Y^{d}=2\;\mathrm{diag}(m_{e},m_{\mu},m_{\tau})/v_{5}$. Since the down-type
quark mass matrix of Eq. (3.17) and the neutrino mass matrix of Eq. (3.25)
share a common Yukawa coupling row matrix $Y^{a}$, a combined fit of these two
sectors is necessary. Our numerical fit accordingly minimizes a $\chi^{2}$
function

$\displaystyle\chi^{2}=\sum_{j=1}^{8}\left(\frac{T_{j}-O_{j}}{E_{j}}\right)^{2},$
(3.36)

where $T_{j}$, $O_{j}$, and $E_{j}$ represent theoretical prediction, measured
central value, and experimental $1\,\sigma$ error for the observable $j$,
respectively. The index $j$ runs over the down-type quark masses and all five
measured observables in the neutrino sector.

We determine the Yukawa coupling matrices $Y^{a}$, $Y^{b}$, and $Y^{c}$ by
fitting them against three down-type quark masses, two neutrino mass-squared
differences, and three mixing angles in the neutrino sector. It is important
to note that the CP-violating Dirac phase and the two Majorana phases in the
neutrino sector have not yet been experimentally measured. These, together
with the entries of $D_{L}$ and $D_{R}$, are actually an output of our fit.

We conduct a comprehensive scan over all viable unification points presented
in Fig. 2, imposing the condition of perturbativity for the relevant
couplings, i.e.,
$\max(|Y^{a}_{i}|),\max(|Y^{b}_{i}|),\max(|Y^{c}_{i}|),|\lambda^{\prime}|\leq
1$. Additionally, we employ the criterion $\chi^{2}/n\leq 1$ to deem a fit as
good, where $n(=8)$ represents the number of fitted observables. It is
noteworthy that not all unification points that allow for a numerically good
fit to the fermion sector successfully pass the proton decay test.

Our comprehensive numerical fit encompassing both the down-type quark and
neutrino sectors highlights a limitation in accommodating the inverted
neutrino mass ordering within this model. Specifically, the hierarchical and
diagonal nature of the down-type quark mass matrix is dictated by the charged
lepton Yukawa coupling matrix $Y^{d}$. Since the matrix elements
$(M_{D})_{ij}$ depend on a linear combination of $(Y^{d})_{ij}$ and
$Y^{c}_{i}Y^{a}_{j}$, it becomes evident that both $Y^{a}$ and $Y^{c}$ must
manifestly be hierarchical row matrices to yield a satisfactory fit to the
data. However, achieving this hierarchical row structure is untenable when
considering the inverted ordering of neutrino masses. In the inverted
scenario, the entries in the first row and the first column of the neutrino
mass matrix $M_{N}$ typically possess comparable magnitudes, while the lower
$2\times 2$ block is required to be somewhat smaller. Working in the mass
eigenstate basis for the charged leptons, this necessity imposes a constraint
that all entries in $Y^{a}$ must be of the same order, contradicting the
hierarchical arrangement needed in the down-type quark sector. This tension
arises directly from the simplicity of our model, leading to the prediction
that neutrinos must exhibit the normal mass ordering.

The fitting procedure, applied under the assumption of the normal ordering of
neutrino masses, enables numerical determination of three unitary rotation
matrices $D_{L}$, $D_{R}$, and $N$, along with the Yukawa couplings for the
charged leptons and the down-type quarks. To comprehensively calculate partial
lifetimes for various proton decay modes arising from both gauge boson and
scalar mediations, it is essential to have knowledge of the unitary matrices
$U_{L}$ and $U_{R}$ that diagonalize the up-type quark mass matrix and its
associated Yukawa couplings. A nice feature of our approach is the ability to
express both $U_{L}$ and $U_{R}$ in terms of $D_{L}$ and $V_{\mathrm{CKM}}$,
as specified in Eqs. (3.32) and (3.33), along with eight additional phases.

In summary, the model accurately accommodates charged lepton masses, up-type
quark masses, and CKM parameters. Additionally, we conduct a combined
numerical fit for the neutrino mass parameters, down-type quark masses, and
PMNS parameters, recognizing their intrinsic interdependence. A pivotal
outcome of the fit for the proton decay considerations lies in the unitary
transformations $U_{L}$, $U_{R}$, $D_{L}$, and $D_{R}$, with the first two
matrices featuring five and three unknown phases, respectively, and numerical
determination of all Yukawa couplings of the model. There are, all in all,
eight additional phases that our fit cannot determine, i.e., $\eta_{1}$,
$\eta_{2}$, $\kappa_{1}$, $\kappa_{2}$, $\kappa_{3}$, $\xi_{1}$, $\xi_{2}$,
and $\xi_{3}$, where, as we show next, only $\eta_{1}$ and $\eta_{2}$ are
relevant for the study of proton decay signatures. This puts us in a perfect
position to produce an in-depth anatomy of proton decay signatures of this
model.

##  4 Proton Decay

The main idea behind our proton decay analysis is to accurately identify the
most dominant channels for both types of mediators that are present in our
model and compare associated predictions for partial lifetimes with current
experimental limits and future expectations based on a ten-year period of data
taking at Hyper-K. A summary of the best experimental limits for all eight
two-body proton decay channels as well as expectations for future
sensitivities at Hyper-K, if and when available, is accordingly given in Table
III. (JUNO Collaboration expects to reach a limit of $9.6\times 10^{33}$ years
for $p\rightarrow K^{+}\overline{\nu}$ after a ten-year period of data taking
JUNO:2022qgr .)

decay channel |  $\tau_{p}$ current bound |  $\tau_{p}$ future sensitivity
---|---|---
$p\rightarrow\pi^{0}e^{+}$ |  $2.4\times 10^{34}$ years Super-Kamiokande:2020wjk  |  $7.8\times 10^{34}$ years Hyper-Kamiokande:2018ofw
$p\rightarrow\pi^{0}\mu^{+}$ |  $1.6\times 10^{34}$ years Super-Kamiokande:2020wjk  |  $7.7\times 10^{34}$ years Hyper-Kamiokande:2018ofw
$p\rightarrow\pi^{+}\overline{\nu}$ |  $3.9\times 10^{33}$ years Super-Kamiokande:2013rwg  | -
$p\rightarrow\eta^{0}e^{+}$ |  $1.0\times 10^{34}$ years Super-Kamiokande:2017gev  |  $4.3\times 10^{34}$ years Hyper-Kamiokande:2018ofw
$p\rightarrow\eta^{0}\mu^{+}$ |  $4.7\times 10^{33}$ years Super-Kamiokande:2017gev  |  $4.9\times 10^{34}$ years Hyper-Kamiokande:2018ofw
$p\rightarrow K^{0}e^{+}$ |  $1.1\times 10^{33}$ years Brock:2012ogj  | -
$p\rightarrow K^{0}\mu^{+}$ |  $3.6\times 10^{33}$ years Super-Kamiokande:2022egr  | -
$p\rightarrow K^{+}\overline{\nu}$ |  $8.0\times 10^{33}$ years Okumura:2020xfs  |  $3.2\times 10^{34}$ years Hyper-Kamiokande:2018ofw
Table III: Current limits on partial proton decay lifetimes for all two-body
decay processes and future expectations for a ten-year period of data taking
at 90 % C.L..

We furthermore provide, in what follows, analytic expressions for partial
decay widths to address phase dependence of our results. We do that separately
for the gauge boson and scalar leptoquark proton decay mediations. Recall,
relevant gauge bosons are
$X_{\mu}^{+4/3},Y_{\mu}^{+1/3}\in(\overline{3},2,+5/6)\in 24_{G}$ with a
common mass $M_{\mathrm{GUT}}$, whereas scalar leptoquark is
$\Lambda_{3}(3,1,-1/3)\in 5_{H}$ with mass $M_{\Lambda_{3}}$.

###  4.1 Proton decay via gauge bosons

To evaluate relevant decay widths we resort to a formalism presented in Refs.
FileviezPerez:2004hn ; Nath:2006ut . The $d=6$ level operators of interest are

$\displaystyle O(e^{C}_{\alpha},d_{\beta})$
$\displaystyle=k^{2}c(e^{C}_{\alpha},d_{\beta})\,\epsilon_{ijk}\,\overline{u^{C}_{i}}\,\gamma^{\mu}\,u_{j}\,\overline{e^{C}_{\alpha}}\,\gamma_{\mu}\,d_{k\beta},$
(4.37) $\displaystyle O(e_{\alpha},d^{C}_{\beta})$
$\displaystyle=k^{2}c(e_{\alpha},d^{C}_{\beta})\,\epsilon_{ijk}\,\overline{u^{C}_{i}}\,\gamma^{\mu}\,u_{j}\,\overline{d^{C}_{k\beta}}\,\gamma_{\mu}\,e_{\alpha},$
(4.38) $\displaystyle O(\nu_{l},d_{\alpha},d^{C}_{\beta})$
$\displaystyle=k^{2}c(\nu_{l},d_{\alpha},d^{C}_{\beta})\,\epsilon_{ijk}\,\overline{u^{C}_{i}}\,\gamma^{\mu}\,d_{j\alpha}\,\overline{d^{C}_{k\beta}}\,\gamma_{\mu}\,\nu_{l},$
(4.39)

where $k^{2}=g^{2}_{\mathrm{GUT}}/M^{2}_{\mathrm{GUT}}$,
$(e_{1},e_{2})\equiv(e,\mu)$, $(d_{1},d_{2})\equiv(d,s)$, and $l=1,2,3$.

We are primarily interested in derivation of explicit expressions for
dimensionless coefficients $c(e^{C}_{\alpha},d_{\beta})$,
$c(e_{\alpha},d^{C}_{\beta})$, and $c(\nu_{l},d_{\alpha},d^{C}_{\beta})$. The
flavor dependent coefficients relevant for proton decays into charged leptons
are

$\displaystyle c(e_{\beta},d^{C})$
$\displaystyle=e^{-i\xi_{1}}(D^{\dagger}_{R})_{1\beta},$ (4.40) $\displaystyle
c(e_{\beta}^{C},d)$ $\displaystyle=e^{-i\xi_{1}}\left[(D^{*}_{L})_{\beta
1}+(V_{\mathrm{CKM}})_{11}(D_{L}^{*}D(\eta)^{*}V^{\dagger}_{\mathrm{CKM}})_{\beta
1}\right],$ (4.41) $\displaystyle c(e_{\beta},s^{C})$
$\displaystyle=e^{-i\xi_{1}}(D^{\dagger}_{R})_{2\beta}$ (4.42) $\displaystyle
c(e^{C}_{\beta},s)$ $\displaystyle=e^{-i\xi_{1}}\left[(D^{*}_{L})_{\beta
2}+e^{i\eta_{1}}(V_{\mathrm{CKM}})_{12}(D^{*}_{L}D(\eta)^{*}V^{\dagger}_{\mathrm{CKM}})_{\beta
1}\right].$ (4.43)

Clearly, a common multiplicative phase factor $e^{-i\xi_{1}}$ will disappear
once one squares the amplitude in order to evaluate physical decay widths of
interest. Also, the only parameters that are not provided by our numerical
study of the gauge coupling unification and fermion mass fit are phases
$\eta_{1}$ and $\eta_{2}$ in
$D(\eta)\equiv\mathrm{diag}(1,e^{i\eta_{1}},e^{i\eta_{2}})$.

To evaluate decay widths associated with $p\rightarrow\pi^{+}\overline{\nu}$
and $p\rightarrow K^{+}\overline{\nu}$ one needs to take into account the fact
that the neutrino flavor is not an observable in proton decay experiments and
thus needs to be summed over. We accordingly obtain

$\displaystyle\sum_{l=1}^{3}|\,c(\nu_{l},d,d^{C})\,|^{2}$
$\displaystyle=\left|\left(D(\xi)^{*}D(\kappa)V_{\mathrm{CKM}}D(\eta)\right)_{11}\right|^{2}=\left|(V_{\mathrm{CKM}})_{11}\right|^{2},$
(4.44) $\displaystyle\sum_{l=1}^{3}\left|c(\nu_{l},d,s^{C})\right|^{2}$
$\displaystyle=\left|\left(D(\xi)^{*}D(\kappa)V_{\mathrm{CKM}}D(\eta)\right)_{11}\right|^{2}=\left|(V_{\mathrm{CKM}})_{11}\right|^{2},$
(4.45) $\displaystyle\sum_{l=1}^{3}\left|c(\nu_{l},s,d^{C})\right|^{2}$
$\displaystyle=\left|(V_{\mathrm{CKM}})_{12}\right|^{2}.$ (4.46)

The outcome of summations in Eqs. (4.44), (4.45), and (4.46) stipulates that
the flavor dependence of $p\rightarrow\pi^{+}\overline{\nu}$ and $p\rightarrow
K^{+}\overline{\nu}$ processes is completely determined by the CKM entries
FileviezPerez:2004hn .

Our analytical expressions show that the only two phases that are featured in
gauge boson mediated proton decay into mesons and charged leptons are
$\eta_{1}$ and $\eta_{2}$. In fact, even the dependence on $\eta_{2}$ can be
neglected for all practical purposes, as we show later on. We stress that
these analytical results will hold in any $SU(5)$ model where charged fermions
are already in the mass eigenstate basis while the up-type quark mass matrix
is symmetric in flavor space.

Once the coefficients $c(e^{C}_{\alpha},d_{\beta})$,
$c(e_{\alpha},d^{C}_{\beta})$, and $c(\nu_{l},d_{\alpha},d^{C}_{\beta})$ are
determined, one can write down two-body proton decay widths and generate
predictions of the model. For example, the most dominat decay width reads

$\displaystyle\Gamma(p\to\pi^{0}e^{+})$
$\displaystyle=\frac{(m_{p}^{2}-m_{\pi}^{2})^{2}}{m_{p}^{3}}\frac{\pi}{2}A_{L}^{2}\frac{\alpha_{\mathrm{GUT}}^{2}}{M_{\mathrm{GUT}}^{4}}$
$\displaystyle\times\left(A^{2}_{SL}\left|c(e^{C},d)\langle\pi^{0}|(ud)_{L}u_{L}|p\rangle\right|^{2}+A^{2}_{SR}\left|c(e,d^{C})\langle\pi^{0}|(ud)_{R}u_{L}|p\rangle\right|^{2}\right)\,,$
(4.47)

where
$\left<\pi^{0}\vphantom{(ud)_{L}u_{L}p}\right|(ud)_{L}u_{L}\left|p\vphantom{\pi^{0}(ud)_{L}u_{L}}\right>=0.134\,\mathrm{GeV}^{2}$
Aoki:2017puj  and
$\left<\pi^{0}\vphantom{(ud)_{R}u_{L}p}\right|(ud)_{R}u_{L}\left|p\vphantom{\pi^{0}(ud)_{R}u_{L}}\right>=-0.131\,\mathrm{GeV}^{2}$
Aoki:2017puj . These matrix elements are evaluated with an error of the order
of 10 % that we do not take into account in our numerical analysis.
$A_{L}=1.2$ represents a long-distance coefficient Nihei:1994tx , whereas
$A_{SL}$ and $A_{SR}$ are short-distance coefficients. $A_{SL}$ and $A_{SR}$
capture the running of the proton decay operators from $M_{\mathrm{GUT}}$ down
to $M_{Z}$ and are evaluated via Buras:1977yy ; Ellis:1979hy ; Wilczek:1979hc

$A_{SL(R)}=\prod_{i=1,2,3}\prod_{I}^{M_{Z}\leq M_{I}\leq
M_{\mathrm{GUT}}}\left[\frac{\alpha_{i}(M_{I+1})}{\alpha_{i}(M_{I})}\right]^{\frac{\gamma_{L(R)i}}{\sum_{J}^{M_{Z}\leq
M_{J}\leq M_{I}}b^{J}_{i}}},\,\gamma_{L(R)i}=(23(11)/20,9/4,2),$

where indices $I$ and $J$ run through all the new physics states that reside
below the unification scale. We evaluate $A_{SL}$ and $A_{SR}$ for every point
in $M_{\mathrm{\Phi_{1}}}$-$M_{\mathrm{\Sigma_{1}}}$ plane for both $M\geq 1$
TeV and $M\geq 10$ TeV scenarios. All eight proton decay widths, due to the
gauge boson exchange, are explicitly written down in Appendix 1.

###  4.2 Proton decay via scalar leptoquark

The relevant $d=6$ operators for proton decay through scalar leptoquark
$\Lambda_{3}$ are Nath:2006ut

$\displaystyle O_{H}(d_{\alpha},e_{\beta})$
$\displaystyle=M^{-2}_{\Lambda_{3}}a(d_{\alpha},e_{\beta})\,u^{T}L\,C^{-1}\,d_{\alpha}\,u^{T}L\,C^{-1}\,e_{\beta},$
(4.48) $\displaystyle O_{H}(d_{\alpha},e^{C}_{\beta})$
$\displaystyle=M^{-2}_{\Lambda_{3}}a(d_{\alpha},e^{C}_{\beta})\,u^{T}L\,C^{-1}\,d_{\alpha}\,e^{C^{\dagger}}_{\beta}L\,C^{-1}\,u^{C^{*}},$
(4.49) $\displaystyle O_{H}(d^{C}_{\alpha},e_{\beta})$
$\displaystyle=M^{-2}_{\Lambda_{3}}a(d^{C}_{\alpha},e_{\beta})\,d^{C^{\dagger}}_{\alpha}L\,C^{-1}\,u^{C^{*}}\,u^{T}L\,C^{-1}\,e_{\beta},$
(4.50) $\displaystyle O_{H}(d^{C}_{\alpha},e^{C}_{\beta})$
$\displaystyle=M^{-2}_{\Lambda_{3}}a(d^{C}_{\alpha},e^{C}_{\beta})\,d^{C^{\dagger}}_{\alpha}L\,C^{-1}\,u^{C^{*}}\,e_{\beta}^{C^{\dagger}}\,L\,C^{-1}\,u^{C^{*}},$
(4.51) $\displaystyle O_{H}(d_{\alpha},d_{\beta},\nu_{l})$
$\displaystyle=M^{-2}_{\Lambda_{3}}a(d_{\alpha},d_{\beta},\nu_{l})\,u^{T}\,L\,C^{-1}\,d_{\alpha}\,d^{T}_{\beta}\,L\,C^{-1}\,\nu_{l},$
(4.52) $\displaystyle O_{H}(d_{\alpha},d^{C}_{\beta},\nu_{l})$
$\displaystyle=M^{-2}_{\Lambda_{3}}a(d_{\alpha},d^{C}_{\beta},\nu_{l})\,d_{\beta}^{C^{\dagger}}\,L\,C^{-1}\,u^{C^{*}}\,d^{T}_{\alpha}\,L\,C^{-1}\,\nu_{l},$
(4.53)

where C is a charge conjugation operator and $L=(1-\gamma_{5})/2$. Again, it
is dimensionless coefficients $a(d_{\alpha},e_{\beta})$,
$a(d_{\alpha},e^{C}_{\beta})$, $a(d^{C}_{\alpha},e_{\beta})$,
$a(d^{C}_{\alpha},e^{C}_{\beta})$, $a(d_{\alpha},d_{\beta},\nu_{l})$, and
$a(d_{\alpha},d^{C}_{\beta},\nu_{l})$ that encapsulate flavor dependence of
proton decay signatures we are interested in.

We find the following expressions Dorsner:2012nq

$\displaystyle a(d_{\alpha},e_{\beta})$
$\displaystyle=e^{-i\xi_{1}}\frac{m^{U}_{1}m^{E}_{\beta}}{v^{2}}\left(V_{\mathrm{CKM}}D(\eta)\right)_{1\alpha}\left(V^{*}_{\mathrm{CKM}}D(\eta)^{*}D^{\dagger}_{L}\right)_{1\beta},$
(4.54) $\displaystyle a(d_{\alpha},e^{C}_{\beta})$
$\displaystyle=\frac{m_{1}^{U}}{m_{\beta}^{E}}a(d_{\alpha},e_{\beta}),$ (4.55)
$\displaystyle a(d_{\alpha}^{C},e_{\beta})$
$\displaystyle=e^{-i\xi_{1}}\frac{m^{E}_{\beta}}{v^{2}}\left(D^{\dagger}_{R}M^{\mathrm{diag}}_{E}D_{L}D(\eta)V^{T}_{\mathrm{CKM}}\right)_{\alpha
1}\left(V^{*}_{\mathrm{CKM}}D(\eta)^{*}D^{\dagger}_{L}\right)_{1\beta},$
(4.56) $\displaystyle a(d_{\alpha}^{C},e^{C}_{\beta})$
$\displaystyle=\frac{m_{1}^{U}}{m_{\beta}^{E}}a(d^{C}_{\alpha},e_{\beta}),$
(4.57) $\displaystyle a(d_{\alpha},d_{\beta},\nu_{l})$
$\displaystyle=e^{-i\xi_{1}+i\kappa_{1}}\frac{m_{1}^{U}}{v^{2}}\left(V_{\mathrm{CKM}}D(\eta)\right)_{1\alpha}\left(D^{\dagger}_{L}M_{E}^{\mathrm{diag}}N^{*}\right)_{\beta
l},$ (4.58) $\displaystyle a(d_{\alpha},d^{C}_{\beta},\nu_{l})$
$\displaystyle=\frac{e^{-i\xi_{1}+i\kappa_{1}}}{v^{2}}\left(D^{\dagger}_{R}M^{\mathrm{diag}}_{E}D_{L}D(\eta)V^{T}_{\mathrm{CKM}}\right)_{\beta
1}\left(D^{\dagger}_{L}M^{\mathrm{diag}}_{E}N^{*}\right)_{\alpha l},$ (4.59)

where $v=v_{5}/\sqrt{2}$, $m^{U}_{1}\equiv m_{u}$, and
$(m^{E}_{1},m^{E}_{2})\equiv(m_{e},m_{\mu})$. Clearly, the overall phase
factors $e^{-i\xi_{1}}$ and $e^{-i\xi_{1}+i\kappa_{1}}$ will not be featured
in any of the decay widths. The only phases that will leave an imprint on the
partial decay lifetimes due to the scalar leptoquark mediation are, once
again, $\eta_{1}$ and $\eta_{2}$.

To obtain decay widths for $p\rightarrow\pi^{+}\overline{\nu}$ and
$p\rightarrow K^{+}\overline{\nu}$ processes due to scalar leptoquark exchange
we need to sum over neutrino flavors. We accordingly find these contractions

$\displaystyle\sum_{l=1}^{3}a(d_{\alpha},d_{\beta},\nu_{l})a^{*}(d_{\gamma},d_{\delta},\nu_{l})$
$\displaystyle=\frac{m_{1}^{U^{2}}}{v^{4}}\left(V_{\mathrm{CKM}}D(\eta)\right)_{1\alpha}$
$\displaystyle\times\left(V^{*}_{\mathrm{CKM}}D(\eta)^{*}\right)_{1\gamma}\left(D^{\dagger}_{L}M^{\mathrm{diag}\,2}_{E}D_{L}\right)_{\beta\delta},$
(4.60)
$\displaystyle\sum_{l=1}^{3}a(d_{\alpha},d^{C}_{\beta},\nu_{l})a^{*}(d_{\gamma},d^{C}_{\delta},\nu_{l})$
$\displaystyle=\frac{1}{v^{4}}\left(D^{\dagger}_{R}M^{\mathrm{diag}}_{E}D_{L}D(\eta)V^{T}_{\mathrm{CKM}}\right)_{\beta
1}$
$\displaystyle\times\left(D^{T}_{R}M^{\mathrm{diag}}_{E}D_{L}^{*}D(\eta)^{*}V^{\dagger}_{\mathrm{CKM}}\right)_{\delta
1}\left(D^{\dagger}_{L}M^{\mathrm{diag}\,2}_{E}D_{L}\right)_{\alpha\gamma},$
(4.61)
$\displaystyle\sum_{l=1}^{3}a(d_{\alpha},d_{\beta},\nu_{l})a^{*}(d_{\gamma},d^{C}_{\delta},\nu_{l})$
$\displaystyle=\frac{m_{1}^{U}}{v^{4}}\left(D^{\dagger}_{L}M^{\mathrm{diag}\,2}_{E}D_{L}\right)_{\beta\gamma}$
$\displaystyle\times\left(V_{\mathrm{CKM}}D(\eta)\right)_{1\alpha}\left(D^{T}_{R}M^{\mathrm{diag}}_{E}D_{L}^{*}D(\eta)^{*}V^{\dagger}_{\mathrm{CKM}}\right)_{\delta
1}.$ (4.62)

It is clear that the parameters associated with the neutrino sector, including
the PMNS matrix, are not featured in proton decay signatures via scalar
leptoquark exchange.

Once all these coefficients are analytically determined we can write down two-
body proton decay widths due to the scalar leptoquark mediation and generate
predictions of our model. For example, the most constraining decay width is

$\displaystyle\Gamma(p\rightarrow K^{+}\bar{\nu})$
$\displaystyle=\frac{(m_{p}^{2}-m^{2}_{K})^{2}}{32\pi
m^{3}_{p}M_{\Lambda_{3}}^{4}}A_{L}^{2}$
$\displaystyle\times\sum_{i=1}^{3}\left|A_{SL}\left(a(s,d,\nu_{i})\langle
K^{+}|(us)_{L}d_{L}|p\rangle+a(d,s,\nu_{i})\langle
K^{+}|(ud)_{L}s_{L}|p\rangle\right)\right.$
$\displaystyle\left.+A_{SR}\left(a(d,s^{C},\nu_{i})\langle
K^{+}|(us)_{R}d_{L}|p\rangle+a(s,d^{C},\nu_{i})\langle
K^{+}|(ud)_{R}s_{L}|p\rangle\right)\right|^{2},$ (4.63)

where the values of matrix elements $\langle K^{+}|(us)_{L,R}d_{L}|p\rangle$
and $\langle K^{+}|(ud)_{L,R}s_{L}|p\rangle$ are given in Table 5. As before,
$A_{L}$ represents a long-distance coefficient, whereas $A_{SL}$ and $A_{SR}$
are short-distance coefficients. The expressions of proton decay widths for
all eight two-body decay modes can be found in Appendix 2.

##  5 Predictions

We are finally in a position to place additional limits on otherwise viable
parameter space of the model using our results for the gauge boson and scalar
leptoquark proton decay mediations.

The most relevant limit originates from the model prediction for the gauge
mediatated $p\to\pi^{0}e^{+}$ process given in Eq. (4.47). We present thus
inferred proton decay limit in both panels of Fig. 2 using a solid black line.
Note that to the left of that line it is not possible to have partial proton
decay lifetime of $p\to\pi^{0}e^{+}$ that is in agreement with the current
experimental limit for any choice of phases $\eta_{1}$ and $\eta_{2}$. In
short, the shaded region between the proton decay bound (solid black line) and
the outermost neutrino mass scale bound (dashed black curve) is viable with
regard to all experimental input. We also show in Fig. 2, using gray dashed
line, what one expects to be excluded with $p\to\pi^{0}e^{+}$ non-observation
after a ten-year period of data taking at Hyper-K Hyper-Kamiokande:2018ofw .

Figure 2:  Viable parameter space of the model (shaded in gray) for the $M\geq
1$ TeV and $M\geq$ 10 TeV scenarios with contours of constant values for
$M_{\mathrm{GUT}}/(10^{15}\,\mathrm{GeV})$ (red solid lines),
$\alpha_{\mathrm{GUT}}$ (blue dot-dashed lines), and $m_{0}/(\sqrt{\Delta
m^{2}_{31}}/2)$ (green solid lines). For additional details, see the text.

Again, Fig. 2 features contours of constant values for
$M_{\mathrm{GUT}}/(10^{15}\,\mathrm{GeV})$ (red solid lines),
$\alpha_{\mathrm{GUT}}$ (blue dot-dashed lines), and $m_{0}/(\sqrt{\Delta
m^{2}_{31}}/2)$ (green solid lines) in $M_{\Phi_{1}}$-$M_{\Sigma_{1}}$ plane
for $M\geq 1$ TeV and $M\geq 10$ TeV scenarios, where
$\log_{10}(M_{\Phi_{1}}/1\,\mathrm{GeV})\in[10.2,12.9]$ and
$\log_{10}(M_{\Sigma_{1}}/1\,\mathrm{GeV})\in[8.4,14.2]$. The lattice spacing
in $M_{\Phi_{1}}$-$M_{\Sigma_{1}}$ plane for our analysis, in both directions,
is 0.1 in units of $\log_{10}(M_{\Phi_{1}}/1\,\mathrm{GeV})$ and
$\log_{10}(M_{\Sigma_{1}}/1\,\mathrm{GeV})$, respectively. Again, at each
lattice point in $M_{\Phi_{1}}$-$M_{\Sigma_{1}}$ plane we evaluate mass
spectrum that leads to maximal possible value of $M_{\mathrm{GUT}}$ and
determine associated $\alpha_{\mathrm{GUT}}$, $A_{SL}$, and $A_{SR}$. The mass
spectrum that generates max ($M_{\mathrm{GUT}}$) is always such that
$M_{\Phi_{3}}=M_{\Phi_{6}}=M_{\phi_{1}}=M_{\phi_{8}}=M$. We furthermore
perform running of the SM charged fermion parameters at each lattice point and
use associated values to perform numerical fit of all fermion masses. The
outcome of the fit are all unitary transformations that provide transition
from flavor to mass-eigenstate basis of the SM fermions and associated Yukawa
coupling constants. Parameters that our procedure cannot determine are phases
$\eta_{1}$, $\eta_{2}$, $\kappa_{1}$, $\kappa_{2}$, $\kappa_{3}$, $\xi_{1}$,
$\xi_{2}$, and $\xi_{3}$, where only $\eta_{1}$ and $\eta_{2}$ are relevant
for the study of proton decay signatures.

In order to accomplish comparative study between gauge boson and scalar
leptoquark mediation signatures for the partial proton decay lifetimes, we
choose one particular point in Fig. 2 for $M\geq$ 1 TeV scenario to be our
starting point. This point, with coordinates
$(M_{\Phi_{1}},M_{\Sigma_{1}})=(10^{11.3}\,\mathrm{GeV},10^{9.5}\,\mathrm{GeV})$,
is denoted Q and is chosen to be near the current proton decay bound rendered
with solid black line in the upper panel of Fig. 2. We subsequently evaluate
$\alpha_{\mathrm{GUT}}^{2}/M^{4}_{\mathrm{GUT}}$ at this point and find all
other points in Fig. 2 that satisfy criteria that the associated value of
$\alpha_{\mathrm{GUT}}^{2}/M^{4}_{\mathrm{GUT}}$ at those points is within
$\pm$2 % with respect to the value extracted for point Q. This procedure
yields ten points of interest for $M\geq$ 1 TeV scenario that are presented in
the upper panel of Fig. 2.

Once these ten points in $M_{\Phi_{1}}$-$M_{\Sigma_{1}}$ plane are known, we
choose among them one point with large value of $\alpha_{\mathrm{GUT}}$
(Q$(10^{11.3}\,\mathrm{GeV},10^{9.5}\,\mathrm{GeV})$) and another one with
small value of $\alpha_{\mathrm{GUT}}$
(R$(10^{11.0}\,\mathrm{GeV},10^{12.7}\,\mathrm{GeV})$) and evaluate at both of
these points gauge boson mediated proton decay widths for all eight channels,
where we also vary $\eta_{1}$ and $\eta_{2}$ phases to find maximal and
minimal values for these decay widths. We furthermore evaluate scalar
leptoquark mediated proton decay widths for all eight channels at both Q and
R, where, once again, we vary $\eta_{1}$ and $\eta_{2}$ to capture phase
dependence. To find decay widths associated with scalar leptoquark mediation,
we fix the mass of $\Lambda_{3}$ to have a partial lifetime for $p\to
K^{+}\overline{\nu}$ as close as possible to the current experimental limit.
We take $M_{\Lambda_{3}}=2.7\times 10^{12}$ GeV for $M\geq$ 1 TeV scenario.
Again, this choice ensures that the scalar mediated proton decay signatures
will be observed with certainty at future proton decay experiments.

Figure 3:  Proton decay signatures via gauge boson and scalar leptoquark
mediations within $M\geq$ 1 TeV and $M\geq$ 10 TeV scenarios for four specific
points and eight channels, as indicated. Black lines are current experimental
limits, blue vertical bars are predictions for gauge boson mediation
signatures, red vertical bars are predictions for the scalar leptoquark
mediations, and gray dashed lines represent future experimental sensitivities
after a ten-year period of data taking at 90 % C.L..

We preform the same procedure of selecting two particular points for $M\geq
10$ TeV scenario of Fig. 2, where we label these points
Q${}^{\prime}(10^{12.1}\,\mathrm{GeV},10^{11.5}\,\mathrm{GeV})$ and
R${}^{\prime}(10^{12.0}\,\mathrm{GeV},10^{12.7}\,\mathrm{GeV})$. To evaluate
scalar mediated proton decay signatures for points Q′ and R′ we set
$M_{\Lambda_{3}}=2.3\times 10^{12}$ GeV. Again, this choice of
$M_{\Lambda_{3}}$ places prediction at point Q′ for $p\to K^{+}\overline{\nu}$
within an imminent reach of Hyper-K.

We show signatures for both types of mediation side by side in Fig. 3 for all
four points of interest, where blue bars are used for the gauge boson
mediation partial lifetime predictions, red bars are used for predictions
associated with scalar leptoquark mediation, current experimental limits are
represented by thin black lines, and gray dashed lines stand for future
expectations after a ten-year period of data taking at 90 % C.L., if and where
available.

Upper panels in Fig. 3 are for proton decay signatures for points Q and R,
whereas lower panels correspond to predictions for points Q′ and R′. Note that
the uncertainties in proton decay widths associated with scalar leptoquark
mediation are much larger than those associated with gauge boson mediation.
Again, in both cases these uncertainties originate solely from variation of
two phases $\eta_{1}$ and $\eta_{2}$. We quantify these flavor uncertainties
in Table IV, where we evaluate
$\Delta=(\max(\tau_{p})-\min(\tau_{p}))/\min(\tau_{p})$ in % at points Q, R,
Q′, and R′ for each process and for both types of mediation.

$\Delta$ (%) | gauge | scalar | gauge | scalar
---|---|---|---|---
decay channel |  $\Delta$(Q) |  $\Delta$(R) |  $\Delta$(Q) |  $\Delta$(R) |  $\Delta$(Q${}^{\prime})$ |  $\Delta$(R${}^{\prime})$ |  $\Delta$(Q${}^{\prime})$ |  $\Delta$(R${}^{\prime})$
$p\rightarrow\pi^{0}e^{+}$ | 2 | 1 | 339 | 85 | 1 | 1 | 142 | 96
$p\rightarrow\pi^{0}\mu^{+}$ | 54 | 6 | 1147 | 132 | 19 | 9 | 214 | 166
$p\rightarrow\pi^{+}\overline{\nu}$ | 0 | 0 | 363 | 87 | 0 | 0 | 145 | 99
$p\rightarrow\eta^{0}e^{+}$ | 3 | 1 | 358 | 90 | 1 | 1 | 146 | 97
$p\rightarrow\eta^{0}\mu^{+}$ | 810 | 59 | 2743 | 382 | 70 | 81 | 513 | 313
$p\rightarrow K^{0}e^{+}$ | 2 | 1 | 386 | 103 | 1 | 1 | 90 | 102
$p\rightarrow K^{0}\mu^{+}$ | 5 | 1 | 1304 | 152 | 1 | 1 | 146 | 171
$p\rightarrow K^{+}\overline{\nu}$ | 0 | 0 | 361 | 87 | 0 | 0 | 145 | 99
Table IV: Phase uncertainty
$\Delta=(\max(\tau_{p})-\min(\tau_{p}))/\min(\tau_{p})$ in % at points Q, R,
Q′, and R′ for all eight channels and for both types of proton decay
mediation.

One can see from Fig. 3 and Table IV that gauge mediated proton decays
$p\to\pi^{+}\overline{\nu}$ and $p\to K^{+}\overline{\nu}$ do not exhibit any
dependence on $\eta_{1}$ and $\eta_{2}$. Also, the prediction uncertainty is
usually smaller for points with smaller value of $\alpha_{\mathrm{GUT}}$ for
both types of mediation. One can furthermore observe that even though points Q
and R have practically the same value of
$\alpha_{\mathrm{GUT}}^{2}/M_{\mathrm{GUT}}^{4}$, the prediction for the
$p\to\pi^{0}e^{+}$ lifetime for point Q is closer to the experimental limit
than for point R. This is due to the fact that the short-distance coefficients
$A_{SL}$ and $A_{SR}$ are larger in the region where $\alpha_{\mathrm{GUT}}$
is larger. This effect can also be observed in the scalar leptoquark mediation
signatures. For example, we find that $A_{SL}=3.41$ and $A_{SR}=3.14$ at point
Q, whereas $A_{SL}=3.08$ and $A_{SR}=2.86$ at point R.

We have already stipulated that variation of $\eta_{2}$ does not affect gauge
mediated proton decay signatures at all. To that end, we present in Fig. 4
contours of constant decay width for $p\to\pi^{0}\mu^{+}$ for both the gauge
(blue) and scalar leptoquark (red) mediation in $\eta_{1}$-$\eta_{2}$ plane
for points Q (upper panels) and Q′ (lower panels). It is clear from Fig. 4
that the impact of $\eta_{2}$ on gauge boson mediated proton decay signatures
is marginal, at best.

Figure 4:  Contours of constant $\Gamma(p\to\pi^{0}\mu^{+})$ for the gauge
boson (blue) and scalar leptoquark (red) mediation for points Q and Q′, as
indicated.

Side by side comparison of two types of proton decay mediation, as shown in
Fig. 3, leaves us with the following conclusions. If this model is realized in
nature, and if proton is observed to decay to $\pi^{0}$ and $e^{+}$, the decay
is certainly due to the gauge boson exchange. If the dominant mode is indeed
the gauge mediated proton decay, the next channel that is closest to being
detected is $p\to\eta^{0}e^{+}$, even though it is predicted to be outside of
the projected sensitivity after a ten-year period of data taking at Hyper-K.
All other gauge mediated proton decay signatures exceed any reasonable
improvement in proton decay bounds to allow for potential discovery in decades
to come.

If, on the other hand, proton is observed to decay to $K^{+}$ and
$\overline{\nu}$, the decay is mediated by scalar leptoquark. The channel that
is then closest to the ten-year sensitivity projection is
$p\to\pi^{0}\mu^{+}$. All other channels have predicted lifetimes that vastly
exceed what can be probed with current detector technologies.

If both $p\to\pi^{0}e^{+}$ and $p\to K^{+}\overline{\nu}$ are observed, we
would have scenario where both gauge bosons and scalar leptoquark
simultaneously contribute towards proton decay signatures. It would then be
possible to have an enhancement of the $p\to\pi^{0}\mu^{+}$ decay width, as
can be seen from Figs. 3 and 4, if the interference between the gauge and
scalar leptoquark contributions at the amplitude level is constructive. Except
for potential observation of $p\to\eta^{0}e^{+}$, the remaining four channels
will be experimentally inaccessible for all practical purposes within this
particular scenario.

The very last points in the parameter space of $M\geq 1$ TeV and $M\geq 10$
TeV scenarios in Fig. 2 that could be, at least in principle, probed by proton
decay experiments are points A and A′, respectively, with common coordinates
$(M_{\Phi_{1}},M_{\Sigma_{1}})=(10^{12.8}\,\mathrm{GeV},10^{12.9}\,\mathrm{GeV})$.
The model yields $\tau_{p}(p\to\pi^{0}e^{+})=1.1\times 10^{36}$ years and
$\tau_{p}(p\to\pi^{0}e^{+})=1.5\times 10^{35}$ years at points A and A′,
respectively. $M$ is thus inversely proportional to predicted lifetime
$\tau_{p}(p\to\pi^{0}e^{+})$ at point
$(M_{\Phi_{1}},M_{\Sigma_{1}})=(10^{12.8}\,\mathrm{GeV},10^{12.9}\,\mathrm{GeV})$.
We can accordingly infer that the current experimental limit on
$\tau_{p}(p\to\pi^{0}e^{+})$ implies that $M\leq 65$ TeV, where $M$ can be
identified, for all practical purposes, with geometric mean of the masses of
$\Phi_{3}$, $\Phi_{6}$, $\phi_{1}$, and $\phi_{8}$. Viable parameter space of
our model is nicely bounded from one side by experimental limit on
$p\to\pi^{0}e^{+}$ and from the other side by neutrino mass scale
perturbativity requirement. The $M=65$ TeV scenario is where these two bounds
meet.

We can also place an accurate limit on $M_{\Lambda_{3}}$ within $M\geq 1$ TeV
and $M\geq 10$ TeV scenarios for any lattice point in
$M_{\Phi_{1}}$-$M_{\Sigma_{1}}$ plane with the data on $p\to
K^{+}\overline{\nu}$. For example, we find that $M_{\Lambda_{3}}\geq 1.8\times
10^{12}$ GeV, $M_{\Lambda_{3}}\geq 1.9\times 10^{12}$ GeV,
$M_{\Lambda_{3}}\geq 1.8\times 10^{12}$ GeV, and $M_{\Lambda_{3}}\geq
1.7\times 10^{12}$ GeV for points Q, R, Q′, R′, respectively.

Final comments are in order.
($a$) We find an additional multiplicative factor of 4 in denominator of
$m_{0}$ parameter with respect to our previous works Dorsner:2019vgf ;
Dorsner:2021qwg . This has accordingly reduced available parameter space of
the model.
($b$) The matrix elements that we use carry around 10 % uncertainty, each,
that we do not include in our study of proton decay signatures. These errors
would translate to an additional 20 % uncertainty for all eight proton decay
widths. Again, the uncertainties presented in Table IV capture flavor
dependence only.
($c$) We remark that, in principle, the fermionic submultiplet $\Sigma_{3}$
does not need to be degenerate in mass with $\Sigma_{6}$ and $\Sigma_{1}$ if
one does not require $M_{\Sigma_{3}},M_{\Sigma_{6}},M_{\Sigma_{1}}>0$. This
might be problematic for our study since, as shown in Ref. Antusch:2023mqe , a
lower mass of $\Sigma_{3}$, by itself, can significantly increase the
unification scale. One might then expect that taking its mass close to the TeV
scale would drastically increase $M_{\mathrm{GUT}}$. This, however, does not
happen in our model due to the presence of additional colored light states.
For example, if one sets $\Sigma_{3}$ to be very light, the maximal
unification scale $M_{\mathrm{GUT}}$ is only a factor of 1.5 larger than what
we show in Fig. 2. Setting $\Sigma_{3}$ light, however, opens up another
issue. To explain it, let us, for example, focus on the third-generation mass.
We get, from Eq. (3.17), that $m_{b}\sim m_{\tau}+\Delta m$, where $\Delta
m\sim(5/4)(174\,\mathrm{GeV})(Y^{c}Y^{a})(v_{24}/M_{\Sigma_{3}})$. To
reproduce correct bottom mass, using the data from Table II, one requires
$\Delta m\sim 1$ GeV. Assuming that $\Sigma_{3}$ state indeed resides at the
TeV scale and $v_{24}\sim 2\times 10^{16}$ GeV, we obtain $Y^{c}\sim
10^{-17}$, where $Y^{a}\sim 1$ is used to maximize the neutrino mass scale. We
present, in this manuscript, scenario with mass-degenerate $\Sigma_{1}$,
$\Sigma_{6}$, and $\Sigma_{3}$ states, which dictates $Y^{c}$ coupling to be
of the order of the usual electron Yukawa coupling and leave thorough
discussion of the light $\Sigma_{3}$ scenario for potential future
publication.
($d$) We find that the lower bound on $M_{\Lambda_{3}}$ is around $10^{12}$
GeV that somewhat exceeds a value of $3\times 10^{11}$ GeV Dorsner:2012uz
found in a simple non-renormalizable $SU(5)$ setting. We attribute this change
to ($i$) use of short-distance coefficients $A_{SL}$ and $A_{SR}$, ($ii$)
implementation of matrix elements instead of more traditional parameters $F$
and $D$ extracted from form factors in semileptonic hyperon decays and nucleon
axial charge Claudson:1981gh ; Aoki:2008ku , and ($iii$) proper derivation of
scalar leptoquark couplings in $SU(5)$ Dorsner:2012nq .

##  6 Conclusion

There are only two possible types of mediators of proton decay within the
$SU(5)$ model in question. The anticipated experimental signal of these decay
processes can, hence, originate solely from exchange of two gauge bosons, or
entirely from a single scalar leptoquark exchange, or from combination of the
two. In the following, we highlight the most notable features of the proton
decay signals of our model.

  * •

Our analysis stipulates that we can conclude, with certainty, that if a proton
is experimentally observed to decay to $\pi^{0}$ and $e^{+}$, the decay is
mediated by the gauge bosons. If gauge mediation indeed dominates, the only
other channel that might potentially be observed is $p\to\eta^{0}e^{+}$, even
though it is predicted to be outside of the projected sensitivity after a ten-
year period of data taking at Hyper-K.

  * •

If, on the other hand, proton is observed to decay to $K^{+}$ and
$\overline{\nu}$, the decay is certainly mediated by a scalar leptoquark. One
might then hope to observe $p\to\pi^{0}\mu^{+}$, even though it is also
predicted to be outside of the projected sensitivity after a ten-year period
of data taking.

  * •

If both $p\to\pi^{0}e^{+}$ and $p\to K^{+}\overline{\nu}$ are observed, the
decay width for the process $p\to\pi^{0}\mu^{+}$ might be enhanced through
fortuitous interference between the gauge and scalar leptoquark contributions
at the amplitude level.

  * •

The model thus predicts that $p\to\pi^{+}\overline{\nu}$,
$p\to\eta^{0}\mu^{+}$, $p\to K^{0}e^{+}$, and $p\to K^{0}\mu^{+}$ will be
experimentally inaccessible in decades to come regardless of the type of
proton decay mediation dominance.

Other prominent features of the model are as follows. The lightest neutrino is
massless and the neutrino mass hierarchy is of the normal ordering. There are
four scalar multiplets, i.e., $\Phi_{3},\Phi_{6}\in 35_{H}$ and
$\phi_{1},\phi_{8}\in 24_{H}$, that need to be light if $M_{\mathrm{GUT}}$ is
to attain maximal possible value, where the geometric mean of the masses of
these fields can be identified with parameter $M$ for all practical purposes.
Current experimental limit on $\tau_{p}(p\to\pi^{0}e^{+})$ implies that $M\leq
65$ TeV, where $M$ and $\tau_{p}(p\to\pi^{0}e^{+})$ are inversely
proportional. Any future experimental improvement on
$\tau_{p}(p\to\pi^{0}e^{+})$ will thus lower allowed value of $M$ and
accordingly help to further reduce available parameter space of the model.

## Acknowledgments

S.F. acknowledges the financial support from the Slovenian Research Agency
(research core funding No. P1-0035 and N1-0321). S.S. would like to thank
Kevin Hinze for discussion.

##  Appendix 1 Gauge Mediated Proton Decay Widths

We show below explicit expressions for all eight two-body proton decay widths
due to the gauge boson exchange for completeness of exposition. The gauge
bosons in question are $X_{\mu}^{+4/3},Y_{\mu}^{+1/3}\in(\overline{3},2,+5/6)$
with a common mass $M_{\mathrm{GUT}}$.

$\begin{split}\Gamma(p\rightarrow\pi^{0}e^{+}_{\beta})&=\frac{(m_{p}^{2}-m^{2}_{\pi})^{2}}{m^{3}_{p}}\frac{\pi}{2}A_{L}^{2}\frac{\alpha_{\mathrm{GUT}}^{2}}{M_{\mathrm{GUT}}^{4}}\\\
&\times\Bigr{\\{}\left|A_{SR}\langle\pi^{0}|(ud)_{R}u_{L}|p\rangle
c(e_{\beta},d^{C})\right|^{2}+\left|A_{SL}\langle\pi^{0}|(ud)_{L}u_{L}|p\rangle
c(e^{C}_{\beta},d)\right|^{2}\Bigr{\\}}\;.\end{split}$ (1.64)

$\Gamma(p\rightarrow\pi^{+}\bar{\nu})=\frac{(m_{p}^{2}-m^{2}_{\pi})^{2}}{m^{3}_{p}}\frac{\pi}{2}A_{L}^{2}\frac{\alpha_{\mathrm{GUT}}^{2}}{M_{\mathrm{GUT}}^{4}}A_{SR}^{2}|\langle\pi^{+}|(ud)_{R}d_{L}|p\rangle|^{2}\sum_{i=1}^{3}|c(\nu_{i},d,d^{C})|^{2}\;.$
(1.65)

$\begin{split}\Gamma(p\rightarrow\eta
e^{+}_{\beta})=&\frac{(m_{p}^{2}-m^{2}_{\eta})^{2}}{m^{3}_{p}}\frac{\pi}{2}A_{L}^{2}\frac{\alpha_{\mathrm{GUT}}^{2}}{M_{\mathrm{GUT}}^{4}}\\\
&\times\Bigr{\\{}\left|A_{SR}\langle\eta|(ud)_{R}u_{L}|p\rangle
c(e_{\beta},d^{C})\right|^{2}+\left|A_{SL}\langle\eta|(ud)_{L}u_{L}|p\rangle
c(e^{C}_{\beta},d)\right|^{2}\Bigr{\\}}\;.\end{split}$ (1.66)

$\begin{split}\Gamma(p\rightarrow
K^{0}e^{+}_{\beta})&=\frac{(m_{p}^{2}-m^{2}_{K})^{2}}{m^{3}_{p}}\frac{\pi}{2}A_{L}^{2}\frac{\alpha_{\mathrm{GUT}}^{2}}{M_{\mathrm{GUT}}^{4}}\\\
&\times\Bigr{\\{}\left|A_{SR}\langle K^{0}|(us)_{R}u_{L}|p\rangle
c(e_{\beta},s^{C})\right|^{2}+\left|A_{SL}\langle K^{0}|(us)_{L}u_{L}|p\rangle
c(e^{C}_{\beta},s)\right|^{2}\Bigr{\\}}\;.\end{split}$ (1.67)

$\begin{split}\Gamma(p\rightarrow
K^{+}\bar{\nu})&=\frac{(m_{p}^{2}-m^{2}_{K})^{2}}{m^{3}_{p}}\frac{\pi}{2}A_{L}^{2}\frac{\alpha_{\mathrm{GUT}}^{2}}{M_{\mathrm{GUT}}^{4}}\\\
&\times\Bigr{\\{}A^{2}_{SR}\left|\langle
K^{+}|(us)_{R}d_{L}|p\rangle\right|^{2}\sum_{i=1}^{3}\left|c(\nu_{i},d,s^{C})\right|^{2}\\\
&+A^{2}_{SL}\left|\langle
K^{+}|(ud)_{R}s_{L}|p\rangle\right|^{2}\sum_{i=1}^{3}\left|c(\nu_{i},s,d^{C})\right|^{2}\Bigr{\\}}\;.\end{split}$
(1.68)

We use $m_{p}=0.9383$ GeV, $m_{\pi}=0.134$ GeV, $m_{\eta}=0.548$ GeV, and
$m_{K}=0.493677$ GeV to generate all our numerical results. The central values
of relevant matrix elements Aoki:2017puj  are specified in Table 5.

Matrix element | Form factor (GeV2)
---|---
$\langle\pi^{0}|(ud)_{R}u_{L}|p\rangle$ | $-0.131$
$\langle\pi^{0}|(ud)_{L}u_{L}|p\rangle$ | $0.134$
$\langle\pi^{+}|(du)_{R}d_{L}|p\rangle$ | $-0.186$
$\langle\pi^{+}|(du)_{L}d_{L}|p\rangle$ | $0.189$
$\langle K^{0}|(us)_{R}u_{L}|p\rangle$ | $0.103$
$\langle K^{0}|(us)_{L}u_{L}|p\rangle$ | $0.057$
$\langle K^{+}|(us)_{R}d_{L}|p\rangle$ | $-0.049$
$\langle K^{+}|(ud)_{R}s_{L}|p\rangle$ | $-0.134$
$\langle K^{+}|(us)_{L}d_{L}|p\rangle$ | $0.041$
$\langle K^{+}|(ud)_{L}s_{L}|p\rangle$ | $0.139$
$\langle\eta|(ud)_{R}u_{L}|p\rangle$ | $0.006$
$\langle\eta|(ud)_{L}u_{L}|p\rangle$ | $0.113$
Table 5: Central values of matrix elements entering the proton decay
computation at 2 GeV scale Aoki:2017puj . For uncertainties associated with
each of the elements, see Ref. Aoki:2017puj .

##  Appendix 2 Scalar Mediated Proton Decay Widths

We show below explicit expressions for all eight two-body proton decay widths
due to the exchange of a scalar leptoquark $\Lambda_{3}(3,1,-1/3)$ of mass
$M_{\Lambda_{3}}$.

$\displaystyle\Gamma(p\rightarrow\pi^{0}e^{+}_{\beta})$
$\displaystyle=\frac{(m^{2}_{p}-m^{2}_{\pi})^{2}}{32\pi
m_{p}^{3}M_{\Lambda_{3}}^{4}}A^{2}_{L}$
$\displaystyle\times\Biggl{\\{}\Bigl{|}A_{SR}a(d^{C},e_{\beta})\langle\pi^{0}|(du)_{R}u_{L}|p\rangle+A_{SL}a(d,e_{\beta})\langle\pi^{0}|(du)_{L}u_{L}|p\rangle\Bigr{|}^{2}$
$\displaystyle+\Bigl{|}A_{SL}a(d,e^{C}_{\beta})\langle\pi^{0}|(du)_{L}u_{L}|p\rangle+A_{SR}a(d^{C},e^{C}_{\beta})\langle\pi^{0}|(du)_{R}u_{L}|p\rangle\Bigr{|}^{2}\Biggr{\\}}\;.$
(2.69)

$\displaystyle\Gamma(p\rightarrow\pi^{+}\bar{\nu})$
$\displaystyle=\frac{(m_{p}^{2}-m^{2}_{\pi})^{2}}{32\pi
m^{3}_{p}M_{\Lambda_{3}}^{4}}A_{L}^{2}$
$\displaystyle\times\sum_{i=1}^{3}\Bigl{|}A_{SR}a(d,d^{C},\nu_{i})\langle\pi^{+}|(du)_{R}d_{L}|p\rangle+A_{SL}a(d,d,\nu_{i})\langle\pi^{+}|(du)_{L}d_{L}|p\rangle\Bigr{|}^{2}\;.$
(2.70)

$\displaystyle\Gamma(p\rightarrow\eta^{0}e^{+}_{\beta})$
$\displaystyle=\frac{(m_{p}^{2}-m^{2}_{\eta})^{2}}{32\pi
m^{3}_{p}M_{\Lambda_{3}}^{4}}A_{L}^{2}$
$\displaystyle\times\Bigg{\\{}\Bigl{|}A_{SL}a(d,e_{\beta})\langle\eta|(ud)_{L}u_{L}|p\rangle+A_{SR}a(d^{C},e_{\beta})\langle\eta|(ud)_{R}u_{L}|p\rangle\Bigr{|}^{2}$
$\displaystyle+\Bigl{|}A_{SL}a(d^{C},e^{C}_{\beta})\langle\eta|(ud)_{L}u_{L}|p\rangle+A_{SR}a(d,e^{C}_{\beta})\langle\eta|(ud)_{R}u_{L}|p\rangle\Bigr{|}^{2}\Biggr{\\}}\;.$
(2.71)

$\displaystyle\Gamma(p\rightarrow K^{0}e^{+}_{\beta})$
$\displaystyle=\frac{(m_{p}^{2}-m^{2}_{K})^{2}}{64\pi
m^{3}_{p}M_{\Lambda_{3}}^{4}}A_{L}^{2}$
$\displaystyle\times\Biggl{\\{}\Bigr{|}A_{SR}a(s^{C},e_{\beta})\langle
K^{0}|(us)_{R}u_{L}|p\rangle+A_{SL}a(s,e_{\beta})\langle
K^{0}|(us)_{L}u_{L}|p\rangle$ $\displaystyle-A_{SR}a(s,e^{C}_{\beta})\langle
K^{0}|(us)_{R}u_{L}|p\rangle-A_{SL}a(s^{C},e^{C}_{\beta})\langle
K^{0}|(us)_{L}u_{L}|p\rangle\Bigr{|}^{2}$
$\displaystyle+\Bigr{|}A_{SR}a(s^{C},e_{\beta})\langle
K^{0}|(us)_{R}u_{L}|p\rangle+A_{SL}a(s,e_{\beta})\langle
K^{0}|(us)_{L}u_{L}|p\rangle$ $\displaystyle+A_{SR}a(s,e^{C}_{\beta})\langle
K^{0}|(us)_{R}u_{L}|p\rangle+A_{SL}a(s,e^{C}_{\beta})\langle
K^{0}|(us)_{L}u_{L}|p\rangle\Bigr{|}^{2}\Biggr{\\}}\;.$ (2.72)

$\displaystyle\Gamma(p\rightarrow K^{+}\bar{\nu})$
$\displaystyle=\frac{(m_{p}^{2}-m^{2}_{K})^{2}}{32\pi
m^{3}_{p}M_{\Lambda_{3}}^{4}}A_{L}^{2}$
$\displaystyle\times\sum_{i=1}^{3}\left|A_{SL}\left(a(s,d,\nu_{i})\langle
K^{+}|(us)_{L}d_{L}|p\rangle+a(d,s,\nu_{i})\langle
K^{+}|(ud)_{L}s_{L}|p\rangle\right)\right.$
$\displaystyle\left.+A_{SR}\left(a(d,s^{C},\nu_{i})\langle
K^{+}|(us)_{R}d_{L}|p\rangle+a(s,d^{C},\nu_{i})\langle
K^{+}|(ud)_{R}s_{L}|p\rangle\right)\right|^{2}\;.$ (2.73)

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