# Constraining scalars of $16_{H}$ through proton decays in non-renormalisable
$SO(10)$ models

Saurabh K. Shukla<EMAIL_ADDRESS>Theoretical Physics Division, Physical
Research Laboratory, Navarangpura, Ahmedabad-380009, India  Indian Institute
of Technology Gandhinagar, Palaj-382055, India

###### Abstract

Non-renormalisable versions of $SO(10)$ based on irreducible representations
with lesser degrees of freedom, are free of running into the catastrophe of
non-perturbativity of standard model gauge couplings in contrast to the
renormalisable versions having tensors with many degrees of freedom. $16_{H}$
is the smallest representation, participates in Yukawa Lagrangian at the non-
renormalisable level, contributing to the charged and neutral fermion masses,
and has six distinct scalars with different $B-L$ charges. We computed the
leptoquark and diquark couplings of different pairs of scalars stemming from
all possible decomposition of the term resulting from the coupling of
$16_{\rm{H}}$ with the $\mathbf{16}$ dimensional fermion multiplet of $SO(10)$
, i.e. $\frac{\mathbf{16}\mathbf{16}\,16_{\rm{H}}\,16_{\rm{H}}}{\Lambda}$.
Computing the tree and loop level contribution of different pairs to the
effective dimension six, $B-L$ conserving operators, it turns out only three
pairs, viz $\sigma\big{(}1,1,0\big{)}-T\big{(}3,1,\frac{1}{3}\big{)}$, and
$H\big{(}1,2,-\frac{1}{2}\big{)}-\Delta\big{(}3,2,\frac{1}{6}\big{)}$, and
$H-T$ can induce proton decay at tree level. Assuming that the Yukawa
couplings of the $16_{\rm{H}}$ are comparable to those of the
$\overline{126}_{\rm{H}}$ of a realistic SO(10) model and setting the cutoff
scale to the Planck scale typically constrains the $B-L$ breaking scale to be
$4\sim 5$ orders of magnitude less than the cutoff scale $(\Lambda)$.
Moreover, analysing the branching pattern of the leading two-body decay modes
of the proton, we observed a preference for the proton to decay into second-
generation mesons due to the hierarchical nature of Yukawa couplings. In a
realistic $SO(10)$ scenario, we find that $M_{T}>10^{8}$ TeV, while
$M_{\Delta}$ could be as light as a few TeV$s$.

##  I Prelude

Even after the experimental verification of the Standard Model (SM)
predictions with unprecedented accuracy, SM is still agnostic about the
neutrino masses, baryon number violation, matter-antimatter asymmetry and many
other observational and phenomenological pieces. Grand Unified Theory (GUT) is
an aesthetically appealing and viable candidate that pragmatically solves some
of the SM’s inconsistencies while preserving its mores. GUT models place
quarks and leptons within the same multiplet, which are otherwise scattered
and independent in the SM. Various grand unification models based on different
gauge symmetries have been known for more than four decades Georgi and Glashow
(1974); Fritzsch and Minkowski (1975); Pati and Salam (1973).

Grand unified models based on $SO(10)$ gauge symmetry accommodates an entire
generation of SM chiral fermions and a right-handed neutrino in a single
sixteen-dimensional irreducible representation (irrep) ($\mathbf{16}$-plet)
(cf. Langacker (1981); Aulakh and Girdhar (2005); Fukuyama  _et al._ (2005);
Nath and Fileviez Perez (2007) for review). The necessity of the presence of
the right-handed neutrino is in alignment with the demand of light-neutrinos
being massive, making the theory left-right symmetric above the so called
seesaw scale. In addition to $\mathbf{16}$-plet, $SO(10)$ GUT also comprises
other scalar irreps pertinent to meet some elementary demands, i.e. 1) to
decompose into the standard model, and 2) to reproduce the observed charged
and neutral fermion mass spectrum and mixing angles. Three different kinds of
irreps can, in principle, contribute to the $SO(10)$ scalar potential i)
$SO(10)$ gauge symmetry breaking irreps which are $45_{\rm{H}},$ $54_{\rm{H}}$
and $210_{\rm{H}}$ Slansky (1981), ii) irreps which induce rank reduction,
i.e. $16_{H},\,\overline{126}_{H}$, and are linked to neutrino mass generation
Georgi and Nanopoulos (1979); Barbieri  _et al._ (1980); Buccella  _et al._
(1980); Yasue (1981)111Neutrino masses originating from the inclusion of
$120_{\rm{H}}$ instead of $16_{\rm{H}}$ and $\overline{126}_{\rm{H}}$ have
been discussed in Barbieri  _et al._ (1979)., and iii) irreps participating in
the Yukawa interactions at renormalisable level, $10_{\rm{H}}$,
$\overline{126}_{\rm{H}}$ and $120_{\rm{H}}$, and at least two of the
aforementioned representations are needed for a realistic fermion mass
spectrum Babu and Mohapatra (1993); Bajc  _et al._ (2006); Joshipura and Patel
(2011); Altarelli and Meloni (2013); Dueck and Rodejohann (2013); Meloni  _et
al._ (2014, 2017); Babu  _et al._ (2017); Ohlsson and Pernow (2018). Any GUT
model utilizes at least one irreps from each case mentioned in (i, ii, and
iii) for a realistic model, and the choice is rather subjective and influenced
by various phenomenological aspects.

Incorporating the typically large tensor representations (LTR)
($120_{\rm{H}}$, $\overline{126}_{\rm{H}}$ and $210_{\rm{H}}$) in the scalar
sector accounts for the associated low-energy phenomenology while some of them
contributing in the Yukawa potential at the renormalisable level. However,
these large tensor representations contribute to the $\beta$-functions,
causing low-energy gauge couplings to increase with energy. Consequently, the
theory becomes strongly coupled in the ultraviolet (UV), potentially leading
to the gauge couplings hitting the Landau pole before the Planck Scale
($M_{p}$). This phenomenon, known as asymptotic slavery (non-asymptotic
freedom) in $SO(10)$ gauge theories, results in low-energy gauge couplings
becoming unrelated to the non-perturbative evolution of gauge couplings in the
UV, a feature that is uncalled for Maiani  _et al._ (1978); Vaughn (1979);
Rubakov and Troitsky (1999); Aulakh (2002a, b); Aulakh  _et al._ (2004); Bajc
and Sannino (2016). In order to evade this concern, one can stick to smaller
tensor representations ($10_{\rm{H}}$, $16_{\rm{H}}$, $45_{\rm{H}}$ and
$54_{\rm{H}}$) and adhere to non-renormalisable interactions Babu and Barr
(1995); Dvali and Pokorski (1996); Barr and Raby (1997); Albright (2003);
Chang  _et al._ (2005). Non-renormalisable interactions could arise from the
presence of another theory valid above a particular cutoff scale, of which the
germane gauge theory could be a low-energy approximation. Likewise, the SM
relies on renormalisable interactions to explain the masses of charged
fermions and resorts to non-renormalisable interactions for the masses of
neutral fermions, suggesting the existence of a more fundamental theory at
higher energy scales. Moreover, incorporating higher-dimensional operators is
also known to yield desired and viable Yukawa ratios among different
generations of fermions Anderson  _et al._ (1994); Antusch  _et al._ (2020),
leading to a minimal model in terms of free parameters.

The smallest representation capable of generating charged and neutral fermion
masses, including the Majorana mass for the light and heavy neutrinos at the
non-renormalisable level, is the $16_{\rm{H}}$ representation. It comprises
six distinct scalar particles, which are the scalar counterparts of the SM
fermions residing in the $\mathbf{16}$-dimensional fermion multiplet. Besides,
these distinct particles are also non-trivially charged under $B-L$ symmetry
and, thereby, are capable of mediating baryon (B) and lepton (L) number-
changing phenomena like proton decay Weinberg (1979); Wilczek and Zee (1979);
Sakai and Yanagida (1982); Golowich (1981); Dorsner and Fileviez Perez (2005a,
b); Kolešová and Malinský (2019); Buchmuller and Patel (2019); Patel and
Shukla (2022). The inclusion of only $16_{\rm{H}}$ irrep along with the
$\mathbf{16}$-plet fermion, when decomposed, yields the interaction of two
scalars with two SM fermions suppressed by a cutoff scale. In the absence of
any LTR, these Yukawa couplings of $16_{\rm{H}}$ can be of a similar order to
those of an irrep coupling at the renormalisable level (preferably
$\overline{126}_{\rm{H}}$) to yield a realistic fermion mass spectrum and thus
one can constrain the spectrum of the pair of scalars from various
phenomenological implications with these determinable Yukawa couplings of
$16_{\rm{H}}$.

This article is a continuation of our previous works in which we have
constrained the spectrum of different scalars stemming from irreps capable of
participating in the renormalisable $SO(10)$ Yukawa interactions. We have
comprehensively classified scalars capable of inducing $B-L$ conserving and
violating two body proton decays Patel and Shukla (2022) and examined the role
played by sextet scalars in different phenomenological implications Patel and
Shukla (2023). In this work, we thoroughly analyse different pairs of scalars
capable of inducing tree and loop-level two body proton decay emanating from
all possible allowed decomposition of
$\mathbf{16}\,\mathbf{16}\,16_{\rm{H}}\,16_{\rm{H}}$ coupling and thereby
constraining various mediators from the same. Further, we determine the extent
to which a pair of scalars could be light while adhering to the constraints
imposed by proton decays in a typical realistic non-renormalisable $SO(10)$
model. A similar study has been performed in the context of supersymmetric
non-renormalisable $SO(10)$ GUT in Wiesenfeldt (2005) and recently, proton
decay, both at the loop level Helo  _et al._ (2019); Gargalionis  _et al._
(2024) and via higher-dimensional operators Beneito  _et al._ (2023), has
gained attention.

This article is organised as follows. In section (II), we explore the
decomposition of the coupling between $16_{\rm{H}}$ and $\mathbf{16}$-plet
fermions occurring at the non-renormalisable level. Additionally, we identify
pairs of scalars within $16_{\rm{H}}$ capable of inducing proton decay. The
computation of the contribution from various scalar pairs to the effective
proton decay operators is presented in section (III). Utilizing these general
results, we analyze the branching pattern of the leading modes of proton decay
and constrain the mediators accordingly in section (IV). Finally, we conclude
our study in section (V).

##  II Couplings of $16_{H}$ in $SO(10)$

A single $\mathbf{16}$-dimensional irreducible representation (irrep) of
$SO(10)$ can accommodate an entire generation of standard model Weyl fermions
along with their hypothetically-missing ally, a conjugated-right handed
neutrino. Once the $SO(10)$ gauge symmetry is broken into one of its
subgroups, i.e. $SU(5)\times U(1)$, the $16$-plet can be decomposed into the
irreps of $SU(5)$ as shown below Gell-Mann  _et al._ (1979).

${}\mathbf{16}=\big{(}\mathbf{1},\mathbf{-5}\big{)}+\big{(}\mathbf{\overline{5}},\mathbf{3}\big{)}+\big{(}\mathbf{10},\mathbf{-1}\big{)}$
(1)

The $SU(5)$ irreps shown in eq. (1) are the matter multiplets and consist of
SM quarks and leptons. The SM Weyl fermions residing in the different $SU(5)$
irreps are shown below. Throughout this article, we denote generation indices
by the uppercase Latin alphabets $\big{(}1\leq A,B,C...\leq 3\big{)}$, $SU(5)$
indices by lowercase Latin letters $\big{(}1\leq n,o,p,...,z\leq 5\big{)}$,
$SU(3)$ indices by lowercase Greek letters
$\big{(}1\leq\alpha,\beta,\gamma...\leq 3\big{)}$, and lowercase Latin
alphabets $\big{(}4\leq a,b,c,...,m\leq 5)$ represent the $SU(2)$ indices,
unless specified explicitly. Moreover, the bold letter denotes the fermion
representation, while the scalar representations are shown with subscript $H$.

$\displaystyle{}\mathbf{1}$ $\displaystyle=$ $\displaystyle\nu^{C},\hskip
28.45274pt\mathbf{\overline{5}}_{a}\;=\;\varepsilon^{ab}\,l^{b},\hskip
28.45274pt\mathbf{\overline{5}}_{\alpha}\;=\;d^{C}_{\alpha},$
$\displaystyle\mathbf{10}^{ab}$ $\displaystyle=$
$\displaystyle\varepsilon^{ab}\,e^{C},\hskip
28.45274pt\mathbf{10}^{a\,\alpha}\;=\;q^{a\alpha},\hskip
28.45274pt\mathbf{10}^{\alpha\beta}\;=\;\varepsilon^{\alpha\beta\gamma}\,u^{C}_{\gamma}.$
(2)

The two rank $SU(2)\;Levi-Civita$ tensor used in eq. (II) is defined as
$\varepsilon_{45}=-\varepsilon_{54}=-\varepsilon^{45}=\varepsilon^{54}=1$,
while the three rank $SU(3)\;Levi-Civita$ tensor is defined as
$\varepsilon_{123}=\varepsilon^{123}=1$, and similarly for all other cyclic
permutations. Additionally, $C$ is the charge conjugation matrix in the
Lorentz space and is defined as $C\equiv i\sigma_{2}$, $\sigma_{2}$ being the
Pauli matrix.

Similarly, the canonically normalised decomposition of scalars residing in the
$16_{H}$-dimensional scalar representation is as follows Patel and Shukla
(2022).

$\displaystyle{}1_{H}$ $\displaystyle=$ $\displaystyle\sigma,\hskip
28.45274pt5^{\dagger}_{a\,\,H}\;=\;H^{\dagger}_{a},\hskip
28.45274pt5^{\dagger}_{\alpha\,\,H}\;=\;T^{\dagger}_{\alpha},$ $\displaystyle
10^{ab}_{H}$ $\displaystyle=$ $\displaystyle t^{ab},\,\hskip
28.45274pt10^{a\,\alpha}\;=\;\frac{1}{\sqrt{2}}\,\Delta^{a\alpha},\hskip
28.45274pt10^{\alpha\beta}\;=\;\Theta^{\alpha\beta}.$ (3)

$16_{H}$ is the smallest representation, contributing to charged fermion
masses and neutrino masses; however, it contributes to the Yukawa Lagrangian
at the non-renormalisable level. Table (1) depicts the different scalar
contained in $16_{\rm{H}}$, together with their SM charge and $B-L$ quantum
number. The $B-L$ of any multiplet is related to subgroups $U_{1}^{X}$, and
$U_{1}^{Y}$ contained in $SO(10)$ by the following relation Buchmuller and
Patel (2019).

${}B-L=\frac{4}{5}\,Y\,-\,\frac{1}{5}X.$ (4)

where, $Y$ and $X$ are respectively the hypercharge and $U_{1}^{X}$ charge of
the multiplet.

SM Charges |  Notation |  $B-L$ Charges
---|---|---
$\left(1,1,0\right)$ | $\sigma$ | $1$
$\left(\overline{3},1,\nicefrac{{1}}{{3}}\right)$ | $T^{\dagger}_{\alpha}$ | $\nicefrac{{-1}}{{3}}$
$\left(1,2,\nicefrac{{-1}}{{2}}\right)$ | $H^{\dagger}_{a}$ | $-1$
$\left(3,2,\nicefrac{{1}}{{6}}\right)$ | $\Delta^{a\alpha}$ | $\nicefrac{{1}}{{3}}$
$\left(\overline{3},1,\nicefrac{{-2}}{{3}}\right)$ | $\Theta^{\alpha\beta}$ | $\nicefrac{{-1}}{{3}}$
$\left(1,1,-1\right)$ | $t^{ab}$ | $1$
Table 1: Classification of scalars present in $16_{H}$ alongwith their SM and
$B-L$ charges

We begin with an $SO(10)$ lagrangian with $\mathbf{16}$ dimensional fermion
multiplet and $16_{\rm{H}}$ scalar representation. The Yukawa Lagrangian of
$\mathbf{16}$ with $16_{H}$ to the first order in $\Lambda$ is shown below,
where $\Lambda$ being the typical cutoff scale.

$\displaystyle{}{\mathcal{L}}$ $\displaystyle\supset$
$\displaystyle\frac{1}{\Lambda}y_{AB}\,\big{(}\mathbf{16}_{A}\,\mathbf{16}_{B}\;16_{H}\,16_{H}\big{)}\,+\,\frac{1}{\Lambda}\bar{y}_{AB}\,\big{(}\mathbf{16}_{A}\,\mathbf{16}_{B}\;16^{\dagger}_{H}\,16^{\dagger}_{H}\big{)}\,\;+\;\rm{H.c.}$
(5)

To decompose the Lagrangian given in eq. (5), one considers all the
effectively generated Yukawa couplings allowed by the integration of all
possible irreps (scalar or fermions) consistent with the assumed gauge
symmetry. The transformation properties of $16$ dimensional irrep under
$SO(10)$ is given as given as follows Slansky (1981).

$\displaystyle{}16\times 16$ $\displaystyle=$ $\displaystyle
10+120+126^{\dagger}$ $\displaystyle 16\times 16^{\dagger}$ $\displaystyle=$
$\displaystyle 1+45+210$ (6)

Eq. (II) suggests that once we have a term like the one in eq. (5), only a few
possibilities of intermediate $SO(10)$ irreps are allowed by $SO(10)$
invariance. The exhaustive ways to generate the coupling given in eq. (5) is
given as follows. In the following, we represent the coupling as
$(...)_{D_{1}}(...)_{D_{2}}$ in a way that the representations inside the
first (second) bracket are contracted to form the $D_{1}\,(D_{2})$, such that
$D_{1}\times D_{2}\supset\mathbf{1}$ where $\mathbf{1}$ transforms trivially
under $SO(10)$. Furthermore, $D_{1}\,(D_{2})$ can be either a scalar or a
fermion representation.

$\displaystyle\bullet$
$\displaystyle\frac{h_{AB}}{\Lambda}\big{(}\mathbf{16}_{A}\,\mathbf{16}_{B}\big{)}_{10_{H}}\,\big{(}16_{H}\,16_{H}\big{)}_{10_{H}},\;\frac{\bar{h}_{AB}}{\Lambda}\big{(}\mathbf{16}_{A}\,\mathbf{16}_{B}\big{)}_{10_{H}}\,\big{(}16^{\dagger}_{H}\,16^{\dagger}_{H}\big{)}_{10^{\dagger}_{H}},$
(7)
$\displaystyle\frac{\tilde{h}_{AB}}{\Lambda}\big{(}\mathbf{16}_{A}\,16_{H}\big{)}_{\mathbf{10}}\,\big{(}\mathbf{16}_{B}\,16_{H}\big{)}_{\mathbf{10}},\text{and}\;\frac{\hat{h}_{AB}}{\Lambda}\Big{(}\mathbf{16}_{A}\,16_{H}\big{)}_{\mathbf{10}}\,\Big{(}\mathbf{16}_{B}\,16_{H}\Big{)}^{*}_{\mathbf{10}^{\dagger}}$
$\displaystyle\bullet$
$\displaystyle\frac{g_{AB}}{\Lambda}\big{(}\mathbf{16}_{A}\,\mathbf{16}_{B}\big{)}_{120_{H}}\,\big{(}16_{H}\,16_{H}\big{)}_{120_{H}},\;\frac{\bar{g}_{AB}}{\Lambda}\big{(}\mathbf{16}_{A}\,\mathbf{16}_{B}\big{)}_{120_{H}}\,\big{(}16^{\dagger}_{H}\,16^{\dagger}_{H}\big{)}_{120^{\dagger}_{H}},$
$\displaystyle\frac{\tilde{g}_{AB}}{\Lambda}\big{(}\mathbf{16}_{A}\,16_{H}\big{)}_{\mathbf{120}}\,\big{(}\mathbf{16}_{B}\,16_{H}\big{)}_{\mathbf{120}}\;\text{and}\;\frac{\hat{g}_{AB}}{\Lambda}\big{(}\mathbf{16}_{A}\,16_{H}\big{)}_{\mathbf{120}}\,\big{(}\mathbf{16}_{B}\,16_{H}\big{)}^{*}_{\mathbf{120^{\dagger}}}$
$\displaystyle\bullet$
$\displaystyle\frac{\bar{f}_{AB}}{\Lambda}\big{(}\mathbf{16}_{A}\,\,\mathbf{16}_{B}\big{)}_{126^{\dagger}_{H}}\,\big{(}16^{\dagger}_{H}\,16^{\dagger}_{H}\big{)}_{126_{H}}\;\text{and}\;\frac{\hat{f}_{AB}}{\Lambda}\big{(}\mathbf{16}_{A}\,\,16_{H}\big{)}_{\mathbf{126}^{\dagger}}\,\big{(}\mathbf{16}_{B}\,16_{H}\big{)}^{*}_{\mathbf{126}}$
$\displaystyle\bullet$
$\displaystyle\frac{k_{AB}}{\Lambda}\,\big{(}\mathbf{16}_{A}\,16^{\dagger}_{H}\big{)}_{\mathbf{1}}\,\big{(}\mathbf{16}_{B}\,16^{\dagger}_{H}\big{)}_{\mathbf{1}},\;\frac{\tilde{k}_{AB}}{\Lambda}\,\big{(}\mathbf{16}_{A}\,16^{\dagger}_{H}\big{)}_{\mathbf{45}}\,\big{(}\mathbf{16}_{B}\,16^{\dagger}_{H}\big{)}_{\mathbf{45}},$
$\displaystyle\text{and}\;\frac{\bar{k}_{AB}}{\Lambda}\,\big{(}\mathbf{16}_{A}\,16^{\dagger}_{H}\big{)}_{\mathbf{210}}\,\big{(}\mathbf{16}_{B}\,16^{\dagger}_{H}\big{)}_{\mathbf{210}}$

where,
$h,\,\bar{h},\tilde{h},\,\hat{h},\,g,\,\bar{g},\tilde{g},\,\hat{g},\,\bar{f},\,\hat{f},\,k,\,\tilde{k}$
and $\bar{k}$ are various Yukawa couplings and $\Lambda$ corresponds to the
energy scale where the intermediate $SO(10)$ irrep has been integrated out.

Having singled out all possible ways for generating the effective term
mentioned in eq. (5), we further shift to decompose these $SO(10)$ invariant
couplings, as given in eq. (II), into the SU(5) framework, starting with the
parameterisation of the coupling between the $\mathbf{16}$-plet fermion and
the $10_{\rm{H}}$ Nath and Syed (2001a).

$\displaystyle{}-{\cal{L}}^{10}_{Y}$ $\displaystyle\supset$ $\displaystyle
H_{AB}\,\mathbf{16}_{A}^{T}\,\,{\mathcal{C}}^{-1}\,\,\mathbf{16}_{B}\,10_{H}$
(8) $\displaystyle=$ $\displaystyle
i2\sqrt{2}\,H_{AB}\,\left({\mathbf{10}^{pq}_{A}}^{T}\,\,{\mathcal{C}}^{-1}\,\,\mathbf{\overline{5}}_{p\,B}\,\tilde{5}^{\dagger}_{q\,H}\,+\,\frac{1}{8}\varepsilon_{pqrst}\,{\mathbf{10}^{pq}}^{T}_{A}\,\,{\mathcal{C}}^{-1}\,\,\mathbf{10}^{rs}_{B}\,\tilde{5}^{t}_{H}\,-\,\mathbf{1}_{A}^{T}\,\,{\mathcal{C}}^{-1}\,\,\mathbf{\overline{5}}_{p\,B}\,\tilde{5}^{p}_{H}\right)$
$\displaystyle+$ $\displaystyle\rm{H.c.}$

where, the $H_{AB}$ is constrained by its symmetricity in the generation
space. In addition, the scalar fields belonging to the $10_{H}$ representation
are denoted by a tilde placed over them. Furthermore, the decomposition of the
interaction between the $16_{\rm{H}}$ and $10_{\rm{H}}$ representations can be
expressed analogously to eq. (8), as illustrated below.

$\displaystyle{}-{\cal{L}}^{10}_{Y}$ $\displaystyle\supset$
$\displaystyle\eta\,16_{H}\,16_{H}\,10_{H}+M^{2}_{10_{\rm{H}}}\,10^{\dagger}_{\rm{H}}\,10_{\rm{H}}\,$
(9) $\displaystyle=$ $\displaystyle
i2\sqrt{2}\,\eta\,\left(10_{H}^{pq}\,5^{\dagger}_{p\,H}\,\tilde{5}^{\dagger}_{q\,H}\,+\,\frac{1}{8}\varepsilon_{pqrst}\,{10_{H}^{pq}}\,5_{H}^{rs}\,\tilde{5}^{t}_{H}\,-\,1_{H}\,5^{\dagger}_{p}\,\tilde{5}^{p}_{H}\right)$
$\displaystyle+$ $\displaystyle
M^{2}_{10_{\rm{H}}}\,\tilde{5}^{\dagger}_{H}\,\tilde{5}_{H}\;+\;\rm{H.c.}$

To integrate out the $10_{H}$ representation, we utilise their couplings
derived in eqs. (8) and (9) in their irreducible $SU(5)$ components. By
utilizing equations of motion, we solve for the various $SU(5)$ components of
$10_{H}$ in eq. (9) and subsequently substitute the results into eq. (8) Nath
and Syed (2001b). Additionally, it is imperative to mention that we integrate
out representations with the same $U_{1}^{X}$ charge. For instance, the
$U_{1}^{X}$ charge of $\tilde{5}^{\dagger}_{H}\subset 16_{\rm{H}}$ is $3$,
while the $U_{1}^{X}$ charge of $5^{\dagger}_{H}\subset 10_{\rm{H}}$ is $-2$
Slansky (1981); Feger  _et al._ (2020). Thus, identical $SU(5)$
representations belonging to different $SO(10)$ representations can differ in
their $U_{1}^{X}$ charges. Consequently, we denote the $SU(5)$ scalars (and
the fermions) intended for integration by placing a tilde ($\sim$) over them.

$\displaystyle{}\mathcal{L}_{\rm{NR}}$ $\displaystyle\supset$
$\displaystyle\frac{h_{AB}}{\Lambda}\Big{(}\mathbf{16}^{T}_{A}\,\,{\mathcal{C}}^{-1}\,\,\mathbf{16}_{B}\Big{)}_{10}\,\Big{(}16_{H}\,16_{H}\Big{)}_{10}$
(10) $\displaystyle=$
$\displaystyle\frac{-8\,h_{AB}}{\Lambda}\Big{[}\big{(}\mathbf{10}^{pq\,T}_{A}\,\,{\mathcal{C}}^{-1}\,\,\mathbf{\overline{5}}_{p\,B}\big{)}\,\big{(}-1_{H}\,5^{\dagger}_{q\,H}+\frac{1}{8}\varepsilon_{qrstu}\,10^{rs}_{H}\,10^{tu}_{H}\big{)}+\big{(}-\mathbf{1}^{T}_{A}\,\,{\mathcal{C}}^{-1}\,\,\mathbf{\overline{5}}_{p\,B}$
$\displaystyle+$
$\displaystyle\frac{1}{8}\,\varepsilon_{qrstp}\,\mathbf{10}^{qr\,T}_{A}\,\,{\mathcal{C}}^{-1}\,\mathbf{10}^{st}_{B}\big{)}\,\big{(}10^{pw}_{H}\,5^{\dagger}_{w\,H}\big{)}\Big{]}\;+\;\rm{H.c.}$

where, we set
$\frac{\eta\,H_{AB}}{M^{2}_{10_{\rm{H}}}}\,\,\equiv\,\frac{h_{AB}}{\Lambda}$.
Similarly, we can derive the decomposition of term associated with $\bar{h}$
coupling of eq. (LABEL:eq:16Fs16Hs) as shown below.

$\displaystyle{}-\mathcal{L}_{\rm{NR}}$ $\displaystyle\supset$
$\displaystyle\frac{\bar{h}_{AB}}{\Lambda}\Big{(}\mathbf{16}^{T}_{A}\,\,{\mathcal{C}}^{-1}\,\,\mathbf{16}_{B}\Big{)}_{10}\,\Big{(}16^{\dagger}_{H}\,16^{\dagger}_{H}\Big{)}_{10^{\dagger}}$
(11) $\displaystyle=$
$\displaystyle\frac{8\,\bar{h}_{AB}}{\Lambda}\Big{[}\big{(}\mathbf{10}^{pq\,T}_{A}\,\,{\mathcal{C}}^{-1}\,\,\mathbf{\overline{5}}_{p\,B}\big{)}\,\big{(}10^{\dagger}_{qw\,H}\,5^{w}_{H}\big{)}$
$\displaystyle+$
$\displaystyle\big{(}-\mathbf{1}^{T}_{A}\,\,{\mathcal{C}}^{-1}\,\,\mathbf{\overline{5}}_{p\,B}+\frac{1}{8}\,\varepsilon_{qrstp}\,\mathbf{10}^{qr\,T}_{A}\,\,{\mathcal{C}}^{-1}\,\mathbf{10}^{st}_{B}\big{)}\,\big{(}-1_{H}\,5^{p}_{H}+\frac{1}{8}\varepsilon^{prstu}\,10^{\dagger}_{rs\,H}\,10^{\dagger}_{tu\,H}\big{)}\Big{]}$
$\displaystyle+$ $\displaystyle\rm{H.c.}$

Further, the decomposition of $\mathbf{16}$-plet fermion with $\mathbf{10}$
dimensional fermion is shown below Nath and Syed (2001a).

$\displaystyle{}\mathcal{L}_{Y}^{10}$ $\displaystyle\supset$
$\displaystyle\tilde{H}_{AB}\,\mathbf{16}^{T}_{A}\,\,{\mathcal{C}}^{-1}\,\,\mathbf{10}_{B}\,16_{H}\,$
(12) $\displaystyle=$ $\displaystyle
i2\sqrt{2}\,\tilde{H}_{AB}\,\Big{(}{\mathbf{10}^{pq}_{A}}^{T}\,\,{\mathcal{C}}^{-1}\,\,\tilde{\mathbf{\overline{5}}}_{p\,B}\,5^{\dagger}_{q\,H}\,+\,\frac{1}{8}\varepsilon_{pqrst}\,{\mathbf{10}^{pq}}^{T}_{A}\,\,{\mathcal{C}}^{-1}\,\,\big{(}\tilde{\mathbf{\overline{5}}}\big{)}^{*\,r}_{B}\,10^{st}_{H}\,$
$\displaystyle-$
$\displaystyle\,\mathbf{1}_{A}^{T}\,\,{\mathcal{C}}^{-1}\,\,\big{(}\tilde{\mathbf{\overline{5}}}\big{)}^{*\,p}_{B}\,5^{\dagger}_{p\,H}\Big{)}\;+\;\rm{H.c.}$

where, the fermion representations belonging to $\mathbf{10}$ are shown by
putting tilde ($\sim$) over them. Utilizing eq. (12), the estimation of
higher-mass dimensional terms with Yukawa coefficients $\tilde{h}$ and
$\hat{h}$, mentioned in eq. (LABEL:eq:16Fs16Hs), is provided as follows:

$\displaystyle{}\mathcal{L}_{NR}$ $\displaystyle\supset$
$\displaystyle\frac{\tilde{h}_{AB}}{\Lambda}\Big{(}\mathbf{16}^{T}_{A}\,\,{\mathcal{C}}^{-1}\,\,16_{H}\Big{)}_{\mathbf{10}}\,\Big{(}\mathbf{16}^{T}_{B}\,\,{\mathcal{C}}^{-1}\,\,16_{H}\Big{)}_{\mathbf{10}}$
(13) $\displaystyle+$
$\displaystyle\frac{\hat{h}_{AB}}{\Lambda}\Big{(}\mathbf{16}^{T}_{A}\,\,{\mathcal{C}}^{-1}\,16_{H}\big{)}_{\mathbf{10}}\,\Big{(}\mathbf{16}^{T}_{B}\,\,{\mathcal{C}}^{-1}\,\,16_{H}\Big{)}^{*}_{\mathbf{10}^{\dagger}}\;+\;\rm{H.c.}$
$\displaystyle=$
$\displaystyle\Big{[}\frac{-8\,\tilde{h}_{AB}}{\Lambda}\big{(}\mathbf{10}^{pq\,T}_{A}\,\,{\mathcal{C}}^{-1}\,\,5^{\dagger}_{p\,H}\big{)}\,\big{(}-\mathbf{1}_{B}\,5^{\dagger}_{q\,H}+\frac{1}{8}\varepsilon_{qrstu}\,\mathbf{10}_{B}^{rs}\,10^{tu}_{H}\big{)}\,+A\leftrightarrow
B\Big{]}\,$ $\displaystyle+$
$\displaystyle\frac{\hat{h}_{AB}}{\Lambda}\Big{[}\big{(}\mathbf{10}^{pq\,T}_{A}\,{\mathcal{C}}^{-1}\,\,5_{r\,H}^{\dagger}\big{)}\,\big{(}\mathbf{10}^{pq}_{B}\,\,5_{r\,H}^{\dagger}\big{)}^{*}+\big{(}\mathbf{\overline{5}}^{T}_{p\,A}\,{\mathcal{C}}^{-1}\,\,10^{qr}_{H}\big{)}\,\big{(}\mathbf{\overline{5}}_{pB}\,10^{qr}_{H}\big{)}^{*}$
$\displaystyle+$
$\displaystyle\big{(}-\mathbf{1}^{T}_{A}\,{\mathcal{C}}^{-1}\,\,5^{\dagger}_{t\,H}+\frac{1}{8}\varepsilon_{pqrst}\,\mathbf{10}^{pq\,T}_{A}\,{\mathcal{C}}^{-1}\,10^{rs}_{H}\big{)}\,\big{(}-\mathbf{1}_{B}\,5^{\dagger}_{t\,H}+\frac{1}{8}\varepsilon_{xywzt}\,\mathbf{10}^{xy}_{B}10^{wz}_{H}\big{)}^{*}\Big{]}\
$ $\displaystyle+$ $\displaystyle\rm{H.c.}$

The expressions provided in eqs. (10), (11), and (13) collectively represent
the decomposition of higher-dimensional terms involving an intermediate
integrated-out irrep of dimension $10$, which can be either scalar or
fermionic. By using the embedding of SM fermions into various irreducible
representations of $SU(5)$, as given in eq. (II), and the embedding of
different scalars into different $SU(5)$ irreps of $16_{\rm{H}}$ as described
in eq. (II), these expressions can be further decomposed into couplings
involving two SM fermions and two scalars, preserving the SM gauge symmetry.

We proceed to calculate the decomposition of effective terms permitted by the
integration of $120$-dimensional irrep. Repeating the exercise conducted for
the case of the $10$-dimensional irrep, we obtain the following expression.

$\displaystyle{}\mathcal{L}_{NR}$ $\displaystyle\supset$
$\displaystyle\frac{g_{AB}}{\Lambda}\Big{(}\mathbf{16}^{T}_{A}\,\,{\mathcal{C}}^{-1}\,\,\mathbf{16}_{B}\Big{)}_{120}\,\Big{(}16_{H}\,16_{H}\Big{)}_{120}+\frac{\bar{g}_{AB}}{\Lambda}\Big{(}\mathbf{16}^{T}_{A}\,\,{\mathcal{C}}^{-1}\,\,\mathbf{16}_{B}\Big{)}_{120}\,\Big{(}16^{\dagger}_{H}\,16^{\dagger}_{H}\Big{)}_{120^{\dagger}}\,$
(14) $\displaystyle+$
$\displaystyle\frac{\tilde{g}_{AB}}{\Lambda}\Big{(}\mathbf{16}^{T}_{A}\,\,{\mathcal{C}}^{-1}\,\,16_{H}\Big{)}_{\mathbf{120}}\,\Big{(}\mathbf{16}_{B}\,\,{\mathcal{C}}^{-1}\,\,16_{H}\Big{)}_{\mathbf{120}}$
$\displaystyle+$
$\displaystyle\frac{\hat{g}_{AB}}{\Lambda}\Big{(}\mathbf{16}^{T}_{A}\,\,{\mathcal{C}}^{-1}\,16_{H}\big{)}_{\mathbf{120}}\,\Big{(}\mathbf{16}_{B}\,\,{\mathcal{C}}^{-1}\,\,16_{H}\Big{)}^{*}_{\mathbf{120}^{\dagger}}\;+\;\rm{H.c.}$
$\displaystyle=$
$\displaystyle\frac{-4\,g_{AB}}{3\Lambda}\,\Big{[}2\big{(}\mathbf{1}_{A}\,\,{\mathcal{C}}^{-1}\,\,\mathbf{\overline{5}}_{p\,B}\big{)}\,\big{(}10^{pq}_{H}\,5^{\dagger}_{q\,H}\big{)}-4\big{(}\mathbf{10}_{A}^{pq\,T}\,\,{\mathcal{C}}^{-1}\,\,\mathbf{\overline{5}}_{p\,B}\big{)}\,\big{(}1_{H}\,5^{\dagger\,q}_{H}\big{)}$
$\displaystyle+$ $\displaystyle
2\big{(}\mathbf{\overline{5}}^{T}_{p\,A}\,\,{\mathcal{C}}^{-1}\,\mathbf{\overline{5}}_{q\,B}\big{)}\,\big{(}1_{H}\,10^{pq}_{H}\big{)}+2\big{(}\mathbf{1}_{A}\,\,{\mathcal{C}}^{-1}\,\,10^{pq}_{B}\big{)}\,\big{(}5^{\dagger}_{p\,H}\,5^{\dagger}_{q\,H}\big{)}$
$\displaystyle-$
$\displaystyle\frac{1}{2}\big{(}\varepsilon_{pqrst}\,\mathbf{10}^{pq\,T}_{A}\,\,{\mathcal{C}}^{-1}\,\,\mathbf{10}^{rn}_{B}\big{)}\,\big{(}5^{\dagger}_{n\,H}\,10^{st}_{H}\big{)}-\frac{1}{2}\big{(}\mathbf{\overline{5}}_{A\,n}\,\mathbf{10}^{pq}_{B}\big{)}\,\big{(}\varepsilon_{pqrst}\,10^{rs}_{H}\,10^{tn}_{H}\big{)}\Big{]}\,$
$\displaystyle+$
$\displaystyle\frac{4\bar{g}_{AB}}{3\Lambda}\,\Big{[}4\big{(}\mathbf{1}_{p\,A}^{T}\,{\mathcal{C}}^{-1}\,\,\mathbf{\overline{5}}_{B}\big{)}\,\big{(}1_{H}\,5^{p}_{H}\big{)}+\big{(}\mathbf{10}^{pq\,T}_{A}\,{\mathcal{C}}^{-1}\,\,\mathbf{\overline{5}}_{r\,B}\big{)}\,\big{(}10^{\dagger}_{pq\,H}\,5^{r}_{H}\big{)}$
$\displaystyle+$ $\displaystyle
2\big{(}\mathbf{\overline{5}}_{p\,A}\,{\mathcal{C}}^{-1}\,\,\mathbf{\overline{5}}_{q\,B}\big{)}\,\big{(}5^{p}_{H}\,5^{q}_{H}\big{)}+2\big{(}\mathbf{1}_{A}^{T}\,{\mathcal{C}}^{-1}\,\,\mathbf{10}^{pq}_{B}\big{)}\,\big{(}1_{H}\,10^{\dagger}_{pq\,H}\big{)}+2\big{(}\mathbf{\overline{5}}_{p\,A}\,{\mathcal{C}}^{-1}\,\,\mathbf{10}_{B}^{qr}\big{)}\,\big{(}5^{p}_{H}\,10^{\dagger}_{qr}\big{)}$
$\displaystyle+$
$\displaystyle\frac{1}{8}\big{(}\varepsilon_{pqrst}\mathbf{10}_{A}^{pq\,T}\,{\mathcal{C}}^{-1}\,\,\mathbf{10}_{B}^{rn}\big{)}\,\big{(}\varepsilon^{xyzst}10^{\dagger}_{xy\,H}\,10^{\dagger}_{zn\,H}\big{)}\Big{]}$
$\displaystyle-$
$\displaystyle\frac{4\tilde{g}_{AB}}{3\,\Lambda}\Big{[}-2\big{(}\mathbf{1}_{A}^{T}\,\,{\mathcal{C}}^{-1}\,5^{\dagger}_{p\,H}\big{)}\,\big{(}\mathbf{10}^{pq}_{B}\,5^{\dagger}_{q\,H}\big{)}-2\big{(}\mathbf{\overline{5}}^{T}_{p\,A}\,{\mathcal{C}}^{-1}\,\,5^{\dagger}_{q\,H}\big{)}\,\big{(}\mathbf{1}_{B}\,10^{pq}_{H}\big{)}$
$\displaystyle-$
$\displaystyle\frac{1}{2}\big{(}\varepsilon_{pqrst}\,\mathbf{10}_{A}^{pq\,T}\,\,{\mathcal{C}}^{-1}\,\,10^{rn}_{H}\big{)}\,\big{(}\mathbf{\overline{5}}_{n\,B}\,10^{st}_{H}\big{)}\Big{]}$
$\displaystyle+$
$\displaystyle\frac{4\hat{g}_{AB}}{3\,\Lambda}\Big{[}\big{(}\mathbf{10}^{pq\,T}_{A}\,{\mathcal{C}}^{-1}\,\,5^{\dagger}_{p\,H}\big{)}\,(\mathbf{10}_{B}^{sq}\,5^{\dagger}_{s\,H}\big{)}^{*}+2\big{(}\mathbf{\overline{5}}^{T}_{p\,A}\,{\mathcal{C}}^{-1}\,\,10^{qr}_{H}\big{)}\,\big{(}\mathbf{\overline{5}}_{p\,B}\,\,10^{qr}_{H}\big{)}^{*}$
$\displaystyle+$
$\displaystyle\big{(}\mathbf{\overline{5}}^{T}_{p\,A}\,{\mathcal{C}}^{-1}\,\,5^{\dagger}_{q\,H}\big{)}\,\big{(}\mathbf{\overline{5}}_{p\,B}\,\,5^{\dagger}_{q\,H}\big{)}^{*}+\frac{1}{8}\,\big{(}\varepsilon_{pqrst}\,\mathbf{10}^{pq\,T}_{A}\,\,{\mathcal{C}}^{-1}\,\,10^{rn}_{H}\big{)}\,\big{(}\varepsilon_{xyzst}\,\mathbf{10}^{xy}_{B}\,\,\,10^{zn}_{H}\big{)}^{*}\Big{]}$
$\displaystyle+$ $\displaystyle\rm{H.c.}$

Eq. (14) provides the coupling of two $SU(5)$ fermion multiplets and two
scalar multiplets resulting from the integration of the $120$ representation.
It recieves contributions from $g$, $\bar{g}$, $\tilde{g}$, and $\hat{g}$,
where the former two couplings are antisymmetric in nature.

We now turn to the computation of terms when the intermediate integrated out
irrep is $126$ dimensional. In contrast to the case of $10_{\rm{H}}$ and
$120_{H}$, fewer possibilities of decomposition of the term mentioned in eq.
(5) mediated by $126$ are allowed. The latter is due to the transformation
property of $126$-dimensional irrep Gell-Mann  _et al._ (1979).

Analogous to the case of $10_{H}$ and $120_{H}$, we calculate the
decomposition of higher-dimensional terms comprising two $\mathbf{16}$ and two
$16_{H}$ representations, mediated by the $126$-dimensional irrep. This
analysis considers both possibilities of the $126$ being either a scalar or a
fermion.

$\displaystyle{}\mathcal{L}_{NR}$ $\displaystyle\supset$
$\displaystyle\frac{\bar{f}_{AB}}{\Lambda}\big{(}\mathbf{16}^{T}_{A}\,\,{\mathcal{C}}^{-1}\,\,\mathbf{16}_{B}\big{)}_{126^{\dagger}_{H}}\,\big{(}16^{\dagger}_{H}\,16^{\dagger}_{H}\big{)}_{126_{H}}$
(15) $\displaystyle+$
$\displaystyle\frac{\hat{f}_{AB}}{\Lambda}\big{(}\mathbf{16}_{A}^{T}\,\,{\mathcal{C}}^{-1}\,\,16_{H}\big{)}_{\mathbf{126}^{\dagger}}\,\big{(}\mathbf{16}_{B}\,16_{H}\big{)}^{*}_{\mathbf{126}}\;+\;\rm{H.c.}$
$\displaystyle=$
$\displaystyle\frac{-4\,\bar{f}_{AB}}{15\,\Lambda}\,\Big{[}\big{(}\mathbf{1}_{A}^{T}\,\,{\mathcal{C}}^{-1}\,\,\mathbf{10}_{B}^{pq}\big{)}\,\big{(}1_{H}\,10^{\dagger}_{pq\,H}\big{)}-\big{(}\mathbf{\overline{5}}_{p\,A}\,{\mathcal{C}}^{-1}\,\,\mathbf{\overline{5}}_{q\,B}\big{)}\,\big{(}5^{p}_{H}\,5^{q}_{H}\big{)}$
$\displaystyle+$
$\displaystyle\frac{3}{2}\big{(}\mathbf{1}_{A}^{T}\,\,{\mathcal{C}}^{-1}\,\,\mathbf{\overline{5}}_{p\,B}+\frac{1}{24}\varepsilon_{qrstp}\,\mathbf{10}^{qr}_{A}\,\,{\mathcal{C}}^{-1}\,\,\mathbf{10}^{st}_{B}\big{)}\,\big{(}1_{H}\,5^{p}_{H}+\frac{1}{24}\varepsilon^{uvwxp}10^{\dagger}_{uv\,H}\,10^{\dagger}_{wx\,H}\big{)}$
$\displaystyle+$
$\displaystyle\big{(}\mathbf{10}_{A}^{pq\,T}\,\,{\mathcal{C}}^{-1}\,\,\mathbf{\overline{5}}_{r\,B}\big{)}\,\big{(}10^{\dagger}_{pq\,H}\,5^{k}_{H}\big{)}-\frac{1}{4\sqrt{3}}\,\big{(}\varepsilon_{pqnuv}\,\mathbf{10}_{A}^{pq\,T}\,\,{\mathcal{C}}^{-1}\,\,\mathbf{10}_{B}^{rs}\big{)}\,\big{(}\varepsilon^{xynuv}\,10^{\dagger}_{xy\,H}\,10^{\dagger}_{rs\,H}\big{)}\Big{]}$
$\displaystyle+$
$\displaystyle\frac{4\hat{f}_{AB}}{15\,\Lambda}\,\Big{[}\big{(}\mathbf{\overline{5}}^{T}_{p\,A}\,\,{\mathcal{C}}^{-1}\,\,5^{\dagger}_{q}\big{)}\,\big{(}\mathbf{\overline{5}}_{p\,B}\,\,{\mathcal{C}}^{-1}\,\,5^{\dagger}_{q}\big{)}^{*}+\big{(}\mathbf{10}^{pq\,T}_{A}\,{\mathcal{C}}^{-1}\,\,5^{\dagger}_{p\,H}\big{)}\,(\mathbf{10}_{B}^{sq}\,5^{\dagger}_{s\,H}\big{)}^{*}$
$\displaystyle+$
$\displaystyle\big{(}\mathbf{\overline{5}}^{T}_{p\,A}\,{\mathcal{C}}^{-1}\,\,10^{qr}_{H}\big{)}\,\big{(}\mathbf{\overline{5}}_{p\,B}\,\,10^{qr}_{H}\big{)}^{*}+\frac{1}{288}\,\big{(}\varepsilon_{pqtuv}\,\mathbf{10}^{pq\,T}_{A}\,\,{\mathcal{C}}^{-1}\,\,10_{H}^{rs}\big{)}\,\big{(}\varepsilon_{wxtuv}\,\mathbf{10}^{wx}_{B}\,10_{H}^{rs}\big{)}^{*}$
$\displaystyle+$
$\displaystyle\frac{1}{288}\,\big{(}\varepsilon_{pqtuv}\,\mathbf{10}^{rs\,T}_{A}\,\,{\mathcal{C}}^{-1}\,\,10_{H}^{pq}\big{)}\,\big{(}\varepsilon_{wxtuv}\,\mathbf{10}^{rs}_{B}\,10_{H}^{wx}\big{)}^{*}\Big{]}\;+\;\rm{H.c.}$

We have, so far, computed the couplings where two alike $16$-plets transform.
Now, we consider the case when one $16$-plet couples to its conjugate, and the
handful of resulting outcomes are possible and mentioned in eq. (II). The
irreducible representations resulting from the decomposition of $16$ and its
conjugate are called the adjoint representations, which are real
representations. Unlike the cases of $10$, $120$, and $126$, only the
fermionic representations of these adjoint irreps will generate the expression
presented in eq. (5). The interaction of $\mathbf{16}$-plet with $\mathbf{45}$
is parameterised below Nath and Syed (2001b).

$\displaystyle{}-\mathcal{L}^{45}_{Y}$ $\displaystyle\supset$
$\displaystyle\tilde{k}_{AB}\,\mathbf{16}^{T}_{A}\,\,{\mathcal{C}}^{-1}\,\,\mathbf{45}_{B}\,16^{\dagger}_{H}$
(16) $\displaystyle=$
$\displaystyle\frac{i}{\sqrt{2}}\,\tilde{k}_{AB}\,\Bigg{[}\sqrt{5}\Big{(}\frac{3}{5}\mathbf{\overline{5}}_{p\,A}^{T}\,\,{\mathcal{C}}^{-1}\,\,5^{p}_{H}+\frac{1}{10}\,\mathbf{10}_{A}^{pq\,T}\,\,{\mathcal{C}}^{-1}\,\,10^{\dagger}_{pq\,H}-\mathbf{1}_{A}^{T}\,\,{\mathcal{C}}^{-1}\,\,1^{\dagger}_{H}\Big{)}\tilde{\mathbf{1}}_{B}$
$\displaystyle+$
$\displaystyle\Big{(}-\mathbf{10}_{A}^{st\,T}\,\,{\mathcal{C}}^{-1}\,\,1_{H}+\frac{1}{2}\varepsilon^{pqrst}\mathbf{\overline{5}}_{p\,A}^{T}\,\,{\mathcal{C}}^{-1}\,\,10^{\dagger}_{qr\,H}\Big{)}\big{(}\tilde{\mathbf{10}}\big{)}^{*}_{st\,B}$
$\displaystyle+$
$\displaystyle\Big{(}-\mathbf{1}^{T}\,\,{\mathcal{C}}^{-1}\,\,10^{\dagger}_{st\,H}+\frac{1}{2}\varepsilon_{pqrst}\,\mathbf{10}^{pq\,T}_{A}\,\,{\mathcal{C}}^{-1}\,\,5^{r}_{H}\Big{)}\tilde{\mathbf{10}}^{st}_{B}$
$\displaystyle+$ $\displaystyle
2\Big{(}\mathbf{10}^{pr\,T}_{A}\,\,{\mathcal{C}}^{-1}\,\,10^{\dagger}_{rq}-\mathbf{\overline{5}}^{T}_{q\,A}\,\,{\mathcal{C}}^{-1}\,\,5^{p}\big{)}\,\tilde{\mathbf{24}}^{q}_{p\,B}\Bigg{]}\;+\;\rm{H.c.}$

Using the decomposition given in eq. (16), the effective coupling between two
$\mathbf{16}$ and $16^{\dagger}_{H}$ mediated by $\mathbf{45}$ is computed as
follows

$\displaystyle{}\mathcal{L}_{\rm{NR}}$ $\displaystyle\supset$
$\displaystyle-\frac{\tilde{k}_{AB}}{\Lambda}\,\big{(}\mathbf{16}^{T}_{A}\,\,{\mathcal{C}}^{-1}\,\,16^{\dagger}_{H}\big{)}_{\mathbf{45}}\,\big{(}\mathbf{16}_{B}\,16^{\dagger}_{H}\big{)}_{\mathbf{45}}$
(17) $\displaystyle=$
$\displaystyle-\frac{\tilde{k}_{AB}}{2\Lambda}\Big{[}5\,\big{(}-\mathbf{1}_{A}^{T}\,\,{\mathcal{C}}^{-1}\,\,1_{H}+\frac{1}{10}\,\mathbf{10}_{A}^{pq\,T}\,\,{\mathcal{C}}^{-1}\,\,10^{\dagger}_{pq\,H}+\frac{3}{5}\,\mathbf{\overline{5}}^{T}_{p\,A}\,\,{\mathcal{C}}^{-1}\,\,5^{p}_{H}\big{)}\,\big{(}-\mathbf{1}_{B}\,\,1_{H}$
$\displaystyle+$
$\displaystyle\frac{1}{10}\,\mathbf{10}_{B}^{st}\,\,\,10^{\dagger}_{st\,H}+\frac{3}{5}\,\mathbf{\overline{5}}_{t\,B}\,\,\,5^{t}_{H}\big{)}+4\big{(}\mathbf{10}_{A}^{pr\,T}\,\,{\mathcal{C}}^{-1}\,10^{\dagger}_{rq\,H}\big{)}\,\big{(}\mathbf{10}^{qn}_{B}\,10^{\dagger}_{np\,H}\big{)}$
$\displaystyle-$ $\displaystyle
4\big{(}\mathbf{10}_{A}^{pr\,T}\,\,{\mathcal{C}}^{-1}\,10^{\dagger}_{rq\,H}\big{)}\,\big{(}\mathbf{\overline{5}}_{p\,B}\,5^{q}_{H}\big{)}-4\big{(}\mathbf{\overline{5}}^{T}_{p\,A}\,\,{\mathcal{C}}^{-1}\,\,5^{q}_{H}\big{)}\,\big{(}\mathbf{10}_{B}^{pr}\,\,{\mathcal{C}}^{-1}\,10^{\dagger}_{rq\,H}\big{)}$
$\displaystyle+$ $\displaystyle
4\big{(}\mathbf{\overline{5}}^{T}_{p\,A}\,\,{\mathcal{C}}^{-1}\,\,5^{q}_{H}\big{)}\,\big{(}\mathbf{\overline{5}}_{q\,B}\,\,{\mathcal{C}}^{-1}\,\,5^{p}_{H}\big{)}$
$\displaystyle+$
$\displaystyle\big{(}-\mathbf{10}_{A}^{pq\,T}\,\,{\mathcal{C}}^{-1}\,\,1_{H}+\frac{1}{2}\,\varepsilon^{rstpq}\,\mathbf{\overline{5}}_{A\,r}^{T}\,\,{\mathcal{C}}^{-1}\,\,10^{\dagger}_{st\,H}\big{)}\,\big{(}-\mathbf{1}_{B}\,10^{\dagger}_{pq\,H}+\frac{1}{2}\varepsilon_{uvwpq}\mathbf{10}^{uv}\,5^{w}_{H}\big{)}\Big{]}$
$\displaystyle+$ $\displaystyle\rm{H.c.}$

Analogous to eq. (17), one can compute the coupling of $\mathbf{16}$ and
$16^{\dagger}_{H}$ with $\mathbf{1}$ and $\mathbf{210}$ and obtain the
effectively generated couplings mediated by them which are shown below.

$\displaystyle{}\mathcal{L}_{NR}$ $\displaystyle\supset$
$\displaystyle\frac{k_{AB}}{\Lambda}\,\big{(}\mathbf{16}^{T}_{A}\,\,{\mathcal{C}}^{-1}\,\,16^{\dagger}_{H}\big{)}_{\mathbf{1}}\,\big{(}\mathbf{16}_{B}\,16^{\dagger}_{H}\big{)}_{\mathbf{1}}+\frac{\bar{k}_{AB}}{\Lambda}\,\big{(}\mathbf{16}^{T}_{A}\,\,{\mathcal{C}}^{-1}\,\,16^{\dagger}_{H}\big{)}_{\mathbf{210}}\,\big{(}\mathbf{16}_{B}\,16^{\dagger}_{H}\big{)}_{\mathbf{210}}\;+\;\rm{H.c.}$
(18) $\displaystyle=$
$\displaystyle-\frac{k_{AB}}{\Lambda}\,\Big{[}\big{(}\mathbf{1}_{A}^{T}\,\,{\mathcal{C}}^{-1}\,\,1_{H}+\frac{1}{2}\,\mathbf{10}_{A}^{pq\,T}\,\,{\mathcal{C}}^{-1}\,\,10^{\dagger}_{pq\,H}-\mathbf{\overline{5}}_{p\,A}\,\,{\mathcal{C}}^{-1}\,\,5^{p}_{H}\big{)}\,\big{(}\mathbf{1}_{A}\,\,1_{H}$
$\displaystyle-$
$\displaystyle\frac{1}{2}\,\mathbf{10}_{B}^{pq}\,\,10^{\dagger}_{pq\,H}-\mathbf{\overline{5}}_{p\,B}\,\,\,5^{p}_{H}\big{)}\Big{]}$
$\displaystyle-$
$\displaystyle\frac{2\bar{k}_{AB}}{3}\,\Big{[}\frac{3}{2}\big{(}\mathbf{\overline{5}}^{T}_{p\,A}\,\,{\mathcal{C}}^{-1}\,\,5^{q}+\frac{1}{3}\mathbf{10}^{qr\,T}_{A}\,\,{\mathcal{C}}^{-1}\,\,10^{\dagger}_{rp\,H}\big{)}\,\big{(}\mathbf{\overline{5}}_{q\,B}\,5^{p}+\frac{1}{3}\mathbf{10}^{pn}_{B}\,10^{\dagger}_{nq\,H}\big{)}$
$\displaystyle+$
$\displaystyle\frac{5}{8}\big{(}\mathbf{1}_{A}^{T}\,\,{\mathcal{C}}^{-1}\,\,1_{H}\frac{1}{10}\mathbf{10}^{pq\,T}\,\,{\mathcal{C}}^{-1}\,\,10^{\dagger}_{pq\,H}+\frac{1}{5}\mathbf{\overline{5}}^{T}_{p\,A}\,\,{\mathcal{C}}^{-1}\,\,5^{p}_{H}\big{)}\,\big{(}\mathbf{1}_{B}\,1_{H}+\frac{1}{10}\mathbf{10}^{st}_{B}\,10^{\dagger}_{st\,H}+\frac{1}{5}\mathbf{\overline{5}}_{s\,B}\,5^{s}_{H}\big{)}$
$\displaystyle+$
$\displaystyle\frac{3}{4}\big{(}\mathbf{10}_{A}^{pq\,T}\,\,{\mathcal{C}}^{-1}\,\,1_{H}+\frac{1}{6}\,\varepsilon^{rstpq}\,\mathbf{\overline{5}}_{A\,r}^{T}\,\,{\mathcal{C}}^{-1}\,\,10^{\dagger}_{st\,H}\big{)}\,\big{(}\mathbf{1}_{B}\,10^{\dagger}_{pq\,H}+\frac{1}{6}\varepsilon_{uvwpq}\mathbf{10}^{uv}\,5^{w}_{H}\big{)}$
$\displaystyle+$
$\displaystyle\frac{1}{36}\,\big{(}\varepsilon_{pqrst}\,\mathbf{10}^{in\,T}_{A}\,\,{\mathcal{C}}^{-1}\,\,5^{q}_{H}\big{)}\,\big{(}\varepsilon^{uvrst}\,\mathbf{\overline{5}}_{u\,B}\,10^{\dagger}_{vn\,H}\big{)}+\frac{1}{16}\big{(}\mathbf{10}^{pq\,T}_{A}\,\,{\mathcal{C}}^{-1}\,\,10^{\dagger}_{st\,H}\big{)}\,\big{(}\mathbf{10}^{st}_{B}\,10^{\dagger}_{pq\,H}\big{)}$
$\displaystyle+$
$\displaystyle\big{(}\mathbf{\overline{5}}_{A}^{T}\,{\mathcal{C}}^{-1}\,\,1_{H}\big{)}\,\big{(}\mathbf{1}_{B}\,5^{i}_{H}\big{)}\Big{]}\;+\;\rm{H.c.}$

The $SU(5)$ invariant expressions given in eqs. (10, 11, 13, 14, 15, 17, and
18) depict the different ways of arriving at the term given in eq. (5). It
demonstrates that incorporating the $SO(10)$ invariant combination of two
$\mathbf{16}$-plets and $16_{H}$, along with its conjugate partner, at the
leading non-renormalisable level, can be decomposed into several $SU(5)$
invariant expressions. The latter arises from the various possibilities of
contracting the same $SO(10)$ invariant term. When further decomposed into
expressions respecting the SM gauge symmetry, they yield the couplings of a
pair of scalars from Table(1) with the SM fermions. The resulting dimension-5
coupling can have various phenomenological implications, ranging from
contributing to fermion masses (charged and neutral) to baryon number-
violating processes (nucleon decay).

We intend to constrain the spectrum of scalars residing in $16_{\rm{H}}$ from
nucleon decay. The respective scalar must have leptoquark and diquark
couplings to mediate proton decay. In Table (2), we gather the couplings of
respective pairs of scalars with SM fermions obtained after the decomposition
of eqs. (10, 11, 13, 14, 15, 17, and 18). Furthermore, we are interested in
pairs of scalars with both diquark and leptoquark vertices. Therefore, we only
consider the coupling of scalar pairs capable of inducing proton decay, while
suppressing the $SU(3)$, $SU(2)$, family indices.

Pair Of Scalar |  Diquark |  Leptoquark
---|---|---
$\sigma-\sigma$ | ✗ | ✗
$\sigma-T$ | $q\,q;u^{C}\,d^{C}$ | $q\,l;u^{C}\,e^{C}$
$\sigma-H$ | $q\,u^{C};\,q\,d^{C}$ | ✗
$\sigma-\Delta$ | ✗ | $d^{C}\,l$
$\sigma-\Theta$ | $d^{C}\,d^{C}$ | ✗
$\sigma-t$ | ✗ | ✗
$T-T$ | $d^{C}\,d^{C}$ | ✗
$T-H$ | $q\,u^{C\,*}$ | $q\,e^{C\,*}$
$T-\Delta$ | $q\,d^{C}$ | ✗
$T-\Theta$ | $q\,q;u^{C}\,d^{C}$ | $q\,l;u^{C}\,e^{C}$
$T-t$ | ✗ | $d^{C}\,e^{C}$
$H-H$ |  $q\,q$; $d^{C}\,d^{C}$ | ✗
$H-\Delta$ | $q\,q;u^{C}\,d^{C}$ | $q\,l;u^{C}\,e^{C}$
$H-\Theta$ | ✗ | $l\,u^{C}$
$H-t$ | $q\,d^{C}$ | ✗
$\Delta-\Delta$ | $q\,q;\;u^{C}\,d^{C}$ | $q\,l;\;u^{C}\,e^{C}$
$\Delta-\Theta$ | $q\,u^{C\,*}$ | $q\,e^{C\,*}$
$\Delta-t$ | $q\,u^{C\,*}$ | $q\,e^{C\,*}$
$\Theta-\Theta$ |  $u^{C}\,d^{C}$; $d^{C}\,d^{C}$ | ✗
$\Theta-t$ | $q\,q;u^{C}\,d^{C}$ | $q\,l;u^{C}\,e^{C}$
$t-t$ | ✗ | ✗
Table 2: Coupling of different pair of scalars with the SM fermions. The “✗”
represent the respective coupling (diquark or leptoquark) is absent.

A few comments are in order regarding table (2).

  * •

The depiction of a pair of scalars in the table 2 stands for all possible ways
the scalars can appear in the eqs. (10, 11, 13, 14, 15, 17, and 18), so
$\sigma-T$ stands for $\sigma-T$, $\sigma-T^{*}$, $\sigma^{*}-T$ and
$\sigma^{*}-T^{*}$.

  * •

A scalar coupling with its conjugate partner, in principle, couples with any
SM fermions with their conjugate partner; hence, those couplings are not
mentioned in the table (2).

  * •

The couplings of $\nu^{C}$ in the table (2) have been omitted as they cannot
contribute to proton decay at dimension 8.

  * •

Out of twenty different pairs of scalars, only eight have the leptoquark and
diquark coupling, hence can induce proton decay. These different pairs of
scalars are $\sigma-T$, $T-H$, $T-\Delta$, $H-\Delta$, $\Delta-\Delta$,
$\Delta-\Theta$, $\Delta-t$ and $\Theta-t$.

  * •

It is apparent that $\Delta$ can couple to any scalar mentioned in Table (1),
and can yield a diquark and leptoquark vertices.

After summarizing the diquark and leptoquark vertices of different pairs of
scalars resulting from the decomposition of various computed $SU(5)$ invariant
expressions, we further focus on their explicit contributions to the various
effective nucleon decay operators in terms of their Wilson coefficients.

##  III Dimension Six Effective Operators

The standard model gauge symmetry preserving and $B-L$ conserving (or $B+L$
violating) effective mass dimension-six ($D=6$) operators capable of inducing
nucleon (free proton or a bound neutron) decay are shown below Weinberg
(1979); Wilczek and Zee (1979); Sakai and Yanagida (1982). As they conserve
$B-L$, these operators are responsible for two-body nucleon decays, in which
the nucleon decays into either charged anti-leptons or neutral ones,
accompanied by mesons. The vertices leading to such operators are forbidden in
the SM due to the apparent conservation of baryon and lepton numbers.
Therefore, they result from the integration of degrees of freedom of particles
(scalar or vector) which are not part of the SM.

$\displaystyle\mathcal{O}_{1}$ $\displaystyle=$ $\displaystyle
c_{1}[A,B,C,D]\,\varepsilon_{\alpha\beta\gamma}\,\varepsilon_{gh}\,\varepsilon_{ab}\,\Big{(}q^{g\alpha\,T}_{A}\,\,{\mathcal{C}}^{-1}\,q_{B}^{h\,\beta}\Big{)}\,\Big{(}q^{a\,\gamma\,T}_{C}\,\,{\mathcal{C}}^{-1}\,l^{b}_{D}\Big{)},$
$\displaystyle\mathcal{O}_{2}$ $\displaystyle=$ $\displaystyle
c_{2}[A,B,C,D]\,\varepsilon_{\alpha\beta\gamma}\,\varepsilon_{gh}\,\Big{(}q^{g\alpha\,T}_{A}\,\,{\mathcal{C}}^{-1}\,\,q_{B}^{h\,\beta}\Big{)}\,\Big{(}u^{C\,T}_{\,\gamma\,C}\,\,{\mathcal{C}}^{-1}\,e^{C}_{D}\Big{)}^{*},$
$\displaystyle\mathcal{O}_{3}$ $\displaystyle=$ $\displaystyle
c_{3}[A,B,C,D]\,\varepsilon^{\alpha\beta\gamma}\,\varepsilon^{ab}\,\Big{(}u^{C\,T}_{\alpha\,A}\,{\mathcal{C}}^{-1}\,\,d^{C}_{\beta\,B}\Big{)}\,\Big{(}q^{a\,\gamma\,T}_{C}\,{\mathcal{C}}^{-1}\,\,l^{b}_{D}\Big{)}^{*},$
$\displaystyle\mathcal{O}_{4}$ $\displaystyle=$ $\displaystyle
c_{4}[A,B,C,D]\,\varepsilon^{\alpha\beta\gamma}\,\Big{(}u^{C\,T}_{\alpha\,A}\,{\mathcal{C}}^{-1}\,\,d^{C}_{\beta\,B}\Big{)}\,\Big{(}u^{C\,T}_{\,\gamma\,C}\,\,{\mathcal{C}}^{-1}\,e^{C}_{D}\Big{)}.$
(19)

The coefficients $c_{1,2,3,4}$ represent various Wilson coefficients
associated with the newly introduced degrees of freedom and are the effective
strength of the respective operator. Typically, these coefficients depend on
the Yukawa couplings and are suppressed by the mass of the mediator.
Additionally, $A$, $B$, $C$, and $D$ represent the generation indices of SM
fermions involved in the operator. Only specific choices of $A$, $B$, $C$, and
$D$ are permitted for tree-level nucleon decays. Specifically, only up-quarks
can participate in the operator and for down-quarks and charged leptons, only
fermions up to the second generation are allowed, while all three generations
are allowed for neutral leptons.

In the present scenario, the Lagrangian contains mass dimension-five terms
that can generate the $D=6$ operators, mentioned in eq. (III), through two
ways: 1) Integrating out a pair of scalars resulting in $D=8$ operators, with
another pair of scalars simultaneously acquiring a vacuum expectation value
(vev), resulting in an effective $D=6$ operator (cf. left diagram of Fig.
(1)); 2) The two pairs of scalars effectively generating $D=6$ operators via
loop mediation (cf. right diagram of Fig. (1)). These $D=6$ effective
operators can arise from specific terms of eqs. (10, 11, 13, 14, 15, 17, and
18), or they can result from a linear combination of various terms.
Integrating various scalars does not necessarily yield all the four operators
mentioned in eq. (III). Moreover, there exists the possibility of other $B-L$
conserving effective operators at $D=8$, in addition to the operators
mentioned in eq.(III) that can lead to proton decays involving multimeson
modes Helo  _et al._ (2019); Heeck and Takhistov (2020). These decays are
inherently phase space suppressed compared to the two-body decays, and hence,
are not explored in this analysis.

Below, we evaluate the contribution to the operators given in Eq. (III)
mediated by various pairs of scalars. While deriving the contributions to the
operators arising from different pairs of scalars, we have assumed only one
particular $SO(10)$ irrep can mediate the decomposition of term given in eq.
(5) at a time. However, in principle, similar operators arising from the same
pair of scalars from the decomposition of different $\text{SO}(10)$-mediated
irreps can mix, a possibility we remain ignorant of.

Figure 1: Tree and loop level proton decay topologies generated by the pair
$\sigma-T$, drawn using Harlander  _et al._ (2020).

We focus on the contribution of the $\sigma-T$ pair, capable of inducing
proton decay by coupling to both leptoquark and diquark (cf. Table (2)). This
pair induces proton decay at both tree and loop levels ( cf. Fig. (1)).
Integrating out the degrees of freedom of $T$ results in a $D=8$ operator. As
$\sigma$ is a singlet under SM gauge symmetry, its vev reduces the $D=8$
effective operator to one of the $D=6$ operators in eq. (III). Moreover, this
pair also mediates proton decay at the loop level with $\sigma$ and $T$ as
scalars propagating inside the loop. The contribution to the operators,
$\mathcal{O}_{1,2,3,4}$, in flavour basis mediated by $\sigma-T$ is as
follows.

$\displaystyle c_{1}\big{[}A,B,C,D\big{]}$ $\displaystyle=$
$\displaystyle\bigg{(}\frac{\langle\sigma\rangle^{2}}{\Lambda^{2}\,M_{T}^{2}}+\frac{D\big{(}M^{2}_{T},M^{2}_{\sigma}\big{)}}{\Lambda^{2}}\bigg{)}\,\Big{[}32\bar{h}_{AB}\,h_{CD}\Big{]},$
$\displaystyle c_{2}\big{[}A,B,C,D\big{]}$ $\displaystyle=$
$\displaystyle\bigg{(}\frac{\langle\sigma\rangle^{2}}{\Lambda^{2}\,M_{T}^{2}}+\frac{D\big{(}M^{2}_{T},M^{2}_{\sigma}\big{)}}{\Lambda^{2}}\bigg{)}\,\Big{[}32\,\bar{h}_{AB}\,\bar{h}^{*}_{CD}-\frac{8}{2025}f_{AB}\,f^{*}_{CD}$
$\displaystyle+$ $\displaystyle
2\,\tilde{k}_{AB}\,\tilde{k}^{*}_{CD}+\frac{16}{9}\bar{k}_{AB}\,\big{(}\bar{k}^{*}_{CD}+\bar{k}^{*}_{DC}\big{)}\Big{]},$
$\displaystyle c_{3}\big{[}A,B,C,D\big{]}$ $\displaystyle+$
$\displaystyle\bigg{(}\frac{\langle\sigma\rangle^{2}}{\Lambda^{2}\,M_{T}^{2}}+\frac{D\big{(}M^{2}_{T},M^{2}_{\sigma}\big{)}}{\Lambda^{2}}\bigg{)}\,\Big{[}64\,h_{AB}\,h^{*}_{CD}+36\,\bar{h}_{AB}\,\bar{h}^{*}_{CD}+\frac{128}{9}\,g_{AB}\,g_{CD}\Big{]},$
$\displaystyle c_{4}\big{[}A,B,C,D\big{]}$ $\displaystyle=$
$\displaystyle\bigg{(}\frac{\langle\sigma\rangle^{2}}{\Lambda^{2}\,M_{T}^{2}}+\frac{D\big{(}M^{2}_{T},M^{2}_{\sigma}\big{)}}{\Lambda^{2}}\bigg{)}\,\big{(}64\
h_{AB}\,\bar{h}_{CD}\big{)}.$ (20)

where, $M_{T}$ and $M_{\sigma}$ are the masses of Triplet and $\sigma$
respectively. The expression inside the parenthesis of eq. (III) is comprised
of two terms: the first term corresponds to the contribution at tree-level,
while the second term arises from the contribution at loop-level. In general,
$\big{\langle}\sigma\big{\rangle}\neq M_{\sigma}$ and the exact relation
between $\sigma$ and $M_{\sigma}$ depend on the contents of the scalar
potential. Moreover, these effective strengths of the various operators depend
on the Yukawa couplings and the mass of the mediators. It is evident from the
above coefficients in eq. (III), $\sigma-T$ can generate all the four
operators mentioned in eq. (III). Despite $\sigma$ being charged under $B-L$,
the resulting operators when $\sigma$ acquires vev conserve $B-L$, as is
expected from the corollary that operators having $B-L=0$ typically appear at
even mass dimensions Kobach (2016); Helset and Kobach (2020).

The loop contribution in eq. (III) depend upon the loop integral, whose
definition is as follows.

$\displaystyle D\big{(}M_{1}^{2},M_{2}^{2}\big{)}$ $\displaystyle\equiv$
$\displaystyle-\frac{1}{16\pi^{2}}\Bigg{(}\frac{M_{1}^{2}\log\Big{(}\frac{M_{1}^{2}}{\mu^{2}}\Big{)}-M_{2}^{2}\log\Big{(}\frac{M_{2}^{2}}{\mu^{2}}\Big{)}}{M_{1}^{2}-M_{2}^{2}}-1\Bigg{)}.$
(21)

The loop function given in the above expression in eq. (21) is symmetric under
the exchange of $M_{1}$ and $M_{2}$, where $M_{1}$ and $M_{2}$ are the masses
of scalars propagating inside the loop. Furthermore, the renormalisation
scale, $\mu$, can be chosen as $1$ GeV, corresponding to the scale at which
the proton at rest can decay. Ideally, all parameters contributing to proton
decay should be renormalised down to this scale ($1$ GeV).

Further, the next pair capable of having diquark and leptoquark couplings is
$T-\Theta$ and the contribution to the effective $D=6$ operators upon its
mediation is computed below.

$\displaystyle c_{1}\big{[}A,B,C,D\big{]}$ $\displaystyle=$
$\displaystyle\frac{1}{\Lambda^{2}}\,D\big{(}M_{T}^{2},M^{2}_{\Theta}\big{)}\,\big{(}-32\,h_{AB}\,h_{CD}\big{)}$
$\displaystyle c_{2}\big{[}A,B,C,D\big{]}$ $\displaystyle=$
$\displaystyle\frac{1}{\Lambda^{2}}D\big{(}M_{T}^{2},M^{2}_{\Theta}\big{)}\,\Big{[}-64\,h_{AB}\,\bar{h}^{*}_{CD}-8\,\tilde{h}_{AB}\,\tilde{h}^{*}_{CD}\Big{]}$
$\displaystyle c_{3}\big{[}A,B,C,D\big{]}$ $\displaystyle=$
$\displaystyle\frac{1}{\Lambda^{2}}D\big{(}M_{T}^{2},M^{2}_{\Theta}\big{)}\,\Big{[}\frac{-64}{9}\,\bar{g}_{AB}\,\bar{g}^{*}_{CD}-2\big{(}\tilde{k}_{AB}+\frac{3}{10}\tilde{k}_{BA}\big{)}\,\big{(}\frac{1}{2}\tilde{k}^{*}_{CD}-\tilde{k}^{*}_{DC}\big{)}$
$\displaystyle-$
$\displaystyle\,\frac{2}{3}\,\left(\bar{k}_{AB}-\frac{5}{4}\bar{k}_{BA}\right)\,\left(\bar{k}^{*}_{CD}-4\bar{k}_{DC}\right)\Big{]}$
$\displaystyle c_{4}\big{[}A,B,C,D\big{]}$ $\displaystyle=$
$\displaystyle\frac{1}{\Lambda^{2}}D\big{(}M_{T}^{2},M^{2}_{\Theta}\big{)}\,\big{(}-\frac{64}{9}\,\bar{g}_{AB}\,g_{CD}\big{)}$
(22)

here, $M_{T},\,\text{and}\,M_{\Theta}$ are the masses of $T$ and $\Theta$
respectively. None of the scalars can assume a vev in this case, as it would
violate $SU_{3}^{\rm{C}}\times U_{1}^{\rm{em}}$ symmetry. Consequently, this
pair can only induce proton decay at the loop level and contribute to all four
operators mentioned in eq. (III).

Next, we consider the case when the pair $T-H$ mediates the proton decay.

$\displaystyle{}c_{2}[A,B,C,D]$ $\displaystyle=$
$\displaystyle-64\,\bigg{(}\frac{\langle
H\rangle^{2}}{\Lambda^{2}\,M_{T}^{2}}+\frac{D\big{(}M^{2}_{H},M^{2}_{T}\big{)}}{\Lambda^{2}}\bigg{)}\,\big{(}\hat{h}_{AC}+\hat{h}^{*}_{CA}\big{)}\,\big{(}\hat{h}_{BD}+\hat{h}^{*}_{DB}\big{)}$
(23)

$M_{H}$ being the mass of the $H$. While $H$ typically transforms as the SM
Higgs, however, it is charged under $B-L$ (cf. Table (1)), limiting $H$ from
assuming a large vev. This pair can mediate tree and loop-level proton decays,
contributing to effective $D=6$ operators. However, it can generate only a
single $D=6$ effective operator, ${\mathcal{O}}_{2}$. Additionally, to arrive
at the above expression of eq. (23), Fierz transformation rules for two-
component spinors have been used Dreiner  _et al._ (2010).

Next, we consider the case when the pair $H-\Delta$ mediates the proton decay.
Its contribution to effective $D=6$ operators is shown below.

$\displaystyle c_{1}\big{[}A,B,C,D\big{]}$ $\displaystyle=$
$\displaystyle\Big{(}\frac{\langle
H\rangle^{2}}{\Lambda^{2}\,M^{2}_{\Delta}}+\frac{D\big{(}M^{2}_{H},M^{2}_{\Delta}\big{)}}{\Lambda^{2}}\Big{)}\,\big{(}32\,h_{AB}\,\bar{h}_{CD}-\frac{64\sqrt{2}}{9}\,g_{AB}\,\bar{g}_{CD}\big{)}$
$\displaystyle c_{2}\big{[}A,B,C,D\big{]}$ $\displaystyle=$
$\displaystyle\Big{(}\frac{\langle
H\rangle^{2}}{\Lambda^{2}\,M^{2}_{\Delta}}+\frac{D\big{(}M^{2}_{H},M^{2}_{\Delta}\big{)}}{\Lambda^{2}}\Big{)}\,\big{(}16\,h_{AB}\,h^{*}_{CD}+\frac{16}{9}g_{AB}\,g^{*}_{CD}+32\,\tilde{h}_{AB}\,\tilde{h}^{*}_{CD}\big{)}$
$\displaystyle c_{3}\big{[}A,B,C,D\big{]}$ $\displaystyle=$
$\displaystyle\Big{(}\frac{\langle
H\rangle^{2}}{\Lambda^{2}\,M^{2}_{\Delta}}+\frac{D\big{(}M^{2}_{H},M^{2}_{\Delta}\big{)}}{\Lambda^{2}}\Big{)}\,\Bigg{[}32\,\bar{h}_{AB}\,\bar{h}^{*}_{CD}+\frac{64}{9}\,\bar{g}_{AB}\,g^{*}_{CD}\big{)}$
$\displaystyle+$
$\displaystyle\left(\tilde{k}_{AB}-2\sqrt{2}\tilde{k}_{BA}\right)\,\left(\left(\frac{1}{\sqrt{2}}-\frac{3}{20}\right)\tilde{k}^{*}_{CD}+\left(\sqrt{2}-\frac{3}{20}\right)\tilde{k}^{*}_{DC}\right)$
$\displaystyle-$ $\displaystyle
2\left(\frac{1}{\sqrt{2}}\bar{k}_{AB}-\frac{2\sqrt{2}}{3}\bar{k}_{BA}\right)\left(\sqrt{2}\big{(}k_{CD}+2k_{DC}\big{)}-\big{(}\frac{\sqrt{2}}{48}\bar{k}_{DC}+\frac{4\sqrt{2}}{3}\bar{k}_{CD}\big{)}\right)\Bigg{]}$
$\displaystyle c_{4}\big{[}A,B,C,D\big{]}$ $\displaystyle=$
$\displaystyle\Big{(}\frac{\langle
H\rangle^{2}}{\Lambda^{2}\,M^{2}_{\Delta}}+\frac{D\big{(}M^{2}_{H},M^{2}_{\Delta}\big{)}}{\Lambda^{2}}\Big{)}\,\big{(}32\bar{h}_{AB}\,h_{CD}-\frac{32}{9}\,g_{AB}\,g_{CD}\big{)}$
(24)

$M_{H}$, and $M_{\Delta}$ being the respective masses of $H$ and $\Delta$.
Analogous to the previous case, this pair also mediate proton decays at tree
and loop levels and contributes to all four effective $D=6$ proton decay
operators.

Further, the pair $\Delta-\Delta$ induces only loop-level proton decay, as
shown below.

$\displaystyle c_{1}\big{[}A,B,C,D\big{]}$ $\displaystyle=$
$\displaystyle\frac{1}{\Lambda^{2}}\,D\big{(}M_{\Delta}^{2},M^{2}_{\Delta}\big{)}\,\big{(}-64\bar{h}_{AB}\,h_{CD}+\frac{10}{9}\,\bar{g}_{AB}\,g_{CD}\big{)}$
$\displaystyle c_{2}\big{[}A,B,C,D\big{]}$ $\displaystyle=$
$\displaystyle\frac{1}{\Lambda^{2}}D\big{(}M_{\Delta}^{2},M^{2}_{\Delta}\big{)}\,\Big{[}-16\sqrt{2}\,\bar{h}_{AB}\,\bar{h}^{*}_{CD}+\frac{10}{9}\,\bar{g}_{AB}\,g^{*}_{CD}+\frac{8}{405}\,f_{AB}\,f^{*}_{CD}\big{)}$
$\displaystyle+$
$\displaystyle\frac{36\sqrt{2}}{5}\,\tilde{k}_{AB}\,\tilde{k}^{*}_{DC}+64\left(\frac{1}{\sqrt{2}}-\frac{16}{3}\right)\bar{k}_{AB}\,\bar{k}^{*}_{DC}\bigg{]}$
$\displaystyle c_{3}\big{[}A,B,C,D\big{]}$ $\displaystyle=$
$\displaystyle\frac{1}{\Lambda^{2}}D\big{(}M_{\Delta}^{2},M^{2}_{\Delta}\big{)}\,\big{(}64h_{AB}\,h^{*}_{CD}+\frac{8}{3}\,g_{AB}\,\bar{g}_{CD}-8g_{AB}\,g^{*}_{CD}+\frac{8}{9}\,\tilde{g}_{AB}\,\tilde{g}^{*}_{CD}\big{)}$
$\displaystyle c_{4}\big{[}A,B,C,D\big{]}$ $\displaystyle=$
$\displaystyle\frac{8}{\Lambda^{2}}D\big{(}M_{\Delta}^{2},M^{2}_{\Delta}\big{)}\,\big{(}16h_{AB}\,\bar{h}_{CD}-\frac{8}{3}\,g_{AB}\,\bar{g}_{CD}\big{)}$
(25)

This particular pair contributes to all the effective $D=6$ operators.

The pair $\Delta-\Theta$ mediates proton decay at loop level and contributes
to only operator $\mathcal{O}_{2}$.

$\displaystyle{}c_{2}[A,B,C,D]$ $\displaystyle=$
$\displaystyle\frac{D\big{(}M^{2}_{\Delta},M^{2}_{\Theta}\big{)}}{\Lambda^{2}}\,\Big{[}\big{(}\hat{h}_{AD}+\hat{h}^{*}_{DA}\big{)}\,\big{(}\hat{h}_{CB}+\hat{h}^{*}_{BC}\big{)}+\frac{16\sqrt{2}}{9}\hat{g}_{AC}\big{(}\hat{g}_{BD}+\hat{g}^{*}_{BD}\big{)}$
(26) $\displaystyle+$
$\displaystyle\frac{16\sqrt{2}}{45}\,\hat{f}_{AD}\,\hat{f}_{BC}\Big{]}$

Similarly, the pair $\Delta-t$ mediates proton decay at loop only and leads to
only operator ${\mathcal{O}_{2}}$.

$\displaystyle{}c_{2}[A,B,C,D]$ $\displaystyle=$
$\displaystyle\frac{D\big{(}M^{2}_{\Delta},M^{2}_{t}\big{)}}{\Lambda^{2}}\,\Big{[}\frac{8\sqrt{2}}{9}\hat{g}_{AC}\big{(}\hat{g}_{BD}+\hat{g}^{*}_{BD}\big{)}-\frac{8\sqrt{2}}{45}\,\hat{f}^{*}_{DA}\,\hat{f}^{*}_{BC}\Big{]}$
(27)

where, $M_{t}$ is the mass of $t$.

Finally, contribution to operators, $\mathcal{O}_{1,2,3,4}$, is shown below
when the proton decay is induced by the pair $\Theta-t$.

$\displaystyle c_{1}\big{[}A,B,C,D\big{]}$ $\displaystyle=$
$\displaystyle\frac{1}{\Lambda^{2}}\,D\big{(}M_{\Theta}^{2},M^{2}_{t}\big{)}\,\big{(}16\bar{h}_{AB}\,h_{CD}+\frac{10}{9}\,\bar{g}_{AB}\,g_{CD}\big{)}$
$\displaystyle c_{2}\big{[}A,B,C,D\big{]}$ $\displaystyle=$
$\displaystyle\frac{1}{\Lambda^{2}}D\big{(}M_{\Theta}^{2},M^{2}_{t}\big{)}\,\bigg{[}16\,\bar{h}_{AB}\,\bar{h}^{*}_{CD}+\frac{10}{9}\,\bar{g}_{AB}\,g^{*}_{CD}+\frac{8}{405}\,f_{AB}\,f^{*}_{CD}\big{)}$
$\displaystyle-$ $\displaystyle\frac{8}{20}\
\tilde{k}_{AB}\,\big{(}\tilde{k}^{*}_{CD}+\tilde{k}^{*}_{DC}\big{)}-\frac{16}{15}\bar{k}_{AB}\,\left(\frac{29}{4}\,\bar{k}^{*}_{CD}+\bar{k}^{*}_{DC}\right)\bigg{]}$
$\displaystyle c_{3}\big{[}A,B,C,D\big{]}$ $\displaystyle=$
$\displaystyle\frac{1}{\Lambda^{2}}D\big{(}M_{\Theta}^{2},M^{2}_{t}\big{)}\,\big{(}64h_{AB}\,h^{*}_{CD}+\frac{8}{3}\,g_{AB}\,\bar{g}^{*}_{CD}+4g_{AB}\,g^{*}_{CD}-\frac{16}{9}\tilde{g}_{AB}\,\tilde{g}^{*}_{CD}\big{)}$
$\displaystyle c_{4}\big{[}A,B,C,D\big{]}$ $\displaystyle=$
$\displaystyle\frac{1}{\Lambda^{2}}D\big{(}M_{\Theta}^{2},M^{2}_{t}\big{)}\,\big{(}32h_{AB}\,\bar{h}_{CD}-\frac{8}{3}\,g_{AB}\,\bar{g}_{CD}\big{)}$
(28)

The expressions derived in eqs. (III, III, 23, III, III, 26, 27 and III)
quantifies the contribution of different pairs of scalars capable of mediating
$B$ and $L$ violating but $B-L$ conserving nucleon decays. These coefficients
depend on the masses of the scalars and determinable Yukawa couplings. The
following points summarise the notable features of these evaluated
coefficients.

  * •

Only three pairs of scalars can mediate nucleon decay at tree level, viz
$\sigma-T$, $H-T$ and $H-\Delta$. $\sigma-T$ and $H-\Delta$ contribute to all
the operators given in eq. (III), while $H-T$ contributes to only
$\mathcal{O}_{2}$. The immediate consequence of the latter is that $H-T$ can
only induce charged antilepton modes accompanied by neutral mesons, which are
$p\to e^{+}\pi^{0}$, $p\to\mu^{+}\,K^{0}$ and others.

  * •

Other five pairs, $T-\Theta$, $\Delta-\Delta$, $\Delta-\Theta$, $\Delta-t$ and
$\Theta-t$, can induce nucleon decay only at the loop level. Of these pairs,
$T-\Theta$, $\Delta-\Delta$, and $\Theta-t$ can contribute to all the four
operators and induce proton decaying into charged and neutral anti-lepton
while $\Delta-\Theta$ and $\Delta-t$ can contribute to $\mathcal{O}_{2}$ only.

  * •

For the case of Yukawa couplings, terms with Yukawa coupling $\hat{h}$,
$\hat{g}$ and $\hat{f}$ can only give rise to those terms which only
contribute to $\mathcal{O}_{2}$ and hence offer a clean signature of
detectability.

  * •

Terms with Yukawa couplings $\tilde{h}$ and $\tilde{g}$ contribute to the
operators $\mathcal{O}_{2}$ and $\mathcal{O}_{3}$.

  * •

Expressions in which the intermediate integrated out irreps are $10_{\rm{H}}$
and $120_{\rm{H}}$ contribute to all four operators of eq. (III) while when
the intermediate integrated out irrep are $45_{\rm{H}}$ and $210_{\rm{H}}$,
contributes to operators $\mathcal{O}_{2}$ and $\mathcal{O}_{3}$.
Additionally, the integration of $126^{\dagger}_{\rm{H}}$ as an intermediate
irrep yields only operator $\mathcal{O}_{2}$ of eq. (III).

  * •

Term with Yukawa coupling $k$ does not contribute to nucleon decay at all as
it only gives rise to either diquark or dilepton vertices.

  * •

In renormalisable GUTs, only two scalars, along with their conjugate partners,
are capable of mediating $B-L$ conserving tree-level proton decay modes with
SM charges $\left(3,1,\frac{1}{3}\right)$ and $\left(3,3,-\frac{1}{3}\right)$.
Additionally, $B-L$ violating modes at the tree level are mediated by the pair
$\left(3,2,\frac{1}{6}\right)$ \- $\left(3,1,\frac{1}{3}\right)$ along with
its conjugate partner Patel and Shukla (2022). However, in the case of non-
renormalisable GUTs, the lowest dimension at which tree-level $B-L$ conserving
proton decay occurs is $D=8$, always mediated by a pair of scalars as
discussed above.

Nevertheless, these pairs of scalars may also contribute to $B-L$ violating
two body proton decays Babu and Mohapatra (2012a, b); Patel and Shukla (2022).
However, if the lowest dimension at which $B-L$ conserving proton decays
appear is $D=8$, we anticipate that $B-L$ violating two-body proton decay
modes would emerge at $D=9$ and beyond, specifically odd dimensions, and thus
will be suppressed. Consequently, we constrain the mediators based on the
$B-L$ conserving two-body proton decay modes.

##  IV Constraints from Proton Decay

So far, we have identified different pairs of scalars capable of inducing tree
as well as loop-level proton decays and analysed their contribution, in terms
of the Wilson coefficient, to the effective $D=6$ proton decay operators. The
estimated coefficients depend on the explicit Yukawa couplings, also
contributing to the masses of charged and neutral fermions. In the current
scenario, these couplings appear at the non-renormalisable level in the
Lagrangian.

The class of renormalisable $SO(10)$ theories featuring the Yukawa sector with
complex $10_{H}$ and $126^{\dagger}_{\rm{H}}$ is known for reproducing the
observed fermion mass spectrum and mixing angles Joshipura and Patel (2011);
Mummidi and Patel (2021), and the Lagrangian is shown below.

$\displaystyle{}-\mathcal{L}_{R}$ $\displaystyle=$
$\displaystyle\mathbf{16}_{A}\,\Big{(}H_{AB}\,10_{H}+F_{AB}\,126_{\rm{H}}^{\dagger}\Big{)}\mathbf{16}_{B}$
(29)

$H$ and $F$ are the symmetric Yukawa couplings in the generation space.

One can evade the coupling of $126^{\dagger}_{\rm{H}}$ with $\mathbf{16}$-plet
fermion and instead include $16_{H}$ which couples to $\mathbf{16}$-plet at
non-renormalisable level. The Lagrangian of such a set-up is shown below.

$\displaystyle{}-{\cal{L}}_{\rm{Y}}$ $\displaystyle=$ $\displaystyle
H_{AB}\,\mathbf{16}_{A}\,\mathbf{16}_{B}\,10_{H}+\frac{y_{AB}}{\Lambda}\left(\mathbf{16}_{A}\,\mathbf{16}_{B}\,16_{H}\,16_{H}\right)+\,\frac{1}{\Lambda}\bar{y}_{AB}\,\big{(}\mathbf{16}_{A}\,\mathbf{16}_{B}\;16^{\dagger}_{H}\,16^{\dagger}_{H}\big{)}\;+\;\rm{H.c.}$

In the absence of $126^{\dagger}_{\rm{H}}$, its role will be typically
performed by $16_{H}$ for a realistic $SO(10)$ model. Further, the non-
renormalisable term mentioned in eq. (IV) can be decomposed only via
$10_{\rm{H}}$-dimensional $SO(10)$ irrep as the coupling of different irreps
with $\mathbf{16}$-plet are denied. Such an absence can be ensured by
introducing a discrete symmetry which only allows coupling of $16$-plet with
$10_{H}$ at the renormalisable level and with $16_{H}$ at non-renormalisable
level. Consequently, $h$ and $\bar{h}$ coupling mentioned in eq.
(LABEL:eq:16Fs16Hs) are allowed, and they could presume the order of Yukawa
coupling of $126_{H}$ with $\mathbf{16}$-plet fermion.

$\displaystyle{}h\frac{\langle\sigma\rangle}{\Lambda}$ $\displaystyle=$
$\displaystyle\bar{h}\frac{\langle\sigma\rangle}{\Lambda}\hskip
14.22636pt\sim\hskip 14.22636ptF.$ (31)

Furthermore, we impose the condition that the entries of $h$ and $\bar{h}$
remain perturbative, i.e. $|h|$ and $|\bar{h}|$ $\leq 4\pi$. This
perturbativity requirement places the lower bound on the parameter $\sigma$ as
$10^{-4.5}\Lambda$, while $\sigma$ itself is not subject to an upper bound. We
set $\Lambda$ as the reduced Planck scale ($\sim\,10^{18.5}$ GeV), making the
$B-L$ scale closer to the typical GUT scale. Opting for a $B-L$ scale lower
than the aforementioned scale renders the couplings $f$ non-perturbative and
thus making $SO(10)$ unrealistic. The best-fit value of the entries of $F$ can
be inferred from Mummidi and Patel (2021).

Further, once the $SO(10)$ gauge symmetry is broken into SM gauge symmetry,
the following expressions of effective Yukawa couplings arise.

$\displaystyle{}Y_{u,d,e,\nu}$ $\displaystyle=$ $\displaystyle
c_{1}^{u,d,e,\nu}H+\frac{\langle\sigma\rangle}{\Lambda}\big{(}c_{2}^{u,d,e,\nu}\,h+c_{3}^{u,d,e,\nu}\,\bar{h}\big{)},$
$\displaystyle M_{R}$ $\displaystyle=$ $\displaystyle
r\frac{\langle\sigma\rangle^{2}}{\Lambda}\big{(}h+\bar{h}\big{)}.$ (32)

where, $c_{1}^{u,d,e,\nu}$ are the various ${\mathcal{O}}\big{(}1\big{)}$
Clebsch Gordan coefficients and can be inferred from Mummidi and Patel (2021).
$c_{2,3}^{u,d,e,\nu}$ and $r$ are also the ${\mathcal{O}}\big{(}1\big{)}$
Clebsh Gordan coefficients. When $\mathbf{\sigma}$, $B-L$ charged singlet-
scalar present in $16_{H}$, assumes vev, Majorana mass associated right-handed
neutrinos is generated, alike the case of renormalisable $SO(10)$. Further,
light neutrino masses can be generated via invoking Type-I seesaw mechanism
Minkowski (1977); Yanagida (1979); Mohapatra and Senjanovic (1980); Schechter
and Valle (1980).

###  IV.1 Branching Pattern

In our scenario, we consider the contribution from the non-renormalisable part
of the Yukawa Lagrangian, as presented in eq. (IV), to proton decay to be
comparable to the renormalisable part, and collectively both these
contributions should adhere to the constraint imposed by proton decay.
Consequently, the Yukawa couplings $h$ and $\bar{h}$ become the primary
contributors to proton decay stemming from the non-renormalisable couplings.
As it is evident from eq.(III) only the pairs $\sigma-T$ and $H-\Delta$ can
potentially induce proton decay at the tree level.

The expressions of partial decay widths of leading proton decay modes are
given in appendix A. Table (3) shows the branching pattern of such proton
decay modes, assuming the contribution of a pair of scalars capable of
inducing proton decay at the tree level.

Branching ratio [%] |  $\sigma-T$ |  $H-\Delta$ |  $\sigma-T\,+H-\Delta$
---|---|---|---
| $M_{\Delta}\to\infty$ | $M_{T}\to\infty$ |  with $M_{T}=M_{\Delta}$
${\rm BR}[p\to e^{+}\pi^{0}]$ | $<1$ | $<1$ | $<1$
${\rm BR}[p\to\mu^{+}\pi^{0}]$ | $2$ | $2$ | $2$
${\rm BR}[p\to\bar{\nu}\pi^{+}]$ | $12$ | $13$ | $13$
${\rm BR}[p\to e^{+}K^{0}]$ | $<1$ | $<1$ | $<1$
${\rm BR}[p\to\mu^{+}K^{0}]$ | $3$ | $2$ | $2$
${\rm BR}[p\to\bar{\nu}K^{+}]$ | $83$ | $84$ | $82$
${\rm BR}[p\to e^{+}\eta]$ | $<1$ | $<1$ | $<1$
${\rm BR}[p\to\mu^{+}\eta]$ | $<1$ | $<1$ | $<1$
Table 3: Proton decay branching fractions estimated for different hierarchies,
assuming $\big{\langle}\sigma\big{\rangle}=10^{14}$ GeV and
$\big{\langle}H\big{\rangle}=246$ GeV.

The crucial observations regarding Table (3) are as follows.

  1. (a)

In the case of scalar-mediated proton decay, the proton favours decay into the
second-generation mesons compared to the first ones. This is due to the
hierarchical nature of Yukawa couplings, which has also been noticed in the
case of renormalisable $SO(10)$ models Patel and Shukla (2022).

The above-mentioned feature of the proton decay spectrum can be understood
from the flavour structure. The realistic fermion mass spectrum in the
underlying model leads to the following hierarchical structure of $h$ and
$\bar{h}$ from which matrices that diagonalises $Y_{u,d,e}$ could be obtained
and are given as follows Mummidi and Patel (2021).

$f=\bar{f}\sim\frac{\Lambda}{\sigma}\,F\sim\frac{\Lambda}{\sigma}\,\lambda^{4}\,\left(\begin{array}[]{ccc}\lambda^{5}&\lambda^{4}&\lambda^{3}\\\
\lambda^{4}&\lambda^{3}&\lambda^{2}\\\
\lambda^{3}&\lambda^{2}&\lambda\end{array}\right),\;\text{and}\;U_{u,d,e}\sim\left(\begin{array}[]{ccc}1&\lambda&\lambda^{3}\\\
\lambda&1&\lambda^{2}\\\
\lambda^{3}&\lambda^{2}&1\end{array}\right),~{}~{}U_{\nu}\sim\left(\begin{array}[]{ccc}&&\\\
&{\cal O}(1)&\\\ &&\end{array}\right)\,$ (33)

where $\lambda=0.23$ is Cabibbo angle. From eq. (33), we find

$\displaystyle~{}~{}U_{f}^{T}\,f\,U_{f^{\prime}}\sim\frac{\Lambda}{\sigma}\,\lambda^{4}\,\left(\begin{array}[]{ccc}\lambda^{5}&\lambda^{4}&\lambda^{3}\\\
\lambda^{4}&\lambda^{3}&\lambda^{2}\\\
\lambda^{3}&\lambda^{2}&\lambda\end{array}\right)\,=U_{d}^{T}\,f\,U_{\nu}$
(37)

where $f,f^{\prime}=u,d,e$. Substitution of the above results of eq. (37) in
the expressions of partial decay widths of the proton as mentioned in the eq.
(A), one finds the following. The definition of various parameters entering in
the following ratios are also defined in the appendix (A).

$\displaystyle\frac{\Gamma[p\to\overline{\nu}\,\pi^{+}]}{\Gamma[p\to\overline{\nu}\,K^{+}]}$
$\displaystyle\simeq$
$\displaystyle\frac{(m_{p}^{2}-m{{}^{2}_{\pi^{+}}})^{2}}{(m_{p}^{2}-m{{}^{2}_{K^{+}}})^{2}}\,\frac{(1+\tilde{D}+\tilde{F})^{2}}{\left(1+\frac{m_{N}}{m_{S}}(\tilde{D}-\tilde{F})+\frac{m_{N}}{m_{\Lambda}}\big{(}\tilde{D}+3\tilde{F}\big{)}\right)^{2}}\,{\lambda}^{2}\simeq
2\lambda^{2}\,.$ $\displaystyle\frac{\Gamma[p\to
e^{+}\,\pi^{0}]}{\Gamma[p\to\mu^{+}\,\pi^{0}]}$ $\displaystyle\simeq$
$\displaystyle\frac{\Gamma[p\to
e^{+}\,K^{0}]}{\Gamma[p\to\mu^{+}\,K^{0}]}\simeq\lambda^{2}\,,$
$\displaystyle\frac{\Gamma[p\to\mu^{+}\,K^{0}]}{\Gamma[p\to\overline{\nu}\,\pi^{+}]}$
$\displaystyle\simeq$
$\displaystyle\frac{(m_{p}^{2}-m{{}^{2}_{K^{0}}})^{2}}{(m_{p}^{2}-m{{}^{2}_{\pi^{+}}})^{2}}\,\frac{\frac{1}{2}\left(\frac{m_{N}64}{m_{S}}(\tilde{D}-\tilde{F})\right)^{2}+\frac{1}{2}64^{2}}{(1+\tilde{D}+\tilde{F})^{2}\,32^{2}}\,\simeq
0.2\,,$
$\displaystyle\frac{\Gamma[p\to\mu^{+}\,K^{0}]}{\Gamma[p\to\overline{\nu}\,K^{+}]}$
$\displaystyle\simeq$
$\displaystyle\frac{(m_{p}^{2}-m{{}^{2}_{K^{0}}})^{2}}{(m_{p}^{2}-m{{}^{2}_{K^{+}}})^{2}}\,\frac{\frac{1}{2}\left(\frac{m_{N}64}{m_{S}}(\tilde{D}-\tilde{F})\right)^{2}+\frac{1}{2}64^{2}}{\left(1+\frac{m_{N}}{m_{S}}(\tilde{D}-\tilde{F})+\frac{m_{N}}{m_{\Lambda}}\big{(}\tilde{D}+3\tilde{F}\big{)}\right)^{2}}\,\simeq
0.2\lambda^{2}\,.$ (38)

The expressions given in eq.(IV.1) explains the claim mentioned in point (a)
and explain the branching pattern of the proton given in the table (3)
mediated by $\sigma-T$. Further, the branching pattern of proton decay induced
by $H-T$ yields similar ratios.

  1. (b)

In the context of a realistic $SO(10)$ scenario, the precise values of the
$B-L$ scale and $\Lambda$ are irrelevant for the tree-level branching pattern
of the proton in the case of $\sigma-T$.

The aforementioned observation can be understood as follows. As indicated by
eq. (III), the tree-level contribution of the coefficient of a specific pair
effectively scales as
$c\sim\frac{\langle\sigma\rangle^{2}}{\Lambda^{2}}\big{|}h\big{|}^{2}$, where
$h$ represents a particular Yukawa coupling. As deduced from eq. (31), for a
realistic $SO(10)$ scenario, $h\sim\frac{\Lambda}{\langle\sigma\rangle}$,
rendering the coefficient seemingly independent of
$\big{\langle}\sigma\big{\rangle}$ and $\Lambda$. The nature of $h$ is
dictated by the requirement to reproduce the observed fermion mass spectrum.
However, opting for an alternative value of $h$ independent of $\sigma$ and
$\Lambda$ introduces a dependence of the coefficient on the latter,
potentially lowering down the mass of the mediator.

  1. (c)

The loop contribution of the pair of scalars to the proton decay is highly
suppressed when the $B-L$ scale is larger than $10^{11}$ GeV, and hence, the
loop contribution has been ignored while computing the branching pattern of
the proton.

Claim (c) can be understood considering the loop contribution of $\sigma-T$ to
the proton decay. Substituting eqs. (III) and (37) into the partial decay
width of proton decay, as provided in eq. (A), resulting in the decay into
$\nu\,K^{+}$, yields the following expression.

$\displaystyle{}\Gamma[p\to\overline{\nu}\,K^{+}]$ $\displaystyle=$
$\displaystyle\frac{\big{(}D\big{(}M_{T}^{2},M_{\sigma}^{2}\big{)}\big{)}^{2}}{\big{\langle}\sigma\big{\rangle}^{4}}\frac{(m_{p}^{2}-m_{K^{\pm}}^{2})^{2}}{32\,\pi\,m_{p}^{3}f_{\pi}^{2}}\left(1+\frac{m_{N}}{m_{S}}(\tilde{D}-\tilde{F})+\frac{m_{N}}{m_{\Lambda}}\big{(}\tilde{D}+3\tilde{F}\big{)}\right)^{2}\,32^{2}\,\lambda^{30}$

where different symbols have their usual meaning provided in the appendix (A)
and $D\big{(}M_{T}^{2},M_{\sigma}^{2}\big{)}$ is the loop function defined in
the eq. (21). The expression given in the above eq. (IV.1) can modified to
yield $\sigma$ as follows.

$\displaystyle{}\big{\langle}\sigma\big{\rangle}$ $\displaystyle=$
$\displaystyle\Bigg{[}\tau_{p\to\overline{\nu}\,K^{+}}\frac{\big{(}D\big{(}M_{T}^{2},M_{\sigma}^{2}\big{)}\big{)}^{2}}{\big{\langle}\sigma\big{\rangle}^{4}}\frac{(m_{p}^{2}-m_{K^{\pm}}^{2})^{2}}{32\,\pi\,m_{p}^{3}f_{\pi}^{2}}$
$\displaystyle\left(1+\frac{m_{N}}{m_{S}}(\tilde{D}-\tilde{F})+\frac{m_{N}}{m_{\Lambda}}\big{(}\tilde{D}+3\tilde{F}\big{)}\right)^{2}\,32^{2}\,\lambda^{30}\Bigg{]}^{0.25}$

Using the current lower bound on the lifetime of $p\to\nu\,K^{+}$ Abe  _et
al._ (2014), we get the following.

$\displaystyle{}\big{\langle}\sigma\big{\rangle}$ $\displaystyle>$
$\displaystyle\Big{[}D\big{(}M^{2}_{T},M^{2}_{\sigma}\big{)}\Big{]}^{0.5}\,2.5*10^{11}\;\rm{GeV}$
(40)

The loop function provided above reaches its maximum value when both masses
are degenerate, and its value is always less than 1, considering the fact that
the maximum mass the scalar can attain is $M_{p}$. Consequently, $\sigma$ must
be greater than $10^{11}$ GeV to satisfy the proton decay bound. In realistic
$SO(10)$ scenarios, $\sigma$ typically lies around $10^{14}$ GeV, and lowering
$\sigma$ below than $10^{14}$ GeV hampers the perturbativity of Yukawa
couplings. As a result, the pair that mediates proton decays at the loop level
cannot be strictly constrained in this scenario.

###  IV.2 Estimation of bound on the mediator

Having obtained the branching pattern of leading proton decay modes by the
pair of scalars able to induce tree-level proton decay modes, we constrain
their masses using the current lower bound on the lifetime of proton Abe  _et
al._ (2014).

The lower bound on the mass of pair $\sigma-T$ from its tree-level
contribution is as follows.

$\displaystyle\tau/{\rm BR}[p\to\overline{\nu}\,K^{+}]$ $\displaystyle=$
$\displaystyle 5.9\times 10^{33}\,{\rm
yrs}\,\times\left(\frac{M_{T}}{2.8\times 10^{11}\,{\rm GeV}}\right)^{4}\,.$
(41)

Similarly, the bound on the mass of $\Delta$ from its tree-level contribution
is the following.

$\displaystyle\tau/{\rm BR}[p\to\overline{\nu}\,K^{+}]$ $\displaystyle=$
$\displaystyle 5.9\times 10^{33}\,{\rm
yrs}\,\times\left(\frac{M_{\Delta}\,\big{\langle}\sigma\big{\rangle}}{4.9\times
10^{13}\,{\rm
GeV}^{2}}\right)^{4}\,\left(\frac{\big{\langle}246\big{\rangle}}{\big{\langle}H\big{\rangle}}\right)^{4}.$
(42)

Eqs. (41) and (42) quantify the bounds on $M_{T}$ and $M_{\Delta}$
respectively. The bound on $M_{T}$ is independent of the $B-L$ scale, while
the bound on $\Delta$ depends on the ratio of
$\frac{\langle\sigma\rangle}{\langle H\rangle}$. Both mediators, $T$ and
$\Delta$, remain independent of the precise value of the cutoff scale
($\Lambda$) due to the consideration of a realistic $SO(10)$ scenario.
Moreover, the mass of $\Delta$ relies on the ratio
$\frac{\langle\sigma\rangle}{\langle H\rangle}$, which can be adjusted to
achieve a lighter mass for $\Delta$, preferably at the TeV scale. This
possibility raises interesting prospects for probing $\Delta$ in future
colliders Doršner  _et al._ (2016); Raj (2017); Padhan  _et al._ (2020).
Additionally, in renormalisable SO(10) models, $\Delta$ also mediates $B-L$
violating proton decay modes. The bound on its mass from $p\to\nu\,K^{+}$
turns out to be greater than $10^{6}$ GeV Patel and Shukla (2022). Thus, the
mass bound on the $\Delta$ gets significantly reduced in our scenario.

##  V Summary and Discussion

Proton decay is one of the crucial predictions of GUTs and is an important
yardstick to constrain the new particles (scalar and vector), which are
inextricable parts of the GUTs. Its detection would discard the baryon number
conservation principle and ultimately provide hope to unify fundamental
interactions.

This study investigates a version of the non-renormalisable $SO(10)$ model
that lacks large tensor representations. The $16_{\rm{H}}$ representation is
the smallest irrep, which couples with the $\mathbf{16}$-plet, generating
charged and neutral fermion masses at the non-renormalisable level. We begin
with decomposing the coupling of $16_{\rm{H}}$ with the
$\mathbf{16}$-fermions, exploring all possible decompositions allowed by
integrating out different $SO(10)$ irreps, into $SU(5)$ invariant expressions.
After that, we calculate the couplings of scalar pairs with SM fermions
stemming from these $SU(5)$ decomposed expressions capable of inducing proton
decay. Subsequently, we assess the tree and loop level contributions of
different pairs of scalars to the effective $D=6$ $B-L$ conserving operators.
We also argued that the $16_{\rm{H}}$ irrep could fulfil the role typically
played by $126^{\dagger}_{\rm{H}}$ in renormalisable $SO(10)$ GUTs, thus
assigning its Yukawa coupling to be similar to that of
$126^{\dagger}_{\rm{H}}$. Finally, we impose constraints on these pairs of
scalars residing in the $16_{\rm{H}}$ representation through proton decays.
The main features of this study are summarised as follows.

  * •

The proton decay in the non-renormalisable $SO(10)$ is always mediated by
pairs of scalars and can happen in two distinct topologies, resulting into
effective $D=6$ operator. Firstly, at the tree level, one scalar is integrated
out while the other acquires a vev, or vice versa. Secondly, both scalars
contribute to proton decay through loops.

  * •

In total, there are eight different pairs capable of inducing proton decay.
Out of them, only three can mediate proton decay at the tree level,
$\sigma-T$, $H-\Delta$ and $H-T$, while other pairs induce proton decay at the
loop level.

  * •

In a non-renormalisable $SO(10)$ model with $10_{\rm{H}}$ and $16_{\rm{H}}$
participating in the Yukawa sector, proton decay at the tree level is mediated
by the pair $\sigma-T$ and $H-\Delta$. Due to the hierarchical Yukawa
couplings, proton favours to decay in the second generation mesons accompanied
with charges or neutral antileptons. For a realistic scenario, the lower bound
on $M_{T}$ is greater than $10^{11}$ GeV, while $M_{\Delta}$ is constrained to
be at least 1 TeV. The bound on $M_{\Delta}$ arises from collider searches.

In the renormalisable version of $SO(10)$ GUTs, proton decay at the lowest
mass dimension, i.e. six, is mediated by a single scalar particle, and these
mediator can typically remain lighter than the conventional GUT scale.
However, in the case of the non-renormalisable version, proton decay at the
lowest dimension is always mediated by a pair of scalars. In some scenarios,
these mediators can remain closer to the electroweak scale than the GUT scale,
thus can also account for the lower-energy associated phenomenology. Non-
renormalisable GUT models offer compelling solutions to the inconsistencies of
renormalisable GUT models in the UV regime and may allow a promising avenue
for the unification of gravitational interactions with the known fundamental
interactions.

## Acknowledgements

I express my gratitude to Namit Mahajan and Ketan M. Patel for valuable
discussions, suggestions, encouragement, and meticulous review of the
manuscript.

##  Appendix A Proton Decay Width Expression

The decay width expressions of leading proton decay modes can be computed
using the chiral perturbation theory Claudson  _et al._ (1982); Chadha and
Daniel (1983) and are given as follows Nath and Fileviez Perez (2007); Beneito
_et al._ (2023).

$\displaystyle{}\Gamma[p\to e_{i}^{+}\pi^{0}]$ $\displaystyle=$
$\displaystyle\frac{(m_{p}^{2}-m_{\pi^{0}}^{2})^{2}}{32\,\pi\,m_{p}^{3}f_{\pi}^{2}}A^{2}\left(\frac{1+\tilde{D}+\tilde{F}}{\sqrt{2}}\right)^{2}\Big{(}\left|\alpha\,y[u_{1},d_{1},e^{C}_{i},u^{C}_{1}]+\beta\,y^{\prime*}[u^{C}_{1},d^{C}_{1},e^{C}_{i},u^{C}_{1}]\right|^{2}\Big{.}$
$\displaystyle+$
$\displaystyle\Big{.}\left|\alpha\,y^{*}[u^{C}_{1},d^{C}_{1},e_{i},u_{1}]+\beta\,y^{\prime}[u_{1},d_{1},e_{i},u_{1}]\right|^{2}\Big{)},$
$\displaystyle\Gamma[p\to\overline{\nu}\pi^{+}]$ $\displaystyle=$
$\displaystyle\frac{(m_{p}^{2}-m_{\pi^{\pm}}^{2})^{2}}{32\,\pi\,m_{p}^{3}f_{\pi}^{2}}A^{2}\left(1+\tilde{D}+\tilde{F}\right)^{2}\sum_{i=1}^{3}\left|\alpha\,y^{*}[u^{C}_{1},d^{C}_{1},\nu_{i},d_{1}]+\beta\,y^{\prime}[u_{1},d_{1},\nu_{i},d_{1}]\right|^{2}\,,$
$\displaystyle\Gamma[p\to e_{i}^{+}K^{0}]$ $\displaystyle=$
$\displaystyle\frac{(m_{p}^{2}-m_{K^{0}}^{2})^{2}}{32\,\pi\,m_{p}^{3}f_{\pi}^{2}}A^{2}\,\frac{1}{2}\Big{[}\Big{|}C_{Li}^{-}-C_{Ri}^{-}+\frac{m_{p}}{m_{B}}(\tilde{D}-\tilde{F})\left(C_{Li}^{+}-C_{Ri}^{+}\right)\Big{|}^{2}\Big{.}$
$\displaystyle+$
$\displaystyle\Big{.}\Big{|}C_{Li}^{-}+C_{Ri}^{-}+\frac{m_{p}}{m_{B}}(\tilde{D}-\tilde{F})\left(C_{Li}^{+}+C_{Ri}^{+}\right)\Big{|}^{2}\Big{]}\,,$
$\displaystyle\Gamma[p\to\overline{\nu}K^{+}]$ $\displaystyle=$
$\displaystyle\frac{(m_{p}^{2}-m_{K^{\pm}}^{2})^{2}}{32\,\pi\,m_{p}^{3}f_{\pi}^{2}}\,A^{2}\,\sum_{i=1}^{3}\Big{|}\frac{2\tilde{D}}{3}\frac{m_{p}}{m_{B}}C^{\nu}_{L1i}+\left(1+\frac{\tilde{D}+3\tilde{F}}{3}\frac{m_{p}}{m_{B}}\right)C^{\nu}_{L2i}\Big{|}^{2},$
$\displaystyle\Gamma[p\to e_{i}^{+}\eta]$ $\displaystyle=$
$\displaystyle\frac{(m_{p}^{2}-m_{\eta}^{2})^{2}}{32\,\pi\,m_{p}^{3}f_{\pi}^{2}}\,A^{2}\,\frac{1}{6}\Big{[}\Big{|}C_{Li}^{+}(1-\tilde{D}+3\tilde{F})-2C_{Li}^{-}\Big{|}^{2}\,$
(43) $\displaystyle+$
$\displaystyle\Big{.}\Big{|}C_{Ri}^{+}(1-\tilde{D}+3\tilde{F})-2C_{Ri}^{-}\Big{|}^{2}\Big{]}\,\Big{.},$

where,

$\displaystyle C^{\pm}_{Li}$ $\displaystyle=$
$\displaystyle\alpha\,c_{3}^{*}[u^{C}_{1},d^{C}_{2},u_{1},e_{i}]\pm\beta\,c_{1}[u_{1},d_{2},u_{1},e_{i}]\,,$
$\displaystyle C^{\pm}_{Ri}$ $\displaystyle=$
$\displaystyle\alpha\,c_{2}^{*}[u_{1},d_{2},u^{C}_{1},e^{C}_{i}]\pm\beta\,c_{4}[u^{C}_{1},d^{C}_{2},u^{C}_{1},e^{C}_{i}]\,,$
$\displaystyle C^{\nu}_{L1i}$ $\displaystyle=$
$\displaystyle-\alpha\,c_{3}^{*}[u_{1}^{C},d_{2}^{C},d_{1},\nu_{i}]-\beta\,c_{1}[u_{1},d_{2},d_{1},\nu_{i}]\,,$
$\displaystyle C^{\nu}_{L2i}$ $\displaystyle=$
$\displaystyle-\alpha\,c_{3}^{*}[u_{1}^{C},d_{1}^{C},d_{2}]-\beta\,c_{1}[u_{1},d_{1},d_{2},\nu_{i}]\,.$
(44)

The various $C^{\prime}s$ defined in the eq. (A) are in the flavour basis and
should be converted appropriately into the mass basis. Also, $m_{H}$
represents the mass of various hadrons denoted by $H$, including protons
($p$), neutral pions ($\pi^{0}$), neutral kaons ($K^{0}$), charged kaons
($K^{\pm}$). $m_{B}$ signifies the average mass of baryons, while $f_{\pi}$
denotes the pion-decay constant. Parameters $\alpha$, $\beta$, $\tilde{D}$,
and $\tilde{F}$ are the input chiral Lagrangian parameter. The parameter $A$
accounts for renormalization effects in hadronic matrix elements, particularly
from the weak scale to the proton mass, $m_{p}$. The numerical values to these
parameters in eq. (A): $\alpha=0.01,{\rm GeV}^{3}$, $\tilde{D}=0.8$, and
$\tilde{F}=0.46$ Aoki  _et al._ (2000); Cabibbo  _et al._ (2003). Furthermore,
we utilize the average baryon mass, $m_{B}=1.15$ GeV, and the pion decay
constant, $f_{\pi}=130$ MeV, alongside various hadron masses sourced from the
Particle Data Group (PDG) Zyla  _et al._ (2020). The parameter $A$ is set to
$1.43$ to accommodate running effects from the electroweak scale ($M_{Z}$) to
the proton mass scale. Although typically, values of $F$ obtained at the
electroweak scale are prone to RGE effects from the GUT scale to $M_{Z}$, we
opt for their GUT-scale values due to minimal changes in Yukawa couplings
during running, which shall have negligible impacts on the results Alonso  _et
al._ (2014).

## References

  * Georgi and Glashow (1974) H. Georgi and S. L. Glashow, “Unity of All Elementary Particle Forces,” Phys. Rev. Lett. 32, 438–441 (1974).
  * Fritzsch and Minkowski (1975) Harald Fritzsch and Peter Minkowski, “Unified Interactions of Leptons and Hadrons,” Annals Phys. 93, 193–266 (1975).
  * Pati and Salam (1973) Jogesh C. Pati and Abdus Salam, “Unified Lepton-Hadron Symmetry and a Gauge Theory of the Basic Interactions,” Phys. Rev. D 8, 1240–1251 (1973).
  * Langacker (1981) Paul Langacker, “Grand Unified Theories and Proton Decay,” Phys. Rept. 72, 185 (1981).
  * Aulakh and Girdhar (2005) Charanjit S. Aulakh and Aarti Girdhar, “SO(10) a la Pati-Salam,” Int. J. Mod. Phys. A 20, 865–894 (2005), arXiv:hep-ph/0204097 .
  * Fukuyama  _et al._ (2005) Takeshi Fukuyama, Amon Ilakovac, Tatsuru Kikuchi, Stjepan Meljanac, and Nobuchika Okada, “SO(10) group theory for the unified model building,” J. Math. Phys. 46, 033505 (2005), arXiv:hep-ph/0405300 .
  * Nath and Fileviez Perez (2007) Pran Nath and Pavel Fileviez Perez, “Proton stability in grand unified theories, in strings and in branes,” Phys. Rept. 441, 191–317 (2007), arXiv:hep-ph/0601023 .
  * Slansky (1981) R. Slansky, “Group Theory for Unified Model Building,” Phys. Rept. 79, 1–128 (1981).
  * Georgi and Nanopoulos (1979) Howard Georgi and Dimitri V. Nanopoulos, “Ordinary Predictions from Grand Principles: T Quark Mass in O(10),” Nucl. Phys. B 155, 52–74 (1979).
  * Barbieri  _et al._ (1980) Riccardo Barbieri, Dimitri V. Nanopoulos, G. Morchio, and F. Strocchi, “Neutrino Masses in Grand Unified Theories,” Phys. Lett. B 90, 91–97 (1980).
  * Buccella  _et al._ (1980) F. Buccella, H. Ruegg, and Carlos A. Savoy, “Spontaneous Symmetry Breaking in O(10),” Phys. Lett. B 94, 491–494 (1980).
  * Yasue (1981) M. Yasue, “Phenomenological Aspect of SO(10) Grand Unified Model,” Prog. Theor. Phys. 65, 708 (1981), [Erratum: Prog.Theor.Phys. 65, 1480 (1981)].
  * Barbieri  _et al._ (1979) Riccardo Barbieri, G. Morchio, and F. Strocchi, “Neutrino Masses and SO(10) Hierarchical Breaking,” Lett. Nuovo Cim. 26, 445 (1979).
  * Babu and Mohapatra (1993) K. S. Babu and R. N. Mohapatra, “Predictive neutrino spectrum in minimal SO(10) grand unification,” Phys. Rev. Lett. 70, 2845–2848 (1993), arXiv:hep-ph/9209215 .
  * Bajc  _et al._ (2006) Borut Bajc, Alejandra Melfo, Goran Senjanovic, and Francesco Vissani, “Yukawa sector in non-supersymmetric renormalizable SO(10),” Phys. Rev. D 73, 055001 (2006), arXiv:hep-ph/0510139 .
  * Joshipura and Patel (2011) Anjan S. Joshipura and Ketan M. Patel, “Fermion Masses in SO(10) Models,” Phys. Rev. D 83, 095002 (2011), arXiv:1102.5148 [hep-ph] .
  * Altarelli and Meloni (2013) Guido Altarelli and Davide Meloni, “A non supersymmetric SO(10) grand unified model for all the physics below $M_{GUT}$,” JHEP 08, 021 (2013), arXiv:1305.1001 [hep-ph] .
  * Dueck and Rodejohann (2013) Alexander Dueck and Werner Rodejohann, “Fits to SO(10) Grand Unified Models,” JHEP 09, 024 (2013), arXiv:1306.4468 [hep-ph] .
  * Meloni  _et al._ (2014) Davide Meloni, Tommy Ohlsson, and Stella Riad, “Effects of intermediate scales on renormalization group running of fermion observables in an SO(10) model,” JHEP 12, 052 (2014), arXiv:1409.3730 [hep-ph] .
  * Meloni  _et al._ (2017) Davide Meloni, Tommy Ohlsson, and Stella Riad, “Renormalization Group Running of Fermion Observables in an Extended Non-Supersymmetric SO(10) Model,” JHEP 03, 045 (2017), arXiv:1612.07973 [hep-ph] .
  * Babu  _et al._ (2017) K. S. Babu, Borut Bajc, and Shaikh Saad, “Yukawa Sector of Minimal SO(10) Unification,” JHEP 02, 136 (2017), arXiv:1612.04329 [hep-ph] .
  * Ohlsson and Pernow (2018) Tommy Ohlsson and Marcus Pernow, “Running of Fermion Observables in Non-Supersymmetric SO(10) Models,” JHEP 11, 028 (2018), arXiv:1804.04560 [hep-ph] .
  * Maiani  _et al._ (1978) L. Maiani, G. Parisi, and R. Petronzio, “Bounds on the Number and Masses of Quarks and Leptons,” Nucl. Phys. B 136, 115–124 (1978).
  * Vaughn (1979) Michael T. Vaughn, “ASYMPTOTIC FREEDOM CONSTRAINTS ON GRAND UNIFIED GAUGE THEORIES,” Z. Phys. C 2, 111 (1979).
  * Rubakov and Troitsky (1999) V. A. Rubakov and Sergey V. Troitsky, “Trends in grand unification: Unification at strong coupling and composite models,” in  _10th International School on Particles and Cosmology_ (1999) arXiv:hep-ph/0001213 .
  * Aulakh (2002a) Charanjit S. Aulakh, “Truly minimal unification: Asymptotically strong panacea?” (2002a), arXiv:hep-ph/0207150 .
  * Aulakh (2002b) Charanjit S. Aulakh, “Taming asymptotic strength,” (2002b), arXiv:hep-ph/0210337 .
  * Aulakh  _et al._ (2004) Charanjit S. Aulakh, Borut Bajc, Alejandra Melfo, Goran Senjanovic, and Francesco Vissani, “The Minimal supersymmetric grand unified theory,” Phys. Lett. B 588, 196–202 (2004), arXiv:hep-ph/0306242 .
  * Bajc and Sannino (2016) Borut Bajc and Francesco Sannino, “Asymptotically Safe Grand Unification,” JHEP 12, 141 (2016), arXiv:1610.09681 [hep-th] .
  * Babu and Barr (1995) K. S. Babu and Stephen M. Barr, “Supersymmetric SO(10) simplified,” Phys. Rev. D 51, 2463–2470 (1995), arXiv:hep-ph/9409285 .
  * Dvali and Pokorski (1996) G. R. Dvali and Stefan Pokorski, “Naturally light Higgs doublet in the spinor representation of SUSY SO(10),” Phys. Lett. B 379, 126–134 (1996), arXiv:hep-ph/9601358 .
  * Barr and Raby (1997) Stephen M. Barr and Stuart Raby, “Minimal SO(10) unification,” Phys. Rev. Lett. 79, 4748–4751 (1997), arXiv:hep-ph/9705366 .
  * Albright (2003) Carl H. Albright, “SO(10) GUT Models and Their Present Success in Explaining Mass and Mixing Data,” Int. J. Mod. Phys. A 18, 3947–3956 (2003), arXiv:hep-ph/0212090 .
  * Chang  _et al._ (2005) Darwin Chang, Takeshi Fukuyama, Yong-Yeon Keum, Tatsuru Kikuchi, and Nobuchika Okada, “Perturbative SO(10) grand unification,” Phys. Rev. D 71, 095002 (2005), arXiv:hep-ph/0412011 .
  * Anderson  _et al._ (1994) G. Anderson, S. Raby, S. Dimopoulos, L. J. Hall, and G. D. Starkman, “A Systematic SO(10) operator analysis for fermion masses,” Phys. Rev. D 49, 3660–3690 (1994), arXiv:hep-ph/9308333 .
  * Antusch  _et al._ (2020) Stefan Antusch, Christian Hohl, and Vasja Susič, “Yukawa ratio predictions in non-renormalizable $\mathrm{SO}(10)$ GUT models,” JHEP 02, 086 (2020), arXiv:1911.12807 [hep-ph] .
  * Weinberg (1979) Steven Weinberg, “Baryon- and lepton-nonconserving processes,” Phys. Rev. Lett. 43, 1566–1570 (1979).
  * Wilczek and Zee (1979) Frank Wilczek and A. Zee, “Operator analysis of nucleon decay,” Phys. Rev. Lett. 43, 1571–1573 (1979).
  * Sakai and Yanagida (1982) N. Sakai and Tsutomu Yanagida, “Proton Decay in a Class of Supersymmetric Grand Unified Models,” Nucl. Phys. B 197, 533 (1982).
  * Golowich (1981) Eugene Golowich, “SCALAR MEDIATED PROTON DECAY,” Phys. Rev. D 24, 2899 (1981).
  * Dorsner and Fileviez Perez (2005a) Ilja Dorsner and Pavel Fileviez Perez, “Could we rotate proton decay away?” Phys. Lett. B 606, 367–370 (2005a), arXiv:hep-ph/0409190 .
  * Dorsner and Fileviez Perez (2005b) Ilja Dorsner and Pavel Fileviez Perez, “How long could we live?” Phys. Lett. B 625, 88–95 (2005b), arXiv:hep-ph/0410198 .
  * Kolešová and Malinský (2019) Helena Kolešová and Michal Malinský, “Flavor structure of GUTs and uncertainties in proton lifetime estimates,” Phys. Rev. D 99, 035005 (2019), arXiv:1612.09178 [hep-ph] .
  * Buchmuller and Patel (2019) Wilfried Buchmuller and Ketan M. Patel, “Proton decay in flux compactifications,” JHEP 05, 196 (2019), arXiv:1904.08810 [hep-ph] .
  * Patel and Shukla (2022) Ketan M. Patel and Saurabh K. Shukla, “Anatomy of scalar mediated proton decays in SO(10) models,” JHEP 08, 042 (2022), arXiv:2203.07748 [hep-ph] .
  * Patel and Shukla (2023) Ketan M. Patel and Saurabh K. Shukla, “Spectrum of color sextet scalars in realistic SO(10) GUT,” Phys. Rev. D 107, 055008 (2023), arXiv:2211.11283 [hep-ph] .
  * Wiesenfeldt (2005) Soren Wiesenfeldt, “Operator analysis for proton decay in SUSY SO(10) GUT models,” Phys. Rev. D 71, 075006 (2005), arXiv:hep-ph/0501223 .
  * Helo  _et al._ (2019) Juan Carlos Helo, Martin Hirsch, and Toshihiko Ota, “Proton decay at one loop,” Phys. Rev. D 99, 095021 (2019), arXiv:1904.00036 [hep-ph] .
  * Gargalionis  _et al._ (2024) John Gargalionis, Juan Herrero-Garcia, and Michael A. Schmidt, “Model-independent estimates for loop-induced baryon-number-violating nucleon decays,” (2024), arXiv:2401.04768 [hep-ph] .
  * Beneito  _et al._ (2023) Arnau Bas Beneito, I, John Gargalionis, Juan Herrero-Garcia, Arcadi Santamaria, and Michael A. Schmidt, “An EFT approach to baryon number violation: lower limits on the new physics scale and correlations between nucleon decay modes,” (2023), arXiv:2312.13361 [hep-ph] .
  * Gell-Mann  _et al._ (1979) Murray Gell-Mann, Pierre Ramond, and Richard Slansky, “Complex Spinors and Unified Theories,” Conf. Proc. C 790927, 315–321 (1979), arXiv:1306.4669 [hep-th] .
  * Nath and Syed (2001a) Pran Nath and Raza M. Syed, “Analysis of couplings with large tensor representations in SO(2N) and proton decay,” Phys. Lett. B 506, 68–76 (2001a), [Erratum: Phys.Lett.B 508, 216–216 (2001)], arXiv:hep-ph/0103165 .
  * Nath and Syed (2001b) Pran Nath and Raza M. Syed, “Complete cubic and quartic couplings of 16 and bar-16 in SO(10) unification,” Nucl. Phys. B 618, 138–156 (2001b), arXiv:hep-th/0109116 .
  * Feger  _et al._ (2020) Robert Feger, Thomas W. Kephart, and Robert J. Saskowski, “LieART 2.0 – A Mathematica application for Lie Algebras and Representation Theory,” Comput. Phys. Commun. 257, 107490 (2020), arXiv:1912.10969 [hep-th] .
  * Heeck and Takhistov (2020) Julian Heeck and Volodymyr Takhistov, “Inclusive Nucleon Decay Searches as a Frontier of Baryon Number Violation,” Phys. Rev. D 101, 015005 (2020), arXiv:1910.07647 [hep-ph] .
  * Harlander  _et al._ (2020) R. V. Harlander, S. Y. Klein, and M. Lipp, “FeynGame,” Comput. Phys. Commun. 256, 107465 (2020), arXiv:2003.00896 [physics.ed-ph] .
  * Kobach (2016) Andrew Kobach, “Baryon Number, Lepton Number, and Operator Dimension in the Standard Model,” Phys. Lett. B 758, 455–457 (2016), arXiv:1604.05726 [hep-ph] .
  * Helset and Kobach (2020) Andreas Helset and Andrew Kobach, “Baryon Number, Lepton Number, and Operator Dimension in the SMEFT with Flavor Symmetries,” Phys. Lett. B 800, 135132 (2020), arXiv:1909.05853 [hep-ph] .
  * Dreiner  _et al._ (2010) Herbi K. Dreiner, Howard E. Haber, and Stephen P. Martin, “Two-component spinor techniques and Feynman rules for quantum field theory and supersymmetry,” Phys. Rept. 494, 1–196 (2010), arXiv:0812.1594 [hep-ph] .
  * Babu and Mohapatra (2012a) K. S. Babu and R. N. Mohapatra, “B-L Violating Proton Decay Modes and New Baryogenesis Scenario in SO(10),” Phys. Rev. Lett. 109, 091803 (2012a), arXiv:1207.5771 [hep-ph] .
  * Babu and Mohapatra (2012b) K. S. Babu and R. N. Mohapatra, “B-L Violating Nucleon Decay and GUT Scale Baryogenesis in SO(10),” Phys. Rev. D 86, 035018 (2012b), arXiv:1203.5544 [hep-ph] .
  * Mummidi and Patel (2021) V. Suryanarayana Mummidi and Ketan M. Patel, “Leptogenesis and fermion mass fit in a renormalizable SO(10) model,” JHEP 12, 042 (2021), arXiv:2109.04050 [hep-ph] .
  * Minkowski (1977) Peter Minkowski, “$\mu\to e\gamma$ at a Rate of One Out of $10^{9}$ Muon Decays?” Phys. Lett. B 67, 421–428 (1977).
  * Yanagida (1979) Tsutomu Yanagida, “Horizontal gauge symmetry and masses of neutrinos,” Conf. Proc. C 7902131, 95–99 (1979).
  * Mohapatra and Senjanovic (1980) Rabindra N. Mohapatra and Goran Senjanovic, “Neutrino Mass and Spontaneous Parity Nonconservation,” Phys. Rev. Lett. 44, 912 (1980).
  * Schechter and Valle (1980) J. Schechter and J. W. F. Valle, “Neutrino Masses in SU(2) x U(1) Theories,” Phys. Rev. D 22, 2227 (1980).
  * Abe  _et al._ (2014) K. Abe _et al._ (Super-Kamiokande), “Search for proton decay via $p\to\nu K^{+}$ using 260 kiloton·year data of Super-Kamiokande,” Phys. Rev. D 90, 072005 (2014), arXiv:1408.1195 [hep-ex] .
  * Doršner  _et al._ (2016) I. Doršner, S. Fajfer, A. Greljo, J. F. Kamenik, and N. Košnik, “Physics of leptoquarks in precision experiments and at particle colliders,” Phys. Rept. 641, 1–68 (2016), arXiv:1603.04993 [hep-ph] .
  * Raj (2017) Nirmal Raj, “Anticipating nonresonant new physics in dilepton angular spectra at the LHC,” Phys. Rev. D 95, 015011 (2017), arXiv:1610.03795 [hep-ph] .
  * Padhan  _et al._ (2020) Rojalin Padhan, Sanjoy Mandal, Manimala Mitra, and Nita Sinha, “Signatures of $\tilde{R}_{2}$ class of Leptoquarks at the upcoming $ep$ colliders,” Phys. Rev. D 101, 075037 (2020), arXiv:1912.07236 [hep-ph] .
  * Claudson  _et al._ (1982) Mark Claudson, Mark B. Wise, and Lawrence J. Hall, “Chiral Lagrangian for Deep Mine Physics,” Nucl. Phys. B 195, 297–307 (1982).
  * Chadha and Daniel (1983) S. Chadha and M. Daniel, “Chiral Lagrangian Calculation of Nucleon Decay Modes Induced by $d=5$ Supersymmetric Operators,” Nucl. Phys. B 229, 105–114 (1983).
  * Aoki  _et al._ (2000) S. Aoki _et al._ (JLQCD), “Nucleon decay matrix elements from lattice QCD,” Phys. Rev. D 62, 014506 (2000), arXiv:hep-lat/9911026 .
  * Cabibbo  _et al._ (2003) Nicola Cabibbo, Earl C. Swallow, and Roland Winston, “Semileptonic hyperon decays,” Ann. Rev. Nucl. Part. Sci. 53, 39–75 (2003), arXiv:hep-ph/0307298 .
  * Zyla  _et al._ (2020) P.A. Zyla _et al._ (Particle Data Group), “Review of Particle Physics,” PTEP 2020, 083C01 (2020).
  * Alonso  _et al._ (2014) Rodrigo Alonso, Hsi-Ming Chang, Elizabeth E. Jenkins, Aneesh V. Manohar, and Brian Shotwell, “Renormalization group evolution of dimension-six baryon number violating operators,” Phys. Lett. B 734, 302–307 (2014), arXiv:1405.0486 [hep-ph] .