# Anatomy of scalar mediated proton decays in $SO(10)$ models

Ketan M. Patel<EMAIL_ADDRESS>Theoretical Physics Division, Physical
Research Laboratory, Navarangpura, Ahmedabad-380009, India  Saurabh K. Shukla
<EMAIL_ADDRESS>Theoretical Physics Division, Physical Research
Laboratory, Navarangpura, Ahmedabad-380009, India  Indian Institute of
Technology Gandhinagar, Palaj-382355, India

###### Abstract

Realistic models based on the renormalizable grand unified theories have
varieties of scalars, many of which are capable of mediating baryon ($B$) and
lepton ($L$) number non-conserving processes. We identify all such scalar
fields residing in ${\bf 10}$, $\overline{\bf 126}$ and ${\bf 120}$
dimensional irreducible representations of $SO(10)$ which can induce baryon
and lepton number violating interactions through the leading order $d=6$ and
$d=7$ operators. Explicitly computing their couplings with the standard model
fermions, we derive the effective operators including the possibility of
mixing between the scalars stemming from a given representation. We find that
such interactions at $d=6$ are mediated by only three sets of scalars:
$T(3,1,-1/3)$, ${\cal T}(3,1,-4/3)$ and $\mathbb{T}(3,3,-1/3)$ and their
conjugates. In the models with ${\bf 10}$ and $\overline{\bf 126}$, only the
first has appropriate couplings to mediate the proton decay. While ${\cal T}$
and $\mathbb{T}$ can induce baryon number violating interactions when ${\bf
120}$ is present, ${\cal T}$ does not contribute to the proton decay at tree
level because of its flavour antisymmetric coupling. Three additional colour
triplets and their conjugates can mediate nucleon decay via $d=7$ operators
which violate also the $B-L$. We give general expressions for partial widths
of proton in terms of the fundamental Yukawa couplings and use these results
to explicitly compute the proton lifetime and branching ratios for the minimal
non-supersymmetric $SO(10)$ model based on ${\bf 10}$ and $\overline{\bf 126}$
Higgs. We find that the proton preferably decays into $\overline{\nu}\,K^{+}$
or $\mu^{+}\,K^{0}$ and list several distinct features of scalar mediated
proton decay. If the latter dominates over the gauge mediated contributions,
the proton decay spectrum provides a direct probe to the flavour structure of
the underlying grand unified theory.

##  I Background

Unification of the Standard Model (SM) gauge symmetries in simple gauge groups
Fritzsch and Minkowski (1975); Georgi and Glashow (1974); Gell-Mann  _et al._
(1979) often implies the existence of a rich spectrum of the scalars beyond
the electroweak doublet Higgs. The latter is just one of the many
submultiplets of an irreducible representation (irrep) of the underlying
unified symmetry group. The dimensions of these representations are typically
larger than the ones which unify, partially or completely, the SM quarks and
leptons of a given generation. This is particularly the case in renormalizable
models in which more than one type of Yukawa interactions are needed to
reproduce the realistic flavour spectrum including neutrino masses and mixing
parameters Dimopoulos  _et al._ (1992); Babu and Mohapatra (1993); Clark  _et
al._ (1982); Aulakh and Mohapatra (1983); Aulakh  _et al._ (2004). For
example, in renormalizable models based on $SO(10)$ Grand Unified Theory
(GUT), one needs at least two of the three, namely ${\bf 10}$, $\overline{\bf
126}$ and ${\bf 120}$, irreps for this purpose Bajc  _et al._ (2006);
Joshipura and Patel (2011). They respectively contain 4, 22 and 24 numbers of
SM scalar multiplets with diverse colour and electroweak charges, see Table 1.
Their non-trivial charges under the SM gauge symmetries and couplings with the
SM quarks and leptons make them interesting from the phenomenological
considerations. Potential new physics effects of many of these scalars have
been explored in the context of the recent flavour anomalies (see for example,
Bordone  _et al._ (2016); Belanger  _et al._ (2022); Perez  _et al._ (2021);
Sahoo  _et al._ (2021)), muon $g-2$ anomaly Bauer and Neubert (2016); Chen
_et al._ (2017); Doršner  _et al._ (2020), some events at the direct search
experiments Dorsner  _et al._ (2010); Patel and Sharma (2011); Doršner  _et
al._ (2016), GUT scale baryogenesis Babu and Mohapatra (2012a, b) and
precision unification of gauge couplings in the absence of low energy
supersymmetry Fileviez Perez  _et al._ (2008); Dorsner  _et al._ (2010); Patel
and Sharma (2011); Babu and Mohapatra (2012a, b). All these require at least a
few scalars at the sub GUT scale.

Some of the scalars with non-zero $B-L$ charges are capable of mediating
baryon number ($B$) and lepton number ($L$) violating decays of baryons Pati
and Salam (1973); Weinberg (1979); Wilczek and Zee (1979) (see also Langacker
(1981); Nath and Fileviez Perez (2007); Babu  _et al._ (2013); Raby (2017);
Fukuyama  _et al._ (2005) for reviews) depending on their couplings with the
SM quarks and leptons. In typical bottom-up approaches, the freedom to choose
their couplings is often exploited to avoid dangerously fast proton decays.
However, if these scalars are part of GUT multiplets which also contains the
SM Higgs like fields, their couplings are often the same couplings that
determine the low energy quark and lepton mass spectrum. Hence, there may
exist strong constraints on the masses of these scalar fields in these models
from the stability of the proton. Such constraints may then forbid them from
being light from the perspective of top-down approaches. Therefore, it is
important to derive explicitly the exact nature of the couplings of various
scalars in the underlying GUT framework and their precise predictions for the
nucleon decay spectrum. Such an analysis has been carried out earlier for
$SU(5)$ and flipped $SU(5)$ GUTs in Golowich (1981); Dorsner  _et al._ (2012).
We perform a comprehensive study for a class of renormalizable $SO(10)$ GUTs
which has a relatively richer spectrum of scalars and more constrained
coupling structure for Yukawa interactions than those in $SU(5)$ GUTs because
of a complete unification of quarks and leptons.

Scalar induced contribution to the proton decay has received less attention in
comparison to the one mediated by vector bosons in GUTs. The latter has been
extensively studied including the flavour effects Dorsner and Fileviez Perez
(2005a); Fileviez Perez (2004); Dorsner and Fileviez Perez (2005b); Kolešová
and Malinský (2019). Firstly, this is because the vector boson induced
contributions dominate over the scalar mediated ones if the mass scale of both
the mediators is of similar magnitude as the latter are suppressed by the
first generation Yukawa couplings. The ratio of scalar to vector induced
contributions in the proton decay width is naively given by

$\frac{y_{u}^{4}/M_{S}^{4}}{g^{4}/M_{V}^{4}}\sim\left({\cal
O}(10^{-4})\,\frac{M_{V}}{M_{S}}\right)^{4}\,,$ (1)

where $g$ is unified gauge coupling, $y_{u}$ is up-type quark Yukawa coupling
and $M_{V,S}$ are masses of vector and scalar mediators, respectively.
Therefore, scalar contributions becomes significant only if
$M_{S}\lesssim{\cal O}(10^{-4})\,M_{V}$. Secondly, the scalar mediated
contributions depend on the Yukawa sector of the theory and are model-
dependent to a large extent111The vector boson induced contributions also
depend on the Yukawa interactions through unitary rotations which relate
physical basis with the field basis. This dependency, however, is indirect and
relatively mild in comparison to the scalar mediated contributions.. Many of
the important aspects of the scalar induced nucleon decays depend on the
nature of the mediator and its exact couplings with the SM quarks and leptons.
These details are not completely captured in the generic model-independent
effective theory based analysis Weinberg (1979); Wilczek and Zee (1979).

This work is aimed to provide a complete classification of the scalar induced
nucleon decays in the models based on renormalizable $SO(10)$ GUTs.
Identifying various scalar fields residing in ${\bf 10}$, $\overline{\bf 126}$
and ${\bf 120}$ which carry $B-L$ current, we determine their explicit
couplings with the SM quarks and leptons with appropriate Clebsch-Gordan
coefficients. Interactions that can give rise to nucleon decays at tree level
through effective dimension-6 ($d=6$) and dimension-7 ($d=7$) are then listed
out and used to determine explicitly the proton decay widths in terms of the
fundamental Yukawa couplings. In the predictive versions of these GUTs, the
latter can be determined, partially or fully, from the observed masses and
mixing parameters of the quarks and leptons. This enables computation of the
proton decay widths with less ambiguity in comparison to the effective theory
based estimations. We demonstrate this by estimating proton decays for the
minimal non-supersymmetric $SO(10)$ GUT which uses ${\bf 10}_{H}$ and
$\overline{\bf 126}_{H}$ to account for realistic fermion spectrum. The proton
decay pattern is found significantly different from the one induced by vector
bosons in this case.

We give a complete spectrum of various scalar sub-multiplets which may arise
in renormalizable $SO(10)$ models in the next section and compute their
couplings relevant for nucleon decays. In sections III and IV, we derive
effective operators after integrating out the relevant scalars and give the
explicit expressions of proton decay widths in terms of these effective
couplings in section V. These results are used to compute the scalar mediated
proton decay spectrum in a minimal model based on two GUT representations in
the section VI. Finally, we summarize in section VII.

##  II Scalar spectrum and couplings

In the renormalizable versions of $SO(10)$ gauge theory, the Yukawa sector, in
general, can consist of scalars which are ${\bf 10}$, $\overline{\bf 126}$ and
${\bf 120}$ dimensional irreps of the underlying gauge group. These irreps,
when decomposed under the SM gauge group, contain several multiplets charged
under $B-L$ which can give rise to baryon and/or lepton number violating
interactions. The SM and $B-L$ charges of these fields and their multiplicity
in ${\bf 10}_{H}$, $\overline{\bf 126}_{H}$ and ${\bf 120}_{H}$ are listed in
Table 1. We use convention in which,

$B-L=\frac{4}{5}Y-\frac{1}{5}X\,,$ (2)

where $X$ and $Y$ are the charges under $U(1)_{X}$ and $U(1)_{Y}$ subgroups of
$SO(10)$, respectively Buchmuller and Patel (2019). The hypercharge generator
is normalized in such a way that the electric charge $Q=T_{3}+Y$.

SM charges |  Notation |  $B-L$ |  ${\bf 10}_{H}$ |  $\overline{\bf 126}_{H}$ |  ${\bf 120}_{H}$
---|---|---|---|---|---
$\left(1,1,0\right)$ | $\sigma$ | $-2$ | $0$ | $1$ | $0$
$\left(1,1,1\right)$ | $s$ | $2$ | $0$ | 0 | $1$
$\left(1,1,-1\right)$ | $\overline{s}$ | $-2$ | 0 | 1 | 1
$\left(1,1,-2\right)$ | $X$ | $-2$ | 0 | 1 | 0
$\left(1,2,-\frac{1}{2}\right)$ | $\overline{D}_{a}$ | 0 | 1 | 1 | 2
$\left(1,2,\frac{1}{2}\right)$ | $D^{a}$ | 0 | 1 | 1 | 2
$\left(1,3,1\right)$ | $t^{ab}$ | 2 | 0 | 1 | 0
$\left(3,1,-\frac{1}{3}\right)$ | $T^{\alpha}$ | $-\frac{2}{3}$ | 1 | 2 | 2
$\left(\overline{3},1,\frac{1}{3}\right)$ | $\overline{T}_{\alpha}$ | $\frac{2}{3}$ | 1 | 1 | 2
$\left(3,1,\frac{2}{3}\right)$ | $\Theta_{\alpha\beta}$ | $-\frac{2}{3}$ | 0 | 1 | 1
$\left(\overline{3},1,-\frac{2}{3}\right)$ | $\overline{\Theta}^{\alpha\beta}$ | $\frac{2}{3}$ | 0 | 0 | 1
$\left(3,1,-\frac{4}{3}\right)$ | $\mathcal{T}^{\alpha}$ | $-\frac{2}{3}$ | 0 | 1 | 1
$\left(\overline{3},1,\frac{4}{3}\right)$ | $\overline{\mathcal{T}}_{\alpha}$ | $\frac{2}{3}$ | 0 | 0 | 1
$\left(3,2,\frac{1}{6}\right)$ | $\Delta^{\alpha a}$ | $\frac{4}{3}$ | 0 | 1 | 1
$\left(\overline{3},2,-\frac{1}{6}\right)$ | $\overline{\Delta}_{\alpha a}$ | $-\frac{4}{3}$ | 0 | 1 | 1
$\left(3,2,\frac{7}{6}\right)$ | $\Omega^{a}_{\alpha\beta}$ | $\frac{4}{3}$ | 0 | 1 | 1
$\left(\overline{3},2,-\frac{7}{6}\right)$ | $\overline{\Omega}_{a}^{\alpha\beta}$ | $-\frac{4}{3}$ | 0 | 1 | 1
$\left(3,3,-\frac{1}{3}\right)$ | $\mathbb{T}{{}^{a\alpha}_{b}}$ | $-\frac{2}{3}$ | 0 | 0 | 1
$\left(\overline{3},3,\frac{1}{3}\right)$ | $\overline{\mathbb{T}}{{}^{a}_{b\alpha}}$ | $\frac{2}{3}$ | 0 | 1 | 1
$\left(6,1,\frac{1}{3}\right)$ | $S{{}^{\alpha}_{\beta\gamma}}$ | $\frac{2}{3}$ | 0 | 1 | 1
$\left(\overline{6},1,-\frac{1}{3}\right)$ | $\overline{S}{{}^{\beta\gamma}_{\alpha}}$ | $-\frac{2}{3}$ | 0 | 0 | 1
$\left(6,1,-\frac{2}{3}\right)$ | $\Sigma^{\alpha\beta}$ | $\frac{2}{3}$ | 0 | 1 | 0
$\left(6,1,\frac{4}{3}\right)$ | ${\cal S}^{\alpha}_{\beta\gamma}$ | $\frac{2}{3}$ | 0 | 1 | 0
$\left(\overline{6},3,-\frac{1}{3}\right)$ | $\mathbb{S}^{\alpha\beta a}_{\gamma b}$ | $-\frac{2}{3}$ | 0 | 1 | 0
$\left(8,2,\frac{1}{2}\right)$ | $O^{\alpha a}_{\beta}$ | 0 | 0 | 1 | 1
$\left(8,2,-\frac{1}{2}\right)$ | $\overline{O}_{\alpha a}^{\beta}$ | 0 | 0 | 1 | 1
Table 1: Classification of different scalar fields residing in ${\bf 10}_{H}$,
$\overline{\bf 126}_{H}$ and ${\bf 120}_{H}$ with their charges under the SM
gauge group ($SU(3)_{C}$, $SU(2)_{L}$, $U(1)_{Y}$), $B-L$ charge and
multiplicities in the given GUT representation.

To derive couplings of these scalars with the SM quarks and leptons, we
decompose each of the $SO(10)$ invariant Yukawa terms into the ones invariant
under the SM gauge symmetry. This is done by first decomposing the underlying
$SO(10)$ interactions into $SU(5)$ and then the latter is further decomposed
in terms of the interactions involving the SM fermions and appropriately
normalised scalars. In the following, we use $i,j,k,...$ to represent $SU(5)$
tensors, $\alpha,\beta,\gamma,...$ and $a,b,c,...$ to represent $SU(3)$ and
$SU(2)$ tensors, respectively. The three generations of fermions are denoted
by $A,B=1,2,3$. The irreps of $SO(10)$ are distinguished from those of $SU(5)$
by indicating the former in bold.

###  II.1 ${\bf 16}_{F}$-${\bf 16}_{F}$-${\bf 10}_{H}$ couplings

The interactions of ${\bf 16}$-plet fermions with ${\bf 10}_{H}$ is
parametrized by

$-{\cal L}^{\bf 10}_{Y}=H_{AB}\,{\bf 16}^{T}_{A}\,C^{-1}\,{\bf 16}_{B}\,{\bf
10}_{H}\,+\,{\rm h.c.}\,,$ (3)

where $H_{AB}=H_{BA}$. $C$ is the Lorentz charge conjugation matrix. For
brevity, we have suppressed the equivalent matrix in the gauge space while
writing the above term. For complex scalar ${\bf 10}_{H}$, one can also have
an additional gauge invariant Yukawa term with ${\bf 10}_{H}^{\dagger}$.
However, to derive the couplings of scalars with SM quarks and leptons, it is
sufficient to use one of these. The $SO(10)$ invariant terms can be translated
into the $SU(5)$ language following a procedure discussed in Mohapatra and
Sakita (1980); Nath and Syed (2001); Syed (2005). Under the $SU(5)$
decomposition, Eq. (3) becomes

$-{\cal L}^{\bf
10}_{Y}=i2\sqrt{2}\,H_{AB}\,\left({10^{ij}_{A}}^{T}\,C^{-1}\,\overline{5}_{iB}\,\overline{5}_{j}\,+\,\frac{1}{8}\epsilon_{ijklm}\,{10^{ij}}^{T}_{A}\,C^{-1}\,10^{kl}_{B}\,5^{m}\,-\,1_{A}^{T}\,C^{-1}\,\overline{5}_{iB}\,5^{i}\right)\,+\,{\rm
h.c.}\,.$ (4)

All the $SU(5)$ representations appearing above are defined in such a way that
their kinetic terms are normalized.

The $SU(5)$ fields are further decomposed in the SM fields as the following.
The fermions residing in $10$, $\overline{5}$ and $1$ dimensional irreps can
be identified with the SM fermions as

$10^{\alpha\beta}\equiv\epsilon^{\alpha\beta\gamma}\,u^{C}_{\gamma}\,,~{}~{}10^{\alpha
a}\equiv q^{\alpha a}\,,~{}~{}10^{ab}\equiv\epsilon^{ab}\,e^{C}\,,$ (5)
$\overline{5}_{\alpha}\equiv
d^{C}_{\alpha}\,,~{}~{}\overline{5}_{a}\equiv\epsilon_{ab}\,l^{b}\,,~{}~{}1\equiv\nu^{C}\,,$
(6)

where all the fermion fields are left-handed Weyl fermions. Similarly, the
scalars identified in $5$\- and $\overline{5}$-plet can be written as

$\displaystyle 5^{\alpha}\equiv T^{\alpha}\,,~{}~{}5^{a}\equiv
D^{a}\,,~{}~{}\overline{5}_{\alpha}\equiv\overline{T}_{\alpha}\,,~{}~{}\overline{5}_{a}\equiv\overline{D}_{a}\,.$
(7)

such that their kinetic terms are also in the canonically normalized form.
Substitution of Eqs. (5,6,7) in Eq. (4) leads to the following terms involving
$B-L$ charged scalar fields.

$\displaystyle-{\cal L}^{{\bf 10}/5}_{Y}$ $\displaystyle\supset$
$\displaystyle i2\sqrt{2}\,H_{AB}\,\left(u^{C\,T}_{\gamma
A}\,C^{-1}\,e^{C}_{B}-\frac{1}{2}\epsilon_{\alpha\beta\gamma}\,\epsilon_{ab}\,q^{\alpha
a\,T}_{A}\,C^{-1}\,q^{\beta b}_{B}\right)T^{\gamma}\,+\,{\rm h.c.}\,,$
$\displaystyle-{\cal L}^{{\bf 10}/\overline{5}}_{Y}$ $\displaystyle\supset$
$\displaystyle
i2\sqrt{2}\,H_{AB}\,\left(\epsilon^{\alpha\beta\gamma}\,u^{C\,T}_{\alpha
A}\,C^{-1}\,d^{C}_{\beta B}-\epsilon_{ab}\,q^{\gamma
a\,T}_{A}\,C^{-1}\,l^{b}_{B}\right)\overline{T}_{\gamma}\,+\,{\rm h.c.}\,.$
(8)

Consequently, both $T$ and $\overline{T}$ have diquark and leptoquark
couplings and each can mediate $B$ and $L$ violating decays of baryons.

###  II.2 ${\bf 16}_{F}$-${\bf 16}_{F}$-$\overline{\bf 126}_{H}$ couplings

The $\overline{\bf 126}_{H}$ has a very rich scalar spectrum. The Yukawa
Lagrangian involving this field is written as

$-{\cal L}^{\overline{\bf 126}}_{Y}=F_{AB}\,{\bf 16}^{T}_{A}\,C^{-1}\,{\bf
16}_{B}\,\overline{\bf 126}_{H}\,+\,{\rm h.c.}\,,$ (9)

where $F_{AB}=F_{BA}$. In the $SU(5)$ notations, the above Lagrangian can be
written as Nath and Syed (2001)

$\displaystyle-{\cal L}^{\overline{\bf 126}}_{Y}$ $\displaystyle=$
$\displaystyle
i\frac{2}{\sqrt{15}}\,F_{AB}\,\Big{[}-1{{}^{T}_{A}}\,C^{-1}\,1_{B}\,1_{0}+1{{}^{T}_{A}}\,C^{-1}\,10_{B}^{ij}\,\overline{10}_{ij}-\overline{5}{{}_{iA}^{T}}\,C^{-1}\,\overline{5}_{jB}\,15^{ij}\,\Big{.}$
(10) $\displaystyle-$
$\displaystyle\sqrt{\frac{3}{2}}\,\left(1{{}^{T}_{A}}\,C^{-1}\,\overline{5}_{iB}\,5^{i}\,+\frac{1}{24}\epsilon_{ijklm}\,10^{ij\,T}_{A}\,C^{-1}\,10^{kl}_{B}\,5^{m}\right)\,$
$\displaystyle+$
$\displaystyle\Big{.}10^{ij\,T}_{A}\,C^{-1}\,\overline{5}_{kB}\,\overline{45}^{k}_{ij}-\frac{1}{4\sqrt{3}}\,\epsilon_{ijklm}\,10^{ij\,T}_{A}\,C^{-1}\,10^{rs}_{B}\,50^{klm}_{rs}\Big{]}\,+\,{\rm
h.c.}\,.$

The first term above does not involve the SM fermions. Similarly, the second
term gives rise to lepto-quark coupling with only the SM singlet leptons which
are typically heavier than the nucleon. Hence, both these terms are not
relevant for the proton decay.

The $\overline{\bf 126}_{H}$ contains a $5$-plet of $SU(5)$ whose
decomposition is given earlier in Eq. (7). Using this and the second line in
Eq. (10), we get

$\displaystyle-{\cal L}^{\overline{\bf 126}/5}_{Y}$ $\displaystyle\supset$
$\displaystyle
i\sqrt{\frac{2}{5}}\,F_{AB}\,\left(-\frac{1}{3}u{{}^{CT}_{\gamma
A}}\,C^{-1}\,e^{C}_{B}+\frac{1}{6}\epsilon_{\alpha\beta\gamma}\,\epsilon_{ab}\,q^{\alpha
aT}_{A}\,C^{-1}\,q^{\beta b}_{B}\right)T_{1}^{\gamma}\,+\,{\rm h.c.}\,.$ (11)

Since $\overline{\bf 126}_{H}$ has more than one color triplet fields, we
distinguish them by assigning a subscript. Decomposition of $15$-plet into the
normalized irreps of $SU(5)$ is given by

$\displaystyle 15^{\alpha\beta}=\Sigma^{\alpha\beta}\,,~{}~{}15^{\alpha
a}=\frac{1}{\sqrt{2}}\Delta^{\alpha a}\,,~{}~{}15^{ab}=t^{ab}\,.$ (12)

$\Sigma^{\alpha\beta}$ ($t^{ab}$) is color sextet (weak triplet) and has only
diquark (dilepton) couplings. Therefore, they do not contribute in the proton
decay. The color triplet weak doublet field $\Delta^{\alpha a}$ has $Y=1/6$
and its couplings with the SM fermions are obtained as

$\displaystyle-{\cal L}^{\overline{\bf 126}/15}_{Y}$ $\displaystyle\supset$
$\displaystyle i2\sqrt{\frac{2}{15}}\,F_{AB}\,\epsilon_{ab}\,d{{}^{CT}_{\alpha
A}}\,C^{-1}\,l^{a}_{B}\,\Delta^{\alpha b}\,+\,{\rm h.c.}\,.$ (13)

$\Delta^{\alpha b}$ has only the lepto-quark coupling and hence it cannot
induce proton decay by itself.

Next, we consider $\overline{45}$-plet which decomposes into various irreps of
$SU(5)$ as:

$\displaystyle\overline{45}^{\gamma}_{\alpha\beta}$ $\displaystyle\equiv$
$\displaystyle\frac{1}{\sqrt{2}}S^{\gamma}_{\alpha\beta}+\frac{1}{2\sqrt{2}}\left(\delta^{\gamma}_{\alpha}\overline{T}_{\beta}-\delta^{\gamma}_{\beta}\overline{T}_{\alpha}\right)\,,~{}~{}\overline{45}^{a}_{\alpha\beta}\equiv\frac{1}{\sqrt{2}}\Omega^{a}_{\alpha\beta}\,\,,$
$\displaystyle\overline{45}^{\beta}_{\alpha a}$ $\displaystyle\equiv$
$\displaystyle\frac{1}{\sqrt{2}}\overline{O}^{\beta}_{\alpha
a}+\frac{1}{2\sqrt{6}}\delta^{\beta}_{\alpha}\overline{D}_{a}\,,~{}~{}\overline{45}^{\beta}_{ab}\equiv\frac{1}{\sqrt{2}}\epsilon_{ab}{\cal
T}^{\beta}\,,$ $\displaystyle\overline{45}^{a}_{b\alpha}$
$\displaystyle\equiv$
$\displaystyle\frac{1}{\sqrt{2}}\overline{\mathbb{T}}^{a}_{b\alpha}-\frac{1}{2\sqrt{2}}\delta^{a}_{b}\overline{T}_{\alpha}\,,~{}~{}\overline{45}^{c}_{ab}\equiv-\frac{\sqrt{3}}{2\sqrt{2}}\left(\delta^{c}_{a}\overline{D}_{b}-\delta^{c}_{b}\overline{D}_{a}\right)\,.$
(14)

The SM and $B-L$ charges of these fields are listed in Table 1. All the fields
other than $\overline{O}^{\beta}_{\alpha a}$ and $\overline{D}_{a}$ carry non-
trivial $B-L$ charges. However, $S_{\alpha\beta}^{\gamma}$ couples to the
quarks only while $\Omega{{}^{a}_{\alpha\beta}}=(3,2,7/6)$ has only lepto-
quark vertex. The couplings of the remaining multiplets with the SM quarks and
leptons are obtained as

$\displaystyle-{\cal L}^{\overline{\bf 126}/\overline{45}}_{Y}$
$\displaystyle\supset$ $\displaystyle
i\sqrt{\frac{2}{15}}\,F_{AB}\,\left[\left(\epsilon^{\alpha\beta\gamma}\,u^{CT}_{\alpha
A}\,C^{-1}\,\,d^{C}_{\beta B}+\epsilon_{ab}\,q^{\gamma
aT}_{A}\,C^{-1}\,l^{b}_{B}\right)\overline{T}_{\gamma}\,\right.$ (15)
$\displaystyle+$ $\displaystyle\left.2\,e^{CT}_{A}\,C^{-1}\,d^{C}_{\alpha
B}\,{\cal T}^{\alpha}\,+\,2\,\epsilon_{bc}\,q^{\alpha
aT}_{A}\,C^{-1}\,l^{b}_{B}\,\overline{\mathbb{T}}^{c}_{a\alpha}\right]\,+{\rm
h.c.}\,.$

Note that the chromo-weak triplet $\mathbb{T}^{a}_{b\alpha}$ and $Y=-4/3$
color triplet ${\cal T}^{\alpha}$ also have only lepto-quark coupling at this
stage. This, along with quark-quark couplings arising from ${\bf 120}_{H}$,
can induce the nucleon decay if there exists a mixing between these scalars in
the underlying model.

Finally, the coupling with $50$-plet of $SU(5)$ can be computed from its
following decomposition:

$\displaystyle 50^{\alpha\beta\gamma}_{\sigma\rho}$ $\displaystyle\equiv$
$\displaystyle\frac{1}{6}\left[\left(\delta^{\alpha}_{\sigma}\delta^{\beta}_{\rho}-\delta^{\alpha}_{\rho}\delta^{\beta}_{\sigma}\right)T^{\gamma}+\left(\delta^{\alpha}_{\rho}\delta^{\gamma}_{\sigma}-\delta^{\alpha}_{\sigma}\delta^{\gamma}_{\rho}\right)T^{\beta}+\left(\delta^{\beta}_{\sigma}\delta^{\gamma}_{\rho}-\delta^{\beta}_{\rho}\delta^{\gamma}_{\sigma}\right)T^{\alpha}\right]\,,$
$\displaystyle 50^{\alpha\beta\gamma}_{\sigma a}$ $\displaystyle\equiv$
$\displaystyle\frac{1}{2\sqrt{3}}\left(\delta^{\alpha}_{\sigma}\overline{\Omega}^{\beta\gamma}_{a}+\delta^{\beta}_{\sigma}\overline{\Omega}^{\gamma\alpha}_{a}+\delta^{\gamma}_{\sigma}\overline{\Omega}^{\alpha\beta}_{a}\right)\,,$
$\displaystyle 50^{\alpha\beta a}_{\sigma\rho}$ $\displaystyle\equiv$
$\displaystyle\frac{1}{2\sqrt{6}}\left(\delta^{\alpha}_{\sigma}O^{\beta
a}_{\rho}-\delta^{\alpha}_{\rho}O^{\beta
a}_{\sigma}+\delta^{\beta}_{\rho}O^{\alpha
a}_{\sigma}-\delta^{\beta}_{\sigma}O^{\alpha a}_{\rho}\right)\,,$
$\displaystyle 50^{\alpha\beta\gamma}_{ab}$ $\displaystyle\equiv$
$\displaystyle\frac{1}{2\sqrt{3}}\epsilon^{\alpha\beta\gamma}\epsilon_{ab}X\,,~{}~{}50^{\alpha\beta
a}_{\sigma b}\equiv\frac{1}{\sqrt{6}}\mathbb{S}^{\alpha\beta a}_{\sigma
b}+\frac{1}{12}\delta^{a}_{b}\left(\delta^{\alpha}_{\sigma}T^{\beta}-\delta^{\beta}_{\sigma}T^{\alpha}\right)\,,$
$\displaystyle 50^{\alpha ab}_{\sigma\rho}$ $\displaystyle\equiv$
$\displaystyle\frac{1}{2\sqrt{3}}\epsilon^{ab}{\cal
S}^{\alpha}_{\sigma\rho}\,,~{}~{}50^{\alpha\beta
a}_{bc}\equiv\frac{1}{2\sqrt{3}}\left(\delta^{a}_{c}\overline{\Omega}^{\alpha\beta}_{b}-\delta^{a}_{b}\overline{\Omega}^{\alpha\beta}_{c}\right)\,,$
$\displaystyle 50^{\alpha ab}_{\sigma c}$ $\displaystyle\equiv$
$\displaystyle\frac{1}{2\sqrt{6}}\left(\delta^{b}_{c}O^{\alpha
a}_{\sigma}-\delta^{a}_{c}O^{\alpha b}_{\sigma}\right)\,,~{}~{}50^{\alpha
ab}_{cd}=\frac{1}{6}\left(\delta^{a}_{c}\delta^{b}_{d}-\delta^{a}_{d}\delta^{b}_{c}\right)T^{\alpha}\,.$
(16)

It is straightforward to see from the above decomposition and Eq. (10) that
the $B-L$ carrying fields ${\cal S}^{\alpha}_{\sigma\rho}$ and
$\mathbb{S}^{\alpha\beta a}_{\sigma b}$ couple to quarks only while the
$\overline{\Omega}^{\alpha\beta}_{a}$ has only a lepto-quark vertex.
Similarly, $X$ couples to the leptons only. As a result, only $T^{\alpha}$ can
contribute to the nucleon decays. The corresponding couplings are given by

$\displaystyle-{\cal L}^{\overline{\bf 126}/50}_{Y}$ $\displaystyle\supset$
$\displaystyle-i\frac{2}{3\sqrt{5}}\,F_{AB}\,\left(u^{CT}_{\gamma
A}\,C^{-1}\,e^{C}_{B}+\frac{1}{2}\epsilon_{\alpha\beta\gamma}\,\epsilon_{ab}\,q^{\alpha
aT}_{A}\,C^{-1}\,q^{\beta b}_{B}\right)T_{2}^{\gamma}\,+\,{\rm h.c.}\,.$ (17)

###  II.3 ${\bf 16}_{F}$-${\bf 16}_{F}$-${\bf 120}_{H}$ couplings

Finally, consider the Yukawa interactions with ${\bf 120}_{H}$ as

$-{\cal L}^{\bf 120}_{Y}=G_{AB}\,{\bf 16}^{T}_{A}\,C^{-1}\,{\bf 16}_{B}\,{\bf
120}_{H}\,+\,{\rm h.c.}\,,$ (18)

where $G_{AB}=-G_{BA}$ is anti-symmetric coupling unlike the previous cases.
One can also have a similar but independent couplings with ${\bf
120}_{H}^{\dagger}$ if the latter is complex scalar. Under $SU(5)$, the above
can be written as

$\displaystyle-{\cal L}^{\bf 120}_{Y}$ $\displaystyle=$ $\displaystyle
i\frac{2}{\sqrt{3}}\,G_{AB}\,\Big{[}-2\,1{{}^{T}_{A}}\,C^{-1}\,\overline{5}_{iB}\,5^{i}\,-10^{ij\,T}_{A}\,C^{-1}\,\overline{5}{{}_{iB}}\,\overline{5}_{j}\,+\sqrt{2}\,\overline{5}{{}^{T}_{iA}}\,C^{-1}\,\overline{5}_{jB}\,10^{ij}\Big{.}$
$\displaystyle-$
$\displaystyle\sqrt{2}\,1{{}^{T}_{A}}\,C^{-1}\,10^{ij}_{B}\,\overline{10}_{ij}\,-\,\frac{1}{2\sqrt{2}}\,\epsilon_{ijklm}\,10^{ij\,T}_{A}\,C^{-1}\,10^{mn}_{B}\,45^{kl}_{n}\,+\sqrt{2}\,\overline{5}{{}^{T}_{iA}}\,C^{-1}\,10^{jk}_{B}\,\overline{45}^{i}_{jk}\Big{]}\,+\,{\rm
h.c.}$

The coupling with $5$-plet involves only the SM singlet leptons. Using the
decompositions of $\overline{5}$, we find that

$\displaystyle-{\cal L}^{{\bf 120}/\overline{5}}_{Y}$ $\displaystyle\supset$
$\displaystyle-i\frac{2}{\sqrt{3}}\,G_{AB}\,\left(\epsilon^{\alpha\beta\gamma}\,u^{CT}_{\alpha
A}\,C^{-1}\,d^{C}_{\beta B}-\epsilon_{ab}\,q^{\gamma
aT}_{A}\,C^{-1}\,l^{b}_{B}\right)\overline{T}_{1\gamma}\,+\,{\rm h.c.}\,.$
(20)

Like in the case of $\overline{126}_{H}$, the $\overline{10}$ belonging to
${\bf 120}_{H}$ also couples to $\nu^{C}$ only and hence does not play any
role in the nucleon decay. The $10$-plet can be decomposed as

$\displaystyle
10^{\alpha\beta}=\overline{\Theta}^{\alpha\beta}\,,~{}~{}10^{\alpha
a}=\frac{1}{\sqrt{2}}\Delta^{\alpha
a}\,,~{}~{}10^{ab}=\epsilon^{ab}\,\overline{s}\,.$ (21)

Apparently, $\overline{s}$ couples to the leptons only. The field
$\Delta^{\alpha a}$ has the same quantum charge as the one in Eq. (12). As it
is evident from Eq. (II.3), $\overline{\Delta}$ and $\Theta$ couple to the
right-handed neutrinos. The interactions with $\Delta^{\alpha a}$ and
$\overline{\Theta}^{\alpha\beta}$ are obtained as

$\displaystyle-{\cal L}^{{\bf 120}/10}_{Y}$ $\displaystyle\supset$
$\displaystyle-i\frac{2}{\sqrt{3}}\,G_{AB}\,\left(2\epsilon_{ab}\,d^{CT}_{\alpha
A}\,C^{-1}\,l^{a}_{B}\,\Delta^{\alpha b}\,-\sqrt{2}\,d^{CT}_{\alpha
A}\,C^{-1}\,d^{C}_{\beta B}\,\overline{\Theta}^{\alpha\beta}\right)\,+\,{\rm
h.c.}\,.$ (22)

The first term is similar to Eq. (13) which has only lepto-quark couplings.
$\overline{\Theta}^{\alpha\beta}$ has only the diquark couplings. Therefore,
none of these scalars residing in $10$ and $\overline{10}$ can induce proton
decay through $d=6$ operators. However, they can contribute to $B-L$ violating
decays of nucleon as we discuss later.

The couplings of remaining fields residing in the $45$\- and
$\overline{45}$-plets can straightforwardly be computed using the
decomposition already given in Eq. (II.2). We find

$\displaystyle-{\cal L}^{{\bf 120}/45}_{Y}$ $\displaystyle\supset$
$\displaystyle i\frac{2}{\sqrt{3}}\,G_{AB}\,\Big{(}2\,u^{CT}_{\alpha
A}\,C^{-1}\,e^{C}_{B}\,T^{\alpha}\,+\,\epsilon^{\alpha\beta\gamma}\,u^{CT}_{\alpha
A}\,C^{-1}\,u^{C}_{\beta B}\,\overline{{\cal T}}_{\gamma}\Big{.}$ (23)
$\displaystyle+$
$\displaystyle\Big{.}\epsilon_{\alpha\beta\gamma}\,\epsilon_{ab}\,q^{\alpha
aT}_{A}\,C^{-1}\,q^{\beta
c}_{B}\,\mathbb{T}^{b\gamma}_{c}\,-\,\epsilon_{\alpha\beta\gamma}\,e^{CT}_{A}\,C^{-1}\,q^{\gamma
a}_{B}\,\overline{\Omega}^{\alpha\beta}_{a}\Big{)}\,+{\rm h.c.}\,.$
$\displaystyle-{\cal L}^{{\bf 120}/\overline{45}}_{Y}$ $\displaystyle\supset$
$\displaystyle-i\frac{2}{\sqrt{3}}\,G_{AB}\,\left[\left(\epsilon^{\alpha\beta\gamma}\,u^{CT}_{\alpha
A}\,C^{-1}\,d^{C}_{\beta B}+\epsilon_{ab}\,q^{\gamma
aT}_{A}\,C^{-1}\,l^{b}_{B}\right)\overline{T}_{2\gamma}\,\right.$ (24)
$\displaystyle+$ $\displaystyle 2\,e^{CT}_{A}\,C^{-1}\,d^{C}_{\alpha B}\,{\cal
T}^{\alpha}\,+\,2\,\epsilon_{bc}\,q^{\alpha
aT}_{A}\,C^{-1}\,l^{b}_{B}\,\overline{\mathbb{T}}^{c}_{a\alpha}$
$\displaystyle-$
$\displaystyle\left.\epsilon^{\alpha\beta\gamma}\,\epsilon_{ab}\,l^{bT}_{A}\,C^{-1}\,u^{C}_{\gamma
B}\,\Omega^{a}_{\alpha\beta}\right]\,+{\rm h.c.}\,.$

Note that the triplet residing in $45$ of $SU(5)$ does not couple to the
quarks because of flavour anti-symmetric couplings. Moreover, unlike in Eq.
(15), the fields ${\cal T}^{\alpha}$ and $\mathbb{T}^{a\alpha}_{b}$ have
quark-quark couplings. Their conjugate fields have lepto-quark couplings.
$\Omega^{a}_{\alpha\beta}$ and its conjugate partner have only the lepto-quark
couplings and therefore they do not induce $d=6$ operator but contribute to
the $d=7$ operators.

All together the couplings in Eqs. (II.1,11,15,17,20,23,24) determine the
amplitude of nucleon decays in the most general renormalizable versions of the
$SO(10)$ GUTs. The noteworthy features are:

  * •

Out of the six pairs of colour triplets with different electroweak charges
(see Table 1), only three pairs have couplings suitable for destabilizing
protons at the tree level through $d=6$ operators. They are
$T$-$\overline{T}$, ${\cal T}$-$\overline{\cal T}$ and
$\mathbb{T}$-$\overline{\mathbb{T}}$. The remaining three pairs, namely
$\Theta$-$\overline{\Theta}$, ${\Delta}$-$\overline{\Delta}$ and
$\Omega$-$\overline{\Omega}$ can induce nucleon decay through $d=7$ operators.

  * •

If the underlying model does not contain ${\bf 120}_{H}$ then nucleon decay is
induced only by pair of colour triplets $T$, $\overline{T}$ with $|Y|=1/3$.
The ${\bf 10}_{H}$ contains a pair of such fields while $\overline{\bf
126}_{H}$ contains a pair and additional $T$. While the other two triplets are
present in $\overline{\bf 126}_{H}$, they have only the lepto-quark couplings.
Moreover, $\Delta$ together with $\overline{T}$ can also induce nucleon decay
through $B-L$ breaking.

  * •

In the presence of ${\bf 120}_{H}$, the nucleon decays can also receive
contributions from ${\cal T}$-$\overline{\cal T}$ and
$\mathbb{T}$-$\overline{\mathbb{T}}$. These fields can contribute by
themselves as well as by mixing with the fields of same charges residing in
$\overline{\bf 126}_{H}$ in the models in which both $\overline{\bf 126}_{H}$
and ${\bf 120}_{H}$ are present.

The results obtained in this section can also be used to derive the couplings
relevant for proton decays in the most general renormalizable $SU(5)$ GUT
models.

##  III Dimension-6 effective operators

We now compute the leading $d=6$ effective operators relevant for the $B$ and
$L$ violating decays of baryons by integrating out the relevant scalar fields.
We treat each case of ${\bf 10}_{H}$, $\overline{\bf 126}_{H}$ and ${\bf
120}_{H}$ separately and comment on the possibility of mixing between the
scalar fields at the end of this section.

###  III.1 From ${\bf 10}$-plet

The contribution to the nucleon decay in models with only ${\bf 10}_{H}$
arises from a pair of color triplet and anti-triplet, $T$ and $\overline{T}$ ,
as can be seen from Eq. (II.1). The most general mass terms for these triplets
can be written as:

$m_{T}^{2}\,T^{\dagger}T+m_{\overline{T}}^{2}\,\overline{T}^{\dagger}\overline{T}+\left(\mu^{2}\,T\,\overline{T}+{\rm
h.c.}\right)\,$ (25)

where the first two terms can originate from $SO(10)$ invariant combination
${\bf 10}_{H}^{\dagger}{\bf 10}_{H}$ and the last term from ${\bf 10}_{H}^{2}$
Slansky (1981). In general, $m_{T}\neq m_{\overline{T}}$ as the mass splitting
can arise if the model also contains ${\bf 45}_{H}$ and/or ${\bf 54}_{H}$.
Because of the presence of the mixing term, the physical states are different
and can be obtained by the following replacements:

$T\to
T\cos\theta_{T}+\overline{T}^{*}\sin\theta_{T}\,,~{}~{}\overline{T}\to\overline{T}\cos\theta_{T}-T^{*}\sin\theta_{T}\,,$
(26)

Here, the mixing angle is obtained from Eq. (25) as

$\tan 2\theta_{T}=\frac{2\mu^{2}}{m_{\overline{T}}^{2}-m_{T}^{2}}\,.$ (27)

With the above replacements, Eq. (25) becomes

$M_{T}^{2}\,T^{\dagger}T+M_{\overline{T}}^{2}\,\overline{T}^{\dagger}\overline{T}\,.$
(28)

Substituting Eq. (26) in Eq. (II.1) and integrating out the physical triplet
and anti-triplet, we obtain the following effective Lagrangian relevant for
$B$ violating baryon decays after some straight-forward algebraic
manipulations:

$\displaystyle{\cal L}^{{\bf 10}}_{\rm eff}$ $\displaystyle=$ $\displaystyle
8\left(\frac{c_{T}^{2}}{M_{T}^{2}}+\frac{s_{T}^{2}}{M_{\overline{T}}^{2}}\right)H_{AB}H^{\dagger}_{CD}\,\,\epsilon^{\alpha\beta\gamma}\,u_{\alpha
A}^{T}C^{-1}d_{\beta B}\,{e^{C}_{C}}^{\dagger}C^{-1}{u^{C}_{\gamma D}}^{*}$
(29) $\displaystyle+$ $\displaystyle
8\left(\frac{s_{T}^{2}}{M_{T}^{2}}+\frac{c_{T}^{2}}{M_{\overline{T}}^{2}}\right)H_{AB}H^{\dagger}_{CD}\,\,\epsilon^{\alpha\beta\gamma}\,u{{}^{CT}_{\alpha
A}}C^{-1}d_{\beta B}^{C}\,\left(e_{C}^{\dagger}C^{-1}u_{\gamma
D}^{*}-\nu_{C}^{\dagger}C^{-1}d_{\gamma D}^{*}\right)$ $\displaystyle+$
$\displaystyle
8s_{T}c_{T}\left(\frac{1}{M_{T}^{2}}-\frac{1}{M_{\overline{T}}^{2}}\right)H_{AB}H^{T}_{CD}\,\,\epsilon^{\alpha\beta\gamma}\,u{{}^{CT}_{\alpha
A}}C^{-1}d^{C}_{\beta B}\,{e^{CT}_{C}}C^{-1}{u^{C}_{\gamma D}}$
$\displaystyle+$ $\displaystyle
8s_{T}c_{T}\left(\frac{1}{M_{T}^{2}}-\frac{1}{M_{\overline{T}}^{2}}\right)H_{AB}H^{T}_{CD}\,\,\epsilon^{\alpha\beta\gamma}\,u_{\alpha
A}^{T}C^{-1}d_{\beta B}\,\left(e_{C}^{T}C^{-1}u_{\gamma
D}-\nu_{C}^{T}C^{-1}d_{\gamma D}\right)$ $\displaystyle+$ $\displaystyle{\rm
h.c.}\,,$

where $c_{T}=\cos\theta_{T}$ and $s_{T}=\sin\theta_{T}$. We have used the
identity $C^{\dagger}=C^{T}=C^{-1}=-C$ in determining Eq. (29).

The effective operators in the physical basis of fermions can be obtain by
replacing $f\to U_{f}f$ in the above ${\cal L}^{{\bf 10}}_{\rm eff}$. The
unitary matrices $U_{f}$ and $U_{f^{C}}$ (with $f=u,d,e,\nu$) can be
explicitly computed from the corresponding fermion mass matrices. In the
physical basis, we get

$\displaystyle{\cal L}^{{\bf 10}}_{\rm eff}$ $\displaystyle=$ $\displaystyle
h[u_{A},d_{B},e^{C}_{C},u^{C}_{D}]\,\,\epsilon^{\alpha\beta\gamma}\,u_{\alpha
A}^{T}\,C^{-1}\,d_{\beta B}\,{e^{C}_{C}}^{\dagger}\,C^{-1}\,u{{}^{C*}_{\gamma
D}}$ (30) $\displaystyle+$ $\displaystyle
h[u^{C}_{A},d^{C}_{B},e_{C},u_{D}]\,\,\epsilon^{\alpha\beta\gamma}\,u{{{}^{CT}_{\alpha
A}}}\,C^{-1}\,d_{\beta B}^{C}\,e_{C}^{\dagger}\,C^{-1}\,u_{\gamma D}^{*}\,$
$\displaystyle+$ $\displaystyle
h[u^{C}_{A},d^{C}_{B},\nu_{C},d_{D}]\,\,\epsilon^{\alpha\beta\gamma}\,u{{}^{CT}_{\alpha
A}}\,C^{-1}\,d_{\beta B}^{C}\,\nu_{C}^{\dagger}\,C^{-1}\,d_{\gamma D}^{*}\,$
$\displaystyle+$ $\displaystyle
h^{\prime}[u^{C}_{A},d^{C}_{B},e^{C}_{C},u^{C}_{D}]\,\,\epsilon^{\alpha\beta\gamma}\,u{{}^{CT}_{\alpha
A}}\,C^{-1}\,d^{C}_{\beta B}\,e{{}^{CT}_{C}}\,C^{-1}\,u^{C}_{\gamma D}$
$\displaystyle+$ $\displaystyle
h^{\prime}[u_{A},d_{B},e_{C},u_{D}]\,\,\epsilon^{\alpha\beta\gamma}\,u_{\alpha
A}^{T}\,C^{-1}\,d_{\beta B}\,e_{C}^{T}\,C^{-1}\,u_{\gamma D}\,$
$\displaystyle+$ $\displaystyle
h^{\prime}[u_{A},d_{B},\nu_{C},d_{D}]\,\,\epsilon^{\alpha\beta\gamma}\,u_{\alpha
A}^{T}\,C^{-1}\,d_{\beta B}\,\nu_{C}^{T}\,C^{-1}\,d_{\gamma D}\,\,+{\rm
h.c.}\,,$

with

$\displaystyle h[u_{A},d_{B},e^{C}_{C},u^{C}_{D}]$ $\displaystyle=$
$\displaystyle
8\left(\frac{c_{T}^{2}}{M_{T}^{2}}+\frac{s_{T}^{2}}{M_{\overline{T}}^{2}}\right)\left(U_{u}^{T}HU_{d}\right)_{AB}\,\left(U_{e^{C}}^{\dagger}H^{\dagger}U_{u^{C}}^{*}\right)_{CD}\,,$
$\displaystyle h[u^{C}_{A},d^{C}_{B},e_{C},u_{D}]$ $\displaystyle=$
$\displaystyle
8\left(\frac{s_{T}^{2}}{M_{T}^{2}}+\frac{c_{T}^{2}}{M_{\overline{T}}^{2}}\right)\left(U_{u^{C}}^{T}HU_{d^{C}}\right){{}_{AB}}\,\left(U_{e}^{\dagger}H^{\dagger}U_{u}^{*}\right){{}_{CD}}\,,$
$\displaystyle h[u^{C}_{A},d^{C}_{B},\nu_{C},d_{D}]$ $\displaystyle=$
$\displaystyle-8\left(\frac{s_{T}^{2}}{M_{T}^{2}}+\frac{c_{T}^{2}}{M_{\overline{T}}^{2}}\right)\left(U_{u^{C}}^{T}HU_{d^{C}}\right){{}_{AB}}\,\left(U_{\nu}^{\dagger}H^{\dagger}U_{d}^{*}\right){{}_{CD}}\,,$
$\displaystyle h^{\prime}[u^{C}_{A},d^{C}_{B},e^{C}_{C},u^{C}_{D}]$
$\displaystyle=$ $\displaystyle
8s_{T}c_{T}\left(\frac{1}{M_{T}^{2}}-\frac{1}{M_{\overline{T}}^{2}}\right)\left(U_{u^{C}}^{T}HU_{d^{C}}\right){{}_{AB}}\,\left(U_{e^{C}}^{T}H^{T}U_{e^{D}}\right){{}_{CD}}\,,$
$\displaystyle h^{\prime}[u_{A},d_{B},e_{C},u_{D}]$ $\displaystyle=$
$\displaystyle
8s_{T}c_{T}\left(\frac{1}{M_{T}^{2}}-\frac{1}{M_{\overline{T}}^{2}}\right)\left(U_{u}^{T}HU_{d}\right)_{AB}\,\left(U_{e}^{T}H^{T}U_{u}\right)_{CD}\,,$
$\displaystyle h^{\prime}[u_{A},d_{B},\nu_{C},d_{D}]$ $\displaystyle=$
$\displaystyle-8s_{T}c_{T}\left(\frac{1}{M_{T}^{2}}-\frac{1}{M_{\overline{T}}^{2}}\right)\left(U_{u}^{T}HU_{d}\right)_{AB}\,\left(U_{\nu}^{T}H^{T}U_{d}\right)_{CD}\,.$
(31)

In the absence of mixing term between $T$ and $\overline{T}$ (i.e.
$\theta_{T}=0$), all $h^{\prime}=0$. Moreover, the coefficients of the first
and the next two operators become independent from each other.

###  III.2 From $\overline{\bf 126}$-plet

Unlike in the case of ${\bf 10}_{H}$, the color triplets and anti-triplet do
not mix with each other in this since $(\overline{\bf 126}_{H})^{2}$ is
forbidden by $SO(10)$ gauge symmetry Slansky (1981). The triplets residing in
$\overline{5}$ and $50$ belonging to $\overline{\bf 126}_{H}$ can mix with
each other in general but such a mixing can arise from the gauge invariant
interactions with other fields and hence is model dependent. Further, as noted
earlier, ${\cal T}^{\alpha}$ and $\overline{\mathbb{T}}^{c}_{a\alpha}$ have
only the lepto-quark couplings. Therefore, the effective operators relevant
for the nucleon decay take simple form in case of $\overline{\bf 126}_{H}$.
Integrating out the triplets and ant-triplet from Eqs.(11,15,17), we find

$\displaystyle{\cal L}^{\overline{\bf 126}}_{\rm eff}$ $\displaystyle=$
$\displaystyle\frac{2}{45}\left(\frac{1}{M_{T_{1}}^{2}}-\frac{2}{M_{T_{2}}^{2}}\right)\,F_{AB}F^{\dagger}_{CD}\,\,\epsilon^{\alpha\beta\gamma}\,u_{\alpha
A}^{T}C^{-1}d_{\beta B}\,{e^{C}_{C}}^{\dagger}C^{-1}{u^{C}_{\gamma D}}^{*}$
(32) $\displaystyle-$
$\displaystyle\frac{2}{15}\frac{1}{M_{\overline{T}}^{2}}\,F_{AB}F^{\dagger}_{CD}\,\,\epsilon^{\alpha\beta\gamma}\,u{{}^{CT}_{\alpha
A}}C^{-1}d_{\beta B}^{C}\,\left(e_{C}^{\dagger}C^{-1}u_{\gamma
D}^{*}-\nu_{C}^{\dagger}C^{-1}d_{\gamma D}^{*}\right)\,+{\rm h.c.}\,,$

where $M_{T_{i}}$ and $M_{\overline{T}}$ are masses of triplets $T_{i}$ and
anti-triplet $\overline{T}$, respectively.

In the physical basis, the above can be rewritten as

$\displaystyle{\cal L}^{\overline{\bf 126}}_{\rm eff}$ $\displaystyle=$
$\displaystyle
f[u_{A},d_{B},e^{C}_{C},u^{C}_{D}]\,\,\epsilon^{\alpha\beta\gamma}\,u_{\alpha
A}^{T}\,C^{-1}\,d_{\beta B}\,{e^{C}_{C}}^{\dagger}\,C^{-1}\,u{{}^{C*}_{\gamma
D}}$ (33) $\displaystyle+$ $\displaystyle
f[u^{C}_{A},d^{C}_{B},e_{C},u_{D}]\,\,\epsilon^{\alpha\beta\gamma}\,u{{}^{CT}_{\alpha
A}}\,C^{-1}\,d_{\beta B}^{C}\,e_{C}^{\dagger}\,C^{-1}\,u_{\gamma D}^{*}\,$
$\displaystyle+$ $\displaystyle
f[u^{C}_{A},d^{C}_{B},\nu_{C},d_{D}]\,\,\epsilon^{\alpha\beta\gamma}\,u{{}^{CT}_{\alpha
A}}\,C^{-1}\,d_{\beta B}^{C}\,\nu_{C}^{\dagger}\,C^{-1}\,d_{\gamma
D}^{*}\,+{\rm h.c.}\,,$

with

$\displaystyle f[u_{A},d_{B},e^{C}_{C},u^{C}_{D}]$ $\displaystyle=$
$\displaystyle\frac{2}{45}\left(\frac{1}{M_{T_{1}}^{2}}-\frac{2}{M_{T_{2}}^{2}}\right)\left(U_{u}^{T}FU_{d}\right)_{AB}\,\left(U_{e^{C}}^{\dagger}F^{\dagger}U_{u^{C}}^{*}\right)_{CD}\,,$
$\displaystyle f[u^{C}_{A},d^{C}_{B},e_{C},u_{D}]$ $\displaystyle=$
$\displaystyle-\frac{2}{15}\frac{1}{M_{\overline{T}}^{2}}\left(U_{u^{C}}^{T}FU_{d^{C}}\right){{}_{AB}}\,\left(U_{e}^{\dagger}F^{\dagger}U_{u}^{*}\right){{}_{CD}}\,,$
$\displaystyle f[u^{C}_{A},d^{C}_{B},\nu_{C},d_{D}]$ $\displaystyle=$
$\displaystyle\frac{2}{15}\frac{1}{M_{\overline{T}}^{2}}\left(U_{u^{C}}^{T}FU_{d^{C}}\right){{}_{AB}}\,\left(U_{\nu}^{\dagger}F^{\dagger}U_{d}^{*}\right){{}_{CD}}\,.$
(34)

The structure of these operators is same as of the ones obtained in case of
${\bf 10}_{H}$ with a noteworthy difference of relative factor between the
contributions mediated by color triplets and anti-triplet.

###  III.3 From ${\bf 120}$-plet

Since $({\bf 120}_{H})^{2}$ is a gauge invariant, $T\in 45$ and
$\overline{T}_{2}\in\overline{45}$ can have a mixing term. Similarly, ${\cal
T}$ and $\mathbb{T}^{a\alpha}_{b}$ can mix with their conjugate partners
appearing in Eqs. (23,24). Adopting the similar procedure discussed in the
first subsection, we obtain the following operators after eliminating the
different colour triplet scalars:

$\displaystyle{\cal L}^{{\bf 120}}_{\rm eff}$ $\displaystyle=$
$\displaystyle\frac{4}{3}\left(\frac{1}{M_{\overline{T}_{1}}^{2}}-\frac{c_{T}^{2}}{M_{\overline{T}_{2}}^{2}}-\frac{s_{T}^{2}}{M_{T}^{2}}\right)\,G_{AB}G^{\dagger}_{CD}\,\,\epsilon^{\alpha\beta\gamma}\,u{{}^{CT}_{\alpha
A}}\,C^{-1}\,d_{\beta B}^{C}\,\left(e_{C}^{\dagger}\,C^{-1}\,u_{\gamma
D}^{*}-\nu_{C}^{\dagger}C^{-1}d_{\gamma D}^{*}\right)\,$ (35) $\displaystyle+$
$\displaystyle\frac{8}{3}s_{T}c_{T}\left(\frac{1}{M{{}^{2}_{T}}}-\frac{1}{M_{\overline{T}_{2}}^{2}}\right)G_{AB}G_{CD}\,\,\epsilon^{\alpha\beta\gamma}\,u{{}^{CT}_{\alpha
A}}\,C^{-1}\,d^{C}_{\beta B}\,e{{}^{CT}_{C}}\,C^{-1}\,{u^{C}_{\gamma D}}$
$\displaystyle-$ $\displaystyle\frac{8}{3}s_{\cal T}c_{\cal
T}\left(\frac{1}{M_{\cal T}^{2}}-\frac{1}{M_{\overline{\cal
T}}^{2}}\right)G_{AB}G_{CD}\,\,\epsilon^{\alpha\beta\gamma}\,u{{}^{CT}_{\alpha
A}}\,C^{-1}\,u^{C}_{\beta B}\,e{{}^{CT}_{C}}\,C^{-1}\,{d^{C}_{\gamma D}}\,$
$\displaystyle-$
$\displaystyle\frac{8}{3}s_{\mathbb{T}}c_{\mathbb{T}}\left(\frac{1}{M_{\mathbb{T}}^{2}}-\frac{1}{M_{\overline{\mathbb{T}}}^{2}}\right)G_{AB}G_{CD}\,\epsilon^{\alpha\beta\gamma}\,\epsilon_{bc}\epsilon_{da}\,q_{\alpha
A}^{aT}\,C^{-1}\,l^{b}_{B}\,q_{\beta C}^{dT}\,C^{-1}\,q^{c}_{\gamma
D}\,+\,{\rm h.c.}\,$

Here, $\theta_{T}$, $\theta_{\cal T}$ and $\theta_{\mathbb{T}}$ are angles
which parametrize mixing term between $T$-$\overline{T}$, ${\cal
T}$-$\overline{\cal T}$ and $\mathbb{T}$-$\overline{\mathbb{T}}$,
respectively, in an analogous way to Eq. (27). Since $T$ has only the lepto-
quark couplings, its contribution to the nucleon decays arises only through
the mixing term.

Using the Fierz rearrangement

$\left(\psi_{1}^{T}C^{-1}\psi_{2}\right)\left(\psi_{3}^{T}C^{-1}\psi_{4}\right)=-\left(\psi_{1}^{T}C^{-1}\psi_{3}\right)\left(\psi_{4}^{T}C^{-1}\psi_{2}\right)-\left(\psi_{1}^{T}C^{-1}\psi_{4}\right)\left(\psi_{2}^{T}C^{-1}\psi_{3}\right)\,,$
(36)

we rewrite the operator given in the third line in Eq. (35) as

$G_{AB}G_{CD}\,\epsilon^{\alpha\beta\gamma}\,u{{}^{CT}_{\alpha
A}}\,C^{-1}\,u^{C}_{\beta B}\,e{{}^{CT}_{C}}\,C^{-1}\,d^{C}_{\gamma
D}=2G_{AD}G_{CB}\,\,\epsilon^{\alpha\beta\gamma}\,u{{}^{CT}_{\alpha
A}}\,C^{-1}\,d^{C}_{\beta B}\,e{{}^{CT}_{C}}\,C^{-1}\,u^{C}_{\gamma D}\,,$
(37)

where we also use $G_{AB}=-G_{BA}$. Further, using
$\epsilon_{bc}\epsilon_{da}=\delta_{bd}\delta_{ca}-\delta_{ba}\delta_{cd}$ and
Eq. (36) the fourth operator can be simplified to

$\displaystyle
G_{AB}G_{CD}\epsilon^{\alpha\beta\gamma}\epsilon_{bc}\epsilon_{da}\,q_{\alpha
A}^{aT}\,C^{-1}\,l^{b}_{B}\,q_{\beta C}^{dT}\,C^{-1}\,q^{c}_{\gamma D}$ (38)
$\displaystyle=$
$\displaystyle-\left(G_{AB}G_{CD}-2G_{AD}G_{CB}\right)\,\epsilon^{\alpha\beta\gamma}\,u_{\alpha
A}^{T}\,C^{-1}\,d_{\beta B}\,e_{C}^{T}\,C^{-1}\,u_{\gamma D}$ $\displaystyle-$
$\displaystyle\left(G_{AB}G_{CD}-2G_{AC}G_{BD}\right)\,\epsilon^{\alpha\beta\gamma}\,u_{\alpha
A}^{T}\,C^{-1}\,d_{\beta B}\,\nu_{C}^{T}\,C^{-1}\,d_{\gamma D}\,.$

Substituting Eqs. (37,38) in Eq. (35) and using the physical basis for
fermions, we get

$\displaystyle{\cal L}^{{\bf 120}}_{\rm eff}$ $\displaystyle=$ $\displaystyle
g[u^{C}_{A},d^{C}_{B},e_{C},u_{D}]\,\,\epsilon^{\alpha\beta\gamma}\,u{{}^{CT}_{\alpha
A}}\,C^{-1}\,d_{\beta B}^{C}\,e_{C}^{\dagger}\,C^{-1}\,u_{\gamma D}^{*}\,$
(39) $\displaystyle+$ $\displaystyle
g[u^{C}_{A},d^{C}_{B},\nu_{C},d_{D}]\,\,\epsilon^{\alpha\beta\gamma}\,u{{}^{CT}_{\alpha
A}}\,C^{-1}\,d_{\beta B}^{C}\,\nu_{C}^{\dagger}\,C^{-1}\,d_{\gamma D}^{*}\,$
$\displaystyle+$ $\displaystyle
g^{\prime}[u^{C}_{A},d^{C}_{B},e^{C}_{C},u^{C}_{D}]\,\,\epsilon^{\alpha\beta\gamma}\,u{{}^{CT}_{\alpha
A}}\,C^{-1}\,d_{\beta B}^{C}\,e{{}^{CT}_{C}}\,C^{-1}\,u^{C}_{\gamma D}\,$
$\displaystyle+$ $\displaystyle
g^{\prime}[u_{A},d_{B},e_{C},u_{D}]\,\,\epsilon^{\alpha\beta\gamma}\,u_{\alpha
A}^{T}\,C^{-1}\,d_{\beta B}\,e_{C}^{T}\,C^{-1}\,u_{\gamma D}\,$
$\displaystyle+$ $\displaystyle
g^{\prime}[u_{A},d_{B},\nu_{C},d_{D}]\,\,\epsilon^{\alpha\beta\gamma}\,u_{\alpha
A}^{T}\,C^{-1}\,d_{\beta B}\,\nu_{C}^{T}\,C^{-1}\,d_{\gamma D}\,\,+{\rm
h.c.}\,,$

with

$\displaystyle g[u^{C}_{A},d^{C}_{B},e_{C},u_{D}]$ $\displaystyle=$
$\displaystyle\frac{4}{3}\left(\frac{1}{M_{\overline{T}_{1}}^{2}}-\frac{c_{T}^{2}}{M_{\overline{T}_{2}}^{2}}-\frac{s_{T}^{2}}{M_{T}^{2}}\right)\,\left(U_{u^{C}}^{T}GU_{d^{C}}\right){{}_{AB}}\,\left(U_{e}^{\dagger}G^{\dagger}U_{u}^{*}\right){{}_{CD}}\,,$
$\displaystyle g[u^{C}_{A},d^{C}_{B},\nu_{C},d_{D}]$ $\displaystyle=$
$\displaystyle-\frac{4}{3}\left(\frac{1}{M_{\overline{T}_{1}}^{2}}-\frac{c_{T}^{2}}{M_{\overline{T}_{2}}^{2}}-\frac{s_{T}^{2}}{M_{T}^{2}}\right)\,\left(U_{u^{C}}^{T}GU_{d^{C}}\right){{}_{AB}}\,\left(U_{\nu}^{\dagger}G^{\dagger}U_{d}^{*}\right){{}_{CD}}\,,$
$\displaystyle g^{\prime}[u^{C}_{A},d^{C}_{B},e^{C}_{C},u^{C}_{D}]$
$\displaystyle=$
$\displaystyle\frac{8}{3}s_{T}c_{T}\left(\frac{1}{M_{T}^{2}}-\frac{1}{M_{\overline{T}_{2}}^{2}}\right)\,\left(U_{u^{C}}^{T}GU_{d^{C}}\right){{}_{AB}}\,\left(U_{e^{C}}^{T}GU_{u^{C}}\right){{}_{CD}}\,$
$\displaystyle-$ $\displaystyle\frac{16}{3}s_{\cal T}c_{\cal
T}\left(\frac{1}{M_{\cal T}^{2}}-\frac{1}{M_{\overline{\cal
T}}^{2}}\right)\left(U_{u^{C}}^{T}GU_{u^{C}}\right)_{AD}\,\left(U_{e^{c}}^{T}GU_{d^{C}}\right)_{CB}\,,$
$\displaystyle g^{\prime}[u_{A},d_{B},e_{C},u_{D}]$ $\displaystyle=$
$\displaystyle\frac{8}{3}s_{\mathbb{T}}c_{\mathbb{T}}\left(\frac{1}{M_{\mathbb{T}}^{2}}-\frac{1}{M_{\overline{\mathbb{T}}}^{2}}\right)\Big{[}\left(U_{u}^{T}GU_{d}\right)_{AB}\,\left(U_{e}^{T}GU_{u}\right)_{CD}\big{.}$
$\displaystyle-$
$\displaystyle\Big{.}2\left(U_{u}^{T}GU_{u}\right)_{AD}\,\left(U_{e}^{T}GU_{d}\right)_{CB}\Big{]}\,,$
$\displaystyle g^{\prime}[u_{A},d_{B},\nu_{C},d_{D}]$ $\displaystyle=$
$\displaystyle\frac{8}{3}s_{\mathbb{T}}c_{\mathbb{T}}\left(\frac{1}{M_{\mathbb{T}}^{2}}-\frac{1}{M_{\overline{\mathbb{T}}}^{2}}\right)\Big{[}\left(U_{u}^{T}GU_{d}\right)_{AB}\,\left(U_{\nu}^{T}GU_{d}\right)_{CD}\big{.}$
(40) $\displaystyle-$
$\displaystyle\Big{.}2\left(U_{u}^{T}GU_{\nu}\right)_{AC}\,\left(U_{d}^{T}GU_{d}\right)_{BD}\Big{]}\,.$

The structure of effective couplings $g^{\prime}$ are very different from
$h^{\prime}$ due to additional contributions supplied by the new triplets.

Altogether, the operators listed in Eqs. (30,33,39) quantify the $B$ and $L$
violating but $B-L$ conserving baryon decays mediated by scalars residing in
${\bf 10}_{H}$, $\overline{\bf 126}_{H}$ and ${\bf 120}_{H}$, respectively,
where we have also considered the possibility of the mixing between the
various scalars arising from the given representation. In general, the scalar
fields of the same SM charges belonging to different GUT representations can
also mix. For example, in the models with at least two of these scalar
representations, different triplets and anti-triplets can mix and the lightest
pair would be a particular linear combination of them. Since the contribution
of the lightest pair is expected to be the most dominant, the operators
obtained by integrating out this pair would be the most relevant in
quantifying the proton decays. In general, such operators can be obtained with
coefficients which are linear combinations of the relevant $h$, $f$ and $g$,
for example. However, the exact quantification of these mixing depends not
only on the specification of the Yukawa sector but also on the full scalar
potential of the underlying model. Therefore, this exercise is highly model-
dependent. Nevertheless, the results obtained above can straightforwardly be
used to compute proton decay in specific models in which mixing between the
various triplets and anti-triplets is deterministic.

##  IV Dimension-7 effective operators

In the previous section, we consider the leading order $d=6$ operators which
violate $B$ and $L$ but conserve $B-L$. A new class of operators which violate
$B-L$ by two units arise at $d=7$ Weinberg (1980); Weldon and Zee (1980). Such
operators are induced by quartic couplings involving the SM singlet field
$\sigma$, one of the electroweak doublets and two color triplet fields. Using
the fields listed in Table 1, the following invariants can be constructed for
this kind of quartic term:

$\displaystyle\sigma\,D^{a}\,T^{\alpha}\,\overline{\Delta}_{\alpha a}\,$ ,
$\displaystyle~{}~{}\sigma\,\overline{D}_{a}\,\overline{T}_{\alpha}\,\Delta^{\alpha
a}\,;$ (41)
$\displaystyle\sigma\,D^{a}\,\Theta_{\alpha\beta}\,\overline{\Omega}^{\alpha\beta}_{a}\,$
,
$\displaystyle~{}~{}\sigma\,\overline{D}_{a}\,\overline{\Theta}^{\alpha\beta}\,\Omega_{\alpha\beta}^{a}\,;$
(42)
$\displaystyle\sigma\,D^{a}\,\overline{\Theta}^{\alpha\beta}\,\Delta^{\gamma
b}\,\epsilon_{\alpha\beta\gamma}\,\epsilon_{ab}\,$ ,
$\displaystyle~{}~{}\sigma\,\overline{D}_{a}\,\Theta_{\alpha\beta}\,\overline{\Delta}_{\gamma
b}\,\epsilon^{\alpha\beta\gamma}\,\epsilon^{ab}\,;$ (43)
$\displaystyle\sigma\,D^{a}\,\overline{\Delta}_{\alpha b}\,\mathbb{T}^{\alpha
b}_{a}\,$ , $\displaystyle~{}~{}\sigma\,\overline{D}_{a}\,\Delta^{\alpha
b}\,\overline{\mathbb{T}}_{\alpha b}^{a}\,.$ (44)

When $\sigma$ acquires a VEV, $B-L$ symmetry is broken and the above quartic
terms can give rise to $d=7$ operators involving four fermions and a Higgs
doublet. After the electroweak symmetry breaking, this generates effective
four-fermion operators with $\Delta(B-L)=-2$. Because of the latter, they give
rise to novel decay channels for the proton and neutron Babu and Mohapatra
(2012a, b).

The quartic terms listed in Eqs. (41-44) can arise from various $SO(10)$
invariant combinations of the scalar irreps, for example $(\overline{\bf
126}_{H})^{4}$, $(\overline{\bf 126}_{H})^{2}({\bf 120}_{H})^{2}$, etc. Such a
term must contain atleast one $\overline{\bf 126}_{H}$ (or ${\bf 126}_{H}$) as
a source of $\sigma$ while the electroweak doublets can come from either of
${\bf 10}_{H}$, $\overline{\bf 126}_{H}$ or ${\bf 120}_{H}$. Various triplets
appearing in Eqs. (41-44) can come from ${\bf 10}_{H}$, $\overline{\bf
126}_{H}$, ${\bf 120}_{H}$ or even from the scalars like ${\bf 126}_{H}$ which
are not part of the Yukawa sector. Although the latter does not directly
couple to quarks and leptons but can still induce nucleon decay by mixing with
the scalar sub-multiplets residing in ${\bf 10}_{H}$, $\overline{\bf 126}_{H}$
and ${\bf 120}_{H}$. The determination of exact operators in this case,
therefore, requires complete specification of the scalar sector of the
underlying model beyond the ones which take part into the Yukawa interactions.
As we remain agnostic about the complete model in this study, we perform the
subsequent analysis assuming that the color triplets fields in Eqs. (41-44)
arise from either of ${\bf 10}_{H}$, $\overline{\bf 126}_{H}$ or ${\bf
120}_{H}$.

###  IV.1 From ${\bf 10}$-plet

The generation of $d=7$ operators requires presence of atleast two different
kinds of color triplets as it can be seen from Eqs. (41-44). Since ${\bf
10}_{H}$ contains only one kind of color triplets, it cannot give rise to such
operators at the leading order by itself.

###  IV.2 From $\overline{\bf 126}$-plet

As it can be seen from Table 1, $\overline{\bf 126}_{H}$ does not contain
$\overline{\Theta}^{\alpha\beta}$ and $\mathbb{T}^{a\alpha}_{b}$. Among the
remaining fields responsible for generating $d=7$ operators,
$\Theta_{\alpha\beta}$ and $\overline{\Delta}_{\alpha a}$ reside in
$\overline{10}$ of $SU(5)$ which couple to the RH neutrinos only, see Eq.
(10). Moreover, $\Delta^{\alpha a}$, $\overline{\Omega}^{\alpha\beta}_{a}$ and
$\overline{\mathbb{T}}^{a}_{\alpha b}$ have only the lepto-quark couplings.
Therefore, only the second term in Eq. (41) can induce the required operator.
Using Eqs. (13,15) and integrating out $\Delta^{\alpha a}$ and
$\overline{T}_{\alpha}$, we get

$\displaystyle{\cal L}^{\overline{\bf 126}}_{\rm eff}$ $\displaystyle=$
$\displaystyle\frac{4\lambda
v_{\sigma}}{15M_{\Delta}^{2}M_{\overline{T}}^{2}}\,F_{AB}^{*}F_{CD}^{*}\,\epsilon^{\alpha\beta\gamma}\,\epsilon^{ab}\overline{D}_{a}\,u^{C\dagger}_{\alpha
A}\,C^{-1}\,{d^{C}_{\beta B}}^{*}\,l_{bC}^{\dagger}\,C^{-1}\,{d^{C}_{\gamma
D}}^{*}\,+\,{\rm h.c.}\,,$ (45)

where $\lambda$ is a coupling of the quartic term and $v_{\sigma}$ is the VEV
that breaks $B-L$. The above can be identified with operator $\tilde{O}_{1}$
listed in Babu and Mohapatra (2012b)222Note that we have assumed absence of
${\bf 126}_{H}$ which forbids $T$-$\overline{T}$ mixing as it was also the
case while deriving $d=6$ operators earlier. If such mixing is allowed, one
also finds an operator similar to $\tilde{O}_{3}$ given in Babu and Mohapatra
(2012b).

After the electroweak symmetry breaking, the above operator reduces to $d=6$
operator containing three quarks and a neutrino field. In the physical basis,
it can be parametrized as

$\displaystyle{\cal L}^{\overline{\bf 126}}_{\rm eff}$ $\displaystyle=$
$\displaystyle\tilde{f}[u^{C}_{A},d^{C}_{B},\nu_{C},d^{C}_{D}]\,\epsilon^{\alpha\beta\gamma}\,u^{CT}_{\alpha
A}\,C^{-1}\,{d^{C}_{\beta B}}\,\nu_{C}^{T}\,C^{-1}\,{d^{C}_{\gamma
D}}\,+\,{\rm h.c.}\,,$ (46)

with

$\displaystyle\tilde{f}[u^{C}_{A},d^{C}_{B},\nu_{C},d^{C}_{D}]$
$\displaystyle=$ $\displaystyle-\frac{4\lambda
v_{\sigma}v_{\overline{D}}}{15M_{\Delta}^{2}M_{\overline{T}}^{2}}\,\left(U_{u^{C}}^{T}FU_{d^{C}}\right)_{AB}\,\left(U_{\nu}^{T}FU_{d^{C}}\right)_{CD}\,,$
(47)

where $v_{\overline{D}}$ is a VEV of $\overline{D}$ residing in $\overline{\bf
126}$.

###  IV.3 From ${\bf 120}$-plet

In comparison to the previous two, ${\bf 120}_{H}$ offers more variety of
$d=7$ operators because of the mixing term allowed between various scalar sub-
multiplets and their conjugates. All four invariants listed in Eqs. (41-44)
give rise to $B-L$ violating operators. Using Eqs. (41-44) and couplings for
various triplet fields evaluated in section II, we find the following leading
order operators after some straight-forward computation:

$\displaystyle{\cal L}^{\bf 120}_{\rm eff}$ $\displaystyle=$
$\displaystyle-\frac{8\lambda
v_{\sigma}}{3}\left[\left(\frac{1}{M{{}^{2}_{\Delta}}\,M{{}^{2}_{\overline{T}_{1}}}}+\frac{c_{\Delta}c_{T}}{M_{\Delta}^{2}M^{2}_{\overline{T}_{2}}}\right)+\frac{s_{\Delta}s_{T}}{M_{\overline{\Delta}}^{2}M_{T}^{2}}\right]*$
(48) $\displaystyle\hskip
10.0pt\,G_{AB}^{*}G_{CD}^{*}\,\epsilon^{\alpha\beta\gamma}\,\epsilon^{ab}\,\overline{D}_{a}\,u^{C\dagger}_{\alpha
A}\,C^{-1}\,{d^{C}_{\beta B}}^{*}\,l^{\dagger}_{bC}\,C^{-1}\,{d^{C}_{\gamma
D}}^{*}\,$ $\displaystyle-$ $\displaystyle\frac{4\sqrt{2}\lambda
v_{\sigma}}{3}\left(\frac{c_{\Theta}c_{\Omega}}{M_{\overline{\Theta}}^{2}M_{\Omega}^{2}}+\frac{s_{\Theta}s_{\Omega}}{M_{\Theta}^{2}M_{\overline{\Omega}}^{2}}\right)\,G_{AB}^{*}G_{CD}^{*}\,\epsilon^{\alpha\beta\gamma}\,\epsilon^{ab}\,\overline{D}_{a}\,d^{C\dagger}_{\alpha
A}\,C^{-1}\,{d^{C}_{\beta B}}^{*}\,l^{\dagger}_{bC}\,C^{-1}\,{u^{C}_{\gamma
D}}^{*}\,$ $\displaystyle-$ $\displaystyle\frac{4\sqrt{2}\lambda
v_{\sigma}}{3}\left(\frac{c_{\Theta}s_{\Omega}}{M_{\overline{\Theta}}^{2}M_{\Omega}^{2}}-\frac{s_{\Theta}c_{\Omega}}{M_{\Theta}^{2}M_{\overline{\Omega}}^{2}}\right)\,G_{AB}^{*}G_{CD}\,\epsilon^{\alpha\beta\gamma}\,\overline{D}_{a}\,d^{C\dagger}_{\alpha
A}\,C^{-1}\,{d^{C}_{\beta B}}^{*}\,e^{CT}_{C}\,C^{-1}\,{q^{a}_{\gamma D}}\,$
$\displaystyle+$ $\displaystyle\frac{8\sqrt{2}\lambda
v_{\sigma}}{3}\left(\frac{c_{\Theta}c_{\Delta}}{M_{\overline{\Theta}}^{2}M_{\Delta}^{2}}+\frac{s_{\Theta}s_{\Delta}}{M_{\Theta}^{2}M{{}^{2}_{\overline{\Delta}}}}\right)\,G_{AB}^{*}G_{CD}^{*}\,\epsilon^{\alpha\beta\gamma}\,D^{a}\,d^{C\dagger}_{\alpha
A}\,C^{-1}\,{d^{C}_{\beta B}}^{*}\,l^{\dagger}_{aC}\,C^{-1}\,{d^{C}_{\gamma
D}}^{*}\,$ $\displaystyle-$ $\displaystyle\frac{8\lambda
v_{\sigma}}{3}\left(\frac{c_{\mathbb{T}}s_{\Delta}}{M_{\overline{\Delta}}^{2}M_{\mathbb{T}}^{2}}-\frac{s_{\mathbb{T}}c_{\Delta}}{M_{\Delta}^{2}M_{\overline{\mathbb{T}}}^{2}}\right)\,G_{AB}^{*}G_{CD}\,\epsilon^{\alpha\beta\gamma}\,D^{a}\,q^{\dagger}_{\alpha
bA}\,C^{-1}\,{q^{*}_{\beta aB}}\,l^{bT}_{C}\,C^{-1}\,d^{C}_{\gamma D}\,$
$\displaystyle+$ $\displaystyle{\rm h.c.}\,,$

where $c_{\chi}=\cos\theta_{\chi}$, $s_{\chi}=\sin\theta_{\chi}$ and
$\theta_{\chi}$ is the angle denoting the mixing between
$\chi$-$\overline{\chi}$ fields analogous to the one defined earlier in Eqs.
(26,27). The operator in the first line in Eq. (48) can be identified with
$\tilde{O}_{1}$, second with $\tilde{O}_{2}$, third with $\tilde{O}_{5}$,
fourth with $\tilde{O}_{6}$ and the last with operator $\tilde{O}_{4}$ as
listed in Babu and Mohapatra (2012b). Note that the operator $\tilde{O}_{3}$
does not arise as coupling of $\mathbb{T}$ with pair of quark doublets are
forbidden by flavour anti-symmetry of $G$.

Using the Fierz rearrangement, Eq. (36), the operator in the second line of
Eq. (48) can be expressed in terms of the one in the first line. Once the
electroweak doublets acquire VEVs, the four-fermion operator arising from Eq.
(48) in the physical basis can be parametrized as the following:

$\displaystyle{\cal L}^{\bf 120}_{\rm eff}$ $\displaystyle=$
$\displaystyle\tilde{g}[u^{C}_{A},d^{C}_{A},\nu_{C},d^{C}_{D}]\,\epsilon^{\alpha\beta\gamma}\,u^{CT}_{\alpha
A}\,C^{-1}\,{d^{C}_{\beta B}}\,\nu_{C}^{T}\,C^{-1}\,d^{C}_{\gamma D}\,$ (49)
$\displaystyle+$
$\displaystyle\tilde{g}^{\prime}[d^{C}_{A},d^{C}_{B},e^{C}_{C},d_{D}]\,\epsilon^{\alpha\beta\gamma}\,d^{C\dagger}_{\alpha
A}\,C^{-1}\,d^{C*}_{\beta B}\,e^{CT}_{C}\,C^{-1}\,d_{\gamma D}\,$
$\displaystyle+$
$\displaystyle\tilde{g}[d^{C}_{A},d^{C}_{B},e_{C},d^{C}_{D}]\,\epsilon^{\alpha\beta\gamma}\,d^{CT}_{\alpha
A}\,C^{-1}\,d^{C}_{\beta B}\,e^{T}_{C}\,C^{-1}\,d^{C}_{\gamma D}\,$
$\displaystyle+$
$\displaystyle\tilde{g}^{\prime}[u_{A},d_{B},\nu_{C},d^{C}_{D}]\,\epsilon^{\alpha\beta\gamma}\,u^{\dagger}_{\alpha
A}\,C^{-1}\,{d^{*}_{\beta B}}\,\nu^{T}_{C}\,C^{-1}\,d^{C}_{\gamma D}\,$
$\displaystyle+$
$\displaystyle\tilde{g}^{\prime}[d_{A},d_{B},e_{C},d^{C}_{D}]\,\epsilon^{\alpha\beta\gamma}\,d^{\dagger}_{\alpha
A}\,C^{-1}\,{d^{*}_{\beta B}}\,e^{T}_{C}\,C^{-1}\,d^{C}_{\gamma D}\,+\,{\rm
h.c.}\,,$

with

$\displaystyle\tilde{g}[u^{C}_{A},d^{C}_{B},\nu_{C},d^{C}_{D}]$
$\displaystyle=$ $\displaystyle\frac{8\lambda
v_{\sigma}v_{\overline{D}}}{3}\left[\left(\frac{1}{M{{}^{2}_{\Delta}}\,M{{}^{2}_{\overline{T}_{1}}}}+\frac{c_{\Delta}c_{T}}{M_{\Delta}^{2}M_{\overline{T}_{2}}^{2}}\right)+\frac{s_{\Delta}s_{T}}{M_{\overline{\Delta}}^{2}M_{T}^{2}}\right]\left(U_{u^{C}}^{T}GU_{d^{C}}\right)_{AB}\,\left(U_{\nu}^{T}GU_{d^{C}}\right)_{CD}$
$\displaystyle+$ $\displaystyle\frac{8\sqrt{2}\lambda
v_{\sigma}v_{\overline{D}}}{3}\left(\frac{c_{\Theta}c_{\Omega}}{M_{\overline{\Theta}}^{2}M_{\Omega}^{2}}+\frac{s_{\Theta}s_{\Omega}}{M_{\Theta}^{2}M_{\overline{\Omega}}^{2}}\right)\,\left(U{{}_{\nu}^{T}}GU_{u^{C}}\right)_{CA}\,\left(U{{}_{d^{C}}^{T}}GU_{d^{C}}\right)_{BD}\,,$
$\displaystyle\tilde{g}^{\prime}[d^{C}_{A},d^{C}_{B},e^{C}_{C},d_{D}]$
$\displaystyle=$ $\displaystyle-\frac{4\sqrt{2}\lambda
v_{\sigma}v_{\overline{D}}}{3}\left(\frac{c_{\Theta}s_{\Omega}}{M_{\overline{\Theta}}^{2}M_{\Omega}^{2}}-\frac{s_{\Theta}c_{\Omega}}{M_{\Theta}^{2}M_{\overline{\Omega}}^{2}}\right)\,\left(U_{d^{C}}^{\dagger}G^{*}U_{d^{C}}^{*}\right)_{AB}\,\left(U_{e^{C}}^{T}GU_{d}\right)_{CD}\,,$
$\displaystyle\tilde{g}[d^{C}_{A},d^{C}_{B},e_{C},d^{C}_{D}]$ $\displaystyle=$
$\displaystyle\frac{8\sqrt{2}\lambda
v_{\sigma}v_{D}}{3}\left(\frac{c_{\Theta}c_{\Delta}}{M_{\overline{\Theta}}^{2}M_{\Delta}^{2}}+\frac{s_{\Theta}s_{\Delta}}{M_{\Theta}^{2}M{{}^{2}_{\overline{\Delta}}}}\right)\,\left(U_{d^{C}}^{T}GU_{d^{C}}\right)_{AB}\,\left(U{{}_{e}^{T}}GU_{d^{C}}\right)_{CD}\,,$
$\displaystyle\tilde{g}^{\prime}[u_{A},d_{B},\nu_{C},d^{C}_{D}]$
$\displaystyle=$ $\displaystyle-\frac{8\lambda
v_{\sigma}v_{D}}{3}\left(\frac{c_{\mathbb{T}}s_{\Delta}}{M_{\overline{\Delta}}^{2}M_{\mathbb{T}}^{2}}-\frac{s_{\mathbb{T}}c_{\Delta}}{M_{\Delta}^{2}M_{\overline{\mathbb{T}}}^{2}}\right)\,\left(U_{u}^{\dagger}G^{*}U_{d}^{*}\right)_{AB}\,\left(U_{\nu}^{T}GU_{d^{C}}\right)_{CD}\,,$
$\displaystyle\tilde{g}^{\prime}[d_{A},d_{B},e_{C},d^{C}_{D}]$
$\displaystyle=$ $\displaystyle-\frac{8\lambda
v_{\sigma}v_{D}}{3}\left(\frac{c_{\mathbb{T}}s_{\Delta}}{M_{\overline{\Delta}}^{2}M_{\mathbb{T}}^{2}}-\frac{s_{\mathbb{T}}c_{\Delta}}{M_{\Delta}^{2}M_{\overline{\mathbb{T}}}^{2}}\right)\,\left(U_{d}^{\dagger}G^{*}U_{d}^{*}\right)_{AB}\,\left(U_{e}^{T}GU_{d^{C}}\right)_{CD}\,.$
(50)

As in the earlier case, primed and unprimed coefficients are defined in such a
way that all the $\tilde{g}^{\prime}=0$ there is no mixing between the
different scalars and their conjugate partners.

In summary, the operators listed in Eqs. (46,49) quantify the $B$, $L$ and
$B-L$ violating baryon decays mediated by scalars residing in $\overline{\bf
126}_{H}$ and ${\bf 120}_{H}$, respectively, at the leading order. Like in the
previous section, we have considered the possibility of the mixing between the
various scalars arising from the given representation. It can be noted that
the masses of $T$, $\overline{T}$, $\mathbb{T}$ and $\overline{\mathbb{T}}$
are already constrained by the leading order $d=6$ operators considered in the
previous section. The $B-L$ violating decays of nucleon can, therefore,
provide lower bounds on the masses of $\Delta$, $\overline{\Delta}$ residing
in $\overline{\bf 126}_{H}$ and $\Delta$, $\overline{\Delta}$, $\Theta$,
$\overline{\Theta}$, $\Omega$ and $\overline{\Omega}$ belonging to ${\bf
120}_{H}$ depending on their couplings with quarks and leptons and the scale
of $B-L$ breaking.

##  V Nucleon decay partial widths

In this section, we give explicit expressions of the proton decay widths in
various channels evaluated from the derived effective operators.

###  V.1 $B-L$ conserving decays

The $d=6$ operators listed previously in Eqs. (30,33,39) can be parametrized
in terms of the following six independent operators:

$\displaystyle{\cal L}_{\rm eff}$ $\displaystyle=$ $\displaystyle
y[u_{A},d_{B},e^{C}_{C},u^{C}_{D}]\,\,\epsilon^{\alpha\beta\gamma}\,u_{\alpha
A}^{T}\,C^{-1}\,d_{\beta B}\,{e^{C}_{C}}^{\dagger}\,C^{-1}\,{u^{C}_{\gamma
D}}^{*}$ (51) $\displaystyle+$ $\displaystyle
y[u^{C}_{A},d^{C}_{B},e_{C},u_{D}]\,\,\epsilon^{\alpha\beta\gamma}\,u{{}^{CT}_{\alpha
A}}\,C^{-1}\,d_{\beta B}^{C}\,e_{C}^{\dagger}\,C^{-1}\,u_{\gamma D}^{*}\,$
$\displaystyle+$ $\displaystyle
y[u^{C}_{A},d^{C}_{B},\nu_{C},d_{D}]\,\,\epsilon^{\alpha\beta\gamma}\,u{{}^{CT}_{\alpha
A}}\,C^{-1}\,d_{\beta B}^{C}\,\nu_{C}^{\dagger}\,C^{-1}\,d_{\gamma D}^{*}\,$
$\displaystyle+$ $\displaystyle
y^{\prime}[u^{C}_{A},d^{C}_{B},e^{C}_{C},u^{C}_{D}]\,\,\epsilon^{\alpha\beta\gamma}\,u{{}^{CT}_{\alpha
A}}\,C^{-1}\,d^{C}_{\beta B}\,e{{}^{CT}_{C}}\,C^{-1}\,u^{C}_{\gamma D}$
$\displaystyle+$ $\displaystyle
y^{\prime}[u_{A},d_{B},e_{C},u_{D}]\,\,\epsilon^{\alpha\beta\gamma}\,u_{\alpha
A}^{T}\,C^{-1}\,d_{\beta B}\,e_{C}^{T}\,C^{-1}\,u_{\gamma D}\,$
$\displaystyle+$ $\displaystyle
y^{\prime}[u_{A},d_{B},\nu_{C},d_{D}]\,\,\epsilon^{\alpha\beta\gamma}\,u_{\alpha
A}^{T}\,C^{-1}\,d_{\beta B}\,\nu_{C}^{T}\,C^{-1}\,d_{\gamma D}\,\,+{\rm
h.c.}\,,$

where $y=h$, $f$ or $g$ if a single GUT scalar representation is considered.
$y$ can also be a linear combination of $h$, $f$ and $g$ when more than one
scalar fields are considered as discussed in the previous section. These
operators match with the most general dimension six operators derived from
effective theory Weinberg (1979); Wilczek and Zee (1979) but now the
coefficient can be explicitly computed in terms of fundamental Yukawa
couplings of a given GUT model.

To write the above operators in the usual left- and right-chiral fields, we
use

$\psi=\psi_{L}\,,~{}~{}\psi^{C}=C\,\psi_{R}^{*}\,,$ (52)

where $C=-i\sigma^{2}$ in Weyl basis. This leads to

$\psi^{T}\,C^{-1}\,\chi=\overline{(\psi_{L})^{C}}\,\chi_{L}\,,~{}~{}{\psi^{C}}^{\dagger}\,C^{-1}\,{\chi^{C}}^{*}=\overline{(\psi_{R})^{C}}\,\chi_{R}\,,$
(53)

where $\overline{\psi^{C}}\equiv\psi^{T}C^{-1}$. Using the above identities
and $\overline{\psi^{C}}\,\chi=\overline{\chi^{C}}\,\psi$, the baryon number
violating operators listed in Eq. (51) can be brought into the following
convenient form:

$\displaystyle{\cal L}_{\rm eff}$ $\displaystyle=$ $\displaystyle
y[u_{A},d_{B},e^{C}_{C},u^{C}_{D}]\,\,\epsilon^{\alpha\beta\gamma}\,\overline{(d_{\beta
BL})^{C}}\,u_{\alpha AL}\,\,\overline{(u_{\gamma DR})^{C}}\,e_{CR}$ (54)
$\displaystyle+$ $\displaystyle
y^{*}[u^{C}_{A},d^{C}_{B},e_{C},u_{D}]\,\,\epsilon^{\alpha\beta\gamma}\,\overline{(d_{\beta
BR})^{C}}\,u_{\alpha AR}\,\,\overline{(u_{\gamma DL})^{C}}\,e_{CL}$
$\displaystyle+$ $\displaystyle
y^{*}[u^{C}_{A},d^{C}_{B},\nu_{C},d_{D}]\,\,\epsilon^{\alpha\beta\gamma}\,\overline{(d_{\beta
BR})^{C}}\,u_{\alpha AR}\,\,\overline{(d_{\gamma DL})^{C}}\,\nu_{CL}$
$\displaystyle+$ $\displaystyle
y^{\prime*}[u^{C}_{A},d^{C}_{B},e^{C}_{C},u^{C}_{D}]\,\,\epsilon^{\alpha\beta\gamma}\,\overline{(d_{\beta
BR})^{C}}\,u_{\alpha AR}\,\,\overline{(u_{\gamma DR})^{C}}\,e_{CR}$
$\displaystyle+$ $\displaystyle
y^{\prime}[u_{A},d_{B},e_{C},u_{D}]\,\,\epsilon^{\alpha\beta\gamma}\,\overline{(d_{\beta
BL})^{C}}\,u_{\alpha AL}\,\,\overline{(u_{\gamma DL})^{C}}\,e_{CL}$
$\displaystyle+$ $\displaystyle
y^{\prime}[u_{A},d_{B},\nu_{C},d_{D}]\,\,\epsilon^{\alpha\beta\gamma}\,\overline{(d_{\beta
BL})^{C}}\,u_{\alpha AL}\,\,\overline{(d_{\gamma DL})^{C}}\,\nu_{CL}+{\rm
h.c.}\,.$

where $y^{*}[A,B,C,D]=\left(y[A,B,C,D]\right)^{*}$ and the same for
$y^{\prime}$.

All the operators listed in Eq. (54) violate both $B$ and $L$ but conserve
$B-L$ leading to decays of baryon into meson and anti-lepton. The hadronic
matrix element between the baryon and meson states can be computed using the
chiral perturbation theory Claudson  _et al._ (1982); Chadha and Daniel
(1983). Using the results from Aoki  _et al._ (2000), one finds the following
expressions for the partial decay widths for the proton decaying into various
mesons Nath and Fileviez Perez (2007):

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

where

$\displaystyle C^{\pm}_{Li}$ $\displaystyle=$
$\displaystyle\alpha\,y^{*}[u^{C}_{1},d^{C}_{2},e_{i},u_{1}]\pm\beta\,y^{\prime}[u_{1},d_{2},e_{i},u_{1}]\,,$
$\displaystyle C^{\pm}_{Ri}$ $\displaystyle=$
$\displaystyle\alpha\,y[u_{1},d_{2},e^{C}_{i},u^{C}_{1}]\pm\beta\,y^{\prime*}[u^{C}_{1},d^{C}_{2},e^{C}_{i},u^{C}_{1}]\,,$
$\displaystyle C^{\nu}_{L1i}$ $\displaystyle=$
$\displaystyle\alpha\,y^{*}[u_{1}^{C},d_{2}^{C},\nu_{i},d_{1}]+\beta\,y^{\prime}[u_{1},d_{2},\nu_{i},d_{1}]\,,$
$\displaystyle C^{\nu}_{L2i}$ $\displaystyle=$
$\displaystyle\alpha\,y^{*}[u_{1}^{C},d_{1}^{C},\nu_{i},d_{2}]+\beta\,y^{\prime}[u_{1},d_{1},\nu_{i},d_{2}]\,.$
(56)

Here, $m_{H}$ denotes the mass of hadron $H$
($H=p,\pi^{0},\pi^{\pm},K^{0},K^{\pm},\eta$), $m_{B}$ is average baryon mass
and $f_{\pi}$ is pion decay constant. $\alpha$, $\beta$, $\tilde{D}$ and
$\tilde{F}$ are the parameters of the chiral Lagrangian. The factor $A$
accounts for the renormalization effects in hadronic matrix elements from the
weak scale to the $m_{p}$.

It can be noticed from Eqs. (V.1,V.1) that the relevant couplings for the
proton decay into the charged leptons are
$y[u_{1},d_{i},e^{C}_{j},u^{C}_{1}]$, $y[u^{C}_{1},d^{C}_{i},e_{j},u_{1}]$,
$y^{\prime}[u^{C}_{1},d^{C}_{i},e^{C}_{j},u^{C}_{1}]$,
$y[u_{1},d_{i},e_{j},u_{1}]$ and for the proton decay into the neutrinos are
$y[u_{1},d_{i},\nu_{j},d_{k}]$ and $y[u^{C}_{1},d^{C}_{i},\nu_{j},d_{k}]$ .
Considering this, the tree-level contribution mediated by ${\cal T}$,
$\overline{\cal T}$ (see Eq. (III.3)) vanishes due to anti-symmetric $G$ as

$\left(U_{u^{C}}^{T}GU_{u^{C}}\right)_{11}=0\,.$ (57)

These fields can induce proton-decay through dimension-6 operators which arise
at loop level. The same result has been found earlier in the context of
$SU(5)$ GUTs in Dorsner  _et al._ (2012) and the 1-loop diagrams which arise
through an additional $W$-boson exchange have also been evaluated. Therefore,
at tree-level through dimension-6 operators, the proton decays are mediated by
only $T$, $\overline{T}$ and $\mathbb{T}$, $\overline{\mathbb{T}}$ in the
models with ${\bf 120}_{H}$.

###  V.2 $B-L$ non-conserving decays

The $B-L$ non-conserving decays of nucleons arise from the operators derived
in Eqs. (46,49). They can be further generalized as

$\displaystyle{\cal L}_{\rm eff}$ $\displaystyle=$
$\displaystyle\tilde{y}[u^{C}_{A},d^{C}_{A},\nu_{C},d^{C}_{D}]\,\epsilon^{\alpha\beta\gamma}\,u^{CT}_{\alpha
A}\,C^{-1}\,{d^{C}_{\beta B}}\,\nu_{C}^{T}\,C^{-1}\,d^{C}_{\gamma D}\,$ (58)
$\displaystyle+$
$\displaystyle\tilde{y}^{\prime}[u_{A},d_{B},\nu_{C},d^{C}_{D}]\,\epsilon^{\alpha\beta\gamma}\,u^{\dagger}_{\alpha
A}\,C^{-1}\,{d^{*}_{\beta B}}\,\nu^{T}_{C}\,C^{-1}\,d^{C}_{\gamma D}\,$
$\displaystyle+$
$\displaystyle\tilde{y}[d^{C}_{A},d^{C}_{B},e_{C},d^{C}_{D}]\,\epsilon^{\alpha\beta\gamma}\,d^{CT}_{\alpha
A}\,C^{-1}\,d^{C}_{\beta B}\,e^{T}_{C}\,C^{-1}\,d^{C}_{\gamma D}\,$
$\displaystyle+$
$\displaystyle\tilde{y}^{\prime}[d_{A},d_{B},e_{C},d^{C}_{D}]\,\epsilon^{\alpha\beta\gamma}\,d^{\dagger}_{\alpha
A}\,C^{-1}\,{d^{*}_{\beta B}}\,e^{T}_{C}\,C^{-1}\,d^{C}_{\gamma D}\,$
$\displaystyle+$
$\displaystyle\tilde{y}^{\prime}[d^{C}_{A},d^{C}_{B},e^{C}_{C},d_{D}]\,\epsilon^{\alpha\beta\gamma}\,d^{C\dagger}_{\alpha
A}\,C^{-1}\,d^{C*}_{\beta B}\,e^{CT}_{C}\,C^{-1}\,d_{\gamma D}\,+\,{\rm
h.c.}\,,$

where $y=f$ or $g$. The above operators can be rewritten in a usual left- and
right-chiral fields using identities given in Eq. (53) and

$\psi^{T}C^{-1}\chi^{C}=\left(\overline{\psi_{L}}\,\chi_{R}\right)^{*}\,~{}~{}\psi^{CT}C^{-1}\chi=\overline{\psi_{R}}\,\chi_{L}\,.$
(59)

The above identities are obtained from the definitions given in Eq. (52). In
the new notation, we find

$\displaystyle{\cal L}_{\rm eff}$ $\displaystyle=$
$\displaystyle\tilde{y}^{*}[u^{C}_{A},d^{C}_{A},\nu_{C},d^{C}_{D}]\,\epsilon^{\alpha\beta\gamma}\,\overline{(d_{\beta
BR})^{C}}\,u_{\alpha AR}\,\,\overline{d_{\gamma DR}}\,\nu_{CL}\,$ (60)
$\displaystyle+$
$\displaystyle\tilde{y}^{\prime*}[u_{A},d_{B},\nu_{C},d^{C}_{D}]\,\epsilon^{\alpha\beta\gamma}\,\overline{(d_{\beta
BL})^{C}}\,u_{\alpha AL}\,\,\overline{d_{\gamma DR}}\,\nu_{CL}\,$
$\displaystyle+$
$\displaystyle\tilde{y}^{*}[d^{C}_{A},d^{C}_{B},e_{C},d^{C}_{D}]\,\epsilon^{\alpha\beta\gamma}\,\overline{(d_{\beta
BR})^{C}}\,d_{\alpha AR}\,\,\overline{d_{\gamma DR}}\,e_{CL}\,$
$\displaystyle+$
$\displaystyle\tilde{y}^{\prime*}[d_{A},d_{B},e_{C},d^{C}_{D}]\,\epsilon^{\alpha\beta\gamma}\,\overline{(d_{\beta
BL})^{C}}\,d_{\alpha AL}\,\,\overline{d_{\gamma DR}}\,e_{CL}\,$
$\displaystyle+$
$\displaystyle\tilde{y}^{\prime}[d^{C}_{A},d^{C}_{B},e^{C}_{C},d_{D}]\,\epsilon^{\alpha\beta\gamma}\,\overline{(d_{\beta
BR})^{C}}\,d_{\alpha AR}\,\,\overline{d_{\gamma DL}}\,e_{CR}\,+\,{\rm
h.c.}\,.$

The operators obtained above violate $B-L$ by two units and lead to processes
in which a nucleon decays into lepton and a meson. The first two operators in
Eq. (60) contribute in the proton decay through channels $p\to\nu\,\pi^{+}$
and $p\to\nu\,K^{+}$.

$\displaystyle\Gamma[p\to\nu\pi^{+}]$ $\displaystyle=$
$\displaystyle\frac{(m_{p}^{2}-m_{\pi^{+}}^{2})^{2}}{32\,\pi\,m_{p}^{3}f_{\pi}^{2}}A^{2}\left(1+\tilde{D}+\tilde{F}\right)^{2}\sum_{i=1}^{3}\left|\alpha\,\tilde{y}^{*}[u^{C}_{1},d^{C}_{1},\nu_{i},d^{C}_{1}]+\beta\,\tilde{y}^{\prime*}[u_{1},d_{1},\nu_{i},d^{C}_{1}]\right|^{2}\,,$
$\displaystyle\Gamma[p\to\nu K^{+}]$ $\displaystyle=$
$\displaystyle\frac{(m_{p}^{2}-m_{K^{+}}^{2})^{2}}{32\,\pi\,m_{p}^{3}f_{\pi}^{2}}\,A^{2}\,\sum_{i=1}^{3}\Big{|}\frac{2\tilde{D}}{3}\frac{m_{p}}{m_{B}}\tilde{C}^{\nu}_{L1i}+\left(1+\frac{\tilde{D}+3\tilde{F}}{3}\frac{m_{p}}{m_{B}}\right)\tilde{C}^{\nu}_{L2i}\Big{|}^{2}\,,$
(61)

where

$\displaystyle\tilde{C}^{\nu}_{L1i}$ $\displaystyle=$
$\displaystyle\alpha\,\tilde{y}^{*}[u_{1}^{C},d_{2}^{C},\nu_{i},d^{C}_{1}]+\beta\,\tilde{y}^{\prime*}[u_{1},d_{2},\nu_{i},d^{C}_{1}]\,,$
$\displaystyle\tilde{C}^{\nu}_{L2i}$ $\displaystyle=$
$\displaystyle\alpha\,\tilde{y}^{*}[u_{1}^{C},d_{1}^{C},\nu_{i},d^{C}_{2}]+\beta\,\tilde{y}^{\prime*}[u_{1},d_{1},\nu_{i},d^{C}_{2}]\,.$
(62)

Experimentally, the above decay modes are indistinguishable from
$p\to\overline{\nu}\,\pi^{+}/K^{+}$.

The remaining three operators in Eq. (60) do not contribute in the proton
decay but induces the $B$ and $B-L$ violating decays of the neutron, for
example $n\to e^{-}\,\pi^{+}$. The decay width for this can be estimated as

$\displaystyle\Gamma[n\to e_{i}^{-}\pi^{+}]$ $\displaystyle=$
$\displaystyle\frac{(m_{n}^{2}-m_{\pi^{+}}^{2})^{2}}{32\,\pi\,m_{n}^{3}f_{\pi}^{2}}A^{2}\left(\frac{1+\tilde{D}+\tilde{F}}{\sqrt{2}}\right)^{2}\Big{(}\left|\beta\,\tilde{y}^{\prime*}[d^{C}_{1},d^{C}_{1},e^{C}_{i},d_{1}]\right|^{2}\Big{.}$
(63) $\displaystyle+$
$\displaystyle\Big{.}\left|\alpha\,\tilde{y}^{*}[d^{C}_{1},d^{C}_{1},e_{i},d^{C}_{1}]+\beta\,\tilde{y}^{\prime*}[d_{1},d_{1},e_{i},d_{1}^{C}]\right|^{2}\Big{)}\,.$

It can be noticed that for $y=g$, i.e. when the operators are induced only
trough the triplets residing in ${\bf 120}_{H}$, one finds vanishing decay
width for $n\to e_{i}^{-}\pi^{+}$ at the leading order due to anti-symmetric
nature of $G$, see Eq. (IV.3).

##  VI Estimation for a ${\bf 10}_{H}+\overline{\bf 126}_{H}$ model

Using the general results obtained above, we now compute the partial decay
widths of the proton in a specific $SO(10)$ model. The Yukawa sector of the
model consists of a complex ${\bf 10}_{H}$ and $\overline{\bf 126}_{H}$ scalar
fields. Each of these contains a pair of colour singlets and $SU(2)$ doublets
with $Y=\pm 1/2$, see Table 1. A pair of linear combinations of these
doublets, namely $h_{u}$ and $h_{d}$, is assumed to remain much lighter than
the GUT scale and induce the electroweak scale. The Vacuum Expectation Values
(VEVs) of $h_{u}$ and $h_{d}$ also generate masses for all the charged
fermions. $\overline{\bf 126}_{H}$ also contains an SM singlet but $B-L$
charged field, VEV of which gives rise to masses for heavy RH neutrinos. This,
along with the Dirac masses generated by the VEVs of $h_{u,d}$ generates
naturally suppressed mass for the SM neutrinos through the type I seesaw
mechanism. The viability of this framework in reproducing the correct spectrum
of fermion masses and mixing parameters has been extensively studied in
several works, see 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); Mummidi and Patel (2021) for example. In the most recent study Mummidi
and Patel (2021), several viable solutions have been obtained for this model
which not only reproduces the known fermion mass spectrum but can also account
for the observed baryon asymmetry through Leptogenesis.

The absence of ${\bf 120}_{H}$ in this model implies that the proton decay in
this class of theories is mediated by only the $Y=\pm 1/3$ colour triplet
scalars as derived in section II. Moreover, the minimal model uses a $U(1)$
Peccei-Quinn symmetry Peccei and Quinn (1977) under which ${\bf 16}_{F}$ has
charge $+1$ while ${\bf 10}_{H}$ and $\overline{\bf 126}_{H}$ each has charge
$-2$. This allows Yukawa couplings with ${\bf 10}_{H}$ but forbids the ones
with ${\bf 10}_{H}^{\dagger}$ leaving only two Yukawa coupling matrices making
the model predictive Joshipura and Patel (2011). As a consequence of Peccei-
Quinn symmetry, the gauge invariant term ${\bf 10}_{H}^{2}$ is forbidden and
hence the components of $T$ and $\overline{T}$ of ${\bf 10}_{H}$ do not mix.
Therefore, we have $\theta=0$ in Eq. (27) and all the $h^{\prime}$ vanish in
Eq. (30). The proton decay, in this case, is governed by only three
independent operators, listed as the first three in Eq. (54).

In general, the interaction terms in the scalar potential which gives rise to
mixing of electroweak doublets can also induce mixing between different $T$
and $\overline{T}$ residing in ${\bf 10}_{H}$ and $\overline{\bf 126}_{H}$. In
this case, the coefficients of the effective operators given in Eq. (54) are
linear combinations of corresponding $h$ and $f$. The determination of the
exact combination however depends on the full scalar potential. To compute the
proton lifetime in this model, we adopt a simplified approach and assume that
the lightest pair of triplets is dominantly arising from either of ${\bf
10}_{H}$ and $\overline{\bf 126}_{H}$. The coefficients $y$, in this case, are
either $h$ or $f$ and can be fully determined from the fundamental Yukawa
coupling matrix $H$ and $F$ and from the flavour rotation matrices $U_{f}$.
These parameters can be extracted from the fermion mass fit performed in
Mummidi and Patel (2021). The parameters corresponding to the best fit
solution are given in Mummidi and Patel (2021) which we reproduce here in the
Appendix A for convenience. We also list the diagonalizing matrix obtained
from the various Yukawa coupling matrices.

The Yukawa coupling matrices $H$ and $F$ are obtained as

$H=\frac{1}{2\sqrt{2}\,\alpha_{1}}\,H^{\prime}\,~{}~{}~{}F=\frac{\sqrt{3}}{4\sqrt{2}\,\alpha_{2}}\,F^{\prime}\,,$
(64)

where $\alpha_{1,2}$ are factors that quantify the mixing of electroweak
doublets with a constraint $|\alpha_{1}|^{2}+|\alpha_{2}|^{2}\leq 1$, see
Mummidi and Patel (2021) for the details. $H^{\prime}$ and $F^{\prime}$ can be
determined from the fermion mass spectrum and an example numerical solution is
given in Appendix A. Note that $H$ can be chosen diagonal and real without the
loss of generality while $F$ is complex symmetric in this basis. The fermion
mass matrices, which are linear combinations of $H$ and $F$, remains complex
symmetric. Hence, one finds $U_{f^{c}}=U_{f}^{*}$ for $f=u,d,e$. The
coefficients $h$ and $f$ then can be determined from the values of
$H^{\prime}$, $F^{\prime}$ and $U_{f}$ given in Appendix A for the best fit
solution. For the parameters in Eq. (V.1), we use $\alpha=0.01\,{\rm
GeV}^{3}$, $\tilde{D}=0.8$, $\tilde{F}=0.46$ Aoki  _et al._ (2000); Cabibbo
_et al._ (2003). Further, the average baryon mass $m_{B}=1.15$ GeV, the pion
decay constant $f_{\pi}=130$ MeV and the values of various hadron masses are
taken from the PDG Zyla  _et al._ (2020). The parameter $A=1.43$ captures
running effects from $M_{Z}$ to $m_{p}$. To account for the running effects
between $M_{\rm GUT}$ to $M_{Z}$ one needs to use the values of $H$ and $F$
extracted at $M_{Z}$. However, we use the values obtained from the fit carried
out at the GUT scale as the change in Yukawa couplings due to running is of
${\cal O}(1)$ and does not change the results significantly Alonso  _et al._
(2014).

###  VI.1 Proton decay pattern

The results obtained for various branching ratios are given in Table 2
assuming that the lightest $T$ and $\overline{T}$ are dominantly the ones
residing in ${\bf 10}_{H}$. Similarly, the branching ratios computed, assuming
that the lightest pair of triplets originates from $\overline{\bf 126}_{H}$,
are listed in Table 3. Note that the branching ratios determined in Table 2
and 3 do not depend on the unknown parameter $\alpha_{1,2}$ appearing in Eq.
(64).

Branching ratio [%] |  $M_{T}\ll M_{\overline{T}}$ |  $M_{T}\gg M_{\overline{T}}$ |  $M_{T}=M_{\overline{T}}$
---|---|---|---
${\rm BR}[p\to e^{+}\pi^{0}]$ | $<1$ | $<1$ | $<1$
${\rm BR}[p\to\mu^{+}\pi^{0}]$ | $7$ | $<1$ | $<1$
${\rm BR}[p\to\bar{\nu}\pi^{+}]$ | $0$ | $15$ | $14$
${\rm BR}[p\to e^{+}K^{0}]$ | $<1$ | $<1$ | $<1$
${\rm BR}[p\to\mu^{+}K^{0}]$ | $93$ | $<1$ | $2$
${\rm BR}[p\to\bar{\nu}K^{+}]$ | $0$ | $84$ | $83$
${\rm BR}[p\to e^{+}\eta]$ | $<1$ | $<1$ | $<1$
${\rm BR}[p\to\mu^{+}\eta]$ | $<1$ | $<1$ | $<1$
Table 2: Proton decay branching fractions estimated for the best fit solution for various hierarchies among the masses of $T$ and $\overline{T}$ residing in ${\bf 10}_{H}$. Branching ratio [%] |  $M_{T_{1,2}}\ll M_{\overline{T}}$ |  $M_{T_{1,2}}\gg M_{\overline{T}}$ |  $M_{T_{1,2}}=M_{\overline{T}}$
---|---|---|---
${\rm BR}[p\to e^{+}\pi^{0}]$ | $<1$ | $<1$ | $<1$
${\rm BR}[p\to\mu^{+}\pi^{0}]$ | $11$ | $<1$ | $<1$
${\rm BR}[p\to\bar{\nu}\pi^{+}]$ | $0$ | $11$ | $11$
${\rm BR}[p\to e^{+}K^{0}]$ | $<1$ | $<1$ | $<1$
${\rm BR}[p\to\mu^{+}K^{0}]$ | $88$ | $10$ | $11$
${\rm BR}[p\to\bar{\nu}K^{+}]$ | $0$ | $78$ | $77$
${\rm BR}[p\to e^{+}\eta]$ | $<1$ | $<1$ | $<1$
${\rm BR}[p\to\mu^{+}\eta]$ | $<1$ | $<1$ | $<1$
Table 3: Proton decay branching fractions estimated for the best fit solution
for various hierarchies among the masses of $T_{1,2}$ and $\overline{T}$
residing in $\overline{{\bf 126}}_{H}$.

The branching ratios of proton decay obtained in Table 2 and 3 are
qualitatively very different from the ones computed assuming the dominant
contribution from the gauge boson mediation. For transparent comparison, we
also compute the latter taking into account the flavour effects for the best
fit solution. The details of computation are given in Appendix B and the
results are listed in Table 4. The important observations are:

  1. (a)

Unlike in the case of gauge boson mediation in which the proton preferably
decays into $\pi^{0}\,e^{+}$ or $\pi^{+}\overline{\nu}$, the scalar induced
proton decay favours the channels involving $K^{0}\,\mu^{+}$ or
$K^{+}\,\overline{\nu}$.

  2. (b)

Typically, one finds ${\rm BR}[p\to e^{+}\pi^{0}]\ll{\rm
BR}[p\to\mu^{+}\pi^{0}]$. This is in contrast to the gauge boson mediated
decays in which one typically finds ${\rm BR}[p\to e^{+}\pi^{0}]\gg{\rm
BR}[p\to\mu^{+}\pi^{0}]$.

  3. (c)

When $\overline{T}$ is lighter than $T$, the proton decays dominantly in the
channels involving charged mesons and neutrinos. As $T$ does not couple to the
neutrinos, proton decays mediated by it results into the neutral mesons and
charged leptons.

  4. (d)

A comparison between Table 2 and 3 suggests that the decay patterns of the
proton do not significantly depend on whether the lightest $T$ and
$\overline{T}$ originate dominantly from ${\bf 10}_{H}$ or $\overline{\bf
126}_{H}$. This indicates that the qualitative results would remain unaltered
even if the lightest pair, $T$ and $\overline{T}$, are general linear
combinations of triplets and anti-triplets, respectively, residing in ${\bf
10}_{H}$ and $\overline{\bf 126}_{H}$.

To check the robustness of the above observations, we evaluate the proton
decay branching ratios for several solutions, with acceptable $\chi^{2}$
values at the minimum, determined in Mummidi and Patel (2021). The results are
displayed in Figs. 1 and 2.

Figure 1: Scalar mediated proton decay spectrum for solutions obtained in case
of the minimal renormalizable non-supersymmetric $SO(10)$ model. Various
branching ratios are evaluated assuming that the lightest pair of colour
triplets, $T$ and $\overline{T}$, arise dominantly from ${\bf 10}_{H}$.

Figure 2: Same as Fig. 1 but assuming that the lightest triplets arise
dominantly from $\overline{\bf 126}_{H}$.

It can be observed from these figures that the predictions of various
branching ratios reported in Tables 2 and 3 do not change significantly even
for the other viable solutions.

The noteworthy features of the proton decay spectrum listed as a, b, c, d can
be understood from the flavour structure. The realistic fermion mass spectrum
in the underlying model leads to the following hierarchical structures for $H$
and $F$:

$H\sim\frac{\lambda^{4}}{\alpha_{1}}\left(\begin{array}[]{ccc}\lambda^{7}&0&0\\\
0&\lambda^{3}&0\\\
0&0&1\end{array}\right)\,,~{}~{}F\sim\frac{\lambda^{4}}{\alpha_{2}}\left(\begin{array}[]{ccc}\lambda^{5}&\lambda^{4}&\lambda^{3}\\\
\lambda^{4}&\lambda^{3}&\lambda^{2}\\\
\lambda^{3}&\lambda^{2}&\lambda\end{array}\right)\,,$ (65)

where $\lambda=0.23$ is Cabibbo angle and we have suppressed ${\cal O}(1)$
coefficients. The Yukawa matrices of the charged fermions, $Y_{u,d,e}$, are
linear combinations of $H$ and $F$. The unitary matrices which diagonalize
$Y_{u,d,e}$ and the neutrino mass matrix have the following generic form

$U_{u}\sim U_{d}\sim
U_{e}\sim\left(\begin{array}[]{ccc}1&\lambda&\lambda^{3}\\\
\lambda&1&\lambda^{2}\\\
\lambda^{3}&\lambda^{2}&1\end{array}\right)\,,~{}~{}U_{\nu}\sim\left(\begin{array}[]{ccc}&&\\\
&{\cal O}(1)&\\\ &&\end{array}\right)\,$ (66)

where $U_{\nu}$ is matrix with all the elements of ${\cal O}(1)$. $U_{u,d,e}$
are CKM-like and lead to small quark mixing while the large mixing in PMNS
matrix arise through $U_{\nu}$. From Eqs. (65,66), we find

$\displaystyle U_{f}^{T}\,H\,U_{f^{\prime}}$ $\displaystyle\sim$
$\displaystyle\frac{\lambda^{4}}{\alpha_{1}}\left(\begin{array}[]{ccc}\lambda^{5}&\lambda^{4}&\lambda^{3}\\\
\lambda^{4}&\lambda^{3}&\lambda^{2}\\\
\lambda^{3}&\lambda^{2}&1\end{array}\right)\,,~{}~{}U_{f}^{T}\,F\,U_{f^{\prime}}\sim\frac{\lambda^{4}}{\alpha_{2}}\left(\begin{array}[]{ccc}\lambda^{5}&\lambda^{4}&\lambda^{3}\\\
\lambda^{4}&\lambda^{3}&\lambda^{2}\\\
\lambda^{3}&\lambda^{2}&\lambda\end{array}\right)\,,$ (73) $\displaystyle
U_{\nu}^{T}\,H\,U_{d}$ $\displaystyle\sim$
$\displaystyle\frac{\lambda^{4}}{\alpha_{1}}\left(\begin{array}[]{ccc}\lambda^{3}&\lambda^{2}&1\\\
\lambda^{3}&\lambda^{2}&1\\\
\lambda^{3}&\lambda^{2}&1\end{array}\right)\,,~{}~{}U_{\nu}^{T}\,F\,U_{d}\sim\frac{\lambda^{4}}{\alpha_{2}}\left(\begin{array}[]{ccc}\lambda^{3}&\lambda^{2}&\lambda\\\
\lambda^{3}&\lambda^{2}&\lambda\\\
\lambda^{3}&\lambda^{2}&\lambda\end{array}\right)\,,$ (80)

where $f,f^{\prime}=u,d,e$.

Substitution of the above results in Eqs. (III.1,III.2,V.1), one finds

$\displaystyle\frac{\Gamma[p\to e_{i}^{+}\,\pi^{0}]}{\Gamma[p\to
e_{i}^{+}\,K^{0}]}$ $\displaystyle\simeq$
$\displaystyle\frac{(m_{p}^{2}-m{{}^{2}_{\pi^{0}}})^{2}}{(m_{p}^{2}-m{{}^{2}_{K^{0}}})^{2}}\,\frac{(1+\tilde{D}+\tilde{F})^{2}}{2\left(1+\frac{m_{p}}{m_{B}}(\tilde{D}-\tilde{F})\right)^{2}}\,{\lambda}^{2}\simeq
3\lambda^{2}\,,$
$\displaystyle\frac{\Gamma[p\to\overline{\nu}\,\pi^{+}]}{\Gamma[p\to\overline{\nu}\,K^{+}]}$
$\displaystyle\simeq$
$\displaystyle\frac{(m_{p}^{2}-m{{}^{2}_{\pi^{+}}})^{2}}{(m_{p}^{2}-m{{}^{2}_{K^{+}}})^{2}}\,\frac{(1+\tilde{D}+\tilde{F})^{2}}{\left(1+\frac{m_{p}}{m_{B}}(\tilde{D}+\tilde{F})\right)^{2}}\,{\lambda}^{2}\simeq
2\lambda^{2}\,.$ (81)

This explains the observation listed as point (a) above. Further,

$\frac{\Gamma[p\to
e^{+}\,\pi^{0}]}{\Gamma[p\to\mu^{+}\,\pi^{0}]}\simeq\frac{\Gamma[p\to
e^{+}\,K^{0}]}{\Gamma[p\to\mu^{+}\,K^{0}]}\simeq\lambda^{2}\,,$ (82)

leads to the result (b). Moreover, (c) can be understood from the fact that

$\frac{\Gamma[p\to\mu^{+}\,K^{0}]}{\Gamma[p\to\overline{\nu}\,K^{+}]}\simeq\frac{(m_{p}^{2}-m{{}^{2}_{K^{0}}})^{2}}{(m_{p}^{2}-m{{}^{2}_{K^{+}}})^{2}}\,\frac{\left(1+\frac{m_{p}}{m_{B}}(\tilde{D}-\tilde{F})\right)^{2}}{\left(1+\frac{m_{p}}{m_{B}}(\tilde{D}+\tilde{F})\right)^{2}}\,\frac{4{\lambda}^{2}}{9}\simeq
0.2\lambda^{2}\,.$ (83)

Also, it can be seen from Eq. (73) that the flavour structure of couplings
relevant for the proton decay are similar in case of ${\bf 10}_{H}$ and
$\overline{\bf 126}_{H}$. This provides justification for the observation made
in (d).

###  VI.2 Limits on the masses of $T$ and $\overline{T}$

Finally, we use the current experimental limits on the lifetime of proton
decay in various channels to obtain the most stringent limit on the masses of
$T$ and $\overline{T}$ in the considered model. When the lightest $T$ is
dominantly given by the one residing in ${\bf 10}_{H}$, the strongest limit on
its mass comes from the decay channels involving $\mu^{+}$. We find from
explicit computation for the best fit solution,

$\displaystyle\tau/{\rm BR}[p\to\mu^{+}\,K^{0}]$ $\displaystyle=$
$\displaystyle 1.6\times 10^{33}\,{\rm
yrs}\,\times\left(\frac{\alpha_{1}}{0.1}\right)^{4}\times\left(\frac{M_{T}}{1.4\times
10^{11}\,{\rm GeV}}\right)^{4}\,,$ $\displaystyle\tau/{\rm
BR}[p\to\mu^{+}\,\pi^{0}]$ $\displaystyle=$ $\displaystyle 1.6\times
10^{34}\,{\rm
yrs}\,\times\left(\frac{\alpha_{1}}{0.1}\right)^{4}\times\left(\frac{M_{T}}{1.3\times
10^{11}\,{\rm GeV}}\right)^{4}\,,$ (84)

where the first factor on the right hand side in the above equations are the
current experimental lower bounds on the lifetime of proton decaying in the
respective channels. Similarly for $\overline{T}$ dominantly arising from
${\bf 10}_{H}$, the most stringent upper bound comes from the proton decaying
into neutrinos and $K^{+}$:

$\displaystyle\tau/{\rm BR}[p\to\overline{\nu}\,K^{+}]$ $\displaystyle=$
$\displaystyle 5.9\times 10^{33}\,{\rm
yrs}\,\times\left(\frac{\alpha_{1}}{0.1}\right)^{4}\times\left(\frac{M_{\overline{T}}}{6.4\times
10^{11}\,{\rm GeV}}\right)^{4}\,,$ (85)

The experimental limits on the lifetimes used for various channels in the
above equations are taken from Takenaka  _et al._ (2020); Regis  _et al._
(2012); Abe  _et al._ (2014).

If the lightest $T$ and $\overline{T}$ arise dominantly from $\overline{\bf
126}_{H}$, we find

$\displaystyle\tau/{\rm BR}[p\to\mu^{+}\,K^{0}]$ $\displaystyle=$
$\displaystyle 1.6\times 10^{33}\,{\rm
yrs}\,\times\left(\frac{\alpha_{2}}{0.1}\right)^{4}\times\left(\frac{M_{T_{i}}}{2.5\times
10^{10}\,{\rm GeV}}\right)^{4}\,,$ $\displaystyle\tau/{\rm
BR}[p\to\mu^{+}\,\pi^{0}]$ $\displaystyle=$ $\displaystyle 1.6\times
10^{34}\,{\rm
yrs}\,\times\left(\frac{\alpha_{2}}{0.1}\right)^{4}\times\left(\frac{M_{T_{i}}}{2.7\times
10^{10}\,{\rm GeV}}\right)^{4}\,,$ (86)

and

$\displaystyle\tau/{\rm BR}[p\to\overline{\nu}\,K^{+}]$ $\displaystyle=$
$\displaystyle 5.9\times 10^{33}\,{\rm
yrs}\,\times\left(\frac{\alpha_{2}}{0.1}\right)^{4}\times\left(\frac{M_{\overline{T}}}{1.1\times
10^{11}\,{\rm GeV}}\right)^{4}\,.$ (87)

It can be noticed that the limits on the masses of $T$ and $\overline{T}$ are
slightly lower than the ones given in Eqs. (VI.2,85). This is due to the fact
that the magnitude of Yukawa couplings with $\overline{\bf 126}_{H}$ are
somewhat smaller than those with ${\bf 10}_{H}$.

###  VI.3 Limit on the mass of $\Delta$

In addition to the $d=6$ operators discussed above, a $d=7$ operator arises
within the model through $B-L$ violating mixing between $\overline{T}$ and
$\Delta$ as discussed in section IV. This induces decay $p\to\nu\,\pi^{+}$ and
$p\to\nu\,K^{+}$. Since the mass of $\overline{T}$ is already constrained by
$d=6$ operators, the lower bound on the mass of $\Delta$ can be inferred from
non-observation of the proton decay. To estimate this, we take $\lambda=1$,
$\alpha_{2}=0.1$, $v_{\overline{D}}=174$ GeV and $M_{\overline{T}}=1.1\times
10^{11}$ GeV from Eq. (87). Substituting these values in Eqs. (47,V.2), we
find

$\displaystyle\tau/{\rm BR}[p\to\nu\,K^{+}]$ $\displaystyle=$ $\displaystyle
5.9\times 10^{33}\,{\rm yrs}\,\times\left(\frac{10^{11}\,{\rm
GeV}}{v_{\sigma}}\right)^{2}\times\left(\frac{M_{\Delta}}{7.0\times
10^{6}\,{\rm GeV}}\right)^{2}\,.$ (88)

The obtained lower bound on $M_{\Delta}$ is two orders of magnitude smaller
than the one obtained in Babu and Mohapatra (2012b) for the same value of
$B-L$ breaking scale. The difference is due to the values of Yukawa couplings
and an additional factor of $(4/15)^{2}$ in Eq. (47) which arise due to
Clebsch-Gordan coefficient.

##  VII Summary and Discussions

Proton decay is a window to peep high energy phenomena from low energy. Its
observation would conclusively discard the conservation of the baryon number
and strengthen the ambition to unify fundamental interactions. Moreover, it
can also provide useful insight into the nature of theory in ultra-violet from
which $B$ and $L$ violations originate. The latter requires the computation of
nucleon decay widths in specific models identifying all possible sources. We
carry out such an exercise for renormalizable $SO(10)$ GUT models. Classifying
the most general scalar spectrum of such theories, we compute explicitly the
couplings of various scalars which can induce baryon and lepton number
violating decays of baryons at the tree level. Effective operators are
computed by integrating out the scalar fields considering also possible mixing
terms between the scalars residing in the given GUT multiplet. We compute
$d=6$ operators which conserve $B-L$ and also derive $d=7$ operators which
violate $B-L$. The latter can lead to nucleon decay modes which are less
studied. We then express the proton decay widths in terms of these operators
and provide a comprehensive analysis of scalar mediated proton decay in a
particular model. The noteworthy features of our general analysis are the
following:

  * •

Even though there exists several multiplets charged under $B-L$ in models with
${\bf 10}_{H}$, $\overline{\bf 126}_{H}$ and ${\bf 120}_{H}$, the $d=6$
operators which can induce $B$ and $L$ non-conserving (but $B-L$ conserving)
baryon decays arise from only three pairs of color triplet fields:
$T(3,1,-\frac{1}{3})$, ${\cal T}(3,1,-\frac{4}{3})$,
$\mathbb{T}(3,3,-\frac{1}{3})$ and their conjugates.

  * •

In the models with ${\bf 10}_{H}$ and/or $\overline{\bf 126}_{H}$, only $T$
and $\overline{T}$ mediate the proton decay. Although $\overline{\bf 126}_{H}$
contains ${\cal T}$ and $\mathbb{T}$ fields, they have only lepto-quark
couplings.

  * •

When ${\bf 120}_{H}$ is present in the model, the contribution to proton decay
from ${\cal T}$-$\overline{\cal T}$ vanishes at tree-level due to the anti-
symmetric nature of Yukawa couplings with ${\bf 120}_{H}$. Therefore, these
fields can contribute to proton decay only at the loop level.

  * •

The $B-L$ non-conserving nucleon decays, which arise through $d=7$ operators
at the leading order, can be mediated in general by $\Theta(3,1,\frac{2}{3})$,
$\Delta(3,2,\frac{1}{6})$, $\Omega(3,2,\frac{7}{6})$ and their conjugate
partners. In the models without ${\bf 120}_{H}$, only $\Delta$ can induce such
decays.

In the context of a minimal model based on ${\bf 10}_{H}$ and $\overline{\bf
126}_{H}$ with Peccei-Quinn symmetry, we find that when scalar mediated
contributions dominate, the proton decay spectrum can be quite different from
the one typically anticipated. Several important aspects of the proton decay
spectrum are listed in the section VI. Proton dominantly decays into
$\overline{\nu}\,K^{+}$ or $\mu^{+}\,K^{0}$ for lighter $\overline{T}$ or $T$,
respectively. Moreover, one finds ${\rm BR}[p\to\mu^{+}\pi^{0}]\gg{\rm
BR}[p\to e^{+}\pi^{0}]$. Both these features are distinct from gauge boson
mediated proton decays in which proton preferably decays into
$\overline{\nu}\,\pi^{+}$ or $e^{+}\,\pi^{0}$ and typically ${\rm
BR}[p\to\mu^{+}\pi^{0}]\ll{\rm BR}[p\to e^{+}\pi^{0}]$. These features mainly
arise from the difference between the Yukawa and gauge couplings. The first is
more sensitive to the flavour structure of the underlying GUT model and hence,
if scalar mediated contributions dominate, the proton decay can provide very
useful insight into the Yukawa structure of the theory.

## Acknowledgements

This work is partially supported under MATRICS project (MTR/2021/000049) by
the Science & Engineering Research Board (SERB), Department of Science and
Technology (DST), Government of India.

##  Appendix A Solution for ${\bf 10}_{H}+\overline{\bf 126}_{H}$ model

In this Appendix, we give necessary parameters of the best fit solution
obtained in the recent work Mummidi and Patel (2021). The solution is obtained
for a minimal non-supersymmetric ${\bf 10}_{H}+\overline{\bf 126}_{H}$ model
with $U(1)$ Peccei-Quinn symmetry. One finds the best fit parameters for the
Yukawa couplings as

$\displaystyle H^{\prime}$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{ccc}0.00023&0&0\\\ 0&-0.04811&0\\\
0&0&-5.79504\\\ \end{array}\right)\times 10^{-3}\,,$ (92) $\displaystyle
F^{\prime}$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{ccc}-0.0088+0.0178i&0.0475\,-0.0889i&0.4635\,+0.6797i\\\
0.0475\,-0.0889i&1.1279\,+0.5108i&-1.2218-2.5921i\\\
0.4635\,+0.6797i&-1.2218-2.5921i&5.4683\,-5.9856i\\\ \end{array}\right)\times
10^{-4}\,.$ (96)

The above with $r=77.4189$, $s=0.3140-0.0282\,i\,$ and
$v_{S}^{\prime}=9.84\times 10^{14}\,{\rm GeV}\,$ leads to the following
effective Yukawa matrices for the quark and leptons:

$\displaystyle Y_{d}$ $\displaystyle=$ $\displaystyle
H^{\prime}+F^{\prime}\,,~{}~{}Y_{u}=r\,(H^{\prime}+sF^{\prime})\,,$
$\displaystyle Y_{e}$ $\displaystyle=$ $\displaystyle
H^{\prime}-3F^{\prime}\,,~{}~{}Y_{\nu}=r\,(H^{\prime}-3sF^{\prime})\,,~{}~{}M_{R}=v_{S}^{\prime}\,F^{\prime}\,.$
(97)

The light neutrino mass matrix is obtained as
$M_{\nu}=-v_{u}^{2}\,Y_{\nu}\,M_{R}^{-1}\,Y_{\nu}^{T}$, where $v_{u,d}=\langle
h_{u,d}\rangle$, $v_{u}^{2}+v_{d}^{2}\equiv v^{2}=(174\,{\rm GeV})^{2}$ and
$v_{u}/v_{d}\equiv\tan\beta=1.5$. The mass matrices for the charged fermions
are given by $M_{d,e}\equiv v_{d}\,Y_{d,e}$ and $M_{u}\equiv v_{u}Y_{u}$.

The unitary matrices $U_{f}$ and $U_{f^{C}}$ diagonalize $M_{f}$
($f=d,u,e,\nu$) such that
$U_{f}^{\dagger}M_{f}M_{f}^{\dagger}U_{f}=U_{f^{C}}^{\dagger}M_{f}^{\dagger}M_{f}U_{f^{C}}\equiv
D^{2}_{f}$. Since $M_{f}$ are symmetric, one finds $U_{f}=U_{f^{C}}^{*}$.
Numerically obtained $U_{f}$ are as the following.

$\displaystyle U_{d}$ $\displaystyle=$ $\displaystyle
U_{d^{C}}^{*}=\left(\begin{array}[]{ccc}-0.9497-0.2658i&-0.0895+0.1388i&-0.0101-0.0116i\\\
-0.1053+0.1272i&-0.5434-0.8213i&0.0276\,+0.0463i\\\ -0.0157&0.0538&0.9984\\\
\end{array}\right)\,,$ (101) $\displaystyle U_{u}$ $\displaystyle=$
$\displaystyle
U_{u^{C}}^{*}=\left(\begin{array}[]{ccc}-0.4028-0.8952i&0.1456\,+0.123i&-0.003-0.0034i\\\
-0.0213-0.1895i&-0.5203-0.8323i&0.0086\,+0.0135i\\\ -0.0016&0.0165&0.9999\\\
\end{array}\right)\,,$ (105) $\displaystyle U_{e}$ $\displaystyle=$
$\displaystyle
U_{e^{C}}^{*}=\left(\begin{array}[]{ccc}-0.3995-0.916i&-0.0093+0.0126i&0.0115\,+0.03i\\\
-0.0161-0.0074i&-0.2511-0.9618i&-0.0268-0.1043i\\\ 0.031&-0.108&0.9937\\\
\end{array}\right)\,,$ (109) $\displaystyle U_{\nu}$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{ccc}-0.4683+0.689i&0.3113\,-0.4343i&0.0721\,-0.1233i\\\
0.151\,-0.2689i&0.3553\,-0.5592i&-0.357+0.5818i\\\ 0.4591&0.5249&0.7168\\\
\end{array}\right)\,.$ (113)

The matrices $H^{\prime}$, $F^{\prime}$ and $U_{f}$ are used to compute the
proton decay spectrum as explained in the section VI.

##  Appendix B Proton decay spectrum from gauge boson mediations

In this Appendix, we compute gauge mediated proton decay partial widths for
quantitative comparison with the scalar induced contributions. The currents
associated with $B-L$ charged gauge bosons, $X\sim(3,2,-5/6)$ and
$Y\sim(3,2,1/6)$ with belong to adjoint representation of $SO(10)$, are given
by 333There also exists vector boson $Z\sim(3,1,2/3)$ with $B-L$ charge $4/3$.
However, it does not induce proton decay by itself Buchmuller and Patel
(2019).Machacek (1979)

$\displaystyle-{\cal L}_{X,Y}$ $\displaystyle=$ $\displaystyle
i\frac{g_{10}}{\sqrt{2}}\,\Big{[}\overline{X}_{\mu}\left(\overline{q}_{A}\gamma^{\mu}u^{C}_{A}+\overline{e^{C}}_{A}\gamma^{\mu}q_{A}-\overline{d^{C}}_{A}\gamma^{\mu}l_{A}\right)\Big{.}$
(114) $\displaystyle+$
$\displaystyle\Big{.}\overline{Y}_{\mu}\left(\overline{q}_{A}\gamma^{\mu}d^{C}_{A}+\overline{\nu^{C}}_{A}\gamma^{\mu}q_{A}-\overline{u^{C}}_{A}\gamma^{\mu}l_{A}\right)\Big{]}\,+\,{\rm
h.c.}\,,$

where $g_{10}$ is unified gauge coupling and we have supressed the $SU(3)$ and
$SU(2)$ indices for simplicity. Eliminating $X$ and $Y$ boson using classical
equations of motions, the effective operators relevant for the proton decay
are derived as:

$\displaystyle{\cal L}_{\rm eff}$ $\displaystyle=$
$\displaystyle\frac{g_{10}^{2}}{2M_{X}^{2}}\left(\overline{u^{C}}_{A}\gamma^{\mu}q_{A}\,\left(\overline{e^{C}}_{B}\gamma_{\mu}q_{B}-\overline{d^{C}}_{B}\gamma_{\mu}l_{B}\right)\right)$
(115) $\displaystyle-$
$\displaystyle\frac{g_{10}^{2}}{2M_{Y}^{2}}\left(\overline{d^{C}}_{A}\gamma^{\mu}q_{A}\,\overline{u^{C}}_{B}\gamma_{\mu}l_{B}\right)\,+\,{\rm
h.c.}\,.$

Using Fierz reordering and fermion field redefinitions, $f\to U_{f}f$, we
obtain (see Buchmuller and Patel (2019) for details)

$\displaystyle{\cal L}_{\rm eff}$ $\displaystyle=$ $\displaystyle
k[u_{A},d_{B},e^{C}_{C},u^{C}_{D}]\left(\overline{e^{C}}_{C}\,\overline{u^{C}}_{D}\,u_{A}\,d_{B}\right)$
(116) $\displaystyle+$ $\displaystyle
k[u^{C}_{A},d^{C}_{B},e_{C},u_{D}]\left(\overline{d^{C}}_{B}\,\overline{u^{C}}_{A}\,u_{D}\,e_{C}\right)$
$\displaystyle+$ $\displaystyle
k[u^{C}_{A},d^{C}_{B},\nu_{C},d_{D}]\left(\overline{d^{C}}_{B}\,\overline{u^{C}}_{A}\,d_{D}\,\nu_{C}\right)\,+\,{\rm
h.c.}\,,$

with

$\displaystyle k[u_{A},d_{B},e^{C}_{C},u^{C}_{D}]$ $\displaystyle=$
$\displaystyle\frac{g_{10}^{2}}{M_{X}^{2}}\left[\left(U_{e^{C}}^{\dagger}U_{d}\right)_{CB}\left(U_{u^{C}}^{\dagger}U_{u}\right)_{DA}+\left(U_{e^{C}}^{\dagger}U_{u}\right)_{CA}\left(U_{u^{C}}^{\dagger}U_{d}\right)_{DB}\right]\,,$
$\displaystyle k[u^{C}_{A},d^{C}_{B},e_{C},u_{D}]$ $\displaystyle=$
$\displaystyle-\frac{g_{10}^{2}}{M_{X}^{2}}\left(U_{d^{C}}^{\dagger}U_{e}\right)_{BC}\left(U_{u^{C}}^{\dagger}U_{u}\right)_{AD}-\frac{g_{10}^{2}}{M_{Y}^{2}}\left(U_{d^{C}}^{\dagger}U_{u}\right)_{BD}\left(U_{u^{C}}^{\dagger}U_{e}\right)_{AC}\,,$
$\displaystyle k[u^{C}_{A},d^{C}_{B},\nu_{C},d_{D}]$ $\displaystyle=$
$\displaystyle\frac{g_{10}^{2}}{M_{X}^{2}}\left(U_{d^{C}}^{\dagger}U_{\nu}\right)_{BC}\left(U_{u^{C}}^{\dagger}U_{d}\right)_{AD}+\frac{g_{10}^{2}}{M_{Y}^{2}}\left(U_{d^{C}}^{\dagger}U_{d}\right)_{BD}\left(U_{u^{C}}^{\dagger}U_{\nu}\right)_{AC}\,.$
(117)

Decay widths of proton in various channels then can be obtained by using
$y[f_{1},f_{2},f_{3},f_{4}]=k[f_{1},f_{2},f_{3},f_{4}]$ and
$y^{\prime}[f_{1},f_{2},f_{3},f_{4}]=0$ in the expressions listed in Eq.
(V.1). Flavour effects arising through various unitary matrices are evaluated
from the best-fit solutions determined in Mummidi and Patel (2021) for the
minimal non-supersymmetric $SO(10)$ model with Peccie-Quinn symmetry. The
results obtained for the best fit solution and for different hierarchy among
the $X$ and $Y$ gauge bosons are given in Table 4. We also give the spectrum
of branching ratios for several solutions with acceptable $\chi^{2}$ in Fig.
3.

Branching ratio [%] |  $M_{X}\ll M_{Y}$ |  $M_{X}\gg M_{Y}$ |  $M_{X}=M_{Y}$
---|---|---|---
${\rm BR}[p\to e^{+}\pi^{0}]$ | $63$ | $33$ | $47$
${\rm BR}[p\to\mu^{+}\pi^{0}]$ | $1$ | $1$ | $<1$
${\rm BR}[p\to\bar{\nu}\pi^{+}]$ | $26$ | $55$ | $46$
${\rm BR}[p\to e^{+}K^{0}]$ | $<1$ | $<1$ | $<1$
${\rm BR}[p\to\mu^{+}K^{0}]$ | $9$ | $<1$ | $4$
${\rm BR}[p\to\bar{\nu}K^{+}]$ | $3$ | $15$ | $1$
${\rm BR}[p\to e^{+}\eta]$ | $<1$ | $<1$ | $<1$
${\rm BR}[p\to\mu^{+}\eta]$ | $<1$ | $<1$ | $<1$
Table 4: Proton decay branching fractions estimated for the best fit solution
for various hierarchies among the masses of $X$ and $Y$ gauge bosons in case
of the minimal renormalizable non-supersymmetric $SO(10)$ model.

Figure 3: Gauge boson mediated proton decay spectrum for several solutions
obtained in case of the minimal renormalizable non-supersymmetric $SO(10)$
model.

It can be seen from Fig. 3 that the results in Table 4 are robust predictions
for the proton decays induced by the vector bosons. Proton decay branching
ratios are qualitatively very different from those obtained by scalar
mediation.

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