††institutetext: Department of Physics, University of Basel,
Klingelbergstrasse 82, CH-4056 Basel, Switzerland

# Quark-lepton Yukawa ratios and nucleon decay in SU(5) GUTs with type-III
seesaw

Stefan Antusch  Kevin Hinze  and Shaikh Saad<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

We consider an extension of the Georgi-Glashow SU(5) GUT model by a
45-dimensional scalar and a 24-dimensional fermionic representation, where the
latter leads to the generation of the observed light neutrino masses via a
combination of a type I and a type III seesaw mechanism. Within this scenario,
we investigate the viability of predictions for the ratios between the charged
lepton and down-type quark Yukawa couplings, focusing on the second and third
family. Such predictions can emerge when the relevant entries of the Yukawa
matrices are generated from single joint GUT operators (i.e. under the
condition of single operator dominance). We show that three combinations are
viable, (i) $y_{\tau}/y_{b}=3/2$, $y_{\mu}/y_{s}=9/2$, (ii)
$y_{\tau}/y_{b}=2$, $y_{\mu}/y_{s}=9/2$, and (iii) $y_{\tau}/y_{b}=2$,
$y_{\mu}/y_{s}=6$. We extend these possibilities to three toy models,
accounting also for the first family masses, and calculate their predictions
for various nucleon decay rates. We also analyse how the requirement of gauge
coupling unification constrains the masses of potentially light relic states
testable at colliders.

##  1 Introduction

Grand Unified Theories (GUTs) Pati:1973rp ; Pati:1974yy ; Georgi:1974sy ;
Georgi:1974yf ; Georgi:1974my ; Fritzsch:1974nn  are arguably one of the most
appealing extensions of the Standard Model (SM) of particle physics. In 1974,
a simple and elegant GUT based on the unifying gauge group SU(5) was proposed
by H. Georgi and S. Glashow (GG model) Georgi:1974sy . However, this model is
incompatible with the current experimental data for three main reasons.
Firstly, the GG model does not allow for gauge coupling unification, which is
a necessary condition for a GUT. Secondly, it predicts massless neutrinos,
which is in conflict with neutrino oscillation experiments requiring that at
least two neutrino should be massive SNO:2002tuh . Thirdly, since the SM Higgs
doublet is embedded into a 5-dimensional Higgs representation of SU(5), the GG
model predicts the GUT scale relation between the charged lepton and down-type
quark Yukawa matrices

$\displaystyle Y_{e}=Y_{d}^{T}.$ (1.1)

This relation in particular implies a GUT scale unification of the tau and
bottom Yukawa couplings $y_{\tau}=y_{b}$, as well as a unification of the muon
and strange Yukawa couplings $y_{\mu}=y_{s}$, which disagrees with the low
energy data.

The first shortcoming requires extending the particle content of the minimal
model by additional GUT representations and suitably splitting the masses of
their component fields such that the running gauge couplings meet. The second
shortcoming can be addressed by introducing SU(5) representations that allow
neutrino mass generation at the tree level Dorsner:2005fq ; Dorsner:2005ii ;
Dorsner:2006hw ; Bajc:2006ia ; Bajc:2007zf ; FileviezPerez:2007bcw ;
Dorsner:2006fx  or at the loop level FileviezPerez:2016sal ; Kumericki:2017sfc
; Saad:2019vjo ; Dorsner:2019vgf ; Dorsner:2021qwg ; Antusch:2023jok .
Finally, the third shortcoming can for instance be resolved by generating the
Yukawa couplings from linear combinations of the renormalisable and higher
dimensional non-renormalisable operators Ellis:1979fg , or at the
renormalizable level by either introducing a 45-dimensional Higgs field and
considering linear combinations of couplings between the SM fermions and both
the 5- as well as the 45-dimensional Higgs field Dorsner:2006dj , or by
introducing vector-like fermions which mix with the SM fermions Babu:2012pb ;
Dorsner:2014wva ; FileviezPerez:2018dyf .

However, historically a first and very aesthetic solution for the third
problem was proposed by H. Georgi and C. Jarlskog (GJ model) in 1979
Georgi:1979df . In their model, the particle content of the GG model is
extended by a 45-dimensional Higgs field (as well as by two 5-dimensional
Higgs fields). If the 45-dimensional Higgs field couples to the SM fermions
this gives rise to the GUT scale relation

$\displaystyle Y_{e}=-3Y_{d}^{T}.$ (1.2)

Considering a linear combination of the operators giving the relations (1.1)
and (1.2) would, on the one hand, solve the shortcoming (as already mentioned
above), but, on the other hand, predictivity in the Yukawa sector would be
lost. Predictivity is however maintained if it is ensured that different
generations of charged leptons and down-type quarks couple to different Higgs
fields (which can, for example, be achieved when a family symmetry is
introduced on top of the gauge symmetry). To achieve predictivity, without
referring to any particular family symmetry, the GJ model hypothesizes the
following textures of the Yukawa coupling matrices,

$\displaystyle Y_{d}=\begin{pmatrix}0&&B&&0\\\ A&&C&&0\\\
0&&0&&D\end{pmatrix},\quad Y_{e}^{T}=\begin{pmatrix}0&&B&&0\\\ A&&-3C&&0\\\
0&&0&&D\end{pmatrix},$ (1.3)

implying the GUT scale relations
$y_{\tau}/y_{b}=1,\,y_{\mu}/y_{s}=-3,\,y_{e}/y_{d}=-1/3$ which were at that
time compatible with the experimental data. However, the current data suggests
(taking only the known SM particles into account in the renormalization group
(RG) evolution) that other ratios such as
$y_{\tau}/y_{b}=3/2,\,y_{\mu}/y_{s}=9/2$ are better suited (see e.g.
Antusch:2021yqe ).

Interestingly, these latter ratios can be obtained from higher dimensional
operators Antusch:2009gu ; Antusch:2013rxa . With these higher dimensional
operators at hand, models similar to the GJ model can be build if the
following two conditions are satisfied: (i) the Yukawa matrices should be
hierarchical, (ii) the 22- and 33- entry should be dominated by a single GUT
operator, a concept which is referred to as single operator dominance
Antusch:2009gu ; Antusch:2013rxa ; Antusch:2019avd .111For models in which the
concept of single operator dominance has been applied, see e.g. Antusch:2011sq
; Antusch:2011qg ; Antusch:2011xz ; Antusch:2012fb ; Antusch:2012gv ;
Antusch:2013wn ; Antusch:2013kna ; Antusch:2013tta ; Antusch:2013eca ;
Antusch:2014poa ; Antusch:2017ano ; Antusch:2017tud ; Antusch:2018gnu ;
Antusch:2022ufb .

Following this approach, non-SUSY GUT scenarios in which neutrino masses are
generated by a type I or a type II seesaw have been investigated in
Antusch:2021yqe , respectively Antusch:2022afk . For GUT scenarios with a type
I seesaw it was shown that the GUT scale ratios $y_{\tau}/y_{b}=3/2$ and
$y_{\mu}/y_{s}=9/2$ are compatible with the experimental data. Moreover, for
GUT scenarios in which neutrino masses are generated by a type II seesaw it
was found, that two combinations of GUT scale relations are viable, namely (i)
$y_{\tau}/y_{b}=3/2$ and $y_{\mu}/y_{s}=9/2$ and (ii) $y_{\tau}/y_{b}=2$ and
$y_{\mu}/y_{s}=6$.

In this paper we will investigate the viability of such GUT scale ratios for
the case that neutrino masses stem from a combination of a type I
Minkowski:1977sc ; Yanagida:1979as ; Gell-Mann:1979vob ; Glashow:1979nm ;
Mohapatra:1980yp  and a type III Foot:1988aq  seesaw mechanism. In this
regard, we will consider a GUT scenario in which the particle content of the
GG model is extended by a fermionic adjoint representation as well as by a
45-dimensional Higgs field.222A non-supersymmetric SU(5) GUT with this
particle content was first considered in FileviezPerez:2007bcw . However, so
far it has not been studied under the assumption of single operator dominance.
The former representation is needed to generate neutrino masses, while the
latter gives rise to operators yielding potentially viable GUT scale Yukawa
ratios. Moreover, both of these representations help to allow for gauge
coupling unification. Using the Mathematica package ProtonDecay
Antusch:2020ztu  and extending the above scenario to “toy models” we also
compute the nucleon decay widths for various decay channels. Finally, we
compute the masses of the added fermion and scalar fields.

The paper is organized as follows: While the GUT scenario as well as the toy
models are introduced in Section 2, the procedure for the numerical analysis
is explained in Section 3. In Section 4 the results are presented and
discussed, before concluding in Section 5. In Appendix A, definitions of the
newly introduced Yukawa couplings are given, while all relevant RGEs that we
have derived are listed in Appendix B.

##  2 GUT scenario

###  2.1 Particle content

The SM fermions are embedded as usual into three generations of
${\bm{\overline{5}_{F}}}_{i}$ and ${\bm{10_{F}}}_{i}$

$\displaystyle{\bm{\overline{5}_{F}}}_{i}=$
$\displaystyle\;d_{i}^{c}(\overline{3},1,\frac{1}{3})\oplus\ell_{i}(1,2,-\frac{1}{2}),$
(2.4) $\displaystyle{\bm{10_{F}}}_{i}=$
$\displaystyle\;q_{i}(3,2,\frac{1}{6})\oplus
u_{i}^{c}(\overline{3},1,-\frac{2}{3})\oplus e_{i}^{c}(1,1,1).$ (2.5)

In the considered scenario, neutrino masses are generated via a combination of
a type I and a type III seesaw mechanism. The corresponding fermionic singlet
$\Sigma_{c}$ and triplet $\Sigma_{b}$ (under $SU(2)_{L}$) are contained in an
adjoint fermionic representation

$\displaystyle\bm{24_{F}}=$
$\displaystyle\;\Sigma_{a}(8,1,0)\oplus\Sigma_{b}(1,3,0)\oplus\Sigma_{c}(1,1,0)\oplus\Sigma_{d}(3,2,-\frac{5}{6})\oplus\Sigma_{e}(\overline{3},2,\frac{5}{6}).$
(2.6)

Moreover, the GUT Higgs fields decompose under the SM gauge group as

$\displaystyle\bm{24_{H}}=$
$\displaystyle\;\Phi_{a}(8,1,0)\oplus\Phi_{b}(1,3,0)\oplus\Phi_{c}(1,1,0)\oplus\Phi_{d}(3,2,-\frac{5}{6})\oplus\Phi_{e}(\overline{3},2,\frac{5}{6}),$
(2.7) $\displaystyle\bm{5_{H}}=$ $\displaystyle\;T_{a}(3,1,-\frac{1}{3})\oplus
H_{a}(1,2,\frac{1}{2}),$ (2.8) $\displaystyle\bm{45_{H}}=$
$\displaystyle\;\phi_{a}(8,2,\frac{1}{2})\oplus\phi_{b}(6,1,-\frac{1}{3})\oplus\phi_{c}(3,3,-\frac{1}{3})\oplus\phi_{d}(\overline{3},2,-\frac{7}{6})\oplus\phi_{e}(\overline{3},1,-\frac{4}{3})$
$\displaystyle\oplus T_{b}(3,1,-\frac{1}{3})\oplus H_{b}(1,2,\frac{1}{2}).$
(2.9)

After the SU(5) breaking, the color triplets $T_{a}$ and $T_{b}$ mix to yield
the mass eigenstates $t_{1}=\cos(\alpha)T_{a}+\sin(\alpha)T_{b}$ and
$t_{2}=-\sin(\alpha)T_{a}+\cos(\alpha)T_{b}$. Similarly, $H_{a}$ and $H_{b}$
mix to form the mass eigenstates $h_{1}=\cos(\beta)H_{a}+\sin(\beta)H_{b}$ and
$h_{2}^{\perp}=-\sin(\beta)H_{a}+\cos(\beta)H_{b}$, where $h_{1}$ is the SM
Higgs doublet.

###  2.2 Neutrino masses

At tree-level the relevant GUT operators for neutrino mass generation
read333After the GUT symmetry breaking these two GUT operators decompose into
19 SM Yukawa interactions. For details see Appendix A.

$\displaystyle\mathcal{L}\supset
Y_{A}\,\bm{\overline{5}_{F}}\bm{24_{F}}\bm{5_{H}}+Y_{B}\,\bm{\overline{5}_{F}}\bm{24_{F}}\bm{45_{H}}.$
(2.10)

After the GUT symmetry breaking the following relevant terms emerge

$\displaystyle\mathcal{L}\supset-
Y_{2}\ell\Sigma_{b}H_{a}-Y_{8}\ell\Sigma_{b}H_{b}-Y_{4}\ell\Sigma_{c}H_{a}-Y_{13}\ell\Sigma_{c}H_{b}-m_{\Sigma_{b}}\Sigma_{b}\Sigma_{b}-m_{\Sigma_{c}}\Sigma_{c}\Sigma_{c},$
(2.11)

where $m_{\Sigma_{b}}$ and $m_{\Sigma_{b}}$ are the respective masses of
$\Sigma_{b}$ and $\Sigma_{c}$, and where the GUT scale relations

$\displaystyle Y_{2}=-\sqrt{\frac{3}{10}}\,Y_{A},\quad Y_{4}=Y_{A},\quad
Y_{8}=\frac{\sqrt{5}}{4}\,Y_{B},\quad\text{and}\quad
Y_{13}=\frac{\sqrt{3}}{4}\,Y_{B}$ (2.12)

hold. After the SU(2) triplet $\Sigma_{b}$ and SU(2) singlet $\Sigma_{c}$ have
been integrated out and the two Higgs fields $H_{a}$ and $H_{b}$ have taken
their vacuum expectation values (vevs) $v_{a}$ and $v_{b}$, where
$v_{a}^{2}+v_{b}^{2}=v^{2}=(246\ \text{GeV})^{2}$, and where
$v_{a}=v\cos(\beta)$ and $v_{b}=v\sin(\beta)$, the neutrino mass matrix
$m_{\nu}$ reads

$\displaystyle
m_{\nu}^{ij}=-\frac{(Y_{2}^{i}\,v_{a}+Y_{8}^{i}\,v_{b})(Y_{2}^{j}\,v_{a}+Y_{8}^{j}\,v_{b})}{4m_{\Sigma_{b}}}-\frac{(Y_{4}^{i}\,v_{a}+Y_{13}^{i}\,v_{b})(Y_{4}^{j}\,v_{a}+Y_{13}^{j}\,v_{b})}{4m_{\Sigma_{c}}}\,.$
(2.13)

Since the neutrino mass matrix $m_{\nu}$ is of rank two, two massive and one
massless neutrino are predicted.

###  2.3 Quark-lepton Yukawa ratios

With $\bm{X}$ and $\bm{Y}$ representing one or multiple Higgs fields, the
charged fermion masses stem from GUT operators of the form

$\displaystyle\bm{Y_{\overline{5}}}^{ij}\,:\quad{\bm{10_{F}}}_{i}{\bm{\overline{5}_{F}}}_{j}\bm{X}\quad$
$\displaystyle\supset\quad Y_{d}^{ij},\ Y_{e}^{ij}$ (2.14)
$\displaystyle\bm{Y_{10}}^{ij}\,:\quad{\bm{10_{F}}}_{i}{\bm{10_{F}}}_{j}\bm{Y}\quad$
$\displaystyle\supset\quad Y_{u}^{ij},$ (2.15)

where $Y_{u}$, $Y_{d}$ and $Y_{e}$ denote the usual SM charged fermion Yukawa
matrices. Assuming in the charged fermion Yukawa sector the concept of single
operator dominance, i.e. that each Yukawa entry is dominated by a singlet GUT
operator, allows to connect the down-type with the charged lepton Yukawa
matrix via group theoretical Clebsch-Gordan (CG) factors $c_{ij}$. In SU(5)
GUTs, and considering up to dimension five operators, the potentially viable
CG factors are
$|c_{ij}|\in\\{1/6,\,1/2,\,2/3,\,1,\,3/2,\,2,\,3,\,9/2,\,6,\,9,\,18\\}$. The
possible GUT operators yielding these ratios are given in Antusch:2009gu ;
Antusch:2013rxa . Moreover, if the matrix $\bm{Y_{\overline{5}}}$ is assumed
to be of hierarchical nature and dominated by its diagonal entries, then the
second and third family down-type quark and charged lepton masses stem
dominantly from the GUT operators $\mathcal{O}_{2}$ and $\mathcal{O}_{3}$
dominating the 22 and 33 positions in $\bm{Y_{\overline{5}}}$. Depending on
which operators are chosen for $\mathcal{O}_{2}$ and $\mathcal{O}_{3}$,
different GUT scale Yukawa ratios $y_{\tau}/y_{b}$ and $y_{\mu}/y_{s}$ are
predicted. Our numerical analysis (cf. Section 4) shows that there are only
two possible choices for the GUT scale ratio $y_{\tau}/y_{b}$, namely 3/2 or
2. The former CG factor can be complemented by a factor 9/2 for the second
family, while for the latter CG factor two different completions,
$y_{\mu}/y_{s}=6$ or $y_{\mu}/y_{s}=9/2$, are possible.

###  2.4 Toy models

We now extend the above motivated scenarios to three toy models which also
include the first family. For simplicity, we chose the matrix
$\bm{Y_{\overline{5}}}$ to be of diagonal nature. The double ratio
$(y_{\mu}y_{d})/(y_{e}y_{s})=10.7^{+1.6}_{-0.9}$, which is nearly constant
under renormalization group running (see e.g. Antusch:2013jca ), suggests,
that the the ratio $y_{\mu}/y_{s}=9/2$ is best complemented by a ratio
$y_{e}/y_{d}=4/9$, while the best completion of the ratio $y_{\mu}/y_{s}=6$ is
given by $y_{e}/y_{d}=1/2$. Utilizing these ratios, our three toy models
relate the down-type with the charged lepton Yukawa matrix via

$\displaystyle\text{Model 1:}\quad
Y_{e}=\text{diag}\left(\frac{4}{9},\frac{9}{2},\frac{3}{2}\right)\cdot
Y_{d}^{T}\,,$ (2.16) $\displaystyle\text{Model 2:}\quad
Y_{e}=\text{diag}\left(\frac{4}{9},\frac{9}{2},2\right)\cdot Y_{d}^{T}\,,$
(2.17) $\displaystyle\text{Model 3:}\quad
Y_{e}=\text{diag}\left(\frac{1}{2},6,2\right)\cdot Y_{d}^{T}\,.$ (2.18)

Moreover, for simplicity444We might consider higher-dimensional operators also
for $\bm{Y_{10}}$, for example to explain the mass hierarchy, however since no
Yukawa ratio predictions arise from this sector, we stick to the simplest case
in our toy models. we assume in each toy model that $\bm{Y_{10}}$ is dominated
by the operator $\bm{10_{F}10_{F}5_{H}}$ in all entries, yielding a symmetric
up-type Yukawa matrix, i.e. $Y_{u}=Y_{u}^{T}$. Finally, in all toy models
neutrino masses stem from a linear combination of the operators
$\bm{\overline{5}_{F}24_{F}5_{H}}$ and $\bm{\overline{5}_{F}24_{F}45_{H}}$.

##  3 Numerical procedure

###  3.1 Implementation of the charged fermion Yukawa sector

We implement all three toy models at the GUT scale as described in Section
2.4. In all three models the down-type Yukawa matrix $Y_{d}$ is simply
implemented as

$\displaystyle Y_{d}=\text{diag}(y_{1}^{d},y_{2}^{d},y_{3}^{d}),$ (3.19)

while the charged lepton Yukawa matrix $Y_{e}$ is implemented according to Eq.
(2.16), (2.17), and (2.18), respectively. Since $Y_{u}$ is symmetric we use a
Takagi decomposition and implement it as

$\displaystyle Y_{u}=U_{u}^{\dagger}Y_{u}^{\text{diag}}U_{u}^{*},$ (3.20)

where555Here we have dropped three unphysical parameters but kept the GUT
phases $\beta_{1}^{u}$ and $\beta_{2}^{u}$ which effect the nucleon decay
widths Ellis:1979hy ; Ellis:2019fwf .

$\displaystyle U_{u}=\begin{pmatrix}1&0&0\\\ 0&c_{23}^{u}&s_{23}^{u}\\\
0&-s_{23}^{u}&c_{23}^{u}\end{pmatrix}\begin{pmatrix}c_{13}^{u}&0&s_{13}^{u}e^{-i\delta^{u}}\\\
0&1&0\\\
-s_{13}^{u}e^{i\delta^{u}}&0&c_{13}^{u}\end{pmatrix}\begin{pmatrix}c_{12}^{u}&s_{12}^{u}&0\\\
-s_{12}^{u}&c_{12}^{u}&0\\\
0&0&1\end{pmatrix}\begin{pmatrix}e^{i\beta_{1}^{u}}&0&0\\\
0&e^{i\beta_{2}^{u}}&0\\\ 0&0&1\end{pmatrix},$ (3.21)

and where $Y_{u}^{\text{diag}}=\text{diag}(y_{1}^{u},y_{2}^{u},y_{3}^{u})$.

###  3.2 Implementation of the neutrino sector

In order to simplify the analysis we assume in the neutrino sector that the
Yukawa matrices $Y_{5}$ and $Y_{6}$ (for the definitions of these couplings,
see Appendix A) are of the form

$\displaystyle Y_{5}=z_{1}\begin{pmatrix}0\\\ 1\\\ 1\end{pmatrix},\quad\quad
Y_{6}=z_{2}\begin{pmatrix}1\\\ 1\\\ 3\end{pmatrix},$ (3.22)

where $z_{1}$ and $z_{2}$ are real parameters. Furthermore, we denote the
relative phase difference between $m_{\Sigma_{b}}$ and $m_{\Sigma_{c}}$ by
$\gamma$ (i.e. $\gamma=\arg(m_{\Sigma_{b}}/m_{\Sigma_{c}})$). This structure
is motivated by CSD3 King:2013iva  which in the case of type I seesaw has been
shown to correctly describe the low-scale neutrino observables together with a
normal neutrino mass hierarchy (see e.g. Antusch:2021yqe  for a recent work).

###  3.3 GUT scale parameters and low energy observables

Each toy model contains 33 input parameters which decompose into the GUT scale
$M_{\text{GUT}}$, the SU(5) gauge coupling $g_{\text{GUT}}$, the masses of the
added particles,666Note that $m_{\Sigma_{d}}=m_{\Sigma_{e}}$. $m_{\Phi_{a}}$,
$m_{\Phi_{b}}$, $m_{\phi_{a}}$, $m_{\phi_{b}}$, $m_{\phi_{c}}$,
$m_{\phi_{d}}$, $m_{\phi_{e}}$, $m_{\Sigma_{a}}$, $m_{\Sigma_{b}}$,
$m_{\Sigma_{c}}$, $m_{\Sigma_{d}}$, $m_{t_{1}}$, $m_{t_{2}}$, $m_{h_{2}}$, the
singular values $y_{1}^{u}$, $y_{2}^{u}$, $y_{3}^{u}$, $y_{1}^{d}$,
$y_{2}^{d}$, $y_{3}^{d}$ and angles $\theta_{12}^{u}$, $\theta_{13}^{u}$,
$\theta_{23}^{u}$ as well as phases $\delta^{u}$, $\beta_{1}^{u}$,
$\beta_{2}^{u}$ of the charged fermion Yukawa matrices, the parameters of the
neutrino Yukawa couplings $z_{1}$, $z_{2}$, and $\gamma$, and the eigenstate
mixing angles $\alpha$ and $\beta$. The respective ranges of these input
parameters are given by777Note that although we do not put any perturbativity
constraints on the neutrino Yukawa couplings $z_{1}$ and $z_{2}$ the fit
automatically choses them to be below 1 (cf. Section 4).

$\displaystyle M_{\text{GUT}}$ $\displaystyle<M_{Pl},$ $\displaystyle
m_{\Phi_{a}},m_{\Phi_{b}},m_{\phi_{a}},m_{\phi_{b}},m_{\phi_{c}},m_{\phi_{d}},m_{\phi_{e}},m_{\Sigma_{a}},m_{\Sigma_{b}},m_{\Sigma_{c}},m_{\Sigma_{d}},m_{h_{2}}$
$\displaystyle\in[1\,\text{TeV},M_{\text{GUT}}],$ $\displaystyle
m_{t_{1}},m_{t_{2}}$ $\displaystyle\in[10^{11}\,\text{GeV},M_{\text{GUT}}],$
$\displaystyle
g_{\text{GUT}},\,y_{1}^{u},\,y_{2}^{u},\,y_{3}^{u},\,y_{1}^{d},\,y_{2}^{d},\,y_{3}^{d}$
$\displaystyle\in[0,1],$ (3.23)
$\displaystyle\theta_{12}^{u},\,\theta_{13}^{u},\,\theta_{23}^{u},\,\alpha,\,\beta$
$\displaystyle\in[0,\pi/2],$
$\displaystyle\delta^{u},\,\beta_{1}^{u},\,\beta_{2}^{u},\,\gamma$
$\displaystyle\in[-\pi,\pi),$ $\displaystyle z_{1},\,z_{2}$ $\displaystyle>0.$

These input parameters are fitted to the 22 low-scale observables (listed in
Eq. (3.3)) and the nucleon decay widths of thirteen decay channels (listed in
Table I).

$\displaystyle g_{1},\,g_{2},\,g_{3},$ $\displaystyle
y_{u},\,y_{c},\,y_{t},\,y_{d},\,y_{s},\,y_{b},\,\theta_{12}^{\text{CKM}},\,\theta_{13}^{\text{CKM}},\,\theta_{23}^{\text{CKM}},\,\delta^{\text{CKM}},\,y_{e},\,y_{\mu},\,y_{\tau},\,\hskip
17.07182pt$ (3.24) $\displaystyle\Delta m_{21}^{2},\,\Delta
m_{31}^{2},\,\theta_{12}^{\text{PMNS}},\,\theta_{13}^{\text{PMNS}},\,\theta_{23}^{\text{PMNS}},\,\delta^{\text{PMNS}}.$

For the SM gauge couplings and Yukawa observables we take the experimental
values from Antusch:2013jca , while the values for the neutrino sector are
taken from NuFIT 5.1 Esteban:2020cvm .

| decay channel |  $\tau/\mathcal{B}$ [year] |  $\Gamma_{\text{partial}}$ [GeV] | Reference
---|---|---|---|---
Proton: | $p\rightarrow\pi^{0}\,e^{+}$ | $>\,2.4\cdot 10^{34}$ | $<\,8.7\cdot 10^{-67}$ | Super-Kamiokande:2020wjk
| $p\rightarrow\pi^{0}\,\mu^{+}$ | $>\,1.6\cdot 10^{34}$ | $<\,1.3\cdot 10^{-66}$ | Super-Kamiokande:2020wjk
| $p\rightarrow\eta^{0}\,e^{+}$ | $>\,1.0\cdot 10^{34}$ | $<\,2.0\cdot 10^{-66}$ | Super-Kamiokande:2017gev
| $p\rightarrow\eta^{0}\,\mu^{+}$ | $>\,4.7\cdot 10^{33}$ | $<\,4.4\cdot 10^{-66}$ | Super-Kamiokande:2017gev
| $p\rightarrow K^{0}\,e^{+}$ | $>\,1.1\cdot 10^{33}$ | $<\,1.9\cdot 10^{-65}$ | Brock:2012ogj
| $p\rightarrow K^{0}\,\mu^{+}$ | $>\,3.6\cdot 10^{33}$ | $<\,5.8\cdot 10^{-66}$ | Super-Kamiokande:2022egr
| $p\rightarrow\pi^{+}\,\overline{\nu}$ | $>\,3.9\cdot 10^{32}$ | $<\,5.3\cdot 10^{-65}$ | Super-Kamiokande:2013rwg
| $p\rightarrow K^{+}\,\overline{\nu}$ | $>\,6.6\cdot 10^{33}$ | $<\,3.2\cdot 10^{-66}$ | Takhistov:2016eqm
Neutron: | $n\rightarrow\pi^{-}\,e^{+}$ | $>\,5.3\cdot 10^{33}$ | $<\,3.9\cdot 10^{-66}$ | Super-Kamiokande:2017gev
| $n\rightarrow\pi^{-}\,\mu^{+}$ | $>\,3.5\cdot 10^{33}$ | $<\,5.9\cdot 10^{-66}$ | Super-Kamiokande:2017gev
| $n\rightarrow\pi^{0}\,\overline{\nu}$ | $>\,1.1\cdot 10^{33}$ | $<\,1.9\cdot 10^{-65}$ | Super-Kamiokande:2013rwg
| $n\rightarrow\eta^{0}\,\overline{\nu}$ | $>\,5.6\cdot 10^{32}$ | $<\,3.7\cdot 10^{-65}$ | Brock:2012ogj
| $n\rightarrow K^{0}\,\overline{\nu}$ | $>\,1.2\cdot 10^{32}$ | $<\,1.7\cdot 10^{-64}$ | Brock:2012ogj
Table I: Current experimental bounds on the decay widths
$\Gamma_{\text{partial}}$, respectively lifetime $\tau/\mathcal{B}$ at 90 %
confidence level, where $\mathcal{B}$ is the branching ratio for the decay
channel. See also Dev:2022jbf  for future projections and sensitivities of
various upcoming detectors.

###  3.4 Fitting procedure

After implementing the input parameters given in Eq. (3.3) at the GUT scale we
compute the RG evolution to the $Z$ scale. For the gauge couplings we use a
2-loop running, while we compute the running of the Yukawa matrices and the
effective neutrino mass operator at 1-loop. The nucleon decay widths are
computed using the Mathematica package Proton Decay Antusch:2020ztu  (for a
description of the calculation see e.g. Antusch:2021yqe ). Taking into account
all observables we compute at the low scale the $\chi^{2}$-function which we
minimize using a differential evolution algorithm giving us a benchmark point.
With a flat prior distribution we calculate $4\times 10^{6}$ data points
performing a Markov-chain-Monte-Carlo (MCMC) analysis using an adaptive
Metropolis-Hastings algorithm Metropolis-Hastings-algorithm  which we start
from this benchmark point. These data points are finally used to compute the
highest posterior density (HPD) ranges of various quantities.

##  4 Results

The results of our numerical analysis are presented in this section. We are in
particular interested in the nucleon decay predictions, the intermediate-scale
particle masses as well as the low scale predictions for the charged lepton
and down-type quark mass ratios. In Section 4.1 we show the results of our
minimization procedure. Starting an MCMC analysis from these benchmark points
allows us to obtain the HPD ranges of various quantities. The results of this
analysis is presented in Section 4.2.

###  4.1 Benchmark points

We obtain for all three models benchmark points through a minimization of the
$\chi^{2}$-function as described in Section 3. In Table II the input
parameters for the respective benchmark points are listed. Moreover, the
dominant pulls $\chi_{i}^{2}$ are presented in Table III. All three models can
be very well fitted to the data. The strongest (though quite small) pull is
given by the first and second family down-type quark masses. The biggest
difference between the three models is the respectively favored GUT scale. For
Models 2 and 3 a GUT scale above $10^{17}$ GeV is favored, while for the
benchmark point of Model 1 a GUT scale below $10^{16}$ GeV is obtained. This
also results in different results for the predicted nucleon decay rates (cf.
Section 4.2). Another difference is the preferred choice of some of the
intermediate-scale particle masses. In the presented benchmark point the mass
of the fermionic field $\Sigma_{a}$ is obtained to be at the GUT scale for
Model 1, at the intermediate scale for Model 2 and at the relatively low scale
(23 TeV) for Model 3. Moreover, a mass of the leptoquark $\phi_{c}$ of 1 TeV,
respectively 4 TeV is obtained for Model 3, respectively Model 2, whereas for
Model 1 the mass of this field is above $10^{6}$ TeV. For the HPD results of
these particle masses confer the subsequent section.

|  | Model 1 |  |  | Model 2 |  |  | Model 3 |
---|---|---|---|---|---|---|---|---|---
$g_{\text{GUT}}\;/\;10^{-1}$ |  | 5.94 |  |  | 6.17 |  |  | 6.33 |
$\log_{10}(M_{\text{GUT}}\;/\;\text{GeV})$ |  | 15.6 |  |  | 17.2 |  |  | 17.3 |
$\log_{10}(m_{\phi_{a}}\;/\;\text{GeV})$ |  | 9.43 |  |  | 14.0 |  |  | 16.7 |
$\log_{10}(m_{\phi_{c}}\;/\;\text{GeV})$ |  | 9.02 |  |  | 3.63 |  |  | 3.00 |
$\log_{10}(m_{\Sigma_{a}}\;/\;\text{GeV})$ |  | 15.6 |  |  | 7.53 |  |  | 4.36 |
$\log_{10}(m_{\Sigma_{b}}\;/\;\text{GeV})$ |  | 14.2 |  |  | 14.9 |  |  | 14.7 |
$\log_{10}(m_{\Sigma_{c}}\;/\;\text{GeV})$ |  | 13.8 |  |  | 12.8 |  |  | 13.2 |
$\log_{10}(m_{\Sigma_{d}}\;/\;\text{GeV})$ |  | 14.2 |  |  | 15.9 |  |  | 14.1 |
$y_{1}^{u}\;/\;10^{-6}$ |  | 2.63 |  |  | 2.11 |  |  | 1.99 |
$y_{2}^{u}\;/\;10^{-3}$ |  | 1.46 |  |  | 1.37 |  |  | 1.18 |
$y_{3}^{u}\;/\;10^{-1}$ |  | 4.54 |  |  | 4.26 |  |  | 3.65 |
$y_{1}^{d}\;/\;10^{-6}$ |  | 6.21 |  |  | 6.30 |  |  | 5.46 |
$y_{2}^{d}\;/\;10^{-4}$ |  | 1.31 |  |  | 1.21 |  |  | 0.99 |
$y_{3}^{d}\;/\;10^{-3}$ |  | 6.64 |  |  | 6.01 |  |  | 5.36 |
$z_{1}\;/\;10^{-1}$ |  | 3.50 |  |  | 9.42 |  |  | 6.86 |
$z_{2}\;/\;10^{-1}$ |  | 1.12 |  |  | 0.32 |  |  | 0.50 |
$\gamma$ |  | 1.85 |  |  | 1.48 |  |  | 1.68 |
$\alpha$ |  | 0.50 |  |  | 1.00 |  |  | 0.50 |
Table II: The GUT scale input parameters of the benchmark points for all three models. | $\chi^{2}$ | $\chi^{2}_{y_{d}}$ | $\chi^{2}_{y_{s}}$ | $\chi^{2}_{y_{b}}$ | $\chi^{2}_{y_{\mu}}$ | $\chi^{2}_{y_{\tau}}$ | $\chi^{2}_{\Gamma({p\rightarrow\pi^{0}e^{+}})}$
---|---|---|---|---|---|---|---
Model 1 | 1.36 | 0.27 | 0.41 | 0.06 | 0.04 | 0.14 | 0.44
Model 2 | 0.31 | 0.23 | 0.02 | 0.01 | 0.00 | 0.05 | 0.00
Model 3 | 0.33 | 0.17 | 0.03 | 0.00 | 0.06 | 0.07 | 0.00
Table III: The total $\chi^{2}$ as well as the dominant pulls $\chi_{i}^{2}$
for the benchmark points of all three models.

###  4.2 Highest posterior densities

As described in Section 3.4 we vary the input parameters listed in Eq. 3.3
around their benchmark points (cf. Table II) using an MCMC analysis. From
these generated points we then compute the HPD intervals of various parameters
and observables.

In Figures 1 – 4 we use the following color coding: For Model 1, 2, and 3 the
HPD intervals of various quantities are colored red, green, and blue,
respectively, while the 1$\sigma$ (2$\sigma$) HPD intervals are colored dark
(light).

####  4.2.1 Quark-lepton mass ratios

The HPD results for the low scale charged lepton and down-type quark mass
ratios are presented in Figure 1. The horizontal dashed line represents the
current experimental central value, whereas the white region shows the current
experimental 1$\sigma$ range. Clearly, all three models are capable of
reproducing viable mass ratios. This strengthens the results of the benchmark
points in the previous subsection (cf. Tables II and III). Compared to Model 2
and 3, Model 1 gives a bit smaller predictions for the mass ratios for all
three generations.

Figure 1: Low scale ($M_{Z}$) HPD intervals for charged lepton and down-type
quark Yukawa ratios of all three families. The 1$\sigma$ (2$\sigma$) HDP
intervals are colored dark (light). Figure 2: HPD intervals of the
intermediate-scale particle masses. The 1$\sigma$ (2$\sigma$) HDP intervals
are colored dark (light).

####  4.2.2 Intermediate-scale particle masses

Figure 2 shows the predicted HPD intervals of the intermediate-scale particle
masses. Most of the masses are predicted to be out of the reach of current and
future colliders, because they would either produce too much proton decay,
spoil gauge coupling unification or because of the fit of the fermion masses.
But interestingly, the fields $\Phi_{b}$, $\phi_{c}$ and $\Sigma_{a}$ are not
only potentially within the reach of future searches, but can also be used to
distinguish between the different models: An observation of the one of the
fields $\Phi_{b}$ or $\Sigma_{a}$ would strongly hint towards Model 3, while
an observation of the field $\phi_{c}$ would disfavor Model 1. In fact, the
most promising lookout could be for the leptoquark $\phi_{c}$. The upper bound
of the HPD 1$\sigma$ range is predicted to be 23 TeV (2.8 TeV) in Model 2 (3),
whereas the upper bound of the 2$\sigma$ intervals is 175 TeV (17 TeV). In the
following, we briefly state the current collider bounds on these particles.

The scalar triplet, $\Phi_{b}$, with zero hypercharge, residing in the
$\bm{24_{H}}$ multiplet is expected to be light in Model 3. Note that
$\Phi_{b}$ contains a neutral $\Phi_{b}^{0}$ and a pair of singly charged
$\Phi_{b}^{\pm}$ states. In the low-energy effective theory, a term of the
form $h_{1}^{\dagger}\Phi_{b}^{2}h_{1}$ is allowed, where $h_{1}$ is the SM
Higgs doublet. As a result of this coupling, the SM Higgs can decay into two
photons $h^{0}\to\gamma\gamma$ via a one-loop diagram mediated by the
$\Phi_{b}^{\pm}$ states. Consistency with the LHC data requires these charged
states to have masses above 250 GeV Chabab:2018ert .

The scalar leptoquark $\phi_{c}$, which is a triplet of SU(2)L, resides around
the TeV scale in Models 2 and 3. In both models, its coupling to the SM
fermions is dominated by the third-generation quark and lepton. Hence, within
our scenarios, its decay branching fraction is dominated by a $b\tau$ final
state. Since leptoquarks carry color, they are efficiently produced at the LHC
through gluon-initiated as well as quark-initiated processes Diaz:2017lit .
LHC searches of $pp\to b\overline{b}\tau\overline{\tau}$ from pair-produced
leptoquarks rule out leptoquark masses below 1400 GeV ATLAS:2019qpq ;
ATLAS:2021oiz .

As can be seen from Eq. (A.26), the color octet fermion $\Sigma_{a}$, which is
expected to be light in Model 3, couples, for example, to a singlet down-quark
(lepton doublet) and a super-heavy colored triplet (octet) scalar.
Consequently, the lifetime of a TeV scale $\Sigma_{a}$ is expected to be
large, and it behaves like a long-lived gluino that typically arises in split-
supersymmetric scenarios Giudice:2004tc ; Arkani-Hamed:2004zhs . Long-lived
colored particles would hadronize, forming so-called R-hadrons Farrar:1978xj .
These bound states are comprised of the long-lived state and light SM quarks
or gluons, and interact with the detector material, typically inside the
calorimeters, via hadronic interactions of the light-quark constituents.
Motivated by split-supersymmetric models, R-hadrons are extensively searched
for at the LHC ATLAS:2019gqq ; CMS:2020iwv . Non-observation of any deviations
of the signal from the expected background puts to a lower limit on the mass
of the long-lived $\Sigma_{a}$ fermion of 2000 GeV ATLAS:2019gqq .

Figure 3: Predicted HPD intervals of the GUT scale. The 1$\sigma$ (2$\sigma$)
HDP intervals are colored dark (light).

Figure 4: Predicted HPD intervals of the nucleon decay widths. The 1$\sigma$
(2$\sigma$) HDP intervals are colored dark (light). For each decay channel the
blue line segments represent the current experimental constraints. The future
Hyper-Kamiokande constraints for the decay channels $p\rightarrow\pi^{0}e^{+}$
and $n\rightarrow\pi^{-}e^{+}$ are indicated by orange line segments.

####  4.2.3 Nucleon decay width and GUT scale

Figure 4 shows the predictions for the HPD intervals of the GUT scale
$M_{\text{GUT}}$. Moreover, the predicted HPD ranges for the nucleon decay
widths of the various decay channels are presented in Figure 4. The blue line
segments in the latter picture indicate the current experimental bounds at 90
% confidence level (cf. Table I). Moreover, the future constraints on the
decay widths for the decay channels $p\rightarrow\pi^{0}e^{+}$ and
$n\rightarrow\pi^{-}e^{+}$ which will be provided by Hyper-Kamiokande Hyper-
Kamiokande:2018ofw  are presented by orange line segments.

In Figure 4 it can be seen that Model 1 clearly predicts the GUT scale to be
below $10^{16}$ GeV. On the other hand, a much larger GUT scale is preferred
by the Models 2 and 3. Since the nucleon decay width is inversely proportional
to the forth power of the GUT scale in the case of gauge boson mediated
nucleon decay, this also results in strongly different prediction for the
nucleon decay widths of the various channels as it can be seen in Figure 4.
The nucleon decay predictions for Model 1 are very close to the current
bounds, the 1$\sigma$ HPD interval of the proton decay channel
$p\rightarrow\pi^{0}e^{+}$ will be fully probed by Hyper-Kamiokande. Moreover,
Hyper-Kamiokande will probe most of the 1$\sigma$ HPD interval of the neutron
decay channel $n\rightarrow\pi^{-}e^{+}$. On the other hand, the gauge boson
mediated nucleon decay is highly suppressed in Models 2 and 3 and cannot be
probed by any planed experiments. Therefore, observation of nucleon decay in
the decay channels $p\rightarrow\pi^{0}e^{+}$ and $n\rightarrow\pi^{-}e^{+}$
would clearly favour Model 1 over the Models 2 and 3.

##  5 Conclusion

In this paper we considered an extension of the Georgi-Glashow SU(5) GUT
scenario by a 45-dimensional scalar and a 24-dimensional fermionic
representation. Neutrino masses in this scenario are generated by a
combination of a type I and a type III seesaw mechanism. Assuming the concept
of single operator dominance we investigated which GUT scale charged lepton
and down-type quark Yukawa ratios can be viable for the second and third
family and found that three combinations work: (i) $y_{\tau}/y_{b}=3/2$,
$y_{\mu}/y_{s}=9/2$, (ii) $y_{\tau}/y_{b}=2$, $y_{\mu}/y_{s}=9/2$, and (iii)
$y_{\tau}/y_{b}=2$, $y_{\mu}/y_{s}=6$. Also taking into account the origin of
the first family masses we extended these possibilities to three toy models
and analyzed various of their predictions. We showed that experimental
discrimination between these models could be possible since they predict
different nucleon decay rates as well as distinct light relics.

##  Appendix A Definition of new Yukawa couplings

The Lagrangian density contains the two terms

$\displaystyle\mathcal{L}\supset
Y_{A}\,\bm{\overline{5}_{F}}^{i}\bm{24_{F}5_{H}}+Y_{B}\,\bm{\overline{5}_{F}}^{i}\bm{24_{F}45_{H}}.$
(A.25)

After the GUT symmetry breaking they decompose into

$\displaystyle\mathcal{L}=\;$
$\displaystyle\sqrt{\frac{2}{15}}Y_{A}\,d^{c}\Sigma_{c}T_{a}-\sqrt{\frac{3}{10}}Y_{A}\,\ell\Sigma_{c}H_{a}+Y_{A}\,d^{c}\Sigma_{a}T_{a}+Y_{A}\,\ell\Sigma_{b}H_{a}+$
$\displaystyle
Y_{A}\,d^{c}\Sigma_{d}H_{a}+Y_{A}\,\ell\Sigma_{e}T_{a}+\sqrt{\frac{5}{12}}Y_{B}\,d^{c}\Sigma_{c}T_{b}+\frac{\sqrt{5}}{4}Y_{B}\,\ell\Sigma_{c}H_{b}+$
$\displaystyle\frac{1}{2\sqrt{2}}Y_{B}\,d^{c}\Sigma_{a}T_{b}+\frac{1}{\sqrt{2}}Y_{B}\,d^{c}\Sigma_{a}\phi_{b}+\frac{1}{\sqrt{2}}Y_{B}\,\ell\Sigma_{a}\phi_{a}+\frac{1}{\sqrt{2}}Y_{B}\,d^{c}\Sigma_{b}\phi_{c}+$
$\displaystyle\frac{\sqrt{3}}{4}Y_{B}\,\ell\Sigma_{b}H_{b}-\frac{1}{\sqrt{2}}Y_{B}\,d^{c}\Sigma_{d}\phi_{a}-\frac{1}{2\sqrt{6}}Y_{B}\,d^{c}\Sigma_{d}H_{b}+\frac{1}{\sqrt{2}}Y_{B}\,\ell\Sigma_{d}\phi_{e}-$
$\displaystyle\frac{1}{\sqrt{2}}Y_{B}\,d^{c}\Sigma_{e}\phi_{d}+\frac{1}{2\sqrt{2}}Y_{B}\,\ell\Sigma_{e}T_{b}-\frac{1}{\sqrt{2}}Y_{B}\,\ell\Sigma_{e}\phi_{c}\,$
$\displaystyle\equiv\;$ $\displaystyle
Y_{1}\,d^{c}\Sigma_{c}T_{a}+Y_{2}\,\ell\Sigma_{c}H_{a}+Y_{3}\,d^{c}\Sigma_{a}T_{a}+Y_{4}\,\ell\Sigma_{b}H_{a}+$
$\displaystyle
Y_{5}\,d^{c}\Sigma_{d}H_{a}+Y_{6}\,\ell\Sigma_{e}T_{a}+Y_{7}\,d^{c}\Sigma_{c}T_{b}+Y_{8}\,\ell\Sigma_{c}H_{b}+$
$\displaystyle
Y_{9}\,d^{c}\Sigma_{a}T_{b}+Y_{10}\,d^{c}\Sigma_{a}\phi_{b}+Y_{11}\,\ell\Sigma_{a}\phi_{a}+Y_{12}\,d^{c}\Sigma_{b}\phi_{c}+$
$\displaystyle
Y_{13}\,\ell\Sigma_{b}H_{b}+Y_{14}\,d^{c}\Sigma_{d}\phi_{a}+Y_{15}\,d^{c}\Sigma_{d}H_{b}+Y_{16}\,\ell\Sigma_{d}\phi_{e}+$
$\displaystyle
Y_{17}\,d^{c}\Sigma_{e}\phi_{d}+Y_{18}\,\ell\Sigma_{e}T_{b}+Y_{19}\,\ell\Sigma_{e}\phi_{c}\;,$
(A.26)

where we defined the Yukawa matrices $Y_{N}$, with $N=1,\dots,19$.

##  Appendix B Renormalization group equations

Here the RGEs for the gauge and Yukawa couplings as well as for the effective
neutrino mass operator are listed. We have used the Mathematica package SARAH
Staub:2008uz ; Staub:2013tta  to obtain the RGEs for the gauge and Yukawa
couplings. The SM contribution for the RGE of the effective neutrino mass
operator is taken from Antusch:2001ck . In order to compute the new
contribution for this RGE we have used the method described therein. We use
the following definition for the Heaviside-Theta function

$\displaystyle\mathcal{H}(\mu,m)=\begin{cases}1,\;\;\textrm{for}\;\mu\geq
m,\\\ 0,\;\;\textrm{for}\;\mu<m.\end{cases}$ (B.27)

###  B.1 Gauge couplings

The RGEs for gauge couplings ($i,k=1-3$) are given by

$\displaystyle\mu\frac{dg_{i}}{d\mu}=\frac{\beta^{g_{i}}_{1-\text{loop}}}{16\pi^{2}}+\frac{\beta^{g_{i}}_{2-\text{loop}}}{(16\pi^{2})^{2}}\,,$
(B.28)

where $\beta^{g_{i}}_{1-\text{loop}}$ is the 1-loop and
$\beta^{g_{i}}_{2-\text{loop}}$ is the 2-loop contribution given by

$\displaystyle\beta^{g_{i}}_{1-\text{loop}}=\bigg{\\{}a_{i}^{\text{SM}}+\mathcal{H}(\mu,m)\Delta
a_{i}\bigg{\\}}g_{i}^{3}\,$ (B.29)
$\displaystyle\beta^{g_{i}}_{2-\text{loop}}=\sum_{k}b^{\text{SM}}_{ik}g^{2}_{k}+\sum_{k}\Delta
b_{ik}g^{2}_{k}\;\mathcal{H}(\mu,m)+\beta^{Y,\text{SM}}_{i}+\Delta\beta^{Y}_{i}.$
(B.30)

Here, $a_{i}^{\text{SM}}$, $b^{SM}_{ik}$ and $\beta^{Y,\text{SM}}_{i}$ are the
well known SM 1-loop and 2-loop coefficients as well as Yukawa contributions
Cheng:1973nv ; Machacek:1983fi . Moreover, the $\Delta\beta^{Y}_{i}$ are given
by

$\displaystyle\Delta\beta^{Y}_{i}=g_{i}^{3}\sum_{k}c_{ik}Y^{T}_{k}Y^{\ast}_{k}\;\mathcal{H}^{2}_{k},$
(B.31)

where we introduced the abbreviation
$\mathcal{H}^{2}_{k}=\mathcal{H}(\mu,m_{F})\mathcal{H}(\mu,m_{H})$ associated
to each of the Yukawa interactions, where, $F$ and $H$ refer to the BSM
fermion and scalar appearing in that interaction, respectively, and where the
$c_{ik}$ are given by

$\displaystyle
c_{1k}=-\bigg{\\{}\frac{1}{5},\frac{3}{10},\frac{8}{15},\frac{9}{20},\frac{29}{10},\frac{17}{5},\frac{1}{5},\frac{3}{10},\frac{8}{15},\frac{8}{15},\frac{12}{5},\frac{3}{5},\frac{9}{20},\frac{116}{15},\frac{29}{10},\frac{17}{5},\frac{29}{5},\frac{17}{5},\frac{51}{10}\bigg{\\}},$
(B.32) $\displaystyle
c_{2k}=-\bigg{\\{}0,\frac{1}{2},0,\frac{11}{4},\frac{3}{2},3,0,\frac{1}{2},0,0,4,6,\frac{11}{4},4,\frac{3}{2},3,3,3,\frac{9}{2}\bigg{\\}},$
(B.33) $\displaystyle
c_{3k}=-\bigg{\\{}\frac{1}{2},0,\frac{13}{3},0,2,1,\frac{1}{2},0,\frac{13}{3},\frac{13}{3},6,\frac{3}{2},0,\frac{16}{3},2,1,4,1,\frac{3}{2}\bigg{\\}}.$
(B.34)

Finally, the $\Delta a_{i}$ and $\Delta b_{i}$ are given as a sum over the
1-loop and 2-loop coefficients of the BSM fermions and scalars, i.e.

$\displaystyle\Delta a_{i}=\sum_{I}\Delta a_{i}^{I}\,,\qquad\Delta
b_{i}=\sum_{I}\Delta b_{i}^{I}\,,$ (B.35)

where $I$ runs over all BSM particles. The 1-loop coefficients are then given
by

$\displaystyle\Delta
a_{i}^{\phi_{a}}=\left\\{\frac{4}{5},\frac{4}{3},2\right\\},\quad\Delta
a_{i}^{\phi_{b}}=\left\\{\frac{2}{15},0,\frac{5}{6}\right\\},\quad\Delta
a_{i}^{\phi_{c}}=\left\\{\frac{1}{5},2,\frac{1}{2}\right\\},$
$\displaystyle\Delta
a_{i}^{\phi_{d}}=\left\\{\frac{49}{30},\frac{1}{2},\frac{1}{3}\right\\},\quad\Delta
a_{i}^{\phi_{e}}=\left\\{\frac{16}{15},0,\frac{1}{6}\right\\},\quad\Delta
a_{i}^{\Phi_{a}}=\\{0,0,\frac{1}{2}\\},$ $\displaystyle\Delta
a_{i}^{\Phi_{b}}=\left\\{0,\frac{1}{3},0\right\\},\quad\Delta
a_{i}^{\Sigma_{a}}=\\{0,0,2\\},\quad\Delta
a_{i}^{\Sigma_{b}}=\left\\{0,\frac{4}{3},0\right\\},$ $\displaystyle\Delta
a_{i}^{\Sigma_{d,e}}=\left\\{\frac{5}{3},1,\frac{2}{3}\right\\},\quad\Delta
a_{i}^{h^{\perp}}=\left\\{\frac{1}{10},\frac{1}{6},0\right\\},\quad\Delta
a_{i}^{t,t^{\perp}}=\left\\{\frac{1}{15},0,\frac{1}{6}\right\\},$ (B.36)

whereas the 2-loop coefficients read

$\displaystyle\Delta
b_{ik}^{\phi_{a}}=\left(\begin{array}[]{ccc}\frac{36}{25}&\frac{36}{5}&\frac{144}{5}\\\
\frac{12}{5}&\frac{52}{3}&48\\\ \frac{18}{5}&18&84\\\
\end{array}\right),\quad\Delta
b_{ik}^{\phi_{b}}=\left(\begin{array}[]{ccc}\frac{8}{75}&0&\frac{16}{3}\\\
0&0&0\\\ \frac{2}{3}&0&\frac{115}{3}\\\ \end{array}\right),\quad\Delta
b_{ik}^{\phi_{c}}=\left(\begin{array}[]{ccc}\frac{4}{25}&\frac{24}{5}&\frac{16}{5}\\\
\frac{8}{5}&56&32\\\ \frac{2}{5}&12&11\\\ \end{array}\right),$ (B.46)
$\displaystyle\Delta
b_{ik}^{\phi_{d}}=\left(\begin{array}[]{ccc}\frac{2401}{150}&\frac{147}{10}&\frac{392}{15}\\\
\frac{49}{10}&\frac{13}{2}&8\\\ \frac{49}{15}&3&\frac{22}{3}\\\
\end{array}\right),\quad\Delta
b_{ik}^{\phi_{e}}=\left(\begin{array}[]{ccc}\frac{1024}{75}&0&\frac{256}{15}\\\
0&0&0\\\ \frac{32}{15}&0&\frac{11}{3}\\\ \end{array}\right),\quad\Delta
b_{ik}^{\Phi_{a}}=\left(\begin{array}[]{ccc}0&0&0\\\ 0&0&0\\\ 0&0&21\\\
\end{array}\right),$ (B.56) $\displaystyle\Delta
b_{ik}^{\Phi_{b}}=\left(\begin{array}[]{ccc}0&0&0\\\ 0&\frac{28}{3}&0\\\
0&0&0\\\ \end{array}\right),\quad\Delta
b_{ik}^{\Sigma_{a}}=\left(\begin{array}[]{ccc}0&0&0\\\ 0&0&0\\\ 0&0&48\\\
\end{array}\right),\quad\Delta
b_{ik}^{\Sigma_{b}}=\left(\begin{array}[]{ccc}0&0&0\\\ 0&\frac{64}{3}&0\\\
0&0&0\\\ \end{array}\right),$ (B.66) $\displaystyle\Delta
b_{ik}^{\Sigma_{d,e}}=\left(\begin{array}[]{ccc}\frac{25}{12}&\frac{15}{4}&\frac{20}{3}\\\
\frac{5}{4}&\frac{49}{4}&4\\\ \frac{5}{6}&\frac{3}{2}&\frac{38}{3}\\\
\end{array}\right),\quad\Delta
b_{ik}^{h^{\perp}}=\left(\begin{array}[]{ccc}\frac{9}{50}&\frac{9}{10}&0\\\
\frac{3}{10}&\frac{13}{6}&0\\\ 0&0&0\\\ \end{array}\right),\quad\Delta
b_{ik}^{t,t^{\perp}}=\left(\begin{array}[]{ccc}\frac{4}{75}&0&\frac{16}{15}\\\
0&0&0\\\ \frac{2}{15}&0&\frac{11}{3}\\\ \end{array}\right).$ (B.76)

###  B.2 Yukawa matrices

The RGEs of the Yukawa matrices read

$\displaystyle\mu\frac{dY_{f}}{d\mu}=\frac{\beta_{f}}{16\pi^{2}},$ (B.77)

where $f=\\{u,d,e,k\\}$ and $(k=1,\dots,19)$. For the SM Yukawa matrices
$Y_{u}$, $Y_{d}$ and $Y_{e}$ (i.e. $f={u,d,e}$) the beta functions are given
by

$\displaystyle\beta_{f}=\beta_{f}^{\text{SM}}+\delta\beta_{f},$ (B.78)

where $\beta_{f}^{\text{SM}}$ is the SM beta function Cheng:1973nv ;
Machacek:1983fi , and where

$\displaystyle\delta\beta_{f}=Y_{f}T_{1}+\sum_{k}a^{f}_{k}(Y_{k})_{j}(Y^{T}_{d}Y^{\ast}_{k})_{i}\;\mathcal{H}^{2}_{k}\,.$
(B.79)

Here, we have defined $T_{1}$ as

$\displaystyle
T_{1}=Y^{T}_{2}Y^{\ast}_{2}\;\mathcal{H}^{2}_{2}+\frac{3}{2}Y^{T}_{4}Y^{\ast}_{4}\;\mathcal{H}^{2}_{4}+3Y^{T}_{5}Y^{\ast}_{5}\;\mathcal{H}^{2}_{5}.$
(B.80)

while the $a_{k}^{f}$ are given by

$\displaystyle
a_{k}^{u}=\Big{\\{}0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0\Big{\\}},$ (B.81)
$\displaystyle
a_{k}^{d}=\left\\{\frac{1}{2},0,\frac{4}{3},0,3,0,\frac{1}{2},0,\frac{4}{3},\frac{4}{3},0,\frac{3}{2},0,\frac{8}{3},1,0,2,0,0\right\\},$
(B.82) $\displaystyle
a_{k}^{e}=\left\\{0,-\frac{3}{2},0,\frac{15}{4},0,\frac{3}{2},0,\frac{1}{2},0,0,4,0,\frac{3}{4},0,0,\frac{3}{2},0,\frac{3}{2},\frac{9}{4}\right\\}.$
(B.83)

In order to simplify the notation, from hereon, associated to each Yukawa
$Y_{i}\to Y_{i}\;\mathcal{H}^{2}_{i}$ must be understood. The beta function of
the Yukawa matrices $Y_{1},\dots,Y_{19}$ then read

$\displaystyle\beta_{1}=$
$\displaystyle\;Y_{1}\left(-\frac{1}{5}g^{2}_{1}-4g^{2}_{3}+\sum_{k}a_{k}^{1}Y^{T}_{k}Y^{\ast}_{k}\right)+\left(Y_{d}Y^{\dagger}_{d}\right)Y_{1}+\sum_{w}b_{w}^{1}\left(Y^{T}_{1}Y^{\ast}_{w}\right)Y_{w}$
$\displaystyle+\frac{8}{3}\left(Y^{T}_{3}Y^{\ast}_{9}\right)Y_{7}+2\left(Y^{T}_{6}Y^{\ast}_{18}\right)Y_{7},$
(B.84) $\displaystyle\beta_{2}=$
$\displaystyle\;Y_{2}\left(-\frac{9}{20}g^{2}_{1}-\frac{9}{4}g^{2}_{2}+\sum_{k}a_{k}^{2}Y^{T}_{k}Y^{\ast}_{k}+T\right)+\left(-\frac{3}{2}Y_{e}^{T}Y^{\ast}_{e}\right)Y_{2}+\sum_{w}b_{w}^{2}\left(Y^{T}_{2}Y^{\ast}_{w}\right)Y_{w}$
$\displaystyle+\frac{3}{2}\left(Y^{T}_{4}Y^{\ast}_{13}\right)Y_{8}+3\left(Y^{T}_{5}Y^{\ast}_{15}\right)Y_{8},$
(B.85) $\displaystyle\beta_{3}=$
$\displaystyle\;Y_{3}\left(-\frac{1}{5}g^{2}_{1}-13g^{2}_{3}+\sum_{k}a_{k}^{3}Y^{T}_{k}Y^{\ast}_{k}\right)+\left(Y_{d}Y^{\dagger}_{d}\right)Y_{3}+\sum_{w}b_{w}^{3}\left(Y^{T}_{3}Y^{\ast}_{w}\right)Y_{w}$
$\displaystyle+\left(Y^{T}_{1}Y^{\ast}_{7}\right)Y_{9}+2\left(Y^{T}_{6}Y^{\ast}_{18}\right)Y_{9},$
(B.86) $\displaystyle\beta_{4}=$
$\displaystyle\;Y_{4}\left(-\frac{9}{20}g^{2}_{1}-\frac{33}{4}g^{2}_{2}+\sum_{k}a_{k}^{4}Y^{T}_{k}Y^{\ast}_{k}+T\right)+\left(\frac{5}{2}Y_{e}^{T}Y^{\ast}_{e}\right)Y_{4}+\sum_{w}b_{w}^{4}\left(Y^{T}_{4}Y^{\ast}_{w}\right)Y_{w}$
$\displaystyle+\left(Y^{T}_{2}Y^{\ast}_{8}\right)Y_{13}+3\left(Y^{T}_{5}Y^{\ast}_{15}\right)Y_{13},$
(B.87) $\displaystyle\beta_{5}=$
$\displaystyle\;Y_{5}\left(-\frac{29}{20}g^{2}_{1}-\frac{9}{4}g^{2}_{2}-8g^{2}_{3}+\sum_{k}a_{k}^{5}Y^{T}_{k}Y^{\ast}_{k}+T\right)+\left(3Y_{d}Y^{\dagger}_{d}\right)Y_{5}+\sum_{w}b_{w}^{5}\left(Y^{T}_{5}Y^{\ast}_{w}\right)Y_{w}$
$\displaystyle+\left(Y^{T}_{2}Y^{\ast}_{8}\right)Y_{15}+\frac{3}{2}\left(Y^{T}_{4}Y^{\ast}_{13}\right)Y_{15},$
(B.88) $\displaystyle\beta_{6}=$
$\displaystyle\;Y_{6}\left(-\frac{17}{10}g^{2}_{1}-\frac{9}{2}g^{2}_{2}-4g^{2}_{3}+\sum_{k}a_{k}^{6}Y^{T}_{k}Y^{\ast}_{k}\right)+\left(\frac{1}{2}Y^{T}_{e}Y^{\ast}_{e}\right)Y_{6}+\sum_{w}b_{w}^{6}\left(Y^{T}_{6}Y^{\ast}_{w}\right)Y_{w}$
$\displaystyle+\left(Y^{T}_{1}Y^{\ast}_{7}\right)Y_{18}+\frac{8}{3}\left(Y^{T}_{3}Y^{\ast}_{9}\right)Y_{18},$
(B.89) $\displaystyle\beta_{7}=$
$\displaystyle\;Y_{7}\left(-\frac{1}{5}g^{2}_{1}-4g^{2}_{3}+\sum_{k}a_{k}^{7}Y^{T}_{k}Y^{\ast}_{k}\right)+\left(Y_{d}Y^{\dagger}_{d}\right)Y_{7}+\sum_{w}b_{w}^{7}\left(Y^{T}_{7}Y^{\ast}_{w}\right)Y_{w}$
$\displaystyle+\left(2Y^{T}_{18}Y^{\ast}_{6}\right)Y_{1}+\frac{8}{3}\left(Y^{T}_{9}Y^{\ast}_{3}\right)Y_{1},$
(B.90) $\displaystyle\beta_{8}=$
$\displaystyle\;Y_{8}\left(-\frac{9}{20}g^{2}_{1}-\frac{9}{4}g^{2}_{2}+\sum_{k}a_{k}^{8}Y^{T}_{k}Y^{\ast}_{k}\right)+\left(Y_{e}^{T}Y^{\ast}_{e}\right)Y_{8}+\sum_{w}b_{w}^{8}\left(Y^{T}_{8}Y^{\ast}_{w}\right)Y_{w}$
$\displaystyle+\frac{3}{2}\left(Y^{T}_{13}Y^{\ast}_{4}\right)Y_{2}+3\left(Y^{T}_{15}Y^{\ast}_{5}\right)Y_{2},$
(B.91) $\displaystyle\beta_{9}=$
$\displaystyle\;Y_{9}\left(-\frac{1}{5}g^{2}_{1}-13g^{2}_{3}+\sum_{k}a_{k}^{9}Y^{T}_{k}Y^{\ast}_{k}\right)+\left(Y_{d}Y^{\dagger}_{d}\right)Y_{9}+\sum_{w}b_{w}^{9}\left(Y^{T}_{9}Y^{\ast}_{w}\right)Y_{w}$
$\displaystyle+2\left(Y^{T}_{18}Y^{\ast}_{6}\right)Y_{3}+\left(Y^{T}_{7}Y^{\ast}_{1}\right)Y_{3},$
(B.92) $\displaystyle\beta_{10}=$
$\displaystyle\;Y_{10}\left(-\frac{1}{5}g^{2}_{1}-13g^{2}_{3}+\sum_{k}a_{k}^{10}Y^{T}_{k_{10}}Y^{\ast}_{k}\right)+\left(Y_{d}Y^{\dagger}_{d}\right)Y_{10}+\sum_{w}b_{w}^{10}\left(Y^{T}_{10}Y^{\ast}_{w}\right)Y_{w},$
(B.93) $\displaystyle\beta_{11}=$
$\displaystyle\;Y_{11}\left(-\frac{9}{20}g^{2}_{1}-\frac{9}{4}g^{2}_{2}-9g^{2}_{3}+\sum_{k}a_{k}^{11}Y^{T}_{k}Y^{\ast}_{k}\right)+\left(Y_{e}^{T}Y^{\ast}_{e}\right)Y_{11}+\sum_{w}b_{w}^{11}\left(Y^{T}_{11}Y^{\ast}_{w}\right)Y_{w},$
(B.94) $\displaystyle\beta_{12}=$
$\displaystyle\;Y_{12}\left(-\frac{1}{5}g^{2}_{1}-6g^{2}_{2}-4g^{2}_{3}+\sum_{k}a_{k}^{12}Y^{T}_{k}Y^{\ast}_{k}\right)+\left(Y_{d}Y^{\dagger}_{d}\right)Y_{12}+\sum_{w}b_{w}^{12}\left(Y^{T}_{12}Y^{\ast}_{w}\right)Y_{w},$
(B.95) $\displaystyle\beta_{13}=$
$\displaystyle\;Y_{13}\left(-\frac{9}{20}g^{2}_{1}-\frac{33}{4}g^{2}_{2}+\sum_{k}a_{k}^{13}Y^{T}_{k}Y^{\ast}_{k}\right)+\left(\frac{1}{2}Y_{e}^{T}Y^{\ast}_{e}\right)Y_{13}+\sum_{w}b_{w}^{13}\left(Y^{T}_{13}Y^{\ast}_{w}\right)Y_{w}$
$\displaystyle+3\left(Y^{T}_{15}Y^{\ast}_{5}\right)Y_{4}+\left(Y^{T}_{8}Y^{\ast}_{2}\right)Y_{4},$
(B.96) $\displaystyle\beta_{14}=$
$\displaystyle\;Y_{14}\left(-\frac{29}{20}g^{2}_{1}-\frac{9}{4}g^{2}_{2}-8g^{2}_{3}+\sum_{k}a_{k}^{14}Y^{T}_{k}Y^{\ast}_{k}\right)+\left(Y_{d}Y^{\dagger}_{d}\right)Y_{14}+\sum_{w}b_{w}^{14}\left(Y^{T}_{14}Y^{\ast}_{w}\right)Y_{w},$
(B.97) $\displaystyle\beta_{15}=$
$\displaystyle\;Y_{15}\left(-\frac{29}{20}g^{2}_{1}-\frac{9}{4}g^{2}_{2}-8g^{2}_{3}+\sum_{k}a_{k}^{15}Y^{T}_{k}Y^{\ast}_{k}\right)+\left(Y_{d}Y^{\dagger}_{d}\right)Y_{15}+\sum_{w}b_{w}^{15}\left(Y^{T}_{15}Y^{\ast}_{w}\right)Y_{w}$
$\displaystyle+\frac{3}{2}\left(Y^{T}_{13}Y^{\ast}_{4}\right)Y_{5}+\left(Y^{T}_{8}Y^{\ast}_{2}\right)Y_{5},$
(B.98) $\displaystyle\beta_{16}=$
$\displaystyle\;Y_{16}\left(-\frac{17}{10}g^{2}_{1}-\frac{9}{2}g^{2}_{2}-4g^{2}_{3}+\sum_{k}a_{k}^{16}Y^{T}_{k}Y^{\ast}_{k}\right)+\left(\frac{1}{2}Y_{e}^{T}Y^{\ast}_{e}\right)Y_{16}+\sum_{w}b_{w}^{16}\left(Y^{T}_{16}Y^{\ast}_{w}\right)Y_{w},$
(B.99) $\displaystyle\beta_{17}=$
$\displaystyle\;Y_{17}\left(-\frac{29}{20}g^{2}_{1}-\frac{9}{4}g^{2}_{2}-8g^{2}_{3}+\sum_{k}a_{k}^{17}Y^{T}_{k}Y^{\ast}_{k}\right)+\left(Y_{d}Y^{\dagger}_{d}\right)Y_{17}+\sum_{w}b_{w}^{17}\left(Y^{T}_{17}Y^{\ast}_{w}\right)Y_{w},$
(B.100) $\displaystyle\beta_{18}=$
$\displaystyle\;Y_{18}\left(-\frac{17}{10}g^{2}_{1}-\frac{9}{2}g^{2}_{2}-4g^{2}_{3}+\sum_{k}a_{k}^{18}Y^{T}_{k}Y^{\ast}_{k}\right)+\left(\frac{1}{2}Y_{e}^{T}Y^{\ast}_{e}\right)Y_{18}+\sum_{w}b_{w}^{18}\left(Y^{T}_{18}Y^{\ast}_{w}\right)Y_{w}$
$\displaystyle+\left(Y^{T}_{7}Y^{\ast}_{1}\right)Y_{6}+\frac{8}{3}\left(Y^{T}_{9}Y^{\ast}_{3}\right)Y_{6},$
(B.101) $\displaystyle\beta_{19}=$
$\displaystyle\;Y_{19}\left(-\frac{17}{20}g^{2}_{1}-\frac{9}{2}g^{2}_{2}-4g^{2}_{3}+\sum_{k}a_{k}^{19}Y^{T}_{k}Y^{\ast}_{k}\right)+\left(\frac{1}{2}Y_{e}^{T}Y^{\ast}_{e}\right)Y_{19}+\sum_{w}b_{w}^{19}\left(Y^{T}_{19}Y^{\ast}_{w}\right)Y_{w},$
(B.102)

where the coefficients $a_{k}^{f}$ are given by

$\displaystyle
a_{k}^{1}=\\{3,1,\frac{8}{3},0,0,2,\frac{3}{2},1,0,0,0,0,0,0,0,0,0,0,0\\},$
(B.103) $\displaystyle
a_{k}^{2}=\\{\frac{3}{2},\frac{5}{2},0,\frac{3}{2},3,0,\frac{3}{2},1,0,0,0,0,0,0,0,0,0,0,0\\},$
(B.104) $\displaystyle
a_{k}^{3}=\\{1,0,\frac{9}{2},0,0,2,0,0,\frac{1}{2},\frac{1}{2},1,0,0,0,0,0,0,0,0\\},$
(B.105) $\displaystyle
a_{k}^{4}=\\{0,1,0,\frac{11}{4},3,0,0,0,0,0,0,\frac{3}{2},\frac{1}{2},0,0,0,0,0,0\\},$
(B.106) $\displaystyle
a_{k}^{5}=\\{0,1,0,\frac{3}{2},\frac{9}{2},0,0,0,0,0,0,0,0,\frac{4}{3},\frac{1}{2},\frac{1}{2},0,0,0\\},$
(B.107) $\displaystyle
a_{k}^{6}=\\{1,0,\frac{8}{3},0,0,4,0,0,0,0,0,0,0,0,0,0,1,\frac{1}{2},\frac{3}{4}\\},$
(B.108) $\displaystyle
a_{k}^{7}=\\{\frac{3}{2},1,0,0,0,0,3,1,\frac{8}{3},0,0,0,0,0,0,0,0,2,0\\},$
(B.109) $\displaystyle
a_{k}^{8}=\\{\frac{3}{2},1,0,0,0,0,\frac{3}{2},\frac{5}{2},0,0,0,0,\frac{3}{2},0,3,0,0,0,0\\},$
(B.110) $\displaystyle
a_{k}^{9}=\\{0,0,\frac{1}{2},0,0,0,1,0,\frac{9}{2},\frac{1}{2},1,0,0,0,0,0,0,2,0\\},$
(B.111) $\displaystyle
a_{k}^{10}=\\{0,0,\frac{1}{2},0,0,0,0,0,\frac{1}{2},\frac{19}{6},1,0,0,0,0,0,0,0,0\\},$
(B.112) $\displaystyle
a_{k}^{11}=\\{0,0,\frac{1}{2},0,0,0,0,0,\frac{1}{2},\frac{1}{2},6,0,0,1,0,0,0,0,0\\},$
(B.113) $\displaystyle
a_{k}^{12}=\\{0,0,0,\frac{1}{2},0,0,0,0,0,0,0,4,\frac{1}{2},0,0,0,0,0,1\\},$
(B.114) $\displaystyle
a_{k}^{13}=\\{0,0,0,\frac{1}{2},0,0,0,1,0,0,0,\frac{3}{2},\frac{11}{4},0,3,0,0,0,0\\},$
(B.115) $\displaystyle
a_{k}^{14}=\\{0,0,0,0,\frac{1}{2},0,0,0,0,0,1,0,0,5,\frac{1}{2},\frac{1}{2},0,0,0\\},$
(B.116) $\displaystyle
a_{k}^{15}=\\{0,0,0,0,\frac{1}{2},0,0,1,0,0,0,0,\frac{3}{2},\frac{4}{3},\frac{9}{2},\frac{1}{2},0,0,0\\},$
(B.117) $\displaystyle
a_{k}^{16}=\\{0,0,0,0,\frac{1}{2},0,0,0,0,0,0,0,0,\frac{4}{3},\frac{1}{2},4,0,0,0\\},$
(B.118) $\displaystyle
a_{k}^{17}=\\{0,0,0,0,0,\frac{1}{2},0,0,0,0,0,0,0,0,0,0,5,\frac{1}{2},\frac{3}{4}\\},$
(B.119) $\displaystyle
a_{k}^{18}=\\{0,0,0,0,0,\frac{1}{2},1,0,\frac{8}{3},0,0,0,0,0,0,0,1,4,\frac{3}{4}\\},$
(B.120) $\displaystyle
a_{k}^{19}=\\{0,0,0,0,0,\frac{1}{2},0,0,0,0,0,1,0,0,0,0,1,\frac{1}{2},4\\},$
(B.121)

while the coefficients $b_{k}^{f}$ read

$\displaystyle
b_{w}^{1}=\\{0,0,\frac{4}{3},0,1,0,\frac{3}{2},0,0,\frac{4}{3},0,\frac{3}{2},0,\frac{8}{3},1,0,2,0,0\\},$
(B.122) $\displaystyle
b_{w}^{2}=\\{0,0,0,\frac{3}{4},0,\frac{3}{2},0,\frac{3}{2},0,0,4,0,\frac{3}{4},0,0,\frac{3}{2},0,\frac{3}{2},\frac{9}{4}\\},$
(B.123) $\displaystyle
b_{w}^{3}=\\{\frac{1}{2},0,0,0,1,0,\frac{1}{2},0,0,\frac{4}{3},0,\frac{3}{2},0,\frac{8}{3},1,0,2,0\\},$
(B.124) $\displaystyle
b_{w}^{4}=\\{0,\frac{1}{2},0,0,0,\frac{3}{2},0,\frac{1}{2},0,0,4,0,\frac{9}{4},0,0,\frac{3}{2},0,\frac{3}{2},\frac{9}{4}\\},$
(B.125) $\displaystyle
b_{w}^{5}=\\{\frac{1}{2},0,\frac{4}{3},0,0,0,\frac{1}{2},0,\frac{4}{3},\frac{4}{3},0,\frac{3}{2},0,\frac{8}{3},4,0,2,0,0\\},$
(B.126) $\displaystyle
b_{w}^{6}=\\{0,\frac{1}{2},0,\frac{3}{4},0,0,0,\frac{1}{2},0,0,4,0,\frac{3}{4},0,0,\frac{3}{2},0,\frac{7}{2},\frac{9}{4}\\},$
(B.127) $\displaystyle
b_{w}^{7}=\\{\frac{3}{2},0,\frac{4}{3},0,1,0,0,0,\frac{4}{3},\frac{4}{3},0,\frac{3}{2},0,\frac{8}{3},1,0,2,0,0\\},$
(B.128) $\displaystyle
b_{w}^{8}=\\{0,\frac{3}{2},0,\frac{3}{4},0,\frac{3}{2},0,0,0,0,4,0,\frac{3}{4},0,0,\frac{3}{2},0,\frac{3}{2},\frac{9}{4}\\},$
(B.129) $\displaystyle
b_{w}^{9}=\\{\frac{1}{2},0,4,0,1,0,\frac{1}{2},0,0,\frac{4}{3},0,\frac{3}{2},0,\frac{8}{3},1,0,2,0,0\\},$
(B.130) $\displaystyle
b_{w}^{10}=\\{\frac{1}{2},0,\frac{4}{3},0,1,0,\frac{1}{2},0,\frac{4}{3},0,0,\frac{3}{2},0,\frac{8}{3},1,0,2,0,0\\},$
(B.131) $\displaystyle
b_{w}^{11}=\\{0,\frac{1}{2},0,\frac{3}{4},0,\frac{3}{2},0,\frac{1}{2},0,0,0,0,\frac{3}{4},0,0,\frac{3}{2},0,\frac{3}{2},\frac{9}{4}\\},$
(B.132) $\displaystyle
b_{w}^{12}=\\{\frac{1}{2},0,\frac{4}{3},0,1,0,\frac{1}{2},0,\frac{4}{3},\frac{4}{3},0,0,0,\frac{8}{3},1,0,2,0,0\\},$
(B.133) $\displaystyle
b_{w}^{13}=\\{0,\frac{1}{2},0,\frac{9}{4},0,\frac{3}{2},0,\frac{1}{2},0,0,4,0,0,0,0,\frac{3}{2},0,\frac{3}{2},\frac{9}{4}\\},$
(B.134) $\displaystyle
b_{w}^{14}=\\{\frac{1}{2},0,\frac{4}{3},0,1,0,\frac{1}{2},0,\frac{4}{3},\frac{4}{3},0,\frac{3}{2},0,0,1,0,2,0,0\\},$
(B.135) $\displaystyle
b_{w}^{15}=\\{\frac{1}{2},0,\frac{4}{3},0,4,0,\frac{1}{2},0,\frac{4}{3},\frac{4}{3},0,\frac{3}{2},0,\frac{8}{3},0,0,2,0,0\\},$
(B.136) $\displaystyle
b_{w}^{16}=\\{0,\frac{1}{2},0,\frac{3}{4},0,\frac{3}{2},0,\frac{1}{2},0,0,4,0,\frac{3}{4},0,0,0,0,\frac{3}{2},\frac{9}{4}\\},$
(B.137) $\displaystyle
b_{w}^{17}=\\{\frac{1}{2},0,\frac{4}{3},0,1,0,\frac{1}{2},0,\frac{4}{3},\frac{4}{3},0,\frac{3}{2},0,\frac{8}{3},1,0,0,0,0\\},$
(B.138) $\displaystyle
b_{w}^{18}=\\{0,\frac{1}{2},0,\frac{3}{4},0,\frac{7}{2},0,\frac{1}{2},0,0,4,0,\frac{3}{4},0,0,\frac{3}{2},0,0,\frac{9}{4}\\},$
(B.139) $\displaystyle
b_{w}^{19}=\\{0,\frac{1}{2},0,\frac{3}{4},0,\frac{3}{2},0,\frac{1}{2},0,0,4,0,\frac{3}{4},0,0,\frac{3}{2},0,\frac{3}{2},0\\}.$
(B.140)

###  B.3 Effective neutrino mass operator

The RGE for the effective neutrino mass operator reads

$\displaystyle 16\pi^{2}\mu\frac{d\kappa}{d\mu}=$
$\displaystyle\;\beta_{\kappa}^{\text{SM}}+\Delta\beta_{\kappa},$ (B.141)

where $\beta_{\kappa}^{\text{SM}}$ is the SM contribution as given in
Antusch:2001ck  and $\Delta\beta_{\kappa}$ is the correction due to the added
BSM particles. For $\Delta\beta_{\kappa}$ we find888To simplify the analysis
we ignore RG induced mixings between different dimension five operators.

$\displaystyle\Delta\beta_{\kappa}=\kappa\bigg{(}$
$\displaystyle\frac{1}{2}Y_{2}^{*}Y_{2}^{T}+\frac{3}{4}Y_{4}^{*}Y_{4}^{T}+\frac{3}{2}Y_{6}^{*}Y_{6}^{T}+\frac{1}{2}Y_{8}^{*}Y_{8}^{T}+4Y_{11}^{*}Y_{11}^{T}+\frac{3}{4}Y_{13}^{*}Y_{13}^{T}+\frac{3}{2}Y_{16}^{*}Y_{16}^{T}$
$\displaystyle+$
$\displaystyle\;\frac{3}{2}Y_{18}^{*}Y_{18}^{T}+\frac{9}{4}Y_{19}^{*}Y_{19}^{T}\bigg{)}+\bigg{(}\frac{1}{2}Y_{2}^{*}Y_{2}^{T}+\frac{3}{4}Y_{4}^{*}Y_{4}^{T}+\frac{3}{2}Y_{6}^{*}Y_{6}^{T}+\frac{1}{2}Y_{8}^{*}Y_{8}^{T}$
$\displaystyle+$
$\displaystyle\;4Y_{11}^{*}Y_{11}^{T}+\frac{3}{4}Y_{13}^{*}Y_{13}^{T}+\frac{3}{2}Y_{16}^{*}Y_{16}^{T}+\frac{3}{2}Y_{18}^{*}Y_{18}^{T}+\frac{9}{4}Y_{19}^{*}Y_{19}^{T}\bigg{)}^{T}\kappa$
$\displaystyle+$
$\displaystyle\;\bigg{(}2Y_{2}^{T}Y_{2}^{*}+3Y_{4}^{T}Y_{4}^{*}+6Y_{5}^{T}Y_{5}^{*}+2Y_{8}^{T}Y_{8}^{*}+3Y_{13}^{T}Y_{13}^{*}+6Y_{15}^{T}Y_{15}^{*}\bigg{)}\kappa\;.$
(B.142)

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