# Sculpting the Standard Model from low-scale Gauge-Higgs-Matter
$\mathrm{E}_{8}$ Grand Unification in ten dimensions

Alfredo Aranda<EMAIL_ADDRESS>Facultad de Ciencias-CUICBAS, Universidad de
Colima, C.P.28045, Colima, México 01000, México  Dual CP Institute of High
Energy Physics, C.P. 28045, Colima, México  Francisco J. de Anda
<EMAIL_ADDRESS>Tepatitlán’s Institute for Theoretical Studies, C.P. 47600,
Jalisco, México  Dual CP Institute of High Energy Physics, C.P. 28045, Colima,
México  António P. Morais<EMAIL_ADDRESS>Departamento de Física,
Universidade de Aveiro, Campus de Santiago, 3810-183 Aveiro, Portugal  Centre
for Research and Development in Mathematics and Applications (CIDMA), Campus
de Santiago, 3810-183 Aveiro, Portugal  Department of Astronomy and
Theoretical Physics, Lund University, 221 00 Lund, Sweden  Roman Pasechnik
<EMAIL_ADDRESS>Department of Astronomy and Theoretical Physics,
Lund University, 221 00 Lund, Sweden

###### Abstract

The construction and general implications of a model with complete
supersymmetric unification of the Standard Model matter content, interactions
and families’ replication into a single $\mathrm{E}_{8}$ gauge superfield in
ten dimensions is presented. The gauge and extended Poincaré symmetries are
broken through compactification of the
$\mathbb{T}^{6}/(\mathbb{Z}_{3}\times\mathbb{Z}_{3})$ orbifold with Wilson
lines, which reduces the original symmetry and matter content into those of
the Standard Model plus additional heavier states. Proton decay can be
suppressed automatically while the compactification scale may be as low as
$10^{9}~{}\rm{GeV}$, so that the corresponding GUT-scale physics may be
potentially accessible and testable by future collider measurements.

## Introduction

The Standard Model (SM) is currently the best theory at disposal to accurately
describe three of the four fundamental interactions in nature. It is based
upon modern Quantum Field Theory (QFT) and its predictions have matched some
of the most stringent tests ATLAS:2012yve ; CMS:2012qbp ; UA1:1983crd ;
GargamelleNeutrino:1973jyy ; Hasert:1973cr ; CDF:1995wbb ; Parker:2018vye .
Despite its success, there are clear phenomenological indications that the SM
is not the ultimate picture, such as neutrinos being massive particles Super-
Kamiokande:1998kpq , the existence of Dark Matter (DM) Bertone:2004pz  or the
unexplained structure of the SM fermion families. Furthermore, both its gauge
structure, mathematically described by the symmetry
$\mathrm{SU}(3)_{\mathrm{C}}\times\mathrm{SU}(2)_{\mathrm{L}}\times\mathrm{U}(1)_{\mathrm{Y}}$,
as well as its matter content, are purely phenomenologically driven and a
consensual first-principles explanation for their origin is still lacking.

In fact, it is a remarkable theoretical challenge to explain and even
eventually to derive the basic properties of the SM from a unified concept at
high-energy scales. Typically, Grand Unified Theories (GUTs) feature
unification of the SM gauge couplings into a single one by means of embedding
the SM symmetry group into a larger simple gauge group at a certain energy
scale, commonly dubbed the GUT scale. In this way, the SM Higgs, matter and
gauge fields, denoted by green, blue and pink labels, respectively, in Tab. 1,
can be a part of larger representations of the GUT symmetry group.

$\mathrm{T}^{4}\ltimes\mathrm{SO}(3,1)_{\mathrm{}}$ | $\mathrm{SU}(3)_{\mathrm{C}}\times\mathrm{SU}(2)_{\mathrm{L}}\times\mathrm{U}(1)_{\mathrm{Y}}$
---|---
$(\bm{1})$ | ${\color[rgb]{0,0.6,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.6,0}\pgfsys@color@cmyk@stroke{0.64}{0}{0.95}{0.40}\pgfsys@color@cmyk@fill{0.64}{0}{0.95}{0.40}(\bm{1},\bm{2},3)}$
$(\bm{2})$ | $3\times{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}(\bm{1},\bm{2},-3)}+3\times{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}(\bm{1},\bm{1},6)}+3\times{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}(\bar{\bm{3}},\bm{1},-4)}+3\times{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}(\bar{\bm{3}},\bm{1},2)}+3\times{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}(\bm{3},\bm{2},1)}$
$(\bm{4})$ | ${\color[rgb]{1,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,1}\pgfsys@color@cmyk@stroke{0}{1}{0}{0}\pgfsys@color@cmyk@fill{0}{1}{0}{0}(\bm{8},\bm{1},0)}+{\color[rgb]{1,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,1}\pgfsys@color@cmyk@stroke{0}{1}{0}{0}\pgfsys@color@cmyk@fill{0}{1}{0}{0}(\bm{1},\bm{3},0)}+{\color[rgb]{1,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,1}\pgfsys@color@cmyk@stroke{0}{1}{0}{0}\pgfsys@color@cmyk@fill{0}{1}{0}{0}(\bm{1},\bm{1},0)}$
Table 1: The transformation properties of the SM fields under the gauge and
Poincaré symmetries. $(\bm{1})$, $(\bm{2})$ and $(\bm{4})$ denote scalars,
spin-$1/2$ Weyl fermions and spin-$1$ vectors in four dimensions,
respectively.

Historically, the first and simplest way to achieve such a gauge unification
is by utilising the $\mathrm{SU}(5)_{\mathrm{}}$ gauge symmetry at the GUT
scale Georgi:1974sy . The three gauge interactions are a remnant of a more
fundamental one where the SM vector bosons are different components of a
single $\bm{24}$-dimensional representation as shown in Tab. 2. However, the
matter sector is only partially unified where the three fermionic generations
belong to three distinct $\bm{\bar{5}}+\bm{10}$ spin-$1/2$ copies. The Higgs
sector, on the other hand, is enlarged to a spinless $\bm{\overline{5}}$
irreducible representation (irrep), bringing along new coloured scalars. It is
well known that the latter need to be rather heavy, typically with masses
above $10^{16}~{}\mathrm{GeV}$ in order to prevent the proton from rapidly
decaying, while the doublet component is required to be at the TeV scale. The
need for such a remarkable hierarchy is known as the doublet-triplet splitting
problem Sakai:1981gr  and is recognized as one of the theoretical challenges
of minimal four-dimensional $\mathrm{SU}(5)_{\mathrm{}}$ GUTs.

$\mathrm{T}^{4}\ltimes\mathrm{SO}(3,1)_{\mathrm{}}$ | $\mathrm{SU}(5)_{\mathrm{}}$
---|---
$(\bm{1})$ | ${\color[rgb]{0,0.6,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.6,0}\pgfsys@color@cmyk@stroke{0.64}{0}{0.95}{0.40}\pgfsys@color@cmyk@fill{0.64}{0}{0.95}{0.40}(\bar{\bm{5}})}$
$(\bm{2})$ | $3\times{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}(\bar{\bm{5}})}+3\times{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}(\bm{10})}$
$(\bm{4})$ | ${\color[rgb]{1,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,1}\pgfsys@color@cmyk@stroke{0}{1}{0}{0}\pgfsys@color@cmyk@fill{0}{1}{0}{0}(\bm{24})}$
Table 2: The SM field content embedded in $\mathrm{SU}(5)_{\mathrm{}}$ with
transformation properties under the gauge and Poincaré symmetries.

Another possibility came with the $\mathrm{SO}(10)_{\mathrm{}}$ GUT
Fritzsch:1974nn  where, besides gauge field unification, one can also unify
all the SM fermions of an entire matter family into a single $16$-dimensional
$\mathrm{SO}(10)_{\mathrm{}}$ representation as shown in Tab. 3.

$\mathrm{T}^{4}\ltimes\mathrm{SO}(3,1)_{\mathrm{}}$ | $\mathrm{SO}(10)_{\mathrm{}}$
---|---
$(\bm{1})$ | ${\color[rgb]{0,0.6,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.6,0}\pgfsys@color@cmyk@stroke{0.64}{0}{0.95}{0.40}\pgfsys@color@cmyk@fill{0.64}{0}{0.95}{0.40}(\bm{10})}$
$(\bm{2})$ | $3\times{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}(\bm{16})}$
$(\bm{4})$ | ${\color[rgb]{1,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,1}\pgfsys@color@cmyk@stroke{0}{1}{0}{0}\pgfsys@color@cmyk@fill{0}{1}{0}{0}(\bm{45})}$
Table 3: The SM field content embedded in $\mathrm{SO}(10)_{\mathrm{}}$ with
its transformation properties under the gauge and Poincaré symmetries.

Furthermore, the three matter $\bm{16}$-plets contain a SM-singlet each
offering three generations of right-handed neutrinos and the possibility for
explaining the observed active neutrino masses via a seesaw mechanism. In this
context, the seesaw scale can be attributed to the
$\mathrm{SO}(10)_{\mathrm{}}$ breaking scale such that the smallness of
neutrino masses results from a large hierarchy between GUT and electroweak
(EW) scales.

A particularly attractive option is the $\mathrm{E}_{6}$ GUT Gursey:1975ki ,
where matter unification is realized in three copies of the fundamental
$\bm{27}$ irrep. However, in its minimal form, the SM Higgs doublet is also
part of a scalar $\bm{27}$-plet suggesting a higher degree of unification if
both the fermion and scalar counterparts are related by means of simple ${\cal
N}=1$ supersymmetry (SUSY) transformations King:2005my . A SUSY
$\mathrm{E}_{6}$ GUT is then the minimal theory featuring Higgs and matter
unification in three distinct copies of $\bm{27}$ supermultiplets as
summarized in Tab. 4.

$S^{1}\ltimes[\mathrm{T}^{4}\ltimes\mathrm{SO}(3,1)_{\mathrm{}}]$ | $\mathrm{E}_{6}$
---|---
$(2\times\bm{1}+\bm{2})$ | $3\times{\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color@cmyk@fill{0.91}{0}{0.88}{0.12}(\bm{27})}$
$(\bm{4}+\bm{2})$ | ${\color[rgb]{1,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,1}\pgfsys@color@cmyk@stroke{0}{1}{0}{0}\pgfsys@color@cmyk@fill{0}{1}{0}{0}(\bm{78})}$
Table 4: The SM field content embedded in SUSY $\mathrm{E}_{6}$ with the
corresponding transformation properties under the gauge and super-Poincaré
symmetries. $S^{1}$ denotes simple ${\cal N}=1$ SUSY.

Note that the unification of matter fields both in
$\mathrm{SO}(10)_{\mathrm{}}$ and $\mathrm{E}_{6}$ GUTs is realized
independently for each generation without offering any fundamental reason to
explain the family replication observed in nature. This can be accomplished at
the level of a SUSY $\mathrm{E}_{7}$ GUT if the entire SM fermion and Higgs
sectors are embedded in a single real $\bm{912}$ supermultiplet. The challenge
that such a picture poses is on how to obtain chiral matter from a real
representation. While this is not realizable in four space-time dimensions,
the presence of orbifolded extra dimensions (EDs) can leave massless chiral
sectors in a low-energy SM-like effective field theory (EFT). While this can
be achieved with one ED, SUSY representations become simpler in the case of
two compact EDs. This enlarges the super-Poincaré symmetry to
$\mathrm{T}^{6}\ltimes\mathrm{SO}(5,1)_{\mathrm{}}$ with the minimal field
content of Tab. 5.

$S^{1}\ltimes[\mathrm{T}^{6}\ltimes\mathrm{SO}(5,1)_{\mathrm{}}]$ | $\mathrm{E}_{7}$
---|---
$(4\times\bm{1}+\bm{4})$ | ${\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color@cmyk@fill{0.91}{0}{0.88}{0.12}(\bm{912})}$
$(\bm{6}+\bm{4})$ | ${\color[rgb]{1,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,1}\pgfsys@color@cmyk@stroke{0}{1}{0}{0}\pgfsys@color@cmyk@fill{0}{1}{0}{0}(\bm{133})}$
Table 5: The SM field content embedded in SUSY $\mathrm{E}_{7}$ with the
corresponding transformation properties under the gauge and super-Poincaré
symmetries. $S^{1}$ denotes simple ${\cal N}=1$ SUSY.

Such a picture offers a possible framework for both gauge unification and
Higgs-matter-family unification where the three fermion families can be
realised together with other observed properties of the SM particle spectra
and interactions.

The pattern of enlarging of the GUT symmetry can be seen through the Dynkin
diagrams and follows the exceptional chain Buchmuller:1985rc ; Koca:1982zi ,

$\mathrm{SU}(3)_{\mathrm{C}}\times\mathrm{SU}(2)_{\mathrm{L}}\times\mathrm{U}(1)_{\mathrm{Y}}\subset\mathrm{SU}(5)_{\mathrm{}}\subset\mathrm{SO}(10)_{\mathrm{}}\subset\mathrm{E}_{6}\subset\mathrm{E}_{7}\subset\mathrm{E}_{8}\,.$
(1)

The largest group in this chain, $\mathrm{E}_{8}$, has an unique property that
its adjoint representation 248 coincides with its fundamental one
Slansky:1981yr . This suggests that all the SM gauge and matter fields can be,
in principle, unified into a single gauge 248 representation provided that a
maximal $\mathcal{N}=4$ SUSY is realised. Furthermore, it also provides an
$\mathrm{SU}(3)_{\mathrm{}}$ family symmetry as a coset of $\mathrm{E}_{8}$ to
$\mathrm{E}_{6}$ reduction as has been thoroughly explored earlier in Refs.
Camargo-Molina:2016yqm ; Camargo-Molina:2017kxd ; Morais:2020ypd ;
Morais:2020odg .

There is a plethora of models that aim to build GUTs including family
symmetries King:2001uz ; King:2017guk ; Hagedorn:2010th ; Antusch:2014poa ;
Bjorkeroth:2015ora ; Bjorkeroth:2015uou ; Bjorkeroth:2017ybg ; deAnda:2017yeb
; CarcamoHernandez:2020owa ; Morais:2020ypd ; Morais:2020odg ; Camargo-
Molina:2016yqm ; Camargo-Molina:2017kxd ; Camargo-Molina:2016yqm ; Camargo-
Molina:2016bwm  and EDs Reig:2017nrz ; Altarelli:2008bg ; Burrows:2009pi ;
Burrows:2010wz ; deAnda:2018oik ; Altarelli:2006kg ; Adulpravitchai:2010na ;
Adulpravitchai:2009id ; Asaka:2001eh ; deAnda:2019jxw ; deAnda:2018ecu ;
deAnda:2018yfp . These models require various independent groups and a large
number of fields. The $\mathrm{E}_{8}$ group is a good bet for a complete
unification of SM vectors, fermions and scalars, as well as both the gauge and
Yukawa interactions, and has indeed been widely studied both in the context of
string theory Ibanez:1987pj ; Parr:2020oar ; Manolakos:2020cco  and within the
framework of QFT Adler:2002yg ; Adler:2004uj ; Garibaldi:2016zgm ;
Thomas:1985be ; Konshtein:1980km ; Baaklini:1980fv ; Baaklini:1980uq ;
Barr:1987pu ; Bars:1980mb ; Koca:1981xd ; Mahapatra:1988gc ; Ong:1984ej ;
Camargo-Molina:2016yqm ; Olive:1982ai .

Similarly to $\mathrm{E}_{7}$, the main challenge posed by $\mathrm{E}_{8}$ is
that it is a real group – the same as the extended ${\cal N}=4$ SUSY theory –
while the SM requires chiral representations. With orbifolded EDs, the
compactification procedure could leave massless chiral sectors in a low-energy
SM-like EFT. Furthermore, extended ${\cal N}=4$ SUSY in four dimensions can be
obtained from a ${\cal N}=1$ SUSY 10d theory Arkani-Hamed:2001vvu ;
Brink:1976bc . The orbifolding mechanism provides a natural way to
geometrically break a GUT symmetry group and to generate large masses for some
or most of the unobserved states. However, a consistent way of getting the
correct flavour structure and interactions in a SM-like low-energy EFT from
the $\mathrm{E}_{8}$ GUT, that is compatible with all existing
phenomenological constraints, has not yet been developed in the literature.

In this paper, we propose a particular realisation of a GUT based upon the
$\mathrm{E}_{8}$ gauge symmetry and $\mathcal{N}=1$ SUSY in ten spacetime
dimensions (10d) where the full SM matter, Higgs and gauge field content is
unified into a single $\mathrm{E}_{8}$ gauge superfield transforming as in
Tab. 6.

$S^{1}\ltimes[\mathrm{T}^{10}\ltimes\mathrm{SO}(9,1)_{\mathrm{}}]$ | $\mathrm{E}_{8}$
---|---
$(\bm{10}+\bm{8})$ | ${\color[rgb]{1,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,1}\pgfsys@color@cmyk@stroke{0}{1}{0}{0}\pgfsys@color@cmyk@fill{0}{1}{0}{0}(\bm{248})}$
Table 6: Full unification of the SM field content in a single adjoint
$\bm{248}$ supermultiplet composed by a 10d real vector and a 10d
Weyl/Majorana fermion.

The $\mathbb{T}^{6}/(\mathbb{Z}_{3}\times\mathbb{Z}_{3})$ orbifold
compactification breaks SUSY and the original gauge symmetry directly to that
of the SM,
$\mathrm{E}_{8}\to\mathrm{SU}(3)_{\mathrm{C}}\times\mathrm{SU}(2)_{\mathrm{L}}\times\mathrm{U}(1)_{\mathrm{Y}}$
(i.e. without any intermediate, typically anomalous, symmetry groups), leaving
only the SM gauge and fermion fields massless. The operators responsible for
the proton decay naturally appear to be strongly suppressed alleviating the
need for a very strong hierarchy between the ED compactification scale and the
EW one. While the Higgs doublets acquire a positive mass at one of the Wilson-
line breaking scales, via the Renormalisation Group (RG) evolution, one of the
Higgs doublet squared mass terms runs negative at a TeV scale, thus,
triggering the EW symmetry breaking radiatively. This is known as the
radiative EW symmetry breaking (REWSB) mechanism (for its particular
realisation in the framework of non-SUSY trinification model, see Ref.
Camargo-Molina:2016bwm ).

The high-energy theory only has one arbitrary parameter, the $\mathrm{E}_{8}$
gauge coupling, while the details of the orbifold compactification procedure
considered in this article provide two real and twelve complex additional
arbitrary parameters coming from two radii, six complex Wilson-line effective
vacuum expectation values (VEVs) and six complex Higgs VEVs. These parameters
fully determine the properties of the SM-like EFT emergent at low energies and
all the related phenomenology at the EW scale and beyond.

The layout of the paper is as follows. In Sec. I.1, the key basics and
definitions of the proposed $\mathrm{E}_{8}$ GUT in 10d are given. In Sec.
I.2, a specific $\mathbb{T}^{6}/(\mathbb{Z}_{3}\times\mathbb{Z}_{3})$ orbifold
is introduced and its compactification mechanism is studied in detail. In Sec.
I.3, anomaly cancellation in the considered model is discussed. In Sec. II,
the low-energy EFT Lagrangian of the $\mathrm{E}_{8}$ GUT is presented, and
its fermion and scalar mass spectra are derived and discussed in detail. In
Sec. III the processes that generate highly suppressed proton decay are
analyzed. In Sec. IV, the RG evolution of the gauge couplings is presented as
well as the relevant phenomenology it entails is reviewed. An outlook is given
in Sec. V elaborating on basic differences of the proposed framework from
string theory. Finally, in Sec. VI the main conclusions of this work are
presented.

##  I Ten-dimensional $\mathrm{E}_{8}$ GUT

###  I.1 Model structure

In this article, the $\mathcal{N}=1$ SUSY model of Tab. 6 based on an
$\mathrm{E}_{8}$ GUT in 10d is introduced. The corresponding spacetime
contains six compact EDs factorizable as
$\mathbb{R}^{4}\times(\mathbb{T}^{2})^{3}$, with $\mathbb{T}^{2}$ being a
2-torus, which are denoted by three complex coordinates $z_{i}$ in what
follows. The non-compact coordinates of the physical four-dimensional (4d)
spacetime are then labeled as $x^{\mu}\equiv x$. This GUT is a pure super
Yang-Mills (SYM) theory that contains a single 10d vector superfield
$\mathcal{V}_{\textbf{248}}(x,z_{i})$ in the adjoint (which is also the
fundamental) representation of $\mathrm{E}_{8}$. The 10d $\mathcal{N}=1$ SUSY
then decomposes into $\mathcal{N}=4$ SUSY in 4d spacetime Arkani-Hamed:2001vvu
; Brink:1976bc  and needs to be broken through the orbifold boundary
conditions to a simple $\mathcal{N}=1$ SUSY theory. In such a scenario, the
single 10d vector superfield $\mathcal{V}$ decomposes into a 4d vector
superfield $V$, and a sector with three 4d chiral superfields $\phi_{i}$, one
per complex ED, supplemented with an infinite tower of Kaluza–Klein (KK)
states for each of these superfields. A thorough description of the
decomposition of the ED Poincaré representations in the current setup is shown
in Appendix A while further details about the procedure of a generic 10d
orbifolding is reviewed in Appendix B. The Lagrangian of the model and its
decomposition are detailed in Appendix C.

Upon orbifolding, the super-Poincaré group on the first column of Tab. 6
becomes incomplete and the full symmetry of the model becomes

$S^{1}\ltimes[(\mathrm{T}^{4}\times\mathrm{T}^{6}/{\Gamma})\ltimes(\mathrm{SO}(9,1)_{\mathrm{}}/{\mathrm{F}})]\ltimes\mathrm{E}_{8}\,,$
(2)

where $S^{1}$ denotes $\mathcal{N}=1$ SUSY, the $\mathrm{T}$ represents the
translation group and $\mathrm{SO}(9,1)_{\mathrm{}}$ is the 10d Lorentz group.
Here, the discrete subgroup of ED translations
$\Gamma\simeq\mathbb{Z}^{6}\subset\mathrm{T}^{6}$ is also called the lattice
group111This is a direct product of six times the group of integers and
compactifies the EDs, while $\mathrm{F}$ is a discrete subgroup of ED
rotations $\mathrm{F}\subset\mathrm{SO}(6)$, which defines the orbifolding. In
this article, the case
$\mathrm{F}\simeq\mathbb{Z}^{\mathrm{F}}_{3}\times\mathbb{Z}^{\mathrm{C}}_{3}$
is explored for the first time (here and below, the superscripts F and C stand
for family and colour, respectively).

Both the lattice group $\Gamma$ and the orbifolding group $\mathrm{F}$
transformations are accompanied by a gauge transformation, so that both
$\Gamma,\mathrm{F}\subset\mathrm{E}_{8}$. Therefore, the spacetime symmetry no
longer commutes with the gauge one, i.e.

$\exists\ \ \ \alpha\in\Gamma,\beta\in\mathrm{F},\gamma\in\mathrm{E}_{8}\ \ \
|\ \ \ [\alpha,\gamma]\neq 0,\ \ [\beta,\gamma]\neq 0\,,$ (3)

such that the gauge sector becomes sensitive to geometrical effects leading to
its breaking.

In the considered model, the EDs are orbifolded as
$\mathbb{T}^{6}/(\mathbb{Z}^{\mathrm{F}}_{3}\times\mathbb{Z}^{\mathrm{C}}_{3})$
with

$\begin{split}\mathbb{Z}^{\mathrm{F}}_{3}:\
(x,z_{1},z_{2},z_{3})&\sim(x,\omega^{2}z_{1},\omega^{2}z_{2},\omega^{2}z_{3}),\
\ \ \mathcal{V}\to e^{-2i\pi q_{8}^{\mathrm{F}}/3}\ \mathcal{V}\,,\\\
\mathbb{Z}^{\mathrm{C}}_{3}:\
(x,z_{1},z_{2},z_{3})&\sim(x,\omega^{3}z_{1},\omega z_{2},\omega^{2}z_{3}),\ \
\ \ \mathcal{V}\to
e^{-2i\pi(q_{8}^{\mathrm{C}}+q_{8}^{\mathrm{F}}-q_{3}^{\mathrm{F}})/3}\
\mathcal{V}\,,\end{split}$ (4)

where $\omega=e^{2i\pi/3}$ denotes the complex cubic root of unity and
$q_{8}^{\mathrm{C,F}}$ are the Abelian charges under the $T^{3,8}$ generators
of $\mathrm{SU}(3)_{\mathrm{C,F}}$ symmetry groups emerging via the following
decomposition Slansky:1981yr

$\mathrm{E}_{8}\supset\mathrm{SU}(3)_{\mathrm{C}}\times\mathrm{SU}(3)_{\mathrm{L}}\times\mathrm{SU}(3)_{\mathrm{R}}\times\mathrm{SU}(3)_{\mathrm{F}}\,,$
(5)

respectively. The lattice of the $\mathbb{T}^{6}$ tori is defined by the
coordinate transformations

$z_{i}\sim z_{i}+\tau^{r}_{i}\quad\text{with}\quad\tau_{i}^{1}=2\pi
R_{i}\quad\text{and}\quad\tau_{i}^{2}=2\pi e^{i\pi/3}R_{i}\,,$ (6)

where the $R_{i}$ are the radii of the different tori. Note that each torus is
built from two circles of equal radii. Each lattice translation is accompanied
by a gauge transformation $U^{r}_{i=1,2,3}$ which is called a Wilson line

$\mathcal{V}(x,z_{i})=U^{r}_{i}\mathcal{V}(x,z_{i}+\tau^{r}_{i})\,.$ (7)

A non-trivial $U^{r}_{i}$ gauge transformation, i.e. a Wilson line, can be
seen as an effective VEV in the chiral superfields Candelas:1985en . For
consistency with the orbifolding group, it must be aligned with a
representation that has a zero mode in its KK tower. One must note that it is
not an usual VEV, as it does not come from the minimization of a potential.
Instead, it comes from the ED profiles of the scalar fields and makes the same
impact on the theory as an ordinary VEV in the Higgs mechanism.

Even though the effective (Wilson-line) VEVs are extracted from the properties
of the orbifold, they acquire radiative corrections. By integrating out all
the fields, except the scalars getting the effective VEVs, one obtains an
effective potential that drives the magnitude of such VEVs. This is the
essence of the Hosotani mechanism Hosotani:1983xw ; Hosotani:1983vn ;
Hosotani:2004wv ; Hosotani:2004ka ; Haba:2004qf ; Haba:2002py . One must bear
in mind that the direction in the space of effective VEVs must be chosen a
priori, as it comes from the translational boundary conditions. Therefore, the
symmetry is already broken by the orbifold while the VEV magnitude is
determined radiatively to be of the required size from the Hosotani mechanism
(as the tadpoles are absent) and is proportional to the radii of the extra
dimensions.

In what follows, a continuous Wilson line will be associated to $U^{r}_{i}$,
whose gauge transformation is consistent with Eq. 4. Choosing the three
different $R_{i}$ and the arbitrary dimensionless parameters associated to the
Wilson line in $U^{r}_{1}$, one completely defines the parameter space of the
considered GUT. As described in detail in the next section, the model is based
on an orbifold and Wilson-line symmetry breaking mechanism that reduce
$\mathrm{E}_{8}$ directly down to
$\mathrm{SU}(3)_{\mathrm{C}}\times\mathrm{SU}(2)_{\mathrm{L}}\times\mathrm{U}(1)_{\mathrm{Y}}$,
where the EW symmetry can further be broken radiatively (see below).

###  I.2 Orbifolding the EDs with
$\mathbb{T}^{6}/(\mathbb{Z}^{\mathrm{F}}_{3}\times\mathbb{Z}^{\mathrm{C}}_{3})$

The model proposed in this article is built upon the orbifold defined by Eq.
4. The first $\mathbb{Z}_{3}$, denoted as $\mathbb{Z}^{\mathrm{F}}_{3}$ in Eq.
4, breaks $\mathrm{E}_{8}\to\mathrm{E}_{6}\times\mathrm{SU}(3)_{\mathrm{F}}$
while the second one, $\mathbb{Z}^{\mathrm{C}}_{3}$, breaks
$\mathrm{E}_{8}\to\mathrm{SU}(9)_{\mathrm{}}$ Aranda:2020noz ; deAnda:2020prd
. When $\mathbb{Z}^{\mathrm{F}}_{3}$ and $\mathbb{Z}^{\mathrm{C}}_{3}$ act
together, they effectively break $\mathrm{E}_{8}$ into an intersection of
$\mathrm{E}_{6}\times\mathrm{SU}(3)_{\mathrm{F}}$ and
$\mathrm{SU}(9)_{\mathrm{}}$ symmetries containing trinification and family
symmetries as

$\displaystyle\mathrm{E}_{8}\to(\mathrm{E}_{6}\times\mathrm{SU}(3)_{\mathrm{F}})\cap\mathrm{SU}(9)_{\mathrm{}}=\mathrm{SU}(3)_{\mathrm{C}}\times\mathrm{SU}(3)_{\mathrm{L}}\times\mathrm{SU}(3)_{\mathrm{R}}\times\mathrm{U}(1)_{\mathrm{F3}}\times\mathrm{U}(1)_{\mathrm{F8}}\,,$
(8)

with labels resembling the symmetries in the low-energy SM-like EFT limit.
Although this is the breaking associated to the rotational boundary conditions
only, it is useful to use this intersection group as a basis for describing
the 248 in terms of the orbifold charges as presented in Tab. 7.

The lattice of $\mathbb{T}^{6}$ is factorized into three tori. Each torus is
built from two circles which must have equal radii. Different tori can have
different radii defined in Eq. 6, where $R_{i}$ denotes the three different
radii. The ED lattice must be invariant under the orbifold action to be
consistent. One can see that the 6d lattice is built from three copies of the
2d lattice generated by the basis $(1,e^{i\pi/3})$ which is just a
normalization of Eq. 6. The orbifold action is generated by multiples of
$\omega=e^{2i\pi/3}$ acting on the lattice as

$e^{2i\pi/3}(1,e^{i\pi/3})=(e^{i\pi/3}-1,-1)\,,$ (9)

which is known as the modular transformation (i.e. the transformed basis is
just an integer linear combination of the original basis). Therefore, indeed,
the considered orbifold is consistent.

One of the features of orbifolds is the existence of singular points which
imply certain boundary conditions responsible for the symmetry breaking. The
singular points are the ones invariant under the orbifold transformations and
are found as solutions of the following equations

$\mathcal{Z}^{\rm C,F}(z_{i})=z_{i}\,,\ \ \ \mathcal{Z}^{\rm
C,F}(z_{i})=z_{i}+\tau^{r}_{i}\,,$ (10)

where $\mathcal{Z}^{\rm C,F}$ denotes the action of the orbifold group
$\mathbb{Z}^{\mathrm{C,F}}_{3}$ on the coordinates. It defines the fixed
points which in this case are

$\left(x,\left\\{0,\frac{1+e^{i\pi/3}}{3},\frac{2+2e^{i\pi/3}}{3}\right\\},\left\\{0,\frac{1+e^{i\pi/3}}{3},\frac{2+2e^{i\pi/3}}{3}\right\\},\left\\{0,\frac{1+e^{i\pi/3}}{3},\frac{2+2e^{i\pi/3}}{3}\right\\}\right)\,.$
(11)

A visualization of such an orbifold is shown in Fig. 5, which, for
illustration, is focused on a single complex ED $z_{i}$. In (a), the unfolded
compactified ED is shown where, to form the torus in (b), one identifies the
green dotted lines with the red dotted lines, as required by Eq. 6. Three
fixed points from Eq. 11 are represented by blue dots and the coordinates must
be multiplied by the corresponding $R_{i}$. The same coloured lines and points
correspond to each other in all the figures. The folding of a $\mathbb{Z}_{3}$
orbifold dictated by Eq. 4 is schematically represented in (c). In particular,
this procedure is detailed as follows:

  1. 1.

First, one separates the whole space into two equilateral triangles.

  2. 2.

Each equilateral triangle is then divided into three equal isosceles
triangles.

  3. 3.

Then, the three of them are rotated and identified together resulting in the
rightmost of the four pictures in Fig. 5 (c). Note that the dotted lines from
(a) and (b) must be identified with the purple one.

Even though the projective space can not be properly visualized, (d) is merely
illustrative, and the $\mathbb{T}^{2}/\mathbb{Z}_{3}$ would appear as shown
here. The three conical singularities correspond to the three fixed points,
and the purple line also corresponds to the one in (a) and (b) as soon as the
identification with the dotted ones is performed.

Figure 1: *

(a) The unfolded compactified ED

space of a single coordinate $z_{i}$.

Figure 2: *

(b) $\mathbb{T}^{2}$ torus visualization.

Colours are as in (a).

Figure 3: *

(c) Folding the orbifold.

Figure 4: *

(d) Illustrative visualization of the orbifold.

Figure 5: Visualization of a single $\mathbb{T}^{2}/\mathbb{Z}_{3}$ orbifold.
The actual
$\mathbb{T}^{6}/(\mathbb{Z}^{\mathrm{F}}_{3}\times\mathbb{Z}^{\mathrm{C}}_{3})$
orbifold employed in this work consists of the direct product of three copies
of this orbifold.

As detailed in Appendix B, each field receives a different phase under the
orbifold action depending on its charges defined in Eq. 4. The chiral
superfields $\phi_{i}$ receive an extra phase contribution coming from the one
multiplying each $z_{i}$. Each field receives two charges, one for each
$\mathbb{Z}_{3}$, and $\mathcal{V}_{\textbf{248}}$ is split into components as
shown in Tab. 7.

| $V$ | $\phi_{1}$ | $\phi_{2}$ | $\phi_{3}$
---|---|---|---|---
$\mathcal{V}_{{\color[rgb]{1,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,1}\pgfsys@color@cmyk@stroke{0}{1}{0}{0}\pgfsys@color@cmyk@fill{0}{1}{0}{0}(\textbf{8},\textbf{1},\textbf{1},0,0)}}$ | $1,1$ | $\omega,1$ | $\omega,\omega^{2}$ | $\omega,\omega$
$\mathcal{V}_{{\color[rgb]{1,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,1}\pgfsys@color@cmyk@stroke{0}{1}{0}{0}\pgfsys@color@cmyk@fill{0}{1}{0}{0}(\textbf{1},\textbf{8},\textbf{1},0,0)}}$ | $1,1$ | $\omega,1$ | $\omega,\omega^{2}$ | $\omega,\omega$
$\mathcal{V}_{{\color[rgb]{1,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,1}\pgfsys@color@cmyk@stroke{0}{1}{0}{0}\pgfsys@color@cmyk@fill{0}{1}{0}{0}(\textbf{1},\textbf{1},\textbf{8},0,0)}}$ | $1,1$ | $\omega,1$ | $\omega,\omega^{2}$ | $\omega,\omega$
$\mathcal{V}_{{\color[rgb]{1,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,1}\pgfsys@color@cmyk@stroke{0}{1}{0}{0}\pgfsys@color@cmyk@fill{0}{1}{0}{0}(\textbf{1},\textbf{1},\textbf{1},0,0)}}$ | $1,1$ | $\omega,1$ | $\omega,\omega^{2}$ | $\omega,\omega$
$\mathcal{V}_{{\color[rgb]{1,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,1}\pgfsys@color@cmyk@stroke{0}{1}{0}{0}\pgfsys@color@cmyk@fill{0}{1}{0}{0}(\textbf{1},\textbf{1},\textbf{1},0,0)}}$ | $1,1$ | $\omega,1$ | $\omega,\omega^{2}$ | $\omega,\omega$
$\mathcal{V}_{{\color[rgb]{.75,.5,.25}\definecolor[named]{pgfstrokecolor}{rgb}{.75,.5,.25}(\textbf{1},\textbf{1},\textbf{1},1,3)}}$ | $1,\omega$ | $\omega,\omega$ | $\omega,1$ | $\omega,\omega^{2}$
$\mathcal{V}_{{\color[rgb]{.75,.5,.25}\definecolor[named]{pgfstrokecolor}{rgb}{.75,.5,.25}(\textbf{1},\textbf{1},\textbf{1},-1,3)}}$ | $1,\omega^{2}$ | $\omega,\omega^{2}$ | $\omega,\omega$ | $\omega,1$
$\mathcal{V}_{{\color[rgb]{.75,.5,.25}\definecolor[named]{pgfstrokecolor}{rgb}{.75,.5,.25}(\textbf{1},\textbf{1},\textbf{1},1,-3)}}$ | $1,\omega$ | $\omega,\omega$ | $\omega,1$ | $\omega,\omega^{2}$
$\mathcal{V}_{{\color[rgb]{.75,.5,.25}\definecolor[named]{pgfstrokecolor}{rgb}{.75,.5,.25}(\textbf{1},\textbf{1},\textbf{1},-1,-3)}}$ | $1,\omega^{2}$ | $\omega,\omega^{2}$ | $\omega,\omega$ | $\omega,1$
$\mathcal{V}_{{\color[rgb]{.75,.5,.25}\definecolor[named]{pgfstrokecolor}{rgb}{.75,.5,.25}(\textbf{1},\textbf{1},\textbf{1},2,0)}}$ | $1,\omega^{2}$ | $\omega,1$ | $\omega,\omega$ | $\omega,1$
$\mathcal{V}_{{\color[rgb]{.75,.5,.25}\definecolor[named]{pgfstrokecolor}{rgb}{.75,.5,.25}(\textbf{1},\textbf{1},\textbf{1},-2,0)}}$ | $1,\omega$ | $\omega,\omega$ | $\omega,1$ | $\omega,\omega^{2}$
$\mathcal{V}_{(\bar{\textbf{3}},\textbf{3},\textbf{3},0,0)}$ | $1,\omega$ | $\omega,\omega$ | $\omega,1$ | $\omega,\omega^{2}$
$\mathcal{V}_{({\textbf{3}},\bar{\textbf{3}},\bar{\textbf{3}},0,0)}$ | $1,\omega^{2}$ | $\omega,\omega^{2}$ | $\omega,\omega$ | $\omega,1$
$\mathcal{V}_{{\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color@cmyk@fill{0.91}{0}{0.88}{0.12}(\textbf{1},\bar{\textbf{3}},\textbf{3},1,1)}}$ | $\omega^{2},1$ | $1,1$ | $1,\omega^{2}$ | $1,\omega$
$\mathcal{V}_{{\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color@cmyk@fill{0.91}{0}{0.88}{0.12}(\textbf{1},\bar{\textbf{3}},\textbf{3},-1,1)}}$ | $\omega^{2},\omega$ | $1,\omega$ | $1,1$ | $1,\omega^{2}$
$\mathcal{V}_{{\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color@cmyk@fill{0.91}{0}{0.88}{0.12}(\textbf{1},\bar{\textbf{3}},\textbf{3},0,-2)}}$ | $\omega^{2},\omega^{2}$ | $1,\omega^{2}$ | $1,\omega$ | $1,1$

| $V$ | $\phi_{1}$ | $\phi_{2}$ | $\phi_{3}$
---|---|---|---|---
$\mathcal{V}_{{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}({\textbf{3}},\textbf{3},\textbf{1},1,1)}}$ | $\omega^{2},\omega^{2}$ | $1,\omega^{2}$ | $1,\omega$ | $1,1$
$\mathcal{V}_{{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}({\textbf{3}},\textbf{3},\textbf{1},-1,1)}}$ | $\omega^{2},1$ | $1,1$ | $1,\omega^{2}$ | $1,\omega$
$\mathcal{V}_{{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}({\textbf{3}},\textbf{3},\textbf{1},0,-2)}}$ | $\omega^{2},\omega$ | $1,\omega$ | $1,1$ | $1,\omega^{2}$
$\mathcal{V}_{{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}(\bar{\textbf{3}},\textbf{1},\bar{\textbf{3}},1,1)}}$ | $\omega^{2},\omega$ | $1,\omega$ | $1,1$ | $1,\omega^{2}$
$\mathcal{V}_{{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}(\bar{\textbf{3}},\textbf{1},\bar{\textbf{3}},-1,1)}}$ | $\omega^{2},\omega^{2}$ | $1,\omega^{2}$ | $1,\omega$ | $1,1$
$\mathcal{V}_{{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}(\bar{\textbf{3}},\textbf{1},\bar{\textbf{3}},0,-2)}}$ | $\omega^{2},1$ | $1,1$ | $1,\omega^{2}$ | $1,\omega$
$\mathcal{V}_{{\color[rgb]{1,.5,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,.5,0}(\textbf{1},\textbf{3},\bar{\textbf{3}},-1,-1)}}$ | $\omega,1$ | $\omega^{2},1$ | $\omega^{2},\omega^{2}$ | $\omega^{2},\omega$
$\mathcal{V}_{{\color[rgb]{1,.5,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,.5,0}(\textbf{1},\textbf{3},\bar{\textbf{3}},1,-1)}}$ | $\omega,\omega^{2}$ | $\omega^{2},\omega^{2}$ | $\omega^{2},\omega$ | $\omega^{2},1$
$\mathcal{V}_{{\color[rgb]{1,.5,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,.5,0}(\textbf{1},\textbf{3},\bar{\textbf{3}},0,2)}}$ | $\omega,\omega$ | $\omega^{2},\omega$ | $\omega^{2},\omega$ | $\omega^{2},\omega^{2}$
$\mathcal{V}_{{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}(\bar{\textbf{3}},\bar{\textbf{3}},\textbf{1},-1,-1)}}$ | $\omega,\omega$ | $\omega^{2},\omega$ | $\omega^{2},\omega$ | $\omega^{2},\omega^{2}$
$\mathcal{V}_{{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}(\bar{\textbf{3}},\bar{\textbf{3}},\textbf{1},1,-1)}}$ | $\omega,1$ | $\omega^{2},1$ | $\omega^{2},\omega^{2}$ | $\omega^{2},\omega$
$\mathcal{V}_{{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}(\bar{\textbf{3}},\bar{\textbf{3}},\textbf{1},0,2)}}$ | $\omega,\omega^{2}$ | $\omega^{2},\omega^{2}$ | $\omega^{2},\omega$ | $\omega^{2},1$
$\mathcal{V}_{{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}({\textbf{3}},\textbf{1},\textbf{3},-1,-1)}}$ | $\omega,\omega^{2}$ | $\omega^{2},\omega^{2}$ | $\omega^{2},\omega$ | $\omega^{2},1$
$\mathcal{V}_{{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}({\textbf{3}},\textbf{1},\textbf{3},1,-1)}}$ | $\omega,\omega^{2}$ | $\omega^{2},\omega^{2}$ | $\omega^{2},\omega$ | $\omega^{2},1$
$\mathcal{V}_{{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}({\textbf{3}},\textbf{1},\textbf{3},0,2)}}$ | $\omega,1$ | $\omega^{2},1$ | $\omega^{2},\omega^{2}$ | $\omega^{2},\omega$

Table 7: Charges of each $\mathcal{N}=1$ superfield being
$\mathrm{SU}(3)_{\mathrm{C}}\times\mathrm{SU}(3)_{\mathrm{L}}\times\mathrm{SU}(3)_{\mathrm{R}}\times\mathrm{U}(1)_{\mathrm{F3}}\times\mathrm{U}(1)_{\mathrm{F8}}$
multiplet under a
$\mathbb{Z}^{\mathrm{F}}_{3}\times\mathbb{Z}^{\mathrm{C}}_{3}$ orbifolding.
Only the fields with both charges equal to unity have zero modes. The
representations are color coded as adjoint fields, Higgs and leptons, quarks,
mirror Higgs and mirror fermions, mirror quarks, and SM exotics.

The zero (massless) modes containing the SM particle spectrum coming from the
purely rotational boundary conditions (i.e. the ones with charge $(1,1)$), for
each chiral multiplet, are then

$\begin{split}V_{\mu}&:{\color[rgb]{1,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,1}\pgfsys@color@cmyk@stroke{0}{1}{0}{0}\pgfsys@color@cmyk@fill{0}{1}{0}{0}(\textbf{8},\textbf{1},\textbf{1},0,0)}+{\color[rgb]{1,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,1}\pgfsys@color@cmyk@stroke{0}{1}{0}{0}\pgfsys@color@cmyk@fill{0}{1}{0}{0}(\textbf{1},\textbf{8},\textbf{1},0,0)}+{\color[rgb]{1,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,1}\pgfsys@color@cmyk@stroke{0}{1}{0}{0}\pgfsys@color@cmyk@fill{0}{1}{0}{0}(\textbf{1},\textbf{1},\textbf{8},0,0)}+2\times{\color[rgb]{1,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,1}\pgfsys@color@cmyk@stroke{0}{1}{0}{0}\pgfsys@color@cmyk@fill{0}{1}{0}{0}(\textbf{1},\textbf{1},\textbf{1},0,0)}\,,\\\
\phi_{1}&:{\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color@cmyk@fill{0.91}{0}{0.88}{0.12}(\textbf{1},\bar{\textbf{3}},\textbf{3},1,1)}+{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}(\bar{\textbf{3}},\textbf{1},\bar{\textbf{3}},0,-2)}+{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}({\textbf{3}},\textbf{3},\textbf{1},-1,1)}\,,\\\
\phi_{2}&:{\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color@cmyk@fill{0.91}{0}{0.88}{0.12}(\textbf{1},\bar{\textbf{3}},\textbf{3},-1,1)}+{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}(\bar{\textbf{3}},\textbf{1},\bar{\textbf{3}},1,1)}+{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}({\textbf{3}},\textbf{3},\textbf{1},0,-2)}\,,\\\
\phi_{3}&:{\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color@cmyk@fill{0.91}{0}{0.88}{0.12}(\textbf{1},\bar{\textbf{3}},\textbf{3},0,-2)}+{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}(\bar{\textbf{3}},\textbf{1},\bar{\textbf{3}},-1,1)}+{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}({\textbf{3}},\textbf{3},\textbf{1},1,1)}\,.\end{split}$
(12)

So far, we have only considered the rotational boundary conditions while
additional mass-splitting effects emerging from the Wilson-line effective VEVs
must also be incorporated. It is very important to note that the spectrum in
Eq. 12 does not correspond to the massless field content of the theory at any
point. It is shown here because it helps visualize the way in which the
symmetry is being broken by the rotational boundary conditions. The complete
$\mathrm{E}_{8}$ symmetry breaking down to the SM gauge symmetry group does
indeed involve also the translation boundary conditions in the form of Wilson
lines together with the rotational boundary conditions. Hence, the latter two
breakings are superimposed and happen simultaneously.

For ease of notation, in the remainder of this article, the representations in
Eq. 12 are named as

$\begin{split}V_{\mu}&:{\color[rgb]{1,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,1}\pgfsys@color@cmyk@stroke{0}{1}{0}{0}\pgfsys@color@cmyk@fill{0}{1}{0}{0}\Delta_{C}}+{\color[rgb]{1,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,1}\pgfsys@color@cmyk@stroke{0}{1}{0}{0}\pgfsys@color@cmyk@fill{0}{1}{0}{0}\Delta_{L}}+{\color[rgb]{1,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,1}\pgfsys@color@cmyk@stroke{0}{1}{0}{0}\pgfsys@color@cmyk@fill{0}{1}{0}{0}\Delta_{R}}+{\color[rgb]{1,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,1}\pgfsys@color@cmyk@stroke{0}{1}{0}{0}\pgfsys@color@cmyk@fill{0}{1}{0}{0}Z_{F3}}+{\color[rgb]{1,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,1}\pgfsys@color@cmyk@stroke{0}{1}{0}{0}\pgfsys@color@cmyk@fill{0}{1}{0}{0}Z_{F8}}\,,\\\
\phi_{1}&:{\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color@cmyk@fill{0.91}{0}{0.88}{0.12}\textbf{L}_{1}}+{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\textbf{Q}_{R3}}+{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\textbf{Q}_{L2}}\,,\\\
\phi_{2}&:{\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color@cmyk@fill{0.91}{0}{0.88}{0.12}\textbf{L}_{2}}+{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\textbf{Q}_{R1}}+{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\textbf{Q}_{L3}}\,,\\\
\phi_{3}&:{\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color@cmyk@fill{0.91}{0}{0.88}{0.12}\textbf{L}_{3}}+{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\textbf{Q}_{R2}}+{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\textbf{Q}_{L1}}\,.\end{split}$
(13)

Expressing the chiral superfields in terms of their gauge indices, where upper
indices correspond to an antitriplet representation of
$\mathrm{SU}(3)_{\mathrm{}}$ while down indices stand for a triplet one, one
can further split them in terms of their components as

$\begin{array}[]{llll}{\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color<EMAIL_ADDRESS>jk}}:&{\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color<EMAIL_ADDRESS>1k}}={\color[rgb]{0,0.6,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.6,0}\pgfsys@color@cmyk@stroke{0.64}{0}{0.95}{0.40}\pgfsys@color@cmyk@fill{0.64}{0}{0.95}{0.40}H_{dk}^{0}}\,,&{\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color<EMAIL_ADDRESS>2k}}={\color[rgb]{0,0.6,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.6,0}\pgfsys@color@cmyk@stroke{0.64}{0}{0.95}{0.40}\pgfsys@color@cmyk@fill{0.64}{0}{0.95}{0.40}H_{uk}^{-}}\,,&{\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color<EMAIL_ADDRESS>3k}}={\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\nu_{k}}\,,\\\
&{\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color<EMAIL_ADDRESS>1k}}={\color[rgb]{0,0.6,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.6,0}\pgfsys@color@cmyk@stroke{0.64}{0}{0.95}{0.40}\pgfsys@color@cmyk@fill{0.64}{0}{0.95}{0.40}H_{dk}^{+}}\,,&{\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color<EMAIL_ADDRESS>2k}}={\color[rgb]{0,0.6,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.6,0}\pgfsys@color@cmyk@stroke{0.64}{0}{0.95}{0.40}\pgfsys@color@cmyk@fill{0.64}{0}{0.95}{0.40}H_{uk}^{0}}\,,&{\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color<EMAIL_ADDRESS>3k}}={\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}e_{k}}\,,\\\
&{\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color<EMAIL_ADDRESS>1k}}={\color[rgb]{.5,0,.5}\definecolor[named]{pgfstrokecolor}{rgb}{.5,0,.5}\nu^{c}_{k}}\,,&{\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color<EMAIL_ADDRESS>2k}}={\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}e^{c}_{k}}\,,&{\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color<EMAIL_ADDRESS>3k}}={\color[rgb]{.75,.5,.25}\definecolor[named]{pgfstrokecolor}{rgb}{.75,.5,.25}\varphi_{k}}\,,\\\
{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\textbf{Q}_{R\
j}^{\ \
i}}:&{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\textbf{Q}_{R\
j}^{\ \
1}}={\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}d^{c}_{j}}\,,&{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\textbf{Q}_{R\
j}^{\ \
2}}={\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}u^{c}_{j}}\,,&{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\textbf{Q}_{R\
j}^{\ \ 3}}=D^{c}_{j}\,,\\\
{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\textbf{Q}_{Lij}}:&{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\textbf{Q}_{L1j}}={\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}d_{j}}\,,&{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\textbf{Q}_{L2j}}={\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}u_{j}}\,,&{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\textbf{Q}_{L3j}}=D_{j}\,,\\\
\end{array}$ (14)

where the color indices are not explicitly shown and the family ones are not
expanded. To complete the notation, the complex conjugated chiral superfield
representations are named as
${\color[rgb]{1,.5,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,.5,0}\overline{\textbf{L}}},{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\overline{\textbf{Q}}_{R}},{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\overline{\textbf{Q}}_{L}}$,
while the remaining superfields transforming as
$(\bar{\textbf{3}},\textbf{3},\textbf{3},\textbf{1})$ and
$({\textbf{3}},\bar{\textbf{3}},\bar{\textbf{3}},\textbf{1})$ are denoted by X
and $\bar{\textbf{X}}$, respectively. The representations are color-coded as
follows: adjoint fields, Higgs, SM fermions, mirror Higgs, mirror fermions,
right handed neutrinos, flavons, and family-neutral exotics.

As stated before in Eq. 7, there are three Wilson lines $U_{i}^{r}\equiv
U_{i}$, where both real degrees of freedom $r=1,2$ form a complex one. They
correspond to a gauge transformation which must comply with the consistency
conditions coming from the Poincaré algebra

$[U_{i},U_{j}]=0\,,\ \ \ [U_{i},U^{\rm F}_{i}]=0\,,\ \ \ [U_{i},U^{\rm
C}_{i}]=0\,,$ (15)

where $U^{\rm C,F}_{i}$ denote the gauge transformation that accompanies the
orbifold action of $\mathbb{Z}^{\mathrm{C,F}}_{3}$ on a chiral superfield

$\mathcal{Z}^{\rm C,F}[\phi_{i}(x,z)]=U^{\rm
C,F}_{i}\phi_{i}(x,\mathcal{Z}^{\rm C,F}[z])\,.$ (16)

There are two distinct ways to solve the last two constraints in Eq. 15. The
first one is by requiring

$U_{i}^{3}=1\,,$ (17)

which corresponds to the case of discrete Wilson lines resulting in additional
boundary conditions on the fixed points. For instance, before one considers
the effect of such discrete Wilson lines, one has the $U^{\rm C,F}_{i}$ gauge
transformation as a boundary condition in Eq. 10 at the zero brane, while at
the $(1+e^{i\pi/3})/3$ brane they read as $U_{i}U^{\rm C,F}_{i}$ and, finally,
at the $2(1+e^{i\pi/3})/3$ brane, one has $U_{i}^{2}U^{\rm C,F}_{i}$. Any
other boundary condition would result in an incomplete SM field content as it
would leave a whole column, row or family from LABEL:eq:fieldnames without
zero modes. This means that, in the considered GUT, a phenomenologically
viable low-scale spectrum is not compatible with the presence of discrete
Wilson lines.

The second possibility is to restrict the Wilson lines to representations
where the orbifold action is trivial, i.e. zero modes. One can choose a
continuous Wilson line which generates an effective VEV in the three
$\phi_{1,2,3}$ zero modes. However, due to invariance under translations
defined in Eq. 6, from where the effective VEVs result, they have to commute
with each other, complying with the first constraint of Eq. 15. The
restrictions on the effective VEVs come from the $\mathrm{E}_{8}$ commutation
relations in Eq. 107 given in Appendix D.

Note, the Wilson line is assumed to be aligned with
${\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color<EMAIL_ADDRESS>3k}}\sim{\color[rgb]{.5,0,.5}\definecolor[named]{pgfstrokecolor}{rgb}{.5,0,.5}\nu^{c}_{k}}$
and
${\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color<EMAIL_ADDRESS>3k}}\sim{\color[rgb]{.75,.5,.25}\definecolor[named]{pgfstrokecolor}{rgb}{.75,.5,.25}\varphi_{k}}$,
where all corresponding generators commute with each other as seen in Eq. 107.
These generate effective VEVs which are naturally scaled as

$\braket{{\color[rgb]{.75,.5,.25}\definecolor[named]{pgfstrokecolor}{rgb}{.75,.5,.25}\varphi_{i}}},\braket{{\color[rgb]{.5,0,.5}\definecolor[named]{pgfstrokecolor}{rgb}{.5,0,.5}{\nu}^{c}_{i}}}\sim\frac{1}{R_{i}}\,.$
(18)

From these components, the
$\braket{{\color[rgb]{.5,0,.5}\definecolor[named]{pgfstrokecolor}{rgb}{.5,0,.5}{\nu}^{c}_{1}}}$
can always be rotated away by an $\mathrm{SU}(3)_{\mathrm{R}}$ rotation while
the
$\braket{{\color[rgb]{.5,0,.5}\definecolor[named]{pgfstrokecolor}{rgb}{.5,0,.5}{\nu}^{c}_{2,3}}}$
can be made real by family symmetry ($\mathrm{U}(1)_{\mathrm{F3}}$ and
$\mathrm{U}(1)_{\mathrm{F8}}$) transformations. There are no other possible
rotations, therefore, the Wilson line is defined by three complex and two real
parameters.

The Higgs states are assumed to obtain a VEV radiatively, hence, generating a
much smaller scale and breaking the EW symmetry of the SM but preserving the
$\mathrm{U}(1)_{\mathrm{e.m.}}$ of electromagnetism, with the corresponding
charge,

$Q_{\rm EM}=\tfrac{1}{6}[\sqrt{3}T^{\rm L}_{8}+\sqrt{3}T^{\rm R}_{8}-3T^{\rm
L}_{3}-3T^{\rm R}_{3}]\,,$ (19)

where the normalization ${\rm Tr}\ T_{a}T_{b}=2\delta_{ab}$ is adopted.

The chosen Wilson line configuration breaks the gauge trinification symmetry
down to that of the SM, i.e.

$\mathrm{SU}(3)_{\mathrm{C}}\times\mathrm{SU}(3)_{\mathrm{L}}\times\mathrm{SU}(3)_{\mathrm{R}}\times\mathrm{U}(1)_{\mathrm{F3}}\times\mathrm{U}(1)_{\mathrm{F8}}\to\mathrm{SU}(3)_{\mathrm{C}}\times\mathrm{SU}(2)_{\mathrm{L}}\times\mathrm{U}(1)_{\mathrm{Y}}\,.$
(20)

Such a breaking leaves the following massless fields in the low-energy SM-like
EFT:

$\begin{split}V_{\mu}&:{\color[rgb]{1,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,1}\pgfsys@color@cmyk@stroke{0}{1}{0}{0}\pgfsys@color@cmyk@fill{0}{1}{0}{0}(\textbf{8},\textbf{1},0)}+{\color[rgb]{1,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,1}\pgfsys@color@cmyk@stroke{0}{1}{0}{0}\pgfsys@color@cmyk@fill{0}{1}{0}{0}(\textbf{1},\textbf{3},0)}+{\color[rgb]{1,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,1}\pgfsys@color@cmyk@stroke{0}{1}{0}{0}\pgfsys@color@cmyk@fill{0}{1}{0}{0}(\textbf{1},\textbf{1},0)}\,,\\\
\phi_{1}&:{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}(\textbf{1},\textbf{2},-3)}+3\times{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}(\textbf{1},\textbf{1},6)}+{\color[rgb]{0,0.6,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.6,0}\pgfsys@color@cmyk@stroke{0.64}{0}{0.95}{0.40}\pgfsys@color@cmyk@fill{0.64}{0}{0.95}{0.40}(\textbf{1},\textbf{2},3)}+{\color[rgb]{0,0.6,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.6,0}\pgfsys@color@cmyk@stroke{0.64}{0}{0.95}{0.40}\pgfsys@color@cmyk@fill{0.64}{0}{0.95}{0.40}(\textbf{1},\textbf{2},-3)}+{\color[rgb]{.75,.5,.25}\definecolor[named]{pgfstrokecolor}{rgb}{.75,.5,.25}(\textbf{1},\textbf{1},0)}+{\color[rgb]{.5,0,.5}\definecolor[named]{pgfstrokecolor}{rgb}{.5,0,.5}(\textbf{1},\textbf{1},0)}\\\
&\
+{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}(\bar{\textbf{3}},\textbf{1},-4)}+{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}(\bar{\textbf{3}},\textbf{1},2)}+(\bar{\textbf{3}},\textbf{1},2)+{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}(\textbf{3},\textbf{2},1)}+(\textbf{3},\textbf{1},-2)\,,\\\
\phi_{2}&:{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}(\textbf{1},\textbf{2},-3)}+3\times{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}(\textbf{1},\textbf{1},6)}+{\color[rgb]{0,0.6,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.6,0}\pgfsys@color@cmyk@stroke{0.64}{0}{0.95}{0.40}\pgfsys@color@cmyk@fill{0.64}{0}{0.95}{0.40}(\textbf{1},\textbf{2},3)}+{\color[rgb]{0,0.6,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.6,0}\pgfsys@color@cmyk@stroke{0.64}{0}{0.95}{0.40}\pgfsys@color@cmyk@fill{0.64}{0}{0.95}{0.40}(\textbf{1},\textbf{2},-3)}+{\color[rgb]{.75,.5,.25}\definecolor[named]{pgfstrokecolor}{rgb}{.75,.5,.25}(\textbf{1},\textbf{1},0)}+{\color[rgb]{.5,0,.5}\definecolor[named]{pgfstrokecolor}{rgb}{.5,0,.5}(\textbf{1},\textbf{1},0)}\\\
&\
+{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}(\bar{\textbf{3}},\textbf{1},-4)}+{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}(\bar{\textbf{3}},\textbf{1},2)}+(\bar{\textbf{3}},\textbf{1},2)+{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}(\textbf{3},\textbf{2},1)}+(\textbf{3},\textbf{1},-2)\,,\\\
\phi_{3}&:{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}(\textbf{1},\textbf{2},-3)}+3\times{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}(\textbf{1},\textbf{1},6)}+{\color[rgb]{0,0.6,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.6,0}\pgfsys@color@cmyk@stroke{0.64}{0}{0.95}{0.40}\pgfsys@color@cmyk@fill{0.64}{0}{0.95}{0.40}(\textbf{1},\textbf{2},3)}+{\color[rgb]{0,0.6,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.6,0}\pgfsys@color@cmyk@stroke{0.64}{0}{0.95}{0.40}\pgfsys@color@cmyk@fill{0.64}{0}{0.95}{0.40}(\textbf{1},\textbf{2},-3)}+{\color[rgb]{.75,.5,.25}\definecolor[named]{pgfstrokecolor}{rgb}{.75,.5,.25}(\textbf{1},\textbf{1},0)}+{\color[rgb]{.5,0,.5}\definecolor[named]{pgfstrokecolor}{rgb}{.5,0,.5}(\textbf{1},\textbf{1},0)}\\\
&\
+{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}(\bar{\textbf{3}},\textbf{1},-4)}+{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}(\bar{\textbf{3}},\textbf{1},2)}+(\bar{\textbf{3}},\textbf{1},2)+{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}(\textbf{3},\textbf{2},1)}+(\textbf{3},\textbf{1},-2)\,,\end{split}$
(21)

which in what follows are respectively denoted as

$\begin{split}V_{\mu}&:{\color[rgb]{1,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,1}\pgfsys@color@cmyk@stroke{0}{1}{0}{0}\pgfsys@color@cmyk@fill{0}{1}{0}{0}G_{\mu}}+{\color[rgb]{1,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,1}\pgfsys@color@cmyk@stroke{0}{1}{0}{0}\pgfsys@color@cmyk@fill{0}{1}{0}{0}W_{\mu}}+{\color[rgb]{1,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,1}\pgfsys@color@cmyk@stroke{0}{1}{0}{0}\pgfsys@color@cmyk@fill{0}{1}{0}{0}B_{\mu}}\,,\\\
\phi_{i}&:{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}L_{i}}+{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}e^{c}_{i}}+{\color[rgb]{0,0.6,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.6,0}\pgfsys@color@cmyk@stroke{0.64}{0}{0.95}{0.40}\pgfsys@color@cmyk@fill{0.64}{0}{0.95}{0.40}h_{ui}}+{\color[rgb]{0,0.6,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.6,0}\pgfsys@color@cmyk@stroke{0.64}{0}{0.95}{0.40}\pgfsys@color@cmyk@fill{0.64}{0}{0.95}{0.40}h_{di}}+{\color[rgb]{.75,.5,.25}\definecolor[named]{pgfstrokecolor}{rgb}{.75,.5,.25}\varphi_{i}}+{\color[rgb]{.5,0,.5}\definecolor[named]{pgfstrokecolor}{rgb}{.5,0,.5}\nu^{c}_{i}}\,\\\
&+{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}u^{c}_{i-1}}+{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}d^{c}_{i-1}}+D^{c}_{i-1}+{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}Q_{i+1}}+D_{i+1}\,,\end{split}$
(22)

with $i=1,2,3$ and the index rules $0\to 3$ and $4\to 1$ are implied. These
are the only massless superfields resulting from the compactification
procedure described above, containing the SM fermions and gauge bosons, as
well as three-right handed neutrinos
${\color[rgb]{.5,0,.5}\definecolor[named]{pgfstrokecolor}{rgb}{.5,0,.5}\nu^{c}_{i}}$,
six Higgs doublets
${\color[rgb]{0,0.6,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.6,0}\pgfsys@color@cmyk@stroke{0.64}{0}{0.95}{0.40}\pgfsys@color@cmyk@fill{0.64}{0}{0.95}{0.40}h_{ui}},{\color[rgb]{0,0.6,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.6,0}\pgfsys@color@cmyk@stroke{0.64}{0}{0.95}{0.40}\pgfsys@color<EMAIL_ADDRESS>three flavons
${\color[rgb]{.75,.5,.25}\definecolor[named]{pgfstrokecolor}{rgb}{.75,.5,.25}\varphi_{i}}$,
three vector-like color triplet pairs $D^{c}_{i},D_{i}$, and the associated
superpartners. Such a field content will be further split in the mass spectrum
as a result of the Wilson line effective VEVs, leaving solely the SM fermions,
color coded in blue, and gauge bosons, color coded in magenta, as the only
massless states in the theory at low energies (i.e. below the Wilson-line
breaking scale). Thus, the considered $\mathrm{E}_{8}$ GUT fully reproduces
the SM structure at low energies as it should.

###  I.3 Gauge anomaly cancellation

The original theory is based on the $\mathrm{E}_{8}$ gauge group with matter
contained in a single adjoint representation. The only 10d Weyl/Majorana
fermion lies in this real representation. Thus, at the fundamental level, the
model is free from gauge anomalies. As mentioned above, the 10d superfield
decomposes into an infinite tower of KK modes of 4d superfields, and each 248
representation is broken into nine different sets – one for each of the
possible orbifold charges given in Tab. 7. One can then expand the chiral
superfields as

$\phi_{i}(x,z_{i})=\sum_{\textbf{s}}\sum_{a,b=0,1,2}\phi^{\textbf{s}ab}_{i}(x)f(z_{j})_{\textbf{s}ab}\,,$
(23)

with $\textbf{s}=(s_{1},s_{2},s_{3},s_{4},s_{5},s_{6})$ denoting the KK level
coming from a schematic Fourier-like decomposition of the 6d states, while
$a,b=0,1,2$ arising from a splitting of each $\mathbb{Z}_{3}^{\rm C,F}$
orbifolding. Note that the decomposition into different s modes does not break
the $E_{8}$ symmetry, while the decomposition into different $a,b$ modes does
actually break it. The zero modes correspond to the configurations with
$s_{i}=a=b=0$. Note that the above series can be separated into an infinite
set of decomposed 248’s, where the anomalies cancel in each such set.

As was emphasised above, the $\mathrm{E}_{8}$ gauge group is broken
simultaneously by the rotation and translation boundary conditions below the
compactification scale, which is defined by $\Lambda=1/R_{l}$, where $R_{l}$
is the largest radius of $R_{1,2,3}$. Specifically, they break
$\mathrm{E}_{8}\to\mathrm{SU}(3)_{\mathrm{C}}\times\mathrm{SU}(2)_{\mathrm{L}}\times\mathrm{U}(1)_{\mathrm{Y}}$
yielding the massless field content shown in Eq. 21 which comprises the SM
fields as well as additional SM singlets, vector-like fermions and their
superpartners. Therefore, the massless sector of the theory is also anomaly
free Aranda:2020zms .

One must remark that the Wilson lines behave like effective VEVs but they are
not related to spontaneous symmetry breaking. They come from boundary
conditions on translations, just as the orbifolding comes from the boundary
conditions on rotations. As alluded to above, both rotation and translation
boundary conditions simultaneously induce the full breaking of
$\mathrm{E}_{8}$ and thus they both contribute to the particles’ masses that
would be integrated out when studying the corresponding low-energy SM-like
EFT. Therefore, even though the model is built in terms of the fields listed
in Eq. 13, there is no point in the theory when this spectrum appears as
entirely massless, and therefore no anomalies are present at any stage.

As this model is determined on an orbifold, which is not a continuous
spacetime, one must pay special attention to the discontinuous points. This
orbifold has 27 fixed ED points listed in Eq. 11. At these points, which are
4d branes, the boundary conditions force some of the 4d fields to vanish (i.e.
their ED profiles are zero in all the fixed points), so that they do not have
the full $\mathrm{E}_{8}$ symmetry nor the full $\mathrm{E}_{8}$
representation. The gauge transformations in a fixed point have less freedom
as the gauge parameters are only functions of the four open dimensions. To
avoid having multiply valued functions, the ED degrees freedom for the gauge
symmetry must be fixed. Their symmetry, which is defined in a 4d spacetime,
can only be studied in a fixed specific gauge (which only fixes the ED degrees
of freedom) where the Wilson line behaves like an effective VEV as they are
disconnected in EDs. In this gauge, all the fields are periodic in the compact
dimensions as described in Eq. 87. In any other gauge, the fields are
discontinuous multiply-valued functions, and their study requires techniques
beyond the standard complex analysis. Most importantly, the gauge symmetry in
these points is
$\mathrm{SU}(3)_{\mathrm{C}}\times\mathrm{SU}(2)_{\mathrm{L}}\times\mathrm{U}(1)_{\mathrm{Y}}$,
with the massless field content specified in Eq. 21, with no extra gauge
freedom. Therefore, there are no anomalies generated in the fixed points
either Scrucca:2004jn . As the considered GUT has no localized fields, there
are no extra contributions to the anomaly, rendering the model anomaly free at
all steps without the need for any additional fields.

##  II Effective Lagrangian at low energies

In order to study phenomenological implications of the considered GUT, the
next step is to build the EFT Lagrangian from the fields given in Eq. 12 that
are relevant below the compactification scale. They must respect the
trification symmetry supplemented by the family symmetry, i.e.
$\mathrm{SU}(3)_{\mathrm{C}}\times\mathrm{SU}(3)_{\mathrm{L}}\times\mathrm{SU}(3)_{\mathrm{R}}\times\mathrm{U}(1)_{\mathrm{F3}}\times\mathrm{U}(1)_{\mathrm{F8}}$,
that can only be broken through the effective, Wilson line, VEVs in
$\braket{{\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color@cmyk@fill{0.91}{0}{0.88}{0.12}\textbf{L}}}\sim\braket{{\color[rgb]{.75,.5,.25}\definecolor[named]{pgfstrokecolor}{rgb}{.75,.5,.25}\varphi_{i}}},\braket{{\color[rgb]{.5,0,.5}\definecolor[named]{pgfstrokecolor}{rgb}{.5,0,.5}\nu^{c}_{i}}}$.
Furthermore, the higher-order operators are mediated by heavy KK modes. In
Appendix C, the basic steps to obtain an effective Lagrangian from a general
superpotential like the one that emerges in the considered GUT are described.
The general superpotentials of the GUT under discussion are obtained in
Appendix E. From Eq. 96, the low-energy EFT Lagrangian is then defined in
terms of the potentials $\mathcal{K},\,\mathcal{W}$ and $\mathcal{H}$.

At leading order, the gauge field strength superpotential $\mathcal{H}$ takes
the standard form, i.e. $\mathcal{H}_{A}^{\ B}W^{A}W_{B}=W^{A}W_{A}$, the
Kähler potential $\mathcal{K}$ is also the standard one, while the chiral
superpotential $\mathcal{W}$ reads as follows:

$\begin{split}\mathcal{W}\sim&\,g\phi_{1}\phi_{2}\phi_{3}\\\
=&\,g[\epsilon^{ijk}{\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color<EMAIL_ADDRESS>mi}}{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\textbf{Q}_{R\
j}^{\
m}}{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\textbf{Q}_{Llk}}+\epsilon_{abc}\epsilon^{mno}{\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color<EMAIL_ADDRESS>m1}}{\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color<EMAIL_ADDRESS>n2}}{\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color<EMAIL_ADDRESS>o3}}+\epsilon_{abc}{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\textbf{Q}_{R\
1}^{\
a}}{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\textbf{Q}_{R\
2}^{\
b}}{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\textbf{Q}_{R\
3}^{\
c}}+\epsilon^{abc}{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\textbf{Q}_{La1}}{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\textbf{Q}_{Lb2}}{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\textbf{Q}_{Lc3}}]\\\
=&\,g\varepsilon^{ijk}\Big{(}{\color[rgb]{0,0.6,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.6,0}\pgfsys@color@cmyk@stroke{0.64}{0}{0.95}{0.40}\pgfsys@color@cmyk@fill{0.64}{0}{0.95}{0.40}h_{ui}}{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}u^{c}_{j}}{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}Q_{k}}+{\color[rgb]{0,0.6,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.6,0}\pgfsys@color@cmyk@stroke{0.64}{0}{0.95}{0.40}\pgfsys@color@cmyk@fill{0.64}{0}{0.95}{0.40}h_{di}}{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}d^{c}_{j}}{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}Q_{k}}+[{\color[rgb]{.75,.5,.25}\definecolor[named]{pgfstrokecolor}{rgb}{.75,.5,.25}\varphi_{i}}+\braket{{\color[rgb]{.75,.5,.25}\definecolor[named]{pgfstrokecolor}{rgb}{.75,.5,.25}\varphi_{i}}}]D^{c}_{j}D_{k}+[{\color[rgb]{.5,0,.5}\definecolor[named]{pgfstrokecolor}{rgb}{.5,0,.5}\nu^{c}_{i}}+\braket{{\color[rgb]{.5,0,.5}\definecolor[named]{pgfstrokecolor}{rgb}{.5,0,.5}\nu^{c}_{i}}}]{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}d^{c}_{j}}D_{k}+{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}e_{i}^{c}}{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}u^{c}_{j}}D_{k}+{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}L_{i}Q_{j}}D^{c}_{k}\\\
+&\,{\color[rgb]{0,0.6,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.6,0}\pgfsys@color@cmyk@stroke{0.64}{0}{0.95}{0.40}\pgfsys@color@cmyk@fill{0.64}{0}{0.95}{0.40}H_{di}}{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}L_{j}}{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}e^{c}_{k}}+{\color[rgb]{0,0.6,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.6,0}\pgfsys@color@cmyk@stroke{0.64}{0}{0.95}{0.40}\pgfsys@color@cmyk@fill{0.64}{0}{0.95}{0.40}H_{ui}}{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}L_{j}}[{\color[rgb]{.5,0,.5}\definecolor[named]{pgfstrokecolor}{rgb}{.5,0,.5}\nu^{c}_{k}}+\braket{{\color[rgb]{.5,0,.5}\definecolor[named]{pgfstrokecolor}{rgb}{.5,0,.5}\nu^{c}_{k}}}]+{\color[rgb]{0,0.6,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.6,0}\pgfsys@color@cmyk@stroke{0.64}{0}{0.95}{0.40}\pgfsys@color@cmyk@fill{0.64}{0}{0.95}{0.40}H_{di}}{\color[rgb]{0,0.6,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.6,0}\pgfsys@color@cmyk@stroke{0.64}{0}{0.95}{0.40}\pgfsys@color@cmyk@fill{0.64}{0}{0.95}{0.40}H_{uj}}[{\color[rgb]{.75,.5,.25}\definecolor[named]{pgfstrokecolor}{rgb}{.75,.5,.25}\varphi_{k}}+\braket{{\color[rgb]{.75,.5,.25}\definecolor[named]{pgfstrokecolor}{rgb}{.75,.5,.25}\varphi_{k}}}]+{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}d^{c}_{i}}{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}u^{c}_{j}}D^{c}_{k}+{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}Q_{i}}{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}Q_{j}}D_{k}\Big{)}\,,\end{split}$
(24)

where $g$ is the gauge coupling of the $\mathrm{E}_{8}$ theory evolved to an
appropriate energy-scale and the corresponding Clebsch-Gordan coefficients
(CGC) are ignored here.

The fermion mass terms are generated from SUSY breaking effects in the Kähler
potential as can be seen in Appendix C. This takes place already at leading
order via the $\mathcal{D}$-term,

$\braket{\mathcal{D}_{A}}=\sum_{ij}\braket{{\color[rgb]{.5,0,.5}\definecolor[named]{pgfstrokecolor}{rgb}{.5,0,.5}\nu^{c}_{i}}}^{\dagger}t_{A}\braket{{\color[rgb]{.5,0,.5}\definecolor[named]{pgfstrokecolor}{rgb}{.5,0,.5}\nu^{c}_{j}}}+\braket{{\color[rgb]{.75,.5,.25}\definecolor[named]{pgfstrokecolor}{rgb}{.75,.5,.25}\varphi_{i}}}^{\dagger}t_{A}\braket{{\color[rgb]{.75,.5,.25}\definecolor[named]{pgfstrokecolor}{rgb}{.75,.5,.25}\varphi_{j}}}+\braket{{\color[rgb]{.75,.5,.25}\definecolor[named]{pgfstrokecolor}{rgb}{.75,.5,.25}\varphi_{i}}}^{\dagger}t_{A}\braket{{\color[rgb]{.5,0,.5}\definecolor[named]{pgfstrokecolor}{rgb}{.5,0,.5}\nu^{c}_{j}}}+\braket{{\color[rgb]{.5,0,.5}\definecolor[named]{pgfstrokecolor}{rgb}{.5,0,.5}\nu^{c}_{i}}}^{\dagger}t_{A}\braket{{\color[rgb]{.75,.5,.25}\definecolor[named]{pgfstrokecolor}{rgb}{.75,.5,.25}\varphi_{j}}}\neq
0\,,$ (25)

at the scale of the effective Wilson-line VEVs.

There are no adjoint scalars as zero modes in this model but there are KK
modes which can mediate the Wilson line effective VEVs to provide a richer
structure of the couplings. They can enter as in diagram Fig. 6. This way one
can have effective adjoint VEVs. Note that even when in the considered model
the gauge family symmetry is
$\mathrm{U}(1)_{\mathrm{F3}}\times\mathrm{U}(1)_{\mathrm{F8}}$, one generate
adjoints
${\color[rgb]{1,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,1}\pgfsys@color@cmyk@stroke{0}{1}{0}{0}\pgfsys@color@cmyk@fill{0}{1}{0}{0}\Delta_{\rm
F}}$, as such fields exist inside $\mathrm{E}_{8}$ and become KK modes.

Figure 6: The leading-order diagram for the effective adjoint superfields as
${\color[rgb]{1,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,1}\pgfsys@color@cmyk@stroke{0}{1}{0}{0}\pgfsys@color@cmyk@fill{0}{1}{0}{0}\Delta_{\rm
L,R,F}}={\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color<EMAIL_ADDRESS>through an interaction with the corresponding KK mode. There are two possible
KK modes here that come from different chiral fields $\phi_{i}$.

In order to obtain the SM-like EFT Lagrangian, one does not need to consider
the full Lagrangian of the GUT since not every term of the latter is relevant
at low-energy scales. In fact, the part of the superpotential that is relevant
at low energies is obtained by truncating the higher-order terms and yielding
Eq. 24, where only the second and third generation obtain their masses through
the superpotential. However, their mass matrix is completely antisymmetric and
can neither generate the observed flavour structure nor the fermion mass
spectra that would fit the SM. The correct physical mass spectra and the
flavour mixing structure of light fermions are then obtained after SUSY
breaking effects take place. Upon compactification of the EDs, the Wilson
lines behave as effective VEVs in L. In turn, one sees from Eq. 103 in
Appendix C that such effective VEVs cannot generate $\mathcal{F}$-term SUSY
breaking, but rather, they do induce $\mathcal{D}$-term SUSY breaking already
at the leading order. One then finds that, using Eq. 101, the fermion masses
are obtained through the terms

$\begin{split}-\frac{1}{2}\left\langle\left.\frac{\partial^{2}\mathcal{W}}{\partial\phi_{i}\partial\phi_{j}}\right|_{S}\right\rangle\bar{\psi}_{i}\psi_{Lj}-i\sqrt{2}\left\langle\left.\frac{\partial^{2}\mathcal{K}}{\partial\phi_{i}^{\dagger}\partial\phi_{j}}\right|_{S}S^{\dagger}_{i}t_{A}\right\rangle\bar{\psi}_{j}\lambda_{AL}+\frac{i}{2\sqrt{2}}\left\langle\left.\frac{\partial\mathcal{H}_{AB}}{\partial\phi_{i}}\right|_{S}\mathcal{D}_{A}\right\rangle\bar{\lambda}_{B}\psi_{Li}\,,\end{split}$
(26)

where the first one encodes the usual SUSY preserving fermion mass term coming
from the superpotential, the second term is the usual SUSY preserving mass
term for gauginos, whereas the last one contains the new $\mathcal{D}$-term
SUSY breaking effects. In the remainder of this section, we use a notation
expressed in terms of usual fields, rather than superfields, charged under the
SM gauge symmetry. Following the standard notation adopted in the literature,
the
${\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}Q},{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}L},{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}u^{c}},{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}d^{c}},{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}e^{c}},{\color[rgb]{.5,0,.5}\definecolor[named]{pgfstrokecolor}{rgb}{.5,0,.5}\nu^{c}},D,D^{c}$
are defined to be fermions, with their scalar partners having an extra tilde
$\sim$ above their labels. The
${\color[rgb]{0,0.6,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.6,0}\pgfsys@color@cmyk@stroke{0.64}{0}{0.95}{0.40}\pgfsys@color@cmyk@fill{0.64}{0}{0.95}{0.40}H_{u,d}},{\color[rgb]{.75,.5,.25}\definecolor[named]{pgfstrokecolor}{rgb}{.75,.5,.25}\varphi}$
are defined to be scalars while their fermion partners are denoted by an extra
tilde $\sim$ above their labels.

###  II.1 Chiral fermions

The last term in Eq. 26 generates corrections to the fermion mass spectra
proportional to $\braket{\mathcal{D}}$ through the one-loop diagrams shown in
Fig. 7.

Figure 7: The one-loop diagrams providing SUSY-breaking contributions to the
light fermion masses built in terms of usual fields, not superfields . Here,
$\psi$ and $S$ denote the fermion and scalar components of the corresponding
superfields, $\lambda$ stands for the gaugino and $\braket{\mathcal{D}}$
labels a adjoint VEV built from the
$\braket{{\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color<EMAIL_ADDRESS>The three-point vertices come from the $\mathcal{D}\mathcal{D}$ and
$\mathcal{F}^{\dagger}\mathcal{F}$ terms in Eqs. 106 and 101, with at least
one scalar $S$ taking a Wilson line effective VEV.

In order to simplify the corresponding expressions, one defines the shorthand
notations

$\begin{split}{\color[rgb]{0.3,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.3,0,0}\pgfsys@color@cmyk@stroke{0}{0.83}{1}{0.70}\pgfsys@color@cmyk@fill{0}{0.83}{1}{0.70}(\textbf{w}^{2})^{i}_{\
j}}\equiv&\braket{{\color[rgb]{.5,0,.5}\definecolor[named]{pgfstrokecolor}{rgb}{.5,0,.5}\tilde{\nu}^{c\dagger
i}\tilde{\nu}^{c}_{j}}}+\braket{{\color[rgb]{.75,.5,.25}\definecolor[named]{pgfstrokecolor}{rgb}{.75,.5,.25}\varphi^{\dagger
i}\varphi_{j}}}\,,\\\
{\color[rgb]{0.3,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.3,0,0}\pgfsys@color@cmyk@stroke{0}{0.83}{1}{0.70}\pgfsys@color@cmyk@fill{0}{0.83}{1}{0.70}(\textbf{w}^{2}_{D})^{i}_{\
j}}\equiv&\delta^{i}_{j}\
(\braket{{\color[rgb]{.5,0,.5}\definecolor[named]{pgfstrokecolor}{rgb}{.5,0,.5}\tilde{\nu}^{c\dagger
j}\tilde{\nu}^{c}_{j}}}+\braket{{\color[rgb]{.75,.5,.25}\definecolor[named]{pgfstrokecolor}{rgb}{.75,.5,.25}\varphi^{\dagger
j}\varphi_{j}}})\,,\quad\quad({\rm no\ sum\ over}\ j)\,,\\\
{\color[rgb]{0.3,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.3,0,0}\pgfsys@color@cmyk@stroke{0}{0.83}{1}{0.70}\pgfsys@color@cmyk@fill{0}{0.83}{1}{0.70}\textbf{w}^{2}}\equiv&{\color[rgb]{0.3,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.3,0,0}\pgfsys@color@cmyk@stroke{0}{0.83}{1}{0.70}\pgfsys@color@cmyk@fill{0}{0.83}{1}{0.70}(\textbf{w}^{2})^{i}_{\
i}}\,.\end{split}$ (27)

The mass terms for quarks and their vector-like counterparts then read as

$\displaystyle\mathcal{L}_{mq}$
$\displaystyle\sim\epsilon^{ijk}\left\\{{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}u^{c}_{a}}{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}Q_{b}}\braket{{\color[rgb]{0,0.6,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.6,0}\pgfsys@color@cmyk@stroke{0.64}{0}{0.95}{0.40}\pgfsys@color@cmyk@fill{0.64}{0}{0.95}{0.40}H_{ui}}}\left[\delta^{a}_{j}\delta^{b}_{k}+\frac{1}{\Lambda^{2}}({\color[rgb]{0.3,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.3,0,0}\pgfsys@color@cmyk@stroke{0}{0.83}{1}{0.70}\pgfsys@color@cmyk@fill{0}{0.83}{1}{0.70}(\textbf{w}^{2})^{a}_{\
j}}\delta^{b}_{k}-{\color[rgb]{0.3,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.3,0,0}\pgfsys@color@cmyk@stroke{0}{0.83}{1}{0.70}\pgfsys@color@cmyk@fill{0}{0.83}{1}{0.70}(\textbf{w}^{2})^{b}_{\
k}}\delta^{a}_{j})\right]\right.$
$\displaystyle+{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}d^{c}_{a}}{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}Q_{b}}\braket{{\color[rgb]{0,0.6,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.6,0}\pgfsys@color@cmyk@stroke{0.64}{0}{0.95}{0.40}\pgfsys@color@cmyk@fill{0.64}{0}{0.95}{0.40}H_{di}}}\left[\left(1+\frac{\braket{{\color[rgb]{.5,0,.5}\definecolor[named]{pgfstrokecolor}{rgb}{.5,0,.5}\tilde{\nu}^{c\dagger
s}\tilde{\nu}^{c}_{s}}}}{\Lambda^{2}}\right)\delta^{a}_{j}\delta^{b}_{k}+\frac{1}{\Lambda^{2}}({\color[rgb]{0.3,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.3,0,0}\pgfsys@color@cmyk@stroke{0}{0.83}{1}{0.70}\pgfsys@color@cmyk@fill{0}{0.83}{1}{0.70}(\textbf{w}^{2})^{a}_{\
j}}\delta^{b}_{k}-{\color[rgb]{0.3,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.3,0,0}\pgfsys@color@cmyk@stroke{0}{0.83}{1}{0.70}\pgfsys@color@cmyk@fill{0}{0.83}{1}{0.70}(\textbf{w}^{2})^{b}_{\
k}}\delta^{a}_{j})\right]$
$\displaystyle+D^{c}_{a}D_{b}\Big{[}\braket{{\color[rgb]{.75,.5,.25}\definecolor[named]{pgfstrokecolor}{rgb}{.75,.5,.25}\varphi_{i}}}\left(1+\frac{{\color[rgb]{0.3,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.3,0,0}\pgfsys@color@cmyk@stroke{0}{0.83}{1}{0.70}\pgfsys@color@cmyk@fill{0}{0.83}{1}{0.70}\textbf{w}^{2}}}{\Lambda^{2}}+\frac{\braket{{\color[rgb]{.75,.5,.25}\definecolor[named]{pgfstrokecolor}{rgb}{.75,.5,.25}\varphi^{\dagger
s}}{\color[rgb]{.75,.5,.25}\definecolor[named]{pgfstrokecolor}{rgb}{.75,.5,.25}\varphi_{s}}}}{\Lambda^{2}}\right)\delta^{a}_{j}\delta^{b}_{k}+\braket{{\color[rgb]{.5,0,.5}\definecolor[named]{pgfstrokecolor}{rgb}{.5,0,.5}\tilde{\nu}^{c}_{i}}}\frac{\braket{{\color[rgb]{.5,0,.5}\definecolor[named]{pgfstrokecolor}{rgb}{.5,0,.5}\tilde{\nu}^{c\dagger
s}}{\color[rgb]{.75,.5,.25}\definecolor[named]{pgfstrokecolor}{rgb}{.75,.5,.25}\varphi_{s}}}}{\Lambda^{2}}\delta^{a}_{j}\delta^{b}_{k}$
$\displaystyle+\frac{1}{\Lambda^{2}}\braket{{\color[rgb]{.75,.5,.25}\definecolor[named]{pgfstrokecolor}{rgb}{.75,.5,.25}\varphi_{i}}}({\color[rgb]{0.3,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.3,0,0}\pgfsys@color@cmyk@stroke{0}{0.83}{1}{0.70}\pgfsys@color@cmyk@fill{0}{0.83}{1}{0.70}(\textbf{w}^{2})^{a}_{\
j}}\delta^{b}_{k}-{\color[rgb]{0.3,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.3,0,0}\pgfsys@color@cmyk@stroke{0}{0.83}{1}{0.70}\pgfsys@color@cmyk@fill{0}{0.83}{1}{0.70}(\textbf{w}^{2})^{b}_{\
k}}\delta^{a}_{j})\Big{]}$ (28)
$\displaystyle+{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}d^{c}_{a}}D_{b}\Big{[}\braket{{\color[rgb]{.5,0,.5}\definecolor[named]{pgfstrokecolor}{rgb}{.5,0,.5}\tilde{\nu}^{c}_{i}}}\left(1+\frac{{\color[rgb]{0.3,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.3,0,0}\pgfsys@color@cmyk@stroke{0}{0.83}{1}{0.70}\pgfsys@color@cmyk@fill{0}{0.83}{1}{0.70}\textbf{w}^{2}}}{\Lambda^{2}}+\frac{\braket{{\color[rgb]{.5,0,.5}\definecolor[named]{pgfstrokecolor}{rgb}{.5,0,.5}\tilde{\nu}^{c\dagger
s}\tilde{\nu}^{c}_{s}}}}{\Lambda^{2}}\right)\delta^{a}_{j}\delta^{b}_{k}+\braket{{\color[rgb]{.75,.5,.25}\definecolor[named]{pgfstrokecolor}{rgb}{.75,.5,.25}\varphi_{i}}}\frac{\braket{{\color[rgb]{.75,.5,.25}\definecolor[named]{pgfstrokecolor}{rgb}{.75,.5,.25}\varphi^{\dagger
s}}{\color[rgb]{.5,0,.5}\definecolor[named]{pgfstrokecolor}{rgb}{.5,0,.5}\tilde{\nu}^{c}_{s}}}}{\Lambda^{2}}\delta^{a}_{j}\delta^{b}_{k}$
$\displaystyle+\frac{1}{\Lambda^{2}}\braket{{\color[rgb]{.5,0,.5}\definecolor[named]{pgfstrokecolor}{rgb}{.5,0,.5}\tilde{\nu}^{c}_{i}}}({\color[rgb]{0.3,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.3,0,0}\pgfsys@color@cmyk@stroke{0}{0.83}{1}{0.70}\pgfsys@color@cmyk@fill{0}{0.83}{1}{0.70}(\textbf{w}^{2})^{a}_{\
j}}\delta^{b}_{k}-{\color[rgb]{0.3,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.3,0,0}\pgfsys@color@cmyk@stroke{0}{0.83}{1}{0.70}\pgfsys@color@cmyk@fill{0}{0.83}{1}{0.70}(\textbf{w}^{2})^{b}_{\
k}}\delta^{a}_{j})\Big{]}\left.+D^{c}_{a}{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}Q_{b}}\braket{{\color[rgb]{0,0.6,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.6,0}\pgfsys@color@cmyk@stroke{0.64}{0}{0.95}{0.40}\pgfsys@color@cmyk@fill{0.64}{0}{0.95}{0.40}H_{di}}}\left[\frac{\braket{{\color[rgb]{.5,0,.5}\definecolor[named]{pgfstrokecolor}{rgb}{.5,0,.5}\tilde{\nu}^{c\dagger
s}}{\color[rgb]{.75,.5,.25}\definecolor[named]{pgfstrokecolor}{rgb}{.75,.5,.25}\varphi_{s}}}}{\Lambda^{2}}\delta^{a}_{j}\delta^{b}_{k}\right]\right\\}+\mathrm{h.c.}+\mathcal{O}(1/\Lambda^{4})\,,$

where dimensionless constants coming from loop factors, gauge couplings and
CGC have been ignored, for simplicity. The first line contains the up-type
quark mass terms which are antisymmetric at the leading order and only
contribute to two physical quark masses. The next-to-leading order corrections
fill up the entire mass matrix generating a hierarchy between the lightest up
quark and the heavier charm and top quarks. The second line contains the down-
type quark masses which depend on up to three different Higgs VEVs and a new
term that differs from the up quarks. This is enough to create a different
mass structure and a viable Cabibbo–Kobayashi–Maskawa (CKM) mixing matrix
similarly to the discussion in Morais:2020ypd . The third line contains
vector-like $\mathrm{SU}(3)_{\mathrm{C}}$-triplet masses generated at the
compactification scale, which is desirable. The fourth line contains a large
mixing between one of the extra vector-like states and the down-type quarks,
meaning that the light down-type quark lies in a linear combination between
them, further contributing to the CKM matrix. Finally, the fifth line contains
an EW scale mixing between the vector-like and the down-type quarks.

The leading contributions to the masses of all leptons, both chiral and
vector-like, can then be expressed as

$\begin{split}\mathcal{L}_{ml}\sim&\epsilon^{ijk}\bigg{\\{}{\color[rgb]{.5,0,.5}\definecolor[named]{pgfstrokecolor}{rgb}{.5,0,.5}\nu^{c}_{a}}{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}L_{b}}\braket{{\color[rgb]{0,0.6,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.6,0}\pgfsys@color@cmyk@stroke{0.64}{0}{0.95}{0.40}\pgfsys@color@cmyk@fill{0.64}{0}{0.95}{0.40}H_{ui}}}\Big{[}\Big{(}1+\frac{1}{\Lambda^{2}}{\color[rgb]{0.3,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.3,0,0}\pgfsys@color@cmyk@stroke{0}{0.83}{1}{0.70}\pgfsys@color@cmyk@fill{0}{0.83}{1}{0.70}\textbf{w}^{2}}+\frac{1}{\Lambda^{2}}\braket{{\color[rgb]{.75,.5,.25}\definecolor[named]{pgfstrokecolor}{rgb}{.75,.5,.25}\varphi^{\dagger
s}}{\color[rgb]{.75,.5,.25}\definecolor[named]{pgfstrokecolor}{rgb}{.75,.5,.25}\varphi_{s}}}\Big{)}\delta^{a}_{j}\delta^{b}_{k}+\frac{1}{\Lambda^{2}}\Big{(}{\color[rgb]{0.3,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.3,0,0}\pgfsys@color@cmyk@stroke{0}{0.83}{1}{0.70}\pgfsys@color@cmyk@fill{0}{0.83}{1}{0.70}(\textbf{w}^{2})^{a}_{\
j}}\delta^{b}_{k}-{\color[rgb]{0.3,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.3,0,0}\pgfsys@color@cmyk@stroke{0}{0.83}{1}{0.70}\pgfsys@color@cmyk@fill{0}{0.83}{1}{0.70}(\textbf{w}^{2})^{b}_{\
k}}\delta^{a}_{j}\Big{)}\Big{]}\\\
&+{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}e^{c}_{a}}{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}L_{b}}\braket{{\color[rgb]{0,0.6,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.6,0}\pgfsys@color@cmyk@stroke{0.64}{0}{0.95}{0.40}\pgfsys@color@cmyk@fill{0.64}{0}{0.95}{0.40}H_{di}}}\Big{[}\Big{(}1+\frac{1}{\Lambda^{2}}{\color[rgb]{0.3,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.3,0,0}\pgfsys@color@cmyk@stroke{0}{0.83}{1}{0.70}\pgfsys@color@cmyk@fill{0}{0.83}{1}{0.70}\textbf{w}^{2}}+\frac{1}{\Lambda^{2}}\braket{{\color[rgb]{.75,.5,.25}\definecolor[named]{pgfstrokecolor}{rgb}{.75,.5,.25}\varphi^{\dagger
s}\varphi_{s}}}\Big{)}\delta^{a}_{j}\delta^{b}_{k}+\frac{1}{\Lambda^{2}}\Big{(}{\color[rgb]{0.3,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.3,0,0}\pgfsys@color@cmyk@stroke{0}{0.83}{1}{0.70}\pgfsys@color@cmyk@fill{0}{0.83}{1}{0.70}(\textbf{w}^{2})^{a}_{\
j}}\delta^{b}_{k}-{\color[rgb]{0.3,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.3,0,0}\pgfsys@color@cmyk@stroke{0}{0.83}{1}{0.70}\pgfsys@color@cmyk@fill{0}{0.83}{1}{0.70}(\textbf{w}^{2})^{b}_{\
k}}\delta^{a}_{j}\Big{)}\Big{]}\\\
&+{\color[rgb]{0,0.6,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.6,0}\pgfsys@color@cmyk@stroke{0.64}{0}{0.95}{0.40}\pgfsys@color@cmyk@fill{0.64}{0}{0.95}{0.40}\tilde{H}_{ua}}{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}L_{b}}\Big{[}\Big{(}\braket{{\color[rgb]{.5,0,.5}\definecolor[named]{pgfstrokecolor}{rgb}{.5,0,.5}\tilde{\nu}^{c}_{i}}}\Big{[}1+\frac{1}{\Lambda^{2}}\braket{{\color[rgb]{.75,.5,.25}\definecolor[named]{pgfstrokecolor}{rgb}{.75,.5,.25}\varphi^{\dagger
s}\varphi_{s}}}\Big{]}+\frac{1}{\Lambda^{2}}\braket{{\color[rgb]{.75,.5,.25}\definecolor[named]{pgfstrokecolor}{rgb}{.75,.5,.25}\phi_{i}}}\braket{{\color[rgb]{.75,.5,.25}\definecolor[named]{pgfstrokecolor}{rgb}{.75,.5,.25}\varphi^{\dagger
s}}{\color[rgb]{.5,0,.5}\definecolor[named]{pgfstrokecolor}{rgb}{.5,0,.5}\tilde{\nu}^{c}_{s}}}\Big{)}\delta^{a}_{j}\delta^{b}_{k}+\frac{1}{\Lambda^{2}}\Big{(}{\color[rgb]{0.3,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.3,0,0}\pgfsys@color@cmyk@stroke{0}{0.83}{1}{0.70}\pgfsys@color@cmyk@fill{0}{0.83}{1}{0.70}(\textbf{w}^{2})^{a}_{\
j}}\delta^{b}_{k}-{\color[rgb]{0.3,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.3,0,0}\pgfsys@color@cmyk@stroke{0}{0.83}{1}{0.70}\pgfsys@color@cmyk@fill{0}{0.83}{1}{0.70}(\textbf{w}^{2})^{b}_{\
k}}\delta^{a}_{j}\Big{)}\Big{]}\\\
&+{\color[rgb]{0,0.6,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.6,0}\pgfsys@color@cmyk@stroke{0.64}{0}{0.95}{0.40}\pgfsys@color@cmyk@fill{0.64}{0}{0.95}{0.40}\tilde{H}_{da}}{\color[rgb]{0,0.6,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.6,0}\pgfsys@color@cmyk@stroke{0.64}{0}{0.95}{0.40}\pgfsys@color@cmyk@fill{0.64}{0}{0.95}{0.40}\tilde{H}_{ub}}\Big{[}\Big{(}\frac{1}{\Lambda^{2}}\braket{{\color[rgb]{.5,0,.5}\definecolor[named]{pgfstrokecolor}{rgb}{.5,0,.5}\tilde{\nu}^{c}_{i}}}\braket{{\color[rgb]{.5,0,.5}\definecolor[named]{pgfstrokecolor}{rgb}{.5,0,.5}\tilde{\nu}^{c\dagger
s}}{\color[rgb]{.75,.5,.25}\definecolor[named]{pgfstrokecolor}{rgb}{.75,.5,.25}\varphi_{s}}}+\braket{{\color[rgb]{.75,.5,.25}\definecolor[named]{pgfstrokecolor}{rgb}{.75,.5,.25}\phi_{i}}}\Big{[}1+\frac{1}{\Lambda^{2}}\braket{{\color[rgb]{.5,0,.5}\definecolor[named]{pgfstrokecolor}{rgb}{.5,0,.5}\tilde{\nu}^{c\dagger
s}\tilde{\nu}^{c}_{s}}}\Big{]}\Big{)}\delta^{a}_{j}\delta^{b}_{k}+\frac{1}{\Lambda^{2}}\Big{(}{\color[rgb]{0.3,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.3,0,0}\pgfsys@color@cmyk@stroke{0}{0.83}{1}{0.70}\pgfsys@color@cmyk@fill{0}{0.83}{1}{0.70}(\textbf{w}^{2})^{a}_{\
j}}\delta^{b}_{k}-{\color[rgb]{0.3,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.3,0,0}\pgfsys@color@cmyk@stroke{0}{0.83}{1}{0.70}\pgfsys@color@cmyk@fill{0}{0.83}{1}{0.70}(\textbf{w}^{2})^{b}_{\
k}}\delta^{a}_{j}\Big{)}\Big{]}\\\
&+{\color[rgb]{.75,.5,.25}\definecolor[named]{pgfstrokecolor}{rgb}{.75,.5,.25}\tilde{\varphi}_{a}}{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}L_{b}}\braket{{\color[rgb]{0,0.6,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.6,0}\pgfsys@color@cmyk@stroke{0.64}{0}{0.95}{0.40}\pgfsys@color@cmyk@fill{0.64}{0}{0.95}{0.40}H_{ui}}}\braket{{\color[rgb]{.75,.5,.25}\definecolor[named]{pgfstrokecolor}{rgb}{.75,.5,.25}\varphi^{\dagger
s}}{\color[rgb]{.5,0,.5}\definecolor[named]{pgfstrokecolor}{rgb}{.5,0,.5}\tilde{\nu}^{c}_{s}}}\frac{1}{\Lambda}\delta^{a}_{j}\delta^{b}_{k}+{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}e^{c}_{a}}{\color[rgb]{0,0.6,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.6,0}\pgfsys@color@cmyk@stroke{0.64}{0}{0.95}{0.40}\pgfsys@color@cmyk@fill{0.64}{0}{0.95}{0.40}\tilde{H}_{db}}\braket{{\color[rgb]{0,0.6,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.6,0}\pgfsys@color@cmyk@stroke{0.64}{0}{0.95}{0.40}\pgfsys@color@cmyk@fill{0.64}{0}{0.95}{0.40}H_{di}}}\braket{{\color[rgb]{.5,0,.5}\definecolor[named]{pgfstrokecolor}{rgb}{.5,0,.5}\tilde{\nu}^{c\dagger
s}}{\color[rgb]{.75,.5,.25}\definecolor[named]{pgfstrokecolor}{rgb}{.75,.5,.25}\varphi_{s}}}\frac{1}{\Lambda}\delta^{a}_{j}\delta^{b}_{k}\\\
&+{\color[rgb]{.5,0,.5}\definecolor[named]{pgfstrokecolor}{rgb}{.5,0,.5}\nu^{c}_{a}}{\color[rgb]{0,0.6,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.6,0}\pgfsys@color@cmyk@stroke{0.64}{0}{0.95}{0.40}\pgfsys@color@cmyk@fill{0.64}{0}{0.95}{0.40}\tilde{H}_{da}}\braket{{\color[rgb]{0,0.6,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.6,0}\pgfsys@color@cmyk@stroke{0.64}{0}{0.95}{0.40}\pgfsys@color@cmyk@fill{0.64}{0}{0.95}{0.40}{H}_{ui}}}\braket{{\color[rgb]{.5,0,.5}\definecolor[named]{pgfstrokecolor}{rgb}{.5,0,.5}\tilde{\nu}^{c\dagger
s}}{\color[rgb]{.75,.5,.25}\definecolor[named]{pgfstrokecolor}{rgb}{.75,.5,.25}\varphi_{s}}}\frac{1}{\Lambda}\delta^{a}_{j}\delta^{b}_{k}+{\color[rgb]{.5,0,.5}\definecolor[named]{pgfstrokecolor}{rgb}{.5,0,.5}\nu^{c}_{a}}{\color[rgb]{0,0.6,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.6,0}\pgfsys@color@cmyk@stroke{0.64}{0}{0.95}{0.40}\pgfsys@color@cmyk@fill{0.64}{0}{0.95}{0.40}\tilde{H}_{ua}}\braket{{\color[rgb]{0,0.6,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.6,0}\pgfsys@color@cmyk@stroke{0.64}{0}{0.95}{0.40}\pgfsys@color@cmyk@fill{0.64}{0}{0.95}{0.40}{H}_{di}}}\braket{{\color[rgb]{.5,0,.5}\definecolor[named]{pgfstrokecolor}{rgb}{.5,0,.5}\tilde{\nu}^{c\dagger
s}}{\color[rgb]{.75,.5,.25}\definecolor[named]{pgfstrokecolor}{rgb}{.75,.5,.25}\varphi_{s}}}\frac{1}{\Lambda}\delta^{a}_{j}\delta^{b}_{k}\\\
&+{\color[rgb]{.75,.5,.25}\definecolor[named]{pgfstrokecolor}{rgb}{.75,.5,.25}\tilde{\varphi}_{a}}{\color[rgb]{0,0.6,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.6,0}\pgfsys@color@cmyk@stroke{0.64}{0}{0.95}{0.40}\pgfsys@color@cmyk@fill{0.64}{0}{0.95}{0.40}\tilde{H}_{ub}}\braket{{\color[rgb]{0,0.6,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.6,0}\pgfsys@color@cmyk@stroke{0.64}{0}{0.95}{0.40}\pgfsys@color@cmyk@fill{0.64}{0}{0.95}{0.40}{H}_{di}}}\Big{[}\Big{(}1+\frac{1}{\Lambda^{2}}{\color[rgb]{0.3,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.3,0,0}\pgfsys@color@cmyk@stroke{0}{0.83}{1}{0.70}\pgfsys@color@cmyk@fill{0}{0.83}{1}{0.70}\textbf{w}^{2}}+\frac{1}{\Lambda^{2}}\braket{{\color[rgb]{.75,.5,.25}\definecolor[named]{pgfstrokecolor}{rgb}{.75,.5,.25}\varphi^{\dagger
s}\varphi_{s}}}\Big{)}\delta^{a}_{j}\delta^{b}_{k}+\frac{1}{\Lambda^{2}}\Big{(}{\color[rgb]{0.3,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.3,0,0}\pgfsys@color@cmyk@stroke{0}{0.83}{1}{0.70}\pgfsys@color@cmyk@fill{0}{0.83}{1}{0.70}(\textbf{w}^{2})^{a}_{\
j}}\delta^{b}_{k}-{\color[rgb]{0.3,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.3,0,0}\pgfsys@color@cmyk@stroke{0}{0.83}{1}{0.70}\pgfsys@color@cmyk@fill{0}{0.83}{1}{0.70}(\textbf{w}^{2})^{b}_{\
k}}\delta^{a}_{j}\Big{)}\Big{]}\\\
&+{\color[rgb]{.75,.5,.25}\definecolor[named]{pgfstrokecolor}{rgb}{.75,.5,.25}\tilde{\varphi}_{a}}{\color[rgb]{0,0.6,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.6,0}\pgfsys@color@cmyk@stroke{0.64}{0}{0.95}{0.40}\pgfsys@color@cmyk@fill{0.64}{0}{0.95}{0.40}\tilde{H}_{db}}\braket{{\color[rgb]{0,0.6,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.6,0}\pgfsys@color@cmyk@stroke{0.64}{0}{0.95}{0.40}\pgfsys@color@cmyk@fill{0.64}{0}{0.95}{0.40}{H}_{ui}}}\Big{[}\Big{(}1+\frac{1}{\Lambda^{2}}{\color[rgb]{0.3,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.3,0,0}\pgfsys@color@cmyk@stroke{0}{0.83}{1}{0.70}\pgfsys@color@cmyk@fill{0}{0.83}{1}{0.70}\textbf{w}^{2}}+\frac{1}{\Lambda^{2}}\braket{{\color[rgb]{.5,0,.5}\definecolor[named]{pgfstrokecolor}{rgb}{.5,0,.5}\tilde{\nu}^{c\dagger
s}\tilde{\nu}^{c}_{s}}}\Big{)}\delta^{a}_{j}\delta^{b}_{k}+\frac{1}{\Lambda^{2}}\Big{(}{\color[rgb]{0.3,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.3,0,0}\pgfsys@color@cmyk@stroke{0}{0.83}{1}{0.70}\pgfsys@color@cmyk@fill{0}{0.83}{1}{0.70}(\textbf{w}^{2})^{a}_{\
j}}\delta^{b}_{k}-{\color[rgb]{0.3,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.3,0,0}\pgfsys@color@cmyk@stroke{0}{0.83}{1}{0.70}\pgfsys@color@cmyk@fill{0}{0.83}{1}{0.70}(\textbf{w}^{2})^{b}_{\
k}}\delta^{a}_{j}\Big{)}\Big{]}\bigg{\\}}\\\
&+\mathrm{h.c.}+\mathcal{O}(1/\Lambda^{6})\,,\end{split}$ (29)

where dimensionless constants coming from loop factors, gauge couplings and
CGC have been again ignored, for simplicity. The first line contains the Dirac
mass terms for neutrinos, the second one provides the charged lepton mass
terms whereas the third one includes R-parity violating Higgsino-lepton
mixing, which can be rotated into the Higgsino (or vector-like lepton) masses.
This means that the light SM-like charged leptons lie in a linear combination
between them. The fourth line contains the Higgsino mass terms and the fifth
one offers a mixing term between the neutrinos and the flavinos. Note that the
latter will behave as three extra right-handed neutrinos. The remaining lines
contain EW scale mixing between leptons and higgsinos.

As described in Appendix B, the rotational boundary conditions preserve a
$\rm{U}(1)_{\mathcal{R}}$ related to simple SUSY. The Wilson line effective
VEVs break the remaining SUSY, and hence also break $\rm{U}(1)_{\mathcal{R}}$
completely. In usual SUSY models a discrete symmetry
$\mathbb{Z}_{2}^{\mathcal{R}}\subset\rm{U}(1)_{\mathcal{R}}$, denoted as
R-parity, is typically preserved, which does not happen in the current model.
The R-parity violating (RPV) terms
$\sim{\color[rgb]{0,0.6,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.6,0}\pgfsys@color@cmyk@stroke{0.64}{0}{0.95}{0.40}\pgfsys@color@cmyk@fill{0.64}{0}{0.95}{0.40}\tilde{H}_{ua}}{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}L_{b}}$
can be rotated away by an unitary transformation which then generates terms
like $\sim y^{ijk}_{\rm
RPV}{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}Q_{i}L_{j}d^{c}_{k}}$,
whose strongest constraint reads as $y^{111}_{\rm RPV}<0.001$ Dreiner:1997uz ;
Barbier:2004ez . As this is proportional to the first-family neutrino Yukawa
coupling, it is expected to be well below this constraint. There will be
additional R-parity violating terms coming from
$\sim{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\textbf{Q}_{R}}\braket{{\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color@cmyk@fill{0.91}{0}{0.88}{0.12}\Delta_{R}}}{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\textbf{Q}_{R}}{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\textbf{Q}_{R}}+{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\textbf{Q}_{L}}\braket{{\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color@cmyk@fill{0.91}{0}{0.88}{0.12}\Delta_{L}}}{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\textbf{Q}_{L}}{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\textbf{Q}_{L}}$,
through
${\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}d}-D$ and
${\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}d^{c}}-D^{c}$
mixing. They will be smaller than the smallest quark Yukawa coupling which is
enough to avoid the experimental constraints Dreiner:1997uz ; Barbier:2004ez .
The terms that may generate proton decay will be studied in Sec. III.

###  II.2 Lepton-gaugino mixing: gaugino and Majorana masses

The effective VEV
$\braket{{\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color<EMAIL_ADDRESS>generates a mixing between chiral fermions and gauginos through the second
term in Eq. 26. Provided that it is singlet under
$\mathrm{SU}(3)_{\mathrm{C}}$, neither quarks nor gluons participate in such
interaction terms as they are all of the form
$\sim\braket{{\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color@cmyk@fill{0.91}{0}{0.88}{0.12}\textbf{L}}^{\dagger}}{\color[rgb]{1,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,1}\pgfsys@color@cmyk@stroke{0}{1}{0}{0}\pgfsys@color@cmyk@fill{0}{1}{0}{0}\lambda_{A}}t_{A}{\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color<EMAIL_ADDRESS>The gauginos are then named as
${\color[rgb]{1,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,1}\pgfsys@color@cmyk@stroke{0}{1}{0}{0}\pgfsys@color@cmyk@fill{0}{1}{0}{0}\lambda^{L,R,F}_{\tilde{a}}}$,
with $\tilde{a}=1,...,8$ and the Gell-Mann matrices are denoted as
$t_{\tilde{a}i}^{\ \ j}$. Thus, the leading terms that mix leptons and
gauginos can be cast as

$\begin{split}\mathcal{L}_{\psi\lambda}\sim&\
{\color[rgb]{1,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,1}\pgfsys@color@cmyk@stroke{0}{1}{0}{0}\pgfsys@color@cmyk@fill{0}{1}{0}{0}\lambda^{F}_{3,8}}\big{[}{\color[rgb]{.5,0,.5}\definecolor[named]{pgfstrokecolor}{rgb}{.5,0,.5}\nu^{c}_{j}}t_{(3,8)i}^{\
\
j}\braket{{\color[rgb]{.5,0,.5}\definecolor[named]{pgfstrokecolor}{rgb}{.5,0,.5}\tilde{\nu}^{c\dagger
i}}}+{\color[rgb]{.75,.5,.25}\definecolor[named]{pgfstrokecolor}{rgb}{.75,.5,.25}\tilde{\varphi}_{j}}t_{(3,8)i}^{\
\
j}\braket{{\color[rgb]{.75,.5,.25}\definecolor[named]{pgfstrokecolor}{rgb}{.75,.5,.25}\varphi^{\dagger
i}}}\big{]}\\\
&+\big{[}{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}e^{c}_{i}}\braket{{\color[rgb]{.5,0,.5}\definecolor[named]{pgfstrokecolor}{rgb}{.5,0,.5}\tilde{\nu}^{c\dagger
i}}}({\color[rgb]{1,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,1}\pgfsys@color@cmyk@stroke{0}{1}{0}{0}\pgfsys@color@cmyk@fill{0}{1}{0}{0}\lambda^{R}_{1}}-i{\color[rgb]{1,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,1}\pgfsys@color@cmyk@stroke{0}{1}{0}{0}\pgfsys@color@cmyk@fill{0}{1}{0}{0}\lambda^{R}_{2}})+{\color[rgb]{.5,0,.5}\definecolor[named]{pgfstrokecolor}{rgb}{.5,0,.5}\nu^{c}_{i}}\braket{{\color[rgb]{.5,0,.5}\definecolor[named]{pgfstrokecolor}{rgb}{.5,0,.5}\tilde{\nu}^{c\dagger
i}}}({\color[rgb]{1,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,1}\pgfsys@color@cmyk@stroke{0}{1}{0}{0}\pgfsys@color@cmyk@fill{0}{1}{0}{0}\lambda^{R}_{3}}+{\color[rgb]{1,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,1}\pgfsys@color@cmyk@stroke{0}{1}{0}{0}\pgfsys@color@cmyk@fill{0}{1}{0}{0}\lambda^{R}_{8}}/\sqrt{3})-2{\color[rgb]{.75,.5,.25}\definecolor[named]{pgfstrokecolor}{rgb}{.75,.5,.25}\tilde{\varphi}_{i}}\braket{{\color[rgb]{.75,.5,.25}\definecolor[named]{pgfstrokecolor}{rgb}{.75,.5,.25}\varphi^{\dagger
i}}}{\color[rgb]{1,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,1}\pgfsys@color@cmyk@stroke{0}{1}{0}{0}\pgfsys@color@cmyk@fill{0}{1}{0}{0}\lambda^{R}_{8}}/\sqrt{3}\\\
&+{\color[rgb]{.75,.5,.25}\definecolor[named]{pgfstrokecolor}{rgb}{.75,.5,.25}\tilde{\varphi_{i}}}\braket{{\color[rgb]{.5,0,.5}\definecolor[named]{pgfstrokecolor}{rgb}{.5,0,.5}\tilde{\nu}^{c\dagger
i}}}({\color[rgb]{1,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,1}\pgfsys@color@cmyk@stroke{0}{1}{0}{0}\pgfsys@color@cmyk@fill{0}{1}{0}{0}\lambda^{R}_{4}}-i{\color[rgb]{1,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,1}\pgfsys@color@cmyk@stroke{0}{1}{0}{0}\pgfsys@color@cmyk@fill{0}{1}{0}{0}\lambda^{R}_{5}})+{\color[rgb]{.5,0,.5}\definecolor[named]{pgfstrokecolor}{rgb}{.5,0,.5}\nu^{c}_{i}}\braket{{\color[rgb]{.75,.5,.25}\definecolor[named]{pgfstrokecolor}{rgb}{.75,.5,.25}\varphi^{\dagger
i}}}({\color[rgb]{1,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,1}\pgfsys@color@cmyk@stroke{0}{1}{0}{0}\pgfsys@color@cmyk@fill{0}{1}{0}{0}\lambda^{R}_{4}}+i{\color[rgb]{1,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,1}\pgfsys@color@cmyk@stroke{0}{1}{0}{0}\pgfsys@color@cmyk@fill{0}{1}{0}{0}\lambda^{R}_{5}})+{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}e^{c}_{i}}\braket{{\color[rgb]{.75,.5,.25}\definecolor[named]{pgfstrokecolor}{rgb}{.75,.5,.25}\varphi^{\dagger
i}}}({\color[rgb]{1,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,1}\pgfsys@color@cmyk@stroke{0}{1}{0}{0}\pgfsys@color@cmyk@fill{0}{1}{0}{0}\lambda^{R}_{6}}+i{\color[rgb]{1,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,1}\pgfsys@color@cmyk@stroke{0}{1}{0}{0}\pgfsys@color@cmyk@fill{0}{1}{0}{0}\lambda^{R}_{7}})\big{]}\\\
&+\big{[}{\color[rgb]{0,0.6,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.6,0}\pgfsys@color@cmyk@stroke{0.64}{0}{0.95}{0.40}\pgfsys@color@cmyk@fill{0.64}{0}{0.95}{0.40}\tilde{H}_{di}}{\color[rgb]{1,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,1}\pgfsys@color@cmyk@stroke{0}{1}{0}{0}\pgfsys@color@cmyk@fill{0}{1}{0}{0}\lambda^{L}}\braket{{\color[rgb]{.5,0,.5}\definecolor[named]{pgfstrokecolor}{rgb}{.5,0,.5}\tilde{\nu}^{c\dagger
i}}}+{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}L_{i}}{\color[rgb]{1,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,1}\pgfsys@color@cmyk@stroke{0}{1}{0}{0}\pgfsys@color@cmyk@fill{0}{1}{0}{0}\lambda^{L}}\braket{{\color[rgb]{.75,.5,.25}\definecolor[named]{pgfstrokecolor}{rgb}{.75,.5,.25}\varphi^{\dagger
i}}}-2({\color[rgb]{.5,0,.5}\definecolor[named]{pgfstrokecolor}{rgb}{.5,0,.5}\nu^{c}_{i}}\braket{{\color[rgb]{.5,0,.5}\definecolor[named]{pgfstrokecolor}{rgb}{.5,0,.5}\tilde{\nu}^{c\dagger
i}}}+{\color[rgb]{.75,.5,.25}\definecolor[named]{pgfstrokecolor}{rgb}{.75,.5,.25}\tilde{\varphi}_{i}}\braket{{\color[rgb]{.75,.5,.25}\definecolor[named]{pgfstrokecolor}{rgb}{.75,.5,.25}\varphi^{\dagger
i}}}){\color[rgb]{1,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,1}\pgfsys@color@cmyk@stroke{0}{1}{0}{0}\pgfsys@color@cmyk@fill{0}{1}{0}{0}\lambda^{L}_{8}}/\sqrt{3}\big{]}\\\
&+{\rm h.c.}\,,\end{split}$ (30)

where
${\color[rgb]{1,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,1}\pgfsys@color@cmyk@stroke{0}{1}{0}{0}\pgfsys@color@cmyk@fill{0}{1}{0}{0}\lambda^{L}}$
is the $\mathrm{SU}(2)_{\mathrm{L}}$ doublet defined as
${\color[rgb]{1,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,1}\pgfsys@color@cmyk@stroke{0}{1}{0}{0}\pgfsys@color@cmyk@fill{0}{1}{0}{0}\lambda^{L}}=({\color[rgb]{1,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,1}\pgfsys@color@cmyk@stroke{0}{1}{0}{0}\pgfsys@color@cmyk@fill{0}{1}{0}{0}\lambda^{L}_{4}}-i{\color[rgb]{1,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,1}\pgfsys@color@cmyk@stroke{0}{1}{0}{0}\pgfsys@color@cmyk@fill{0}{1}{0}{0}\lambda^{L}_{5}},\
{\color[rgb]{1,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,1}\pgfsys@color@cmyk@stroke{0}{1}{0}{0}\pgfsys@color@cmyk@fill{0}{1}{0}{0}\lambda^{L}_{6}}-i{\color[rgb]{1,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,1}\pgfsys@color@cmyk@stroke{0}{1}{0}{0}\pgfsys@color@cmyk@fill{0}{1}{0}{0}\lambda^{L}_{7}})^{\top}$,
and the dimensionless couplings coming from the extra CGC and gauge couplings
have been omitted. This Lagrangian generates Majorana masses for gauginos,
flavinos and right-handed neutrinos. They are allowed as
$\braket{{\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color<EMAIL_ADDRESS>breaks the $U(1)_{\mathcal{R}}$ symmetry as discussed in Appendix C.

From Eq. 30 one can obtain the gaugino masses that correspond to the masses of
the heavy vector bosons associated with the broken gauge symmetry generators,
${\color[rgb]{1,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,1}\pgfsys@color@cmyk@stroke{0}{1}{0}{0}\pgfsys@color@cmyk@fill{0}{1}{0}{0}\lambda_{A}}M^{\dagger}_{A}M_{B}{\color[rgb]{1,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,1}\pgfsys@color@cmyk@stroke{0}{1}{0}{0}\pgfsys@color@cmyk@fill{0}{1}{0}{0}\lambda_{B}}={\color[rgb]{1,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,1}\pgfsys@color@cmyk@stroke{0}{1}{0}{0}\pgfsys@color@cmyk@fill{0}{1}{0}{0}\lambda_{A}\lambda_{B}}(\braket{{\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color@cmyk@fill{0.91}{0}{0.88}{0.12}\textbf{L}}^{\dagger}}t_{A}t_{B}\braket{{\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color@cmyk@fill{0.91}{0}{0.88}{0.12}\textbf{L}}})$,
with the $A,B$ indices defined as
$\lambda_{A}\equiv\lambda^{L,R,F}_{\tilde{a}}$. The vector boson and gaugino
masses for a given broken generator are proportional to
$\braket{{\color[rgb]{.75,.5,.25}\definecolor[named]{pgfstrokecolor}{rgb}{.75,.5,.25}\varphi^{\dagger
i}\varphi_{i}}},\braket{{\color[rgb]{.5,0,.5}\definecolor[named]{pgfstrokecolor}{rgb}{.5,0,.5}\tilde{\nu}^{c\dagger
i}}{\color[rgb]{.5,0,.5}\definecolor[named]{pgfstrokecolor}{rgb}{.5,0,.5}\tilde{\nu}^{c}_{i}}},\braket{{\color[rgb]{.75,.5,.25}\definecolor[named]{pgfstrokecolor}{rgb}{.75,.5,.25}{\varphi}^{\dagger
i}}{\color[rgb]{.5,0,.5}\definecolor[named]{pgfstrokecolor}{rgb}{.5,0,.5}\tilde{\nu}^{c}_{i}}},\braket{{\color[rgb]{.75,.5,.25}\definecolor[named]{pgfstrokecolor}{rgb}{.75,.5,.25}\varphi_{i}}}+\braket{{\color[rgb]{.5,0,.5}\definecolor[named]{pgfstrokecolor}{rgb}{.5,0,.5}\tilde{\nu}^{c}_{i}}}$,
which are, in turn, given at the compactification scale. This is discussed in
Appendix E and detailed in Eq. 115.

The leading SUSY breaking contributions to the non-diagonal gaugino mass terms
from $M^{\dagger}_{A}M_{B}$ are of the order
$\sim\braket{\mathcal{D}_{A}\mathcal{D}_{B}}$ and have the same mass structure
as the SUSY preserving one, so they may be ignored. This comes from the fact
that the same effective VEV that breaks the GUT symmetry is the one that
breaks SUSY such that they are aligned.

It can be seen from Eq. 30 that leptons and higgsinos mix with gauginos
providing a contribution to their masses. More importantly, the mixing
$\sim\braket{{\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color@cmyk@fill{0.91}{0}{0.88}{0.12}\textbf{L}}^{\dagger}}{\color[rgb]{1,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,1}\pgfsys@color@cmyk@stroke{0}{1}{0}{0}\pgfsys@color@cmyk@fill{0}{1}{0}{0}\lambda_{A}}t_{A}{\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color<EMAIL_ADDRESS>of leptons with neutral gauginos (with Majorana mass $M_{AB}$) induces
Majorana mass terms for leptons
$\sim{\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color@cmyk@fill{0.91}{0}{0.88}{0.12}\textbf{L}}(t_{A}\braket{{\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color@cmyk@fill{0.91}{0}{0.88}{0.12}\textbf{L}}^{\dagger}})^{T}M_{AB}^{-1}\braket{{\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color@cmyk@fill{0.91}{0}{0.88}{0.12}\textbf{L}}^{\dagger}}t_{B}{\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color<EMAIL_ADDRESS>which can be decomposed as

$\begin{split}\mathcal{L}_{mml}\sim&\frac{1}{\Lambda}{\color[rgb]{.5,0,.5}\definecolor[named]{pgfstrokecolor}{rgb}{.5,0,.5}\nu^{c}_{i}}{\color[rgb]{.5,0,.5}\definecolor[named]{pgfstrokecolor}{rgb}{.5,0,.5}\nu^{c}_{j}}\braket{{\color[rgb]{.5,0,.5}\definecolor[named]{pgfstrokecolor}{rgb}{.5,0,.5}\tilde{\nu}^{c\dagger
i}{\color[rgb]{.5,0,.5}\definecolor[named]{pgfstrokecolor}{rgb}{.5,0,.5}\tilde{\nu}^{c\dagger
j}}}}+\frac{1}{\Lambda}{\color[rgb]{.75,.5,.25}\definecolor[named]{pgfstrokecolor}{rgb}{.75,.5,.25}\tilde{\varphi}_{i}}{\color[rgb]{.75,.5,.25}\definecolor[named]{pgfstrokecolor}{rgb}{.75,.5,.25}\tilde{\varphi}_{j}}\braket{{\color[rgb]{.75,.5,.25}\definecolor[named]{pgfstrokecolor}{rgb}{.75,.5,.25}\tilde{\varphi}^{\dagger
i}{\color[rgb]{.75,.5,.25}\definecolor[named]{pgfstrokecolor}{rgb}{.75,.5,.25}\tilde{\varphi}^{\dagger
j}}}}+\frac{1}{\Lambda}{\color[rgb]{.5,0,.5}\definecolor[named]{pgfstrokecolor}{rgb}{.5,0,.5}\nu^{c}_{i}}{\color[rgb]{.75,.5,.25}\definecolor[named]{pgfstrokecolor}{rgb}{.75,.5,.25}\tilde{\varphi}_{j}}\braket{{\color[rgb]{.5,0,.5}\definecolor[named]{pgfstrokecolor}{rgb}{.5,0,.5}\tilde{\nu}^{c\dagger
i}{\color[rgb]{.75,.5,.25}\definecolor[named]{pgfstrokecolor}{rgb}{.75,.5,.25}\tilde{\varphi}^{\dagger
j}}}}\,,\end{split}$ (31)

where the extra CGC and gauge couplings have been omitted. Note that both
right-handed neutrinos and flavinos get large Majorana masses providing a
seesaw mechanism with three very light neutrinos and six heavy ones. However,
as
$\braket{{\color[rgb]{.5,0,.5}\definecolor[named]{pgfstrokecolor}{rgb}{.5,0,.5}\tilde{\nu}^{c}_{1}}}=0$,
then
${\color[rgb]{.5,0,.5}\definecolor[named]{pgfstrokecolor}{rgb}{.5,0,.5}{\nu}^{c}_{1}}$
remains massless and corresponds to the goldstino. Therefore, this model
offers five right-handed neutrinos with masses of the order of the
compactification scale while the remaining one is an EW-scale sterile
neutrino. The masses of the right-handed neutrinos are at the Wilson line
scales which, as will be seen below, they are sufficiently large to generate
small left-handed neutrino masses through the standard seesaw mechanism.

Finally, one can remark that the only fermions that obtain a mass through the
terms
$\sim{\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color@cmyk@fill{0.91}{0}{0.88}{0.12}\textbf{L}}\braket{{\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color@cmyk@fill{0.91}{0}{0.88}{0.12}\textbf{L}}^{\dagger}{\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color@cmyk@fill{0.91}{0}{0.88}{0.12}\textbf{L}}^{\dagger}}{\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color<EMAIL_ADDRESS>are the right-handed neutrinos and flavinos. There are no invariant mass terms
$\sim{\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color@cmyk@fill{0.91}{0}{0.88}{0.12}\textbf{L}}\braket{{\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color@cmyk@fill{0.91}{0}{0.88}{0.12}\textbf{L}}^{\dagger}{\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color@cmyk@fill{0.91}{0}{0.88}{0.12}\textbf{L}}}{\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color<EMAIL_ADDRESS>therefore the leptons do not receive a large mass from the gaugino mixing.

###  II.3 Unbroken gauginos

In the previous section, it has been discussed how the gauginos, corresponding
to broken gauge symmetry generators, get a mass. This mechanism leaves the
gluino, winos and binos massless with eigenvectors,

${\color[rgb]{1,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,1}\pgfsys@color@cmyk@stroke{0}{1}{0}{0}\pgfsys@color@cmyk@fill{0}{1}{0}{0}\tilde{g}_{\tilde{a}}}={\color[rgb]{1,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,1}\pgfsys@color@cmyk@stroke{0}{1}{0}{0}\pgfsys@color@cmyk@fill{0}{1}{0}{0}\lambda^{C}_{\tilde{a}}}\,,\
\ \
{\color[rgb]{1,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,1}\pgfsys@color@cmyk@stroke{0}{1}{0}{0}\pgfsys@color@cmyk@fill{0}{1}{0}{0}\tilde{W}_{1,2,3}}={\color[rgb]{1,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,1}\pgfsys@color@cmyk@stroke{0}{1}{0}{0}\pgfsys@color@cmyk@fill{0}{1}{0}{0}\lambda^{L}_{1,2,3}}\,,\
\ \
{\color[rgb]{1,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,1}\pgfsys@color@cmyk@stroke{0}{1}{0}{0}\pgfsys@color@cmyk@fill{0}{1}{0}{0}\tilde{B}}=(\sqrt{3}{\color[rgb]{1,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,1}\pgfsys@color@cmyk@stroke{0}{1}{0}{0}\pgfsys@color@cmyk@fill{0}{1}{0}{0}\lambda^{L}_{8}}-3{\color[rgb]{1,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,1}\pgfsys@color@cmyk@stroke{0}{1}{0}{0}\pgfsys@color@cmyk@fill{0}{1}{0}{0}\lambda^{R}_{3}}+\sqrt{3}{\color[rgb]{1,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,1}\pgfsys@color@cmyk@stroke{0}{1}{0}{0}\pgfsys@color@cmyk@fill{0}{1}{0}{0}\lambda^{R}_{8}})/\sqrt{15}\,.$
(32)

Figure 8: The two-loop diagrams for SUSY-breaking gaugino masses. The
$\braket{\mathcal{D}}$ does not have any SM charges so it must couple via a
loop diagram.

The unbroken gauginos, i.e. those corresponding to unbroken symmetry
generators, on the other hand, cannot acquire a tree-level SUSY breaking mass.
As $\braket{\mathcal{D}}$ preserves the SM, it does not communicate SUSY
breaking directly. Instead, such a communication has to take place through the
two loop diagrams shown in Fig. 8, providing the mass terms for the remaining
gauginos as

$\mathcal{L}_{mg}\sim\frac{{\color[rgb]{0.3,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.3,0,0}\pgfsys@color@cmyk@stroke{0}{0.83}{1}{0.70}\pgfsys@color@cmyk@fill{0}{0.83}{1}{0.70}\textbf{w}}^{5}}{\Lambda^{4}}\left({\color[rgb]{1,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,1}\pgfsys@color@cmyk@stroke{0}{1}{0}{0}\pgfsys@color@cmyk@fill{0}{1}{0}{0}\tilde{g}\tilde{g}}+{\color[rgb]{1,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,1}\pgfsys@color@cmyk@stroke{0}{1}{0}{0}\pgfsys@color@cmyk@fill{0}{1}{0}{0}\tilde{W}\tilde{W}}+{\color[rgb]{1,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,1}\pgfsys@color@cmyk@stroke{0}{1}{0}{0}\pgfsys@color@cmyk@fill{0}{1}{0}{0}\tilde{B}\tilde{B}}\right)\,,$
(33)

where, as usual, the effective couplings absorb the CGCs and loop factors,
which are different for each gaugino as distinct fields run through the loops.
This effectively defines the usual soft gaugino mass as

$m_{1/2}\sim
g^{4}l^{2}\frac{{\color[rgb]{0.3,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.3,0,0}\pgfsys@color@cmyk@stroke{0}{0.83}{1}{0.70}\pgfsys@color@cmyk@fill{0}{0.83}{1}{0.70}\textbf{w}}^{5}}{\Lambda^{4}}\,,$
(34)

where $l^{2}$ is the two-loop suppression factor, and $g$ is the corresponding
gauge coupling.

###  II.4 Scalar boson masses

The scalar mass terms come from the scalar potential which is obtained by
considering Eq. 101 and substituting the truncated series from Eq. 106 in

$\mathcal{L}_{S}=-\left.\frac{\partial^{2}\mathcal{K}}{\partial\phi_{j}^{\dagger}\partial\phi_{b}}\right|_{S}\left.\frac{\partial\mathcal{W}}{\partial\phi_{j}}\right|_{S}\left.\frac{\partial\mathcal{W}}{\partial\phi_{b}^{\dagger}}\right|_{S}-\frac{1}{2}\left[\left.\frac{\partial\mathcal{K}}{\partial\phi_{i}}\right|_{S}t_{A}S_{i}+\left.\frac{\partial\mathcal{K}}{\partial\phi_{i}^{\dagger}}\right|_{S}t_{A}S_{i}^{\dagger}\right]\left[\left.\frac{\partial\mathcal{K}}{\partial\phi_{j}}\right|_{S}t_{A}S_{j}+\left.\frac{\partial\mathcal{K}}{\partial\phi_{j}^{\dagger}}\right|_{S}t_{A}S_{j}^{\dagger}\right]\,.$
(35)

The effective VEVs
$\braket{{\color[rgb]{.5,0,.5}\definecolor[named]{pgfstrokecolor}{rgb}{.5,0,.5}\tilde{\nu}^{c}}},\braket{{\color[rgb]{.75,.5,.25}\definecolor[named]{pgfstrokecolor}{rgb}{.75,.5,.25}\varphi}}$
enter the potential a priori, as they come from the Wilson lines, i.e. the ED
profiles of the fields, and are not determined by its minimization. The SUSY
preserving mass terms, coming from the first term in Eq. 35, read as

$\begin{split}\mathcal{L}_{mss}\sim&(\delta^{a}_{i}\delta^{b}_{j}-\delta^{a}_{j}\delta^{b}_{i})\left\\{{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\tilde{u}^{c}_{a}\tilde{u}^{c\dagger
i}}\braket{{\color[rgb]{0,0.6,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.6,0}\pgfsys@color@cmyk@stroke{0.64}{0}{0.95}{0.40}\pgfsys@color<EMAIL_ADDRESS>j}}}+({\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\tilde{Q}_{a}}\braket{{\color[rgb]{0,0.6,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.6,0}\pgfsys@color@cmyk@stroke{0.64}{0}{0.95}{0.40}\pgfsys@color@cmyk@fill{0.64}{0}{0.95}{0.40}H_{ub}}})({\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\tilde{Q}^{\dagger
i}}\braket{{\color[rgb]{0,0.6,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.6,0}\pgfsys@color@cmyk@stroke{0.64}{0}{0.95}{0.40}\pgfsys@color<EMAIL_ADDRESS>j}}})\right.\\\
&+{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\tilde{d}^{c}_{a}\tilde{d}^{c\dagger
i}}(\braket{{\color[rgb]{0,0.6,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.6,0}\pgfsys@color@cmyk@stroke{0.64}{0}{0.95}{0.40}\pgfsys@color<EMAIL_ADDRESS>j}}}+\braket{{\color[rgb]{.5,0,.5}\definecolor[named]{pgfstrokecolor}{rgb}{.5,0,.5}\tilde{\nu}^{c}_{b}\tilde{\nu}^{c\dagger
j}}})+({\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\tilde{Q}_{a}}\braket{{\color[rgb]{0,0.6,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.6,0}\pgfsys@color@cmyk@stroke{0.64}{0}{0.95}{0.40}\pgfsys@color@cmyk@fill{0.64}{0}{0.95}{0.40}H_{db}}})({\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\tilde{Q}^{\dagger
i}}\braket{{\color[rgb]{0,0.6,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.6,0}\pgfsys@color@cmyk@stroke{0.64}{0}{0.95}{0.40}\pgfsys@color<EMAIL_ADDRESS>j}}})\\\ &+\tilde{D}^{c}_{a}\tilde{D}^{c\dagger
i}\braket{{\color[rgb]{.75,.5,.25}\definecolor[named]{pgfstrokecolor}{rgb}{.75,.5,.25}\varphi_{b}\varphi^{\dagger
j}}}+\tilde{D}_{a}\tilde{D}^{\dagger
i}(\braket{{\color[rgb]{.75,.5,.25}\definecolor[named]{pgfstrokecolor}{rgb}{.75,.5,.25}\varphi_{b}\varphi^{\dagger
j}}}+\braket{{\color[rgb]{.5,0,.5}\definecolor[named]{pgfstrokecolor}{rgb}{.5,0,.5}\tilde{\nu}^{c}_{b}\tilde{\nu}^{c\dagger
j}}})\\\
&+{\color[rgb]{.5,0,.5}\definecolor[named]{pgfstrokecolor}{rgb}{.5,0,.5}\tilde{\nu}^{c}_{a}\tilde{\nu}^{c\dagger
i}}\braket{{\color[rgb]{0,0.6,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.6,0}\pgfsys@color@cmyk@stroke{0.64}{0}{0.95}{0.40}\pgfsys@color<EMAIL_ADDRESS>j}_{u}}}+{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\tilde{e}^{c}_{a}\tilde{e}^{c\dagger
i}}\braket{{\color[rgb]{0,0.6,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.6,0}\pgfsys@color@cmyk@stroke{0.64}{0}{0.95}{0.40}\pgfsys@color<EMAIL_ADDRESS>j}_{d}}}+{\color[rgb]{.75,.5,.25}\definecolor[named]{pgfstrokecolor}{rgb}{.75,.5,.25}\varphi_{a}\varphi^{\dagger
i}}(\braket{{\color[rgb]{0,0.6,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.6,0}\pgfsys@color@cmyk@stroke{0.64}{0}{0.95}{0.40}\pgfsys@color<EMAIL_ADDRESS>j}_{u}}}+\braket{{\color[rgb]{0,0.6,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.6,0}\pgfsys@color@cmyk@stroke{0.64}{0}{0.95}{0.40}\pgfsys@color<EMAIL_ADDRESS>j}_{d}}})\\\
&+({\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\tilde{L}_{a}}\braket{{\color[rgb]{0,0.6,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.6,0}\pgfsys@color@cmyk@stroke{0.64}{0}{0.95}{0.40}\pgfsys@color<EMAIL_ADDRESS>j}}})({\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\tilde{L}^{\dagger
b}}\braket{{\color[rgb]{0,0.6,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.6,0}\pgfsys@color@cmyk@stroke{0.64}{0}{0.95}{0.40}\pgfsys@color@cmyk@fill{0.64}{0}{0.95}{0.40}H_{di}}})+({\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\tilde{L}_{a}}\braket{{\color[rgb]{0,0.6,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.6,0}\pgfsys@color@cmyk@stroke{0.64}{0}{0.95}{0.40}\pgfsys@color<EMAIL_ADDRESS>j}}})({\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\tilde{L}^{\dagger
b}}\braket{{\color[rgb]{0,0.6,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.6,0}\pgfsys@color@cmyk@stroke{0.64}{0}{0.95}{0.40}\pgfsys@color@cmyk@fill{0.64}{0}{0.95}{0.40}H_{ui}}})\\\
&+{\color[rgb]{0,0.6,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.6,0}\pgfsys@color@cmyk@stroke{0.64}{0}{0.95}{0.40}\pgfsys@color<EMAIL_ADDRESS>i}}(\braket{{\color[rgb]{.75,.5,.25}\definecolor[named]{pgfstrokecolor}{rgb}{.75,.5,.25}\varphi_{b}\varphi^{\dagger
j}}}+\braket{{\color[rgb]{.5,0,.5}\definecolor[named]{pgfstrokecolor}{rgb}{.5,0,.5}\tilde{\nu}^{c}_{b}\tilde{\nu}^{c\dagger
j}}})+{\color[rgb]{0,0.6,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.6,0}\pgfsys@color@cmyk@stroke{0.64}{0}{0.95}{0.40}\pgfsys@color<EMAIL_ADDRESS>i}}\braket{{\color[rgb]{.75,.5,.25}\definecolor[named]{pgfstrokecolor}{rgb}{.75,.5,.25}\varphi_{b}\varphi^{\dagger
j}}}\\\
&\left.+({\color[rgb]{0,0.6,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.6,0}\pgfsys@color@cmyk@stroke{0.64}{0}{0.95}{0.40}\pgfsys@color@cmyk@fill{0.64}{0}{0.95}{0.40}H_{ua}}\braket{{\color[rgb]{0,0.6,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.6,0}\pgfsys@color@cmyk@stroke{0.64}{0}{0.95}{0.40}\pgfsys@color@cmyk@fill{0.64}{0}{0.95}{0.40}H_{db}}})({\color[rgb]{0,0.6,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.6,0}\pgfsys@color@cmyk@stroke{0.64}{0}{0.95}{0.40}\pgfsys@color<EMAIL_ADDRESS>i}}\braket{{\color[rgb]{0,0.6,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.6,0}\pgfsys@color@cmyk@stroke{0.64}{0}{0.95}{0.40}\pgfsys@color<EMAIL_ADDRESS>j}}})+({\color[rgb]{0,0.6,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.6,0}\pgfsys@color@cmyk@stroke{0.64}{0}{0.95}{0.40}\pgfsys@color@cmyk@fill{0.64}{0}{0.95}{0.40}H_{da}}\braket{{\color[rgb]{0,0.6,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.6,0}\pgfsys@color@cmyk@stroke{0.64}{0}{0.95}{0.40}\pgfsys@color@cmyk@fill{0.64}{0}{0.95}{0.40}H_{ub}}})({\color[rgb]{0,0.6,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.6,0}\pgfsys@color@cmyk@stroke{0.64}{0}{0.95}{0.40}\pgfsys@color<EMAIL_ADDRESS>i}}\braket{{\color[rgb]{0,0.6,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.6,0}\pgfsys@color@cmyk@stroke{0.64}{0}{0.95}{0.40}\pgfsys@color<EMAIL_ADDRESS>j}}})\right\\}\,.\end{split}$ (36)

The second term in Eq. 35, on the other hand, contains the SUSY breaking
scalar masses through the $\braket{\mathcal{D}}$-term contribution which, in
variance to the gaugino sector, can couple to every scalar at leading order,

$\begin{split}\mathcal{L}_{msd}\sim&\
\Big{(}{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\tilde{Q}_{a}\tilde{Q}^{\dagger
i}}+{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\tilde{u}^{c}_{a}\tilde{u}^{c\dagger
i}}+{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\tilde{e}^{c}_{a}\tilde{e}^{c\dagger
i}}\Big{)}{\color[rgb]{0.3,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.3,0,0}\pgfsys@color@cmyk@stroke{0}{0.83}{1}{0.70}\pgfsys@color@cmyk@fill{0}{0.83}{1}{0.70}(\textbf{w}^{2}_{D})^{a}_{\
i}}+\Big{(}{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\tilde{d}^{c}_{a}\tilde{d}^{c\dagger
i}}+{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\tilde{L}_{a}\tilde{L}^{\dagger
i}}\Big{)}\Big{[}{\color[rgb]{0.3,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.3,0,0}\pgfsys@color@cmyk@stroke{0}{0.83}{1}{0.70}\pgfsys@color@cmyk@fill{0}{0.83}{1}{0.70}(\textbf{w}^{2}_{D})^{a}_{\
i}}-\delta^{a}_{i}\braket{{\color[rgb]{.5,0,.5}\definecolor[named]{pgfstrokecolor}{rgb}{.5,0,.5}\tilde{\nu}^{c\dagger
j}\tilde{\nu}^{c}_{j}}}\Big{]}\\\ &+\Big{(}\tilde{D}_{a}\tilde{D}^{\dagger
i}+{\color[rgb]{0,0.6,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.6,0}\pgfsys@color@cmyk@stroke{0.64}{0}{0.95}{0.40}\pgfsys@color<EMAIL_ADDRESS>i}}\Big{)}\Big{[}{\color[rgb]{0.3,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.3,0,0}\pgfsys@color@cmyk@stroke{0}{0.83}{1}{0.70}\pgfsys@color@cmyk@fill{0}{0.83}{1}{0.70}(\textbf{w}^{2}_{D})^{a}_{\
i}}-\delta^{a}_{i}{\color[rgb]{0.3,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.3,0,0}\pgfsys@color@cmyk@stroke{0}{0.83}{1}{0.70}\pgfsys@color@cmyk@fill{0}{0.83}{1}{0.70}\textbf{w}^{2}}\Big{]}+\Big{(}\tilde{D}_{a}^{c}\tilde{D}^{c\dagger
i}+{\color[rgb]{0,0.6,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.6,0}\pgfsys@color@cmyk@stroke{0.64}{0}{0.95}{0.40}\pgfsys@color<EMAIL_ADDRESS>i}}\Big{)}\Big{[}{\color[rgb]{0.3,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.3,0,0}\pgfsys@color@cmyk@stroke{0}{0.83}{1}{0.70}\pgfsys@color@cmyk@fill{0}{0.83}{1}{0.70}(\textbf{w}^{2}_{D})^{a}_{\
i}}-\delta^{a}_{i}\braket{{\color[rgb]{.75,.5,.25}\definecolor[named]{pgfstrokecolor}{rgb}{.75,.5,.25}\varphi^{\dagger
j}\varphi_{j}}}\Big{]}\\\
&+{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\tilde{d}^{c}_{a}}\tilde{D}^{c\dagger
a}\braket{{\color[rgb]{.5,0,.5}\definecolor[named]{pgfstrokecolor}{rgb}{.5,0,.5}\tilde{\nu}^{c\dagger
j}}{\color[rgb]{.75,.5,.25}\definecolor[named]{pgfstrokecolor}{rgb}{.75,.5,.25}\varphi_{j}}}+\tilde{D}^{c}_{a}{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\tilde{d}^{c\dagger
a}}\braket{{\color[rgb]{.75,.5,.25}\definecolor[named]{pgfstrokecolor}{rgb}{.75,.5,.25}\varphi^{\dagger
j}}{\color[rgb]{.5,0,.5}\definecolor[named]{pgfstrokecolor}{rgb}{.5,0,.5}\tilde{\nu}^{c}_{j}}}+{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\tilde{L}_{a}}{\color[rgb]{0,0.6,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.6,0}\pgfsys@color@cmyk@stroke{0.64}{0}{0.95}{0.40}\pgfsys@color<EMAIL_ADDRESS>a}_{d}}\braket{{\color[rgb]{.5,0,.5}\definecolor[named]{pgfstrokecolor}{rgb}{.5,0,.5}\tilde{\nu}^{c\dagger
j}}{\color[rgb]{.75,.5,.25}\definecolor[named]{pgfstrokecolor}{rgb}{.75,.5,.25}\varphi_{j}}}+{\color[rgb]{0,0.6,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.6,0}\pgfsys@color@cmyk@stroke{0.64}{0}{0.95}{0.40}\pgfsys@color@cmyk@fill{0.64}{0}{0.95}{0.40}H_{da}}{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\tilde{L}^{\dagger
a}}\braket{{\color[rgb]{.75,.5,.25}\definecolor[named]{pgfstrokecolor}{rgb}{.75,.5,.25}\varphi^{\dagger
j}}{\color[rgb]{.5,0,.5}\definecolor[named]{pgfstrokecolor}{rgb}{.5,0,.5}\tilde{\nu}^{c}_{j}}}\\\
&+{\color[rgb]{.5,0,.5}\definecolor[named]{pgfstrokecolor}{rgb}{.5,0,.5}\tilde{\nu}^{c}_{a}\tilde{\nu}^{c\dagger
i}}\Big{[}{\color[rgb]{0.3,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.3,0,0}\pgfsys@color@cmyk@stroke{0}{0.83}{1}{0.70}\pgfsys@color@cmyk@fill{0}{0.83}{1}{0.70}(\textbf{w}^{2}_{D})^{a}_{\
i}}+\delta^{a}_{i}\braket{{\color[rgb]{.5,0,.5}\definecolor[named]{pgfstrokecolor}{rgb}{.5,0,.5}\tilde{\nu}^{c\dagger
j}\tilde{\nu}^{c}_{j}}}\Big{]}+{\color[rgb]{.75,.5,.25}\definecolor[named]{pgfstrokecolor}{rgb}{.75,.5,.25}\varphi_{a}\varphi^{\dagger
i}}\Big{[}{\color[rgb]{0.3,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.3,0,0}\pgfsys@color@cmyk@stroke{0}{0.83}{1}{0.70}\pgfsys@color@cmyk@fill{0}{0.83}{1}{0.70}(\textbf{w}^{2}_{D})^{a}_{\
i}}+\delta^{a}_{i}\braket{{\color[rgb]{.75,.5,.25}\definecolor[named]{pgfstrokecolor}{rgb}{.75,.5,.25}\varphi^{\dagger
j}\varphi_{j}}}\Big{]}\\\
&+{\color[rgb]{.5,0,.5}\definecolor[named]{pgfstrokecolor}{rgb}{.5,0,.5}\tilde{\nu}^{c}_{a}}{\color[rgb]{.75,.5,.25}\definecolor[named]{pgfstrokecolor}{rgb}{.75,.5,.25}\varphi^{\dagger
a}}\braket{{\color[rgb]{.5,0,.5}\definecolor[named]{pgfstrokecolor}{rgb}{.5,0,.5}\tilde{\nu}^{c\dagger
j}}{\color[rgb]{.75,.5,.25}\definecolor[named]{pgfstrokecolor}{rgb}{.75,.5,.25}\varphi_{j}}}+{\color[rgb]{.75,.5,.25}\definecolor[named]{pgfstrokecolor}{rgb}{.75,.5,.25}\varphi_{a}}{\color[rgb]{.5,0,.5}\definecolor[named]{pgfstrokecolor}{rgb}{.5,0,.5}\tilde{\nu}^{c\dagger