# Asymptotic Ultraviolet-safe Unification of Gauge and Yukawa Couplings:
The exceptional case

Giacomo Cacciapaglia<EMAIL_ADDRESS>Université de Lyon,
Université Claude Bernard Lyon 1, CNRS/IN2P3, IP2I Lyon, UMR 5822,
Villeurbanne F-69622, France  Aldo Deandrea<EMAIL_ADDRESS>Université
de Lyon, Université Claude Bernard Lyon 1, CNRS/IN2P3, IP2I Lyon, UMR 5822,
Villeurbanne F-69622, France  Department of Physics, University of
Johannesburg, PO Box 524, Auckland Park 2006, South Africa  Roman Pasechnik
<EMAIL_ADDRESS>Department of Physics, Lund University, Sölvegatan
14A, SE-223 62 Lund, Sweden  Zhi-Wei Wang<EMAIL_ADDRESS>School of
Physics, University of Electronic Science and Technology of China, 88 Tian-run
Road, Chengdu, China

###### Abstract

The ultimate dream of unification models consists in combining both gauge and
Yukawa couplings into one unified coupling. This is achieved by using a
supersymmetric exceptional ${\rm E}_{6}$ gauge symmetry together with
asymptotic unification in compact five-dimensional space-time. The ultraviolet
fixed point requires _exactly three_ fermion generations: one in the bulk, and
the two light ones localised on the ${\rm SO}(10)$ boundary in order to cancel
gauge anomalies. A second option allows to preserve baryon number and to lower
the compactification scale down to the typical scales of the intermediate
Pati-Salam gauge theory.

Ultraviolet fixed point; ${\rm E}_{6}$ grand unification; Pati-Salam theory

Coupling constants quantify the strength of interactions among particles and
evolve with the energy of the collisions via the renormalisation group [1, 2].
The term unification usually indicates the crossing of the gauge couplings at
a specific energy scale, assuming that a new larger gauge symmetry emerges
after that point. This mechanism yields a unified description of the
electroweak and strong interactions as a single force, within the so-called
Grand Unified Theories (GUTs) [3, 4, 5]. GUTs feature the assembling of matter
as well as gauge degrees of freedom within multiplets in a rather elegant and
dangerous way. In fact, gauge unification typically implies relations among
the Standard Model (SM) couplings and the presence of new gauge states, often
bringing embarrassing predictions, such as fast proton decay [6]. While
countermeasures to these shortcomings are well-known [6, 7], they typically
tend to partially obscure the beauty and the simplicity of the original
models: the unification scale is pushed close to the Planck mass and fields in
large representations of the unified group are often a must [8]. The latter
leads to an unresolved issue: GUTs, especially based on supersymmetry, feature
Landau poles for the unified coupling evolution soon after unification, hence
maiming the validity of the theory.

The idea of asymptotic unification solves this problem [9], as it is based on
the requirement that the theory should flow to a non-trivial ultraviolet (UV)
fixed point at high energy, without the need of a specific unification scale.
This is possible in models with a single extra space dimension [10, 11], which
yields a power running of the gauge couplings [12, 13]. The extra dimension is
compactified on an interval, where the bulk could be flat or warped [14] as
the high-energy behaviour is the same. The model features a five-dimensional
unified gauge symmetry in the bulk, so that at high energies (small distances)
the unified behaviour emerges. The bulk gauge group is broken by boundary
conditions, and the SM fields are identified with the zero modes of the bulk
fields, implying a completely different arrangement of matter and gauge fields
within multiplets as compared to traditional unification.

Extra-dimensional GUTs were widely explored [15, 16, 17, 18, 19], where the
unification is due to a sub–leading logarithmic running of the gauge couplings
[20]. Asymptotic unification, instead, is driven by the contributions of
complete multiplets, not by incomplete ones as in the standard GUT. Hence, it
is the contribution of the bulk resonances that drives it [12, 21]. Asymptotic
unification is a more natural choice than the standard one in extra
dimensional models: in fact, at energies above the inverse radius of the
compact dimension, the theory approaches the extra-dimensional evolution,
showing that the electroweak and strong interactions have always been a single
force at small distances. They appear different at low energies only due to
the breaking effect of the compactification. A concrete model of asymptotic
Grand Unification (aGUT) was recently proposed, based on the minimal
$\text{SU}(5)$ gauge group [22, 23]111Note that alternative asymptotically
safe GUT models may be realised [48, 49, 50] via large $N_{f}$ methods [51,
52].. Interestingly, the running of the Yukawa couplings imposes critical
conditions on these models, for instance ruling out the traditional
$\text{SO}(10)$ symmetry [25].

A more ambitious programme would require the unification of all the couplings
of the SM, namely gauge and Yukawa couplings, at least for one generation. In
extra dimensions, this has been explored in the context of gauge-Higgs
unification models [26]. Within the asymptotic unification paradigm, this can
be achieved if the UV fixed points for all the couplings coincide.
Supersymmetry comes in handy, as it naturally relates couplings of bosons
(gauge bosons) to those of fermions (gauginos). Within this framework, the Lie
group ${\rm E}_{6}$ [27, 28] offers an exceptional opportunity, as the SM
fermions can be embedded within the adjoint representation, hence stemming
from ${\rm E}_{6}$ gauginos [29]. As the Yukawa couplings are generated from
the ${\rm E}_{6}$ gauge interactions, they share the same fixed point in the
UV.

The model, therefore, consists of a supersymmetric ${\rm E}_{6}$ gauge theory
in five dimensions (5D) with matter fields in the fundamental ${\bf 27}$. The
fifth dimension is compactified on the orbifold
$S^{1}/\mathbb{Z}_{2}\times\mathbb{Z}_{2}^{\prime}$. From the four-dimensional
(4D) point of view, this theory has $\mathcal{N}=2$ supersymmetry [30], hence
the gauge and matter superfields consist of

Gauge : $\displaystyle\Rightarrow$ $\displaystyle
W^{\alpha}_{78}+\Phi_{78}\,,$ (1) Matter : $\displaystyle\Rightarrow$
$\displaystyle\Phi_{27}+\Phi^{c}_{27}\,,$ (2)

where $W^{\alpha}$ is the vector superfield, while $\Phi$ are the chiral
superfields, with $\Phi^{c}_{27}$ having conjugate quantum numbers as compared
to $\Phi_{27}$. The model Lagrangian can be constructed in terms of these
components [30, 31, 32]. The orbifold projection, encoded in the two parities
$\mathbb{Z}_{2}$ and $\mathbb{Z}_{2}^{\prime}$ centred on the two endpoints of
the interval, break both the $\mathcal{N}=2$ supersymmetry to $\mathcal{N}=1$
in 4D and the gauge symmetry. For the latter, there are three possible
patterns [33]:

A: $\displaystyle\text{E}_{6}\to\text{SO}(10)\times\text{U}(1)_{\psi}\,,$ B:
$\displaystyle\text{E}_{6}\to\text{SU}(6)_{\text{L}}\times\text{SU}(2)_{\text{R}}\,,$
(3) C:
$\displaystyle\text{E}_{6}\to\text{SU}(6)_{\text{R}}\times\text{SU}(2)_{\text{L}}\,,$

where the subscript “L” and “R” refer to the custodial symmetry in the SM.
Choosing any pair of the above patterns for the two orbifold projections leads
to a 4D gauge theory based on Pati-Salam (PS) [34] times an additional abelian
$\text{U}(1)$ symmetry:
$\text{SU}(4)\times\text{SU}(2)_{\text{L}}\times\text{SU}(2)_{\text{R}}\times\text{U}(1)_{\psi}$.
However, only one choice is phenomenologically viable. For the combination
B-C, it is not possible to obtain chiral zero modes for the SM fermions, while
for A-C the zero mode spectrum does not allow for the breaking of the PS
symmetry. Hence, the only viable model is based on A-B as illustrated in Fig.
1. The field content is summarised in Fig. 2, where we highlight the
decomposition with respect to the various subgroups of ${\rm E}_{6}$, and we
include two matter fields in the fundamental representation, ${\bf 27}$ and
${\bf 27^{\prime}}$, with different orbifold parities. As we will see below
this is the minimal and unique embedding of one SM generation in the bulk.

Figure 1:  Illustration of the ${\rm E}_{6}$ aGUT with 5D ${\rm E}_{6}$
hypermultiplets in the bulk and ${\rm SO}(10)$ $\mathcal{N}=1$ supermultiplets
on the $x^{5}=0$ boundary, as required by gauge anomaly cancellation. Figure
2:  Illustration of the three bulk fields in terms of
PS$\times\text{U}_{\psi}$ components of the ${\rm E}_{6}$ representations. The
position on the grid indicates the quantum numbers under
$\text{SU}(2)_{\text{L}}\times\text{SU}(2)_{\text{R}}$, while the symbols
represent $\text{SU}(4)$ representations. The $\text{SO}(10)$ components are
linked by dashed lines and labelled. Finally, the colours indicate the
presence of a zero mode: red for the $W^{\alpha}_{78}$ component and blue for
the $\Phi_{78}$, green for the matter $\Phi$ and orange for the $\Phi^{c}$.
The black symbols correspond to components without a zero mode.

A similar set-up was first proposed as a string-inspired standard GUT [29].
Our proposal differs in two crucial features: a) the unification is driven
asymptotically by the UV fixed point; b) all non-SM zero mode fields receive
mass without the need for additional bulk fields. The second feature is
crucial to maintain the UV fixed point of the theory.

The UV fixed point emerges from the power law corrections to the
renormalisation group evolution of the effective 4D gauge coupling [12]. The
beta function can be expressed as an effective 5D ’t Hooft coupling [21],
defined in terms of the 4D coupling $\alpha=g^{2}/4\pi$ as

$\tilde{\alpha}(\mu)=\alpha(\mu)\ \frac{\mu}{m_{\text{KK}}}\,,$ (4)

where $\mu$ is the renormalisation scale and $m_{\text{KK}}$ is the mass of
the first Kaluza-Klein (KK) state. At one loop, the beta function for
$\tilde{\alpha}$ reads

$\frac{d\tilde{\alpha}}{d\ln\mu}=\tilde{\alpha}-\frac{b_{5}}{2\pi}\tilde{\alpha}^{2}\,,$
(5)

which is valid for $\mu\gg m_{\text{KK}}$ and has a UV zero at
$\tilde{\alpha}^{\ast}_{\text{UV}}=2\pi/b_{5}$ for $b_{5}>0$. For our model,
we find [35]

$b_{5}=\frac{\pi}{2}\left(C(G)-\sum_{i}T_{i}(R_{i})\right)=3\pi\,,$ (6)

where $C(G)=12$ and $T(27)=3$ for ${\rm E}_{6}$ 222This result stems from a
direct 5D loop computation. More commonly, $b_{5}$ is computed in terms of KK
states, leading to  $b_{5}=2\left(C(G)-\sum_{i}T(R_{i})\right)=12\,.$ The
difference is mainly due to the phase space integration in 4D versus 5D, and
it can be considered as an uncertainty on the numerical predictions.. Hence,
only one more fundamental can be added in the bulk before the UV fixed point
disappears. As a consequence, the model in Ref.[29] does not have such UV
fixed point. In our model, instead, the UV fixed point remains perturbative,
hence endorsing the model with enhanced calculability. We remark that this
behaviour remains the same for both flat and warped extra dimensions 333The
two geometries have the same short distance behaviour. Hence, the only
difference will be encoded in threshold effects near the compactification
scale..

From the 4D point of view, the theory is invariant under
PS$\times\text{U}(1)_{\psi}$, which we use to classify the relevant states. At
the zero mode level, besides the gauge superfields in $W^{\alpha}_{78}$, the
chiral superfields contain the following states:

$\displaystyle\Phi_{78}$ $\displaystyle\supset$ $\displaystyle(\bar{\bf
4},{\bf 1},{\bf 2})_{-3}+({\bf 4},{\bf 1},{\bf 2})_{3}\,,$ (7)
$\displaystyle\Phi_{27}$ $\displaystyle\supset$ $\displaystyle({\bf 4},{\bf
2},{\bf 1})_{1}\,,$ (8) $\displaystyle\Phi^{c}_{27}$ $\displaystyle\supset$
$\displaystyle({\bf 1},{\bf 2},{\bf 2})_{2}\,,$ (9)
$\displaystyle\Phi_{27^{\prime}}$ $\displaystyle\supset$
$\displaystyle(\bar{\bf 4},{\bf 1},{\bf 2})_{1}\,,$ (10)
$\displaystyle\Phi_{27^{\prime}}^{c}$ $\displaystyle\supset$
$\displaystyle({\bf 6},{\bf 1},{\bf 1})_{2}\,+\,({\bf 1},{\bf 1},{\bf
1})_{-4}\,,$ (11)

shown as coloured symbols in Fig. 2. We can already see that the ${\bf 27}$
contains the left-handed SM fermions $({\bf 4},{\bf 2},{\bf 1})_{1}$ and two
Higgs doublets in $({\bf 1},{\bf 2},{\bf 2})_{2}$. The PS Yukawa couplings
stem from the gauge interactions of the ${\bf 27}$ with the $\Phi_{78}$
component of the gauge supermultiplet [38, 39], leading to:

$g\ \Phi_{27}^{c}\Phi_{78}\Phi_{27}\supset\frac{g}{\sqrt{2}}({\bf 1},{\bf
2},{\bf 2})_{2}\ (\bar{\bf 4},{\bf 1},{\bf 2})_{-3}\ ({\bf 4},{\bf 2},{\bf
1})_{1}\,,$ (12)

thus the right-handed SM fermions must be the $(\bar{\bf 4},{\bf 1},{\bf
2})_{-3}$ component of the gauge $\Phi_{78}$. The gauge interactions of the
${\bf 27^{\prime}}$ contain the following zero-mode terms:

$g\
\Phi_{27^{\prime}}^{c}\Phi_{78}\Phi_{27^{\prime}}\supset-\frac{g}{\sqrt{2}}({\bf
1},{\bf 1},{\bf 1})_{-4}\ ({\bf 4},{\bf 1},{\bf 2})_{3}\ (\bar{\bf 4},{\bf
1},{\bf 2})_{1}\\\ +\frac{g}{\sqrt{2}}({\bf 6},{\bf 1},{\bf 1})_{2}\ (\bar{\bf
4},{\bf 1},{\bf 2})_{-3}\ (\bar{\bf 4},{\bf 1},{\bf 2})_{1}\,.$ (13)

The importance of the couplings above is related to the breaking of the
PS$\times\text{U}(1)_{\psi}$ gauge group down to the SM one. In fact, the
$({\bf 1},{\bf 1},{\bf 1})_{-4}$ from the $\bf 27^{\prime}$ can effectively
break $\text{U}(1)_{\psi}$ and also give a mass to the $({\bf 4},{\bf 1},{\bf
2})_{3}$ gauginos. Moreover, the presence of a $({\bf 4},{\bf 1},{\bf 2})_{3}$
state in the gauge multiplet allows for the breaking of the PS symmetry via
the Scherk-Schwarz mechanism [40]. The breaking of the gauge symmetry down to
the SM, therefore, does not require additional states in this theory. The zero
modes in the $({\bf 6},{\bf 1},{\bf 1})_{2}$ component, which behave like a
vector-like bottom quark singlet, can only receive a mass via a superpotential
localised on the $\text{SO}(10)$ boundary with parity A 444In terms of
$\text{SO}(10)\times\text{U}(1)_{\psi}$ components, the superpotential term
reads:
$\left.\Phi_{27^{\prime}}^{c}\right|_{10_{2}}\left.\Phi_{27^{\prime}}^{c}\right|_{10_{2}}\left.\Phi_{27^{\prime}}^{c}\right|_{1_{-4}}\,.$
.

At the UV fixed point, the matching between ${\rm E}_{6}$ and the
PS$\times\text{U}(1)_{\psi}$ couplings reads:

$g_{4}=g_{L}=g_{R}\equiv g\,,\quad g_{\psi}=\frac{g}{\sqrt{2}}\,.$ (14)

For the SM Yukawa couplings in Eq. (12), we have

$\quad y_{\uparrow}=y_{\downarrow}\equiv\frac{g}{\sqrt{2}}\,,$ (15)

where $y_{\uparrow}=y_{\text{t}}=y_{\nu_{\tau}}$ is the Yukawa of up-type
fermions, while $y_{\downarrow}=y_{\text{b}}=y_{\tau}$ for down-type ones. The
identification of up and down-type Yukawas occurs at the PS-breaking scale,
which is typically close to $m_{\text{KK}}$, while the relation between top
and bottom mass also depends on the ratio of Higgs vacuum expectation values,
as typical in supersymmetric models 555We recall that
$y_{\text{t}}=\frac{m_{\text{t}}}{v\cot\beta}\,,\quad
y_{\text{b},\tau}=\frac{m_{\text{b},\tau}}{v\sin\beta}\,,$ where $v=246$ GeV
is the Higgs vacuum expectation value in the SM.. This ratio, expressed in
terms of $\tan\beta$, requires

$\tan\beta=\frac{\langle H_{d}\rangle}{\langle
H_{u}\rangle}=\frac{m_{\text{t}}(m_{\text{KK}})}{m_{\text{b}}(m_{\text{KK}})}\sim
40\,,$ (16)

where the masses are evaluated at the KK scale. In Fig. 3 we show a schematic
plot of the renormalisation group evolution of the SM couplings in the ${\rm
E}_{6}$ model. For simplicity, we identify the PS and $\text{U}(1)_{\psi}$
breaking scales to $m_{\text{KK}}$, and fix the supersymmetry breaking scale
to $10$ TeV, above which the minimal supersymmetric SM (MSSM) is a good
description. The couplings correspond to the usual 4D ones up to the scale
$m_{\text{KK}}$, above which they are replaced by the corresponding 5D ’t
Hooft couplings, defined in Eq. (4). Also, we plot the couplings rescaled to
the ${\rm E}_{6}$ values, as in Eqs. (14) and (15), while the usual PS
matching is applied above $m_{\text{KK}}$. This plot clearly demonstrates that
the gauge and Yukawa couplings of the third generation do unify to a single
value thanks to the UV fixed point, independently on the value of
$m_{\text{KK}}$. However, due to the constrained bulk structure, the light
generations must be localised on one of the two boundaries. Before addressing
this issue, there are two related features of the bulk interactions: baryon
number conservation and the cancellation of 4D gauge anomalies.

Figure 3:  Schematic running of the SM gauge couplings and third generation
Yukawa couplings for two values of $m_{\text{KK}}$, where the Yukawa values
correspond to $\tan\beta=40$. The couplings are rescaled to match the ${\rm
E}_{6}$ unification: $\alpha_{x}=2\frac{y_{x}^{2}}{4\pi}$ for Yukawas and
$\alpha_{1}=\frac{5}{3}\alpha^{\prime}$ for hypercharge, where
$\alpha_{\text{R}}$ comes from the usual PS matching.

Regarding the former, we recall that the theory, at the level of the SM gauge
invariance, features five $\text{U}(1)$ symmetries. Besides the three gauged
ones, $\text{U}(1)_{\text{B-L}}\supset\text{SU}(4)$,
$\text{U}(1)_{\text{R}}\supset\text{SU}(2)_{\text{R}}$ and
$\text{U}(1)_{\psi}$, there are two global charges associated to the matter
fields, $\text{U}(1)_{27}$ and $\text{U}(1)_{27^{\prime}}$. Among the first
three, the SM hypercharge is defined as usual in PS models as
$2Q_{\text{Y}}=Q_{\text{B-L}}+Q_{\text{R}}$. There is a single global charge
that remains after the breaking of PS and $\text{U}(1)_{\psi}$, and that is
not carried by the Higgs fields:

$Q_{\text{B}}=\frac{1}{4}Q_{\text{B-L}}+\frac{1}{6}Q_{27}+\frac{1}{12}Q_{\psi}-\frac{1}{3}Q_{27^{\prime}}\,,$
(17)

where the global charges are normalised to unity. On the SM fields, this
charge matches baryon number. Being respected by all bulk interactions, it
protects the proton from decaying. As in the minimal $\text{SU}(5)$ aGUT model
[22], the components of the bulk fields with opposite parities on the two
boundaries, shown by the black symbols in Fig. 2, have unusual $Q_{\text{B}}$
assignments, hence the lightest state among them is stable. All zero modes
have standard baryon number assignments, including the non-SM states. A
detailed description of the components is presented in the supplementary
material.

Another feature of the bulk model is the presence of 4D anomalies at the level
of the gauge symmetry PS$\times\text{U}(1)_{\psi}$, which can only stem from
the matter fields as the ${\bf 78}$ is real. As shown in Fig. 2, however, the
zero modes of the ${\bf 27}$ and ${\bf 27^{\prime}}$ form effectively complete
representations of $\text{SO}(10)\supset\text{PS}$, namely
$16_{1}+10_{2}+1_{-4}$. Hence, the PS gauge symmetry is anomaly-free. For
instance, the $16_{1}$ is formed by the $({\bf 4},{\bf 2},{\bf 1})_{1}$ zero
mode in $\Phi_{27}$ and the $(\bar{\bf 4},{\bf 1},{\bf 2})_{1}$ in
$\Phi_{27^{\prime}}^{c}$. The 4D anomalies, therefore, only involve the
$\text{U}(1)_{\psi}$ charges, where the coefficients respect
$\mathcal{A}(16_{1})=\mathcal{A}(10_{2}+1_{-4})$. The $\text{U}(1)_{\psi}$
gauge anomalies can only be cancelled by adding fields on the $\text{SO}(10)$
boundary. This leads to two possible models:

  * i)

Two $16_{-1}$ multiplets, whose components match the two SM light generations.
Hence, the total number of SM generations is _predicted to be three_ by gauge
anomaly cancellation. However, baryon number is violated by the localised
gauge and Yukawa interactions.

  * ii)

Two $10_{-2}+1_{4}$, corresponding to massive states, while the light
generations are localised on the other boundary.

In the first case i), the two localised superfields $\Phi^{i}_{16_{-1}}$ allow
for the following couplings to the bulk field containing the Higgs doublets:

$y^{ij}\
\Phi^{i}_{16_{-1}}\Phi^{j}_{16_{-1}}\left.\Phi^{c}_{27}\right|_{10_{2}}\,.$
(18)

This term, and the $\text{SO}(10)$ gauge interactions of the localised fields,
violate baryon and lepton numbers, as in traditional GUT models, via couplings
with the first KK resonances. For flat extra dimensions, this would lead to a
direct bound $m_{\text{KK}}\gtrsim 10^{16}$ GeV [43]. We remark that warping
the extra space can lead to a mild suppression of these coupling [44], hence
it would be feasible to lower the KK scale by one or two orders of magnitude.
Furthermore, additional localised superfields can be added in order to explain
the different values of the Yukawa couplings, as done in minimal
$\text{SO}(10)$ GUTs [8]: the presence of large representations does not spoil
the UV fixed point as they only contribute to logarithmic running, which is
overpowered by the bulk power-law running. As an example, in Fig. 3 we show a
case with $m_{\text{KK}}=10^{15}$ GeV. The mixing between the third generation
and the light ones, however, is forbidden by the $\text{U}(1)_{\psi}$ gauge
symmetry, and it requires additional localised fields, see supplementary
material.

In the second case ii), one can add two copies of the following set of
superfields on the $\text{SU}(6)_{\text{L}}\times\text{SU}(2)_{\text{R}}$
boundary:

$\Phi^{j}_{(15,1)}\,,\;\Phi^{j}_{(\bar{6},2)}\,,\;\Phi^{j}_{(20,2)}\,,\;\;j=1,2\,.$
(19)

The above fields match the bulk field components containing the third
generation, hence the three generations share similar localised couplings. The
localised Yukawas are written in terms of the following $3\times 3$ matrices

$y^{ij}\
\Phi^{i}_{(15,1)}\Phi^{i}_{(20,2)}\left.\Phi_{27}^{c}\right|_{(6,2)}+\lambda^{ij}\
\Phi^{i}_{(\bar{6},2)}\Phi^{j}_{(20,2)}\left.\Phi_{27^{\prime}}^{c}\right|_{(\bar{15},1)}\,,$
(20)

where $i=3$ corresponds to the bulk fields and the second term gives mass to
the unwanted components. As the above couplings have the same structure of the
bulk ones, the baryon number in Eq. (17) remains conserved and the
$m_{\text{KK}}$ scale can be lowered compared to case i). In this model, it
can be as low as the lowest scale allowed by PS to be around $2000$ TeV, which
is obtained by using the current limit on
$\rm{Br}\left(K_{L}\rightarrow\mu^{\pm}e^{\mp}\right)<4.7\times
10^{-12}$[45](see also e.g. [46, 47, 48]).

In conclusion, in this letter we presented a new aGUT based on a
supersymmetric exceptional ${\rm E}_{6}$ gauge symmetry. Its uniqueness stands
in the presence of a single UV fixed point for gauge and Yukawa couplings of
the third generation. The number of SM generations is predicted by gauge
anomaly cancellation. We highlighted a second option, which preserves baryon
number and allows to lower the compactification scale down to a few thousand
TeV. This exceptional model has far reaching implications both for low energy
phenomenology, for instance in the flavour sector, and at high energies, via
new model building opportunities for UV completions.

Acknowledgements – We acknowledge useful suggestions on the ${\rm E}_{6}$
contractions from B. Bajć. R.P. is supported in part by the Swedish Research
Council grant, contract number 2016-05996, as well as by the European Research
Council (ERC) under the European Union’s Horizon 2020 research and innovation
programme (grant agreement No. 668679). Z.W.W. acknowledges “Hundred Talents
Program” supported by UESTC.

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##  Appendix A Supplementary material

###  A.1 5D supersymmetric Lagrangian

The gauge fields consist of one 5D vector $V^{M}$ ($M=1,\dots 5$), one 5D
spinor $(\lambda,\lambda^{\prime})$ and a real scalar $\sigma$, transforming
as the adjoint irreducible representation (irrep) of $\text{E}_{6}$. They can
be expressed in terms of the following $\mathcal{N}=1$ superfields:

$W^{\alpha}_{78}=\\{V^{\mu},\ \lambda\\}\,,\quad\Phi_{78}=\\{\Sigma,\
\lambda^{\prime}\\}\,,$ (21)

where $\Sigma=(\sigma+iV^{5})$. Here, $W^{\alpha}$ is a vector superfield,
while $\Phi$ is a chiral one. Together, they form a 5D hypermultiplet, which
resembles that of $\mathcal{N}=2$ supersymmetry in 4D. The 5D action can be
written as:

$S_{5D}=\frac{1}{4kg^{2}}\
\mbox{Tr}\;\left[\left(\frac{1}{4}W^{\alpha}W_{\alpha}\
\delta^{2}(\bar{\theta})+\mbox{h.c.}\right)+\left(e^{-2gV}\nabla_{y}e^{2gV}\right)^{2}\right]\,,$
(22)

where

$\displaystyle W_{\alpha}$ $\displaystyle=$
$\displaystyle-\frac{1}{4}\bar{D}^{2}e^{-2gV}D_{\alpha}e^{2gV}\,,$ (23)
$\displaystyle\nabla_{y}e^{2gV}$ $\displaystyle=$
$\displaystyle\partial_{y}e^{2gV}-g\bar{\Phi}e^{2gV}-e^{2gV}g\Phi\,.$ (24)

A 5D matter multiplet in the irrep $R$ consists of a Dirac spinor $\psi_{R}$
and two scalar states $\phi_{R1}$ and $\phi_{R2}$. Writing the Dirac spinor in
terms of 4D Weyl spinors as

$\psi_{R}=\begin{pmatrix}\chi_{R}\\\ \bar{\xi}_{R}\end{pmatrix}\,,$ (25)

the fields can be organised into two chiral superfields

$\Phi_{R}=\\{\phi_{R1},\
\chi_{R}\\}\,,\qquad\Phi_{R}^{c}=\\{\phi^{\dagger}_{R2},\ \xi_{R}\\}\,,$ (26)

where the second one transforms as conjugate irrep of $\text{E}_{6}$. In other
words, $\Phi$ contains the left-handed spinor, while $\Phi^{c}$ contains the
charge-conjugate right-handed one. The 5D action, including interactions with
the gauge fields, can be written as

$S_{5D}=\bar{\Phi}_{R}e^{2gV}\Phi_{R}+\bar{\Phi}_{R}^{c}e^{-2gV}\Phi_{R}^{c}+\left(\Phi_{R}^{c}(\nabla_{y}+M)\Phi\
\delta(\bar{\theta})+\mbox{h.c.}\right)\,.$ (27)

The third term above takes care of the 5D Lorentz invariance by introducing a
coupling between the two chiral supermultiplets. Also, the above Lagrangian
has a global U(1) symmetry under which $\Phi_{R}\to e^{iQ_{R}\theta}\
\Phi_{R}$ and $\Phi_{R}^{c}\to e^{-iQ_{R}\theta}\Phi_{R}^{c}$ .

###  A.2 $\text{E}_{6}$ breaking and orbifold boundary conditions

In general, the components of an $\text{E}_{6}$ multiplet will receive
different parity assignments. Let us consider a parity $P$ that breaks
$\text{E}_{6}\to\mathcal{H}$, so that an irrep $R$ of $\text{E}_{6}$
decomposes in components $\chi_{j}$. The parity assignments of each component
$\chi_{j}$ will read:

$P(\chi_{j})=\eta_{R}\ \mathcal{P}_{j}\,,$ (28)

where $\eta_{R}$ is an overall sign that can be chosen different for different
fields in the irrep $R$, while the relative parities $\mathcal{P}_{j}$ only
depend on the $\text{E}_{6}$ irrep (and are the same for all fields).

We consider the following breaking patterns:

$\displaystyle\text{A}:\;\text{E}_{6}$ $\displaystyle\to$
$\displaystyle\text{SO}(10)\times\text{U}(1)_{\psi}\,,$ (29)
$\displaystyle\text{B}:\;\text{E}_{6}$ $\displaystyle\to$
$\displaystyle\text{SU}(6)_{\text{L}}\times\text{SU}(2)_{\text{R}}\,,$ (30)
$\displaystyle\text{C}:\;\text{E}_{6}$ $\displaystyle\to$
$\displaystyle\text{SU}(6)_{\text{R}}\times\text{SU}(2)_{\text{L}}\,.$ (31)

The intersection of any pair of them leaves the Pati-Salam group unbroken,
$\text{SU}(4)\times\text{SU}(2)_{\text{L}}\times\text{SU}(2)_{\text{R}}$, plus
$\text{U}(1)_{\psi}$.

For the fundamental, ${\bf 27}$, we have the following decompositions:

$\displaystyle\begin{split}\text{E}_{6}\rightarrow\text{SO}(10)_{Q_{\psi}}&\rightarrow(\text{SU}(4)\times\text{SU}(2)_{\text{L}}\times\text{SU}(2)_{\text{R}})_{Q_{\psi}}\\\
(16)_{1}\,&=\,\left(4,2,1\right)_{1}\,+\,\left(\bar{4},1,2\right)_{1}\\\
(10)_{-2}\,&=\,\left(6,1,1\right)_{-2}\,+\,\left(1,2,2\right)_{-2}\\\
(1)_{4}\,&=\,\left(1,1,1\right)_{4}\,;\end{split}$ (32)
$\displaystyle\begin{split}\text{E}_{6}\rightarrow\text{SU}(6)_{\text{L}}\times\text{SU}(2)_{\text{R}}&\rightarrow(\text{SU}(4)\times\text{SU}(2)_{\text{L}}\times\text{SU}(2)_{\text{R}})_{Q_{\psi}}\\\
\left(\bar{6},2\right)\,&=\,\left(\bar{4},1,2\right)_{1}\,+\,\left(1,2,2\right)_{-2}\\\
\left(15,1\right)\,&=\,\left(4,2,1\right)_{1}\,+\,\left(6,1,1\right)_{-2}\,+\,\left(1,1,1\right)_{4}\,;\end{split}$
(33)
$\displaystyle\begin{split}\text{E}_{6}\rightarrow\text{SU}(6)_{\text{R}}\times\text{SU}(2)_{\text{L}}&\rightarrow(\text{SU}(4)\times\text{SU}(2)_{\text{L}}\times\text{SU}(2)_{\text{R}})_{Q_{\psi}}\\\
\left(6,2\right)\,&=\,\left(4,2,1\right)_{1}\,+\,\left(1,2,2\right)_{-2}\\\
\left(\bar{15},1\right)\,&=\,\left(\bar{4},1,2\right)_{1}\,+\,\left(6,1,1\right)_{-2}\,+\,\left(1,1,1\right)_{4}\,.\end{split}$
(34)

The internal parities $\mathcal{P}_{j}$ are listed in Table 1.

$\bf 27$ | $\text{SO}(10)\times\text{U}(1)_{\psi}$ | $\text{SU}(6)_{\text{L}}\times\text{SU}(2)_{\text{R}}$ | $\text{SU}(6)_{\text{R}}\times\text{SU}(2)_{\text{L}}$
---|---|---|---
$(4,2,1)_{1}$ | even | even | odd
$(\bar{4},1,2)_{1}$ | even | odd | even
$(6,1,1)_{-2}$ | odd | even | even
$(1,2,2)_{-2}$ | odd | odd | odd
$(1,1,1)_{4}$ | odd | even | even
Table 1:  Intrinsic parities for the fundamental $\bf 27$ of $\text{E}_{6}$.
The colours label PS fields in the same irrep of $\mathcal{H}$.

For the adjoint, $\bf 78$, we have the following decompositions:

$\displaystyle\begin{split}\text{E}_{6}\rightarrow\text{SO}(10)_{Q_{\psi}}&\rightarrow(\text{SU}(4)\times\text{SU}(2)_{\text{L}}\times\text{SU}(2)_{\text{R}})_{Q_{\psi}}\\\
(45)_{0}\,&=\,\left(15,1,1\right)_{0}\,+\,\left(1,3,1\right)_{0}\,+\,\left(1,1,3\right)_{0}\,+\,\left(6,2,2\right)_{0}\\\
(16)_{-3}\,&=\,\left(4,2,1\right)_{-3}\,+\,\left(\bar{4},1,2\right)_{-3}\\\
(\bar{16})_{3}\,&=\,\left(\bar{4},2,1\right)_{3}\,+\,\left(4,1,2\right)_{3}\\\
(1)_{0}\,&=\,\left(1,1,1\right)_{0}\,;\end{split}$ (35)
$\displaystyle\begin{split}\text{E}_{6}\rightarrow\text{SU}(6)_{\text{L}}\times\text{SU}(2)_{\text{R}}&\rightarrow(\text{SU}(4)\times\text{SU}(2)_{\text{L}}\times\text{SU}(2)_{\text{R}})_{Q_{\psi}}\\\
\left(35,1\right)\,&=\,\left(15,1,1\right)_{0}\,+\,\left(1,3,1\right)_{0}\,+\,\left(4,2,1\right)_{-3}\,+\,\left(\bar{4},2,1\right)_{3}\,+\,\left(1,1,1\right)_{0}\\\
\left(1,3\right)\,&=\,\left(1,1,3\right)_{0}\\\
\left(20,2\right)\,&=\,\left(6,2,2\right)_{0}\,+\,\left(\bar{4},1,2\right)_{-3}\,+\,\left(4,1,2\right)_{3}\,;\end{split}$
(36)
$\displaystyle\begin{split}\text{E}_{6}\rightarrow\text{SU}(6)_{\text{R}}\times\text{SU}(2)_{\text{L}}&\rightarrow(\text{SU}(4)\times\text{SU}(2)_{\text{L}}\times\text{SU}(2)_{\text{R}})_{Q_{\psi}}\\\
\left(35,1\right)\,&=\,\left(15,1,1\right)_{0}\,+\,\left(1,1,3\right)_{0}\,+\,\left(4,1,2\right)_{3}\,+\,\left(\bar{4},1,2\right)_{-3}\,+\,\left(1,1,1\right)_{0}\\\
\left(3,1\right)\,&=\,\left(1,3,1\right)_{0}\\\
\left(20,2\right)\,&=\,\left(6,2,2\right)_{0}\,+\,\left(\bar{4},2,1\right)_{3}\,+\,\left(4,2,1\right)_{-3}\,.\end{split}$
(37)

The internal parities $\mathcal{P}_{j}$ are listed in Table 2. Note that for
the gauge fields, the overall signs are fixed to be $\eta=+1$ for $W^{\alpha}$
and $\eta=-1$ for $\Phi$ for all parities. Hence, for the choice of parities
B–C, it is not possible to obtain zero modes in the fourplets of
$\text{SU}(4)$ in $\phi$, which could play the role of gaugino matter fields
in the SM. Instead, for the choice A–C, the gauginos contain the left-handed
$\text{SU}(2)_{\text{L}}$ doublets, hence they cannot be used for the breaking
of the PS gauge symmetry via the Scherk-Schwarz mechanism.

$\bf 78$ | $\text{SO}(10)\times\text{U}(1)_{\psi}$ | $\text{SU}(6)_{\text{L}}\times\text{SU}(2)_{\text{R}}$ | $\text{SU}(6)_{\text{R}}\times\text{SU}(2)_{\text{L}}$
---|---|---|---
$(15,1,1)_{0}$ | even | even | even
$(1,3,1)_{0}$ | even | even | even
$(1,1,3)_{0}$ | even | even | even
$(6,2,2)_{0}$ | even | odd | odd
$(4,2,1)_{-3}$ | odd | even | odd
$(\bar{4},1,2)_{-3}$ | odd | odd | even
$(\bar{4},2,1)_{3}$ | odd | even | odd
$(4,1,2)_{3}$ | odd | odd | even
$(1,1,1)_{0}$ | even | even | even
Table 2:  Intrinsic parities for the adjoint $\bf 78$ of $\text{E}_{6}$. The
colours label PS fields in the same irrep of $\mathcal{H}$.

A detailed list of the bulk field components with their assigned orbifold
parities can be found in Tables 3, 4 and 5, for the choice A–B, which is
presented in the main text.

###  A.3 Baryon number is conserved

At the level of the SM gauge symmetry, the model has 5 U(1) factors:
$\text{U}(1)_{\text{B-L}}$ in $\text{SU}(4)$, $\text{U}(1)_{\text{R}}$ in
$\text{SU}(2)_{\text{R}}$, $\text{U}(1)_{\psi}$, $\text{U}(1)_{27}$ and
$\text{U}(1)_{27^{\prime}}$. The two last ones are global, while the first 3
are gauged.

Two combinations are broken by the vacuum expectation values that break
PS$\times U(1)_{\psi}$ down to the SM:

$\displaystyle\langle({\bf 1},{\bf 1},{\bf
1})_{-4}\rangle\;\;\mbox{in}\;\;{\bf 27^{\prime}}$ $\displaystyle\Rightarrow$
$\displaystyle-4Q_{\psi}-Q_{27^{\prime}}=0$ (38) $\displaystyle\langle({\bf
4},{\bf 1},{\bf 2})_{3}\rangle\;\;\mbox{in}\;\;{\bf 78}$
$\displaystyle\Rightarrow$ $\displaystyle
Q_{\text{R}}-Q_{\text{B-L}}+3Q_{\psi}=0$ (39)

where we assume that the PS breaking is due to the neutrino singlets zero
modes contained in the adjoint (via the Scherk-Scharz mechanism). We can
further define the hypercharge as follows:

$Q_{\text{Y}}=\frac{1}{2}(Q_{\text{B-L}}+Q_{\text{R}})\,.$ (40)

Henceforth, it remains the freedom of defining two unbroken global symmetries.
There is a unique charge which vanishes on both Higgs doublets, hence it will
not be broken by the Higgs vacuum expectation values:

$Q_{\text{B}}=\frac{1}{4}Q_{\text{B-L}}+\frac{1}{6}Q_{27}+\frac{1}{12}Q_{\psi}-\frac{1}{3}Q_{27^{\prime}}\,,$
(41)

where the normalisation matches the baryon number of quarks. The second
combination must be non-zero on at least one of the Higgs doublets: choosing
to normalise the charge to be 1 on $\varphi_{h2}$, we obtain

$Q_{\text{K}}=\frac{1}{2}Q_{\text{R}}+\frac{1}{6}Q_{27}-\frac{1}{6}Q_{\psi}+\frac{2}{3}Q_{27^{\prime}}\,.$
(42)

The charge assignments are listed in Tables 3, 4 and 5.

Note that the PS symmetry may also be broken by the $(\bar{\bf 4},{\bf 1},{\bf
2})_{1}$ in ${\bf 27^{\prime}}$, however this leads to the same charge
definition as above.

###  A.4 Localised superpotential on the SO(10) boundary

In the model with two localised generations on the SO$(10)$ boundary, mixings
with the bulk third generation are not possible, as they are forbidden by the
$Q_{\psi}$ charges. As a solution, one can introduce a minimal set of
localised fields, such as a set of localised Higgs states in the $10_{0}$,
which allows for

$\tilde{y}^{i}\
\Phi^{i}_{16_{-1}}\left.\Phi_{27}\right|_{16_{1}}\Phi_{10_{0}}\,,$ (43)

hence generating a mixing in the left-handed sector of the theory. Note that
no additional gauge anomalies are generated by such localised Higgs states. A
mixing between the localised and bulk Higgs fields could be generated by SUSY-
breaking quartic couplings, or by introducing additional singlets charged
under U$(1)_{\psi}$ yielding the following superpotential:

$a\ \Phi_{1_{2}}\Phi_{1_{-2}}+b\
\Phi_{10_{0}}\left.\Phi_{27}^{c}\right|_{10_{2}}\Phi_{1_{-2}}+c\
\Phi_{1_{2}}\Phi_{1_{2}}\left.\Phi_{27^{\prime}}^{c}\right|_{1_{-4}}\,.$ (44)

$\text{E}_{6}$ fields | PS fields | $(\mathbb{Z}_{2},\mathbb{Z}_{2}^{\prime})$ | SM | name | $Q_{\text{B}}$ | $Q_{\text{K}}$ | $Q_{\text{R}}$ | $Q_{\text{B-L}}$ | $Q_{\psi}$ | $Q_{27}$
---|---|---|---|---|---|---|---|---|---|---
$\Phi_{27}$ | $(\bf{4},\bf{2},\bf{1})_{1}$ | $(+,+)$ | $(3,2)_{1/6}$ | $q_{L}$ | $1/3$ | $0$ | $0$ | $1/3$ | $1$ | $1$
|  |  | $(1,2)_{-1/2}$ | $l_{L}$ | $0$ | $0$ | $0$ | $-1$ | $1$ | $1$
| $(\bar{\bf 4},{\bf 1},{\bf 2})_{1}$ | $(+,-)$ | $(\bar{3},1)_{1/3}$ |  | $1/6$ | $1/2$ | $1$ | $-1/3$ | $1$ | $1$
|  |  | $(\bar{3},1)_{-2/3}$ |  | $1/6$ | $-1/2$ | $-1$ | $-1/3$ | $1$ | $1$
|  |  | $(1,1)_{1}$ |  | $1/2$ | $1/2$ | $1$ | $1$ | $1$ | $1$
|  |  | $(1,1)_{0}$ |  | $1/2$ | $-1/2$ | $-1$ | $1$ | $1$ | $1$
| $(\bf{6},\bf{1},\bf{1})_{-2}$ | $(-,+)$ | $(\bar{3},1)_{1/3}$ |  | $1/6$ | $1/2$ | $0$ | $2/3$ | $-2$ | $1$
|  |  | $(3,1)_{-1/3}$ |  | $-1/6$ | $1/2$ | $0$ | $-2/3$ | $-2$ | $1$
| $(\bf{1},\bf{2},\bf{2})_{-2}$ | $(-,-)$ | $(1,2)_{1/2}$ |  | $0$ | $1$ | $1$ | $0$ | $-2$ | $1$
|  |  | $(1,2)_{-1/2}$ |  | $0$ | $0$ | $-1$ | $0$ | $-2$ | $1$
| $(\bf{1},\bf{1},\bf{1})_{4}$ | $(-,+)$ | $(1,1)_{0}$ |  | $1/2$ | $-1/2$ | $0$ | $0$ | $4$ | $1$
$\Phi^{c}_{27}$ | $(\bar{\bf{4}},\bf{2},\bf{1})_{-1}$ | $(-,-)$ | $(\bar{3},1)_{-1/6}$ |  | $-1/3$ | $0$ | $0$ | $-1/3$ | $-1$ | $-1$
|  |  | $(1,2)_{1/2}$ |  | $0$ | $0$ | $0$ | $1$ | $-1$ | $-1$
| $(\bf 4,{\bf 1},{\bf 2})_{-1}$ | $(-,+)$ | $(3,1)_{-1/3}$ |  | $-1/6$ | $-1/2$ | $-1$ | $1/3$ | $-1$ | $-1$
|  |  | $(3,1)_{2/3}$ |  | $-1/6$ | $1/2$ | $1$ | $-1/3$ | $-1$ | $-1$
|  |  | $(1,1)_{-1}$ |  | $-1/2$ | $-1/2$ | $-1$ | $-1$ | $-1$ | $-1$
|  |  | $(1,1)_{0}$ |  | $-1/2$ | $1/2$ | $1$ | $-1$ | $-1$ | $-1$
| $(\bf{6},\bf{1},\bf{1})_{2}$ | $(+,-)$ | $(\bar{3},1)_{1/3}$ |  | $1/6$ | $-1/2$ | $0$ | $2/3$ | $2$ | $-1$
|  |  | $(3,1)_{-1/3}$ |  | $-1/6$ | $-1/2$ | $0$ | $-2/3$ | $2$ | $-1$
| $(\bf{1},\bf{2},\bf{2})_{2}$ | $(+,+)$ | $(1,2)_{1/2}$ | $\varphi_{h1}$ | $0$ | $0$ | $1$ | $0$ | $2$ | $-1$
|  |  | $(1,2)_{-1/2}$ | $\varphi_{h2}^{\ast}$ | $0$ | $-1$ | $-1$ | $0$ | $2$ | $-2$
| $(\bf{1},\bf{1},\bf{1})_{-4}$ | $(+,-)$ | $(1,1)_{0}$ |  | $-1/2$ | $1/2$ | $0$ | $0$ | $-4$ | $-1$
Table 3:  Field decomposition for the Matter multiplet $R={\bf 27}$. $\text{E}_{6}$ fields | PS fields | $(\mathbb{Z}_{2},\mathbb{Z}_{2}^{\prime})$ | SM | name | $Q_{\text{B}}$ | $Q_{\text{K}}$ | $Q_{\text{R}}$ | $Q_{\text{B-L}}$ | $Q_{\psi}$ | $Q_{27^{\prime}}$
---|---|---|---|---|---|---|---|---|---|---
$\Phi_{27^{\prime}}$ | $(\bf{4},\bf{2},\bf{1})_{1}$ | $(+,-)$ | $(3,2)_{1/6}$ |  | $-1/6$ | $1/2$ | $0$ | $1/3$ | $1$ | $1$
|  |  | $(1,2)_{-1/2}$ |  | $-1/2$ | $-1/2$ | $0$ | $-1$ | $1$ | $1$
| $(\bar{\bf 4},{\bf 1},{\bf 2})_{1}$ | $(+,+)$ | $(\bar{3},1)_{1/3}$ | ${b^{\prime}}_{R}^{c}$ | $-1/3$ | $1$ | $1$ | $-1/3$ | $1$ | $1$
|  |  | $(\bar{3},1)_{-2/3}$ | ${t^{\prime}}_{R}^{c}$ | $-1/3$ | $0$ | $-1$ | $-1/3$ | $1$ | $1$
|  |  | $(1,1)_{1}$ | ${\tau^{\prime}}_{R}^{c}$ | $0$ | $1$ | $1$ | $1$ | $1$ | $1$
|  |  | $(1,1)_{0}$ | ${\nu^{\prime}}_{R}^{c}$ | $0$ | $0$ | $-1$ | $1$ | $1$ | $1$
| $(\bf{6},\bf{1},\bf{1})_{-2}$ | $(-,-)$ | $(\bar{3},1)_{1/3}$ |  | $-1/3$ | $1$ | $0$ | $2/3$ | $-2$ | $1$
|  |  | $(3,1)_{-1/3}$ |  | $1/3$ | $1$ | $0$ | $-2/3$ | $-2$ | $1$
| $(\bf{1},\bf{2},\bf{2})_{-2}$ | $(-,+)$ | $(1,2)_{1/2}$ |  | $-1/2$ | $3/2$ | $1$ | $0$ | $-2$ | $1$
|  |  | $(1,2)_{-1/2}$ |  | $-1/2$ | $1/2$ | $-1$ | $0$ | $-2$ | $1$
| $(\bf{1},\bf{1},\bf{1})_{4}$ | $(-,-)$ | $(1,1)_{0}$ |  | $0$ | $0$ | $0$ | $0$ | $4$ | $1$
$\Phi^{c}_{27^{\prime}}$ | $(\bar{\bf{4}},\bf{2},\bf{1})_{-1}$ | $(-,+)$ | $(\bar{3},2)_{-1/6}$ |  | $1/6$ | $-1/2$ | $0$ | $-1/3$ | $-1$ | $-1$
|  |  | $(1,2)_{1/2}$ |  | $1/2$ | $1/2$ | $0$ | $1$ | $-1$ | $-1$
| $(\bf 4,{\bf 1},{\bf 2})_{-1}$ | $(-,-)$ | $(3,1)_{-1/3}$ |  | $1/3$ | $-1$ | $-1$ | $1/3$ | $-1$ | $-1$
|  |  | $(3,1)_{2/3}$ |  | $1/3$ | $0$ | $1$ | $1/3$ | $-1$ | $-1$
|  |  | $(1,1)_{-1}$ |  | $0$ | $-1$ | $-1$ | $-1$ | $-1$ | $-1$
|  |  | $(1,1)_{0}$ |  | $0$ | $0$ | $1$ | $-1$ | $-1$ | $-1$
| $(\bf{6},\bf{1},\bf{1})_{2}$ | $(+,+)$ | $(\bar{3},1)_{1/3}$ | $B_{R}^{c}$ | $-1/3$ | $-1$ | $0$ | $2/3$ | $2$ | $-1$
|  |  | $(3,1)_{-1/3}$ | $B_{R}$ | $1/3$ | $-1$ | $0$ | $-2/3$ | $2$ | $-1$
| $(\bf{1},\bf{2},\bf{2})_{2}$ | $(+,-)$ | $(1,2)_{1/2}$ |  | $1/2$ | $-1/2$ | $1$ | $0$ | $2$ | $-1$
|  |  | $(1,2)_{-1/2}$ |  | $1/2$ | $-3/2$ | $-1$ | $0$ | $2$ | $-1$
| $(\bf{1},\bf{1},\bf{1})_{-4}$ | $(+,+)$ | $(1,1)_{0}$ | $\varphi_{\psi}$ | $0$ | $0$ | $0$ | $0$ | $-4$ | $-1$
Table 4:  Field decomposition for the Matter multiplet $R={\bf 27^{\prime}}$. $\text{E}_{6}$ fields | PS fields | $(\mathbb{Z}_{2},\mathbb{Z}_{2}^{\prime})$ | SM | name | $Q_{\text{B}}$ | $Q_{\text{K}}$ | $Q_{\text{R}}$ | $Q_{\text{B-L}}$ | $Q_{\psi}$
---|---|---|---|---|---|---|---|---|---
$W^{\alpha}_{78}$/$\Phi_{78}$ | $(\bf{15},\bf{1},\bf{1})_{0}$ | $(+,+)$ | $(8,1)_{0}$ | $G_{\mu}$ | $0$ | $0$ | $0$ | $0$ | $0$
|  |  | $(1,1)_{0}$ | $A^{B-L}_{\mu}$ | $0$ | $0$ | $0$ | $0$ | $0$
|  |  | $(3,1)_{2/3}$ | $X_{\mu}$ | $1/3$ | $0$ | $0$ | $4/3$ | $0$
|  |  | $(\bar{3},1)_{-2/3}$ | $X^{\dagger}_{\mu}$ | $-1/3$ | $0$ | $0$ | $-4/3$ | $0$
| $(\bf{1},\bf{3},\bf{1})_{0}$ | $(+,+)$ | $(1,3)_{0}$ | $W_{\mu}$ | $0$ | $0$ | $0$ | $0$ | $0$
| $(\bf{1},\bf{1},\bf{3})_{0}$ | $(+,+)$ | $(1,1)_{1}$ | $W^{R+}_{\mu}$ | $0$ | $0$ | $2$ | $0$ | $0$
|  |  | $(1,1)_{-1}$ | $W_{\mu}^{R-}$ | $0$ | $0$ | $-2$ | $0$ | $0$
|  |  | $(1,1)_{0}$ | $A^{R}_{\mu}$ | $0$ | $0$ | $0$ | $0$ | $0$
| $(\bf{1},\bf{1},\bf{1})_{0}$ | $(+,+)$ | $(1,1)_{0}$ | $A^{\psi}_{\mu}$ | $0$ | $0$ | $0$ | $0$ | $0$
| $(\bf{6},\bf{2},\bf{2})_{0}$ | $(+,-)$ | $(\bar{3},2)_{5/6}$ |  | $1/6$ | $1/2$ | $1$ | $2/3$ | $0$
|  |  | $(\bar{3},2)_{-1/6}$ |  | $1/6$ | $-1/2$ | $-1$ | $2/3$ | $0$
|  |  | $(3,2)_{1/6}$ |  | $-1/6$ | $1/2$ | $1$ | $-2/3$ | $0$
|  |  | $(\bar{3},2)_{-5/6}$ |  | $-1/6$ | $-1/2$ | $-1$ | $-2/3$ | $0$
| $(\bf{4},\bf{2},\bf{1})_{-3}$ | $(-,+)$ | $(3,2)_{1/6}$ |  | $-1/6$ | $1/2$ | $0$ | $1/3$ | $-3$
|  |  | $(1,2)_{-1/2}$ |  | $-1/2$ | $1/2$ | $0$ | $-1$ | $-3$
| $(\bar{\bf{4}},\bf{2},\bf{1})_{3}$ | $(-,+)$ | $(\bar{3},2)_{-1/6}$ |  | $1/6$ | $-1/2$ | $0$ | $-1/3$ | $3$
|  |  | $(1,2)_{1/2}$ |  | $1/2$ | $-1/2$ | $0$ | $1$ | $3$
| $(\bar{\bf{4}},\bf{1},\bf{2})_{-3}$ | $(-,-)$ | $(\bar{3},1)_{1/3}$ | $b_{R}^{c}$ | $-1/3$ | $1$ | $1$ | $-1/3$ | $-3$
|  |  | $(\bar{3},1)_{-2/3}$ | $t_{R}^{c}$ | $-1/3$ | $0$ | $-1$ | $-1/3$ | $-3$
|  |  | $(1,1)_{1}$ | $\tau_{R}^{c}$ | $0$ | $1$ | $1$ | $1$ | $-3$
|  |  | $(1,1)_{0}$ | $\nu_{R}^{c}$ | $0$ | $0$ | $-1$ | $1$ | $-3$
| $(\bf{4},\bf{1},\bf{2})_{3}$ | $(-,-)$ | $(3,1)_{-1/3}$ | $b^{\prime}_{L}$ | $1/3$ | $-1$ | $-1$ | $1/3$ | $3$
|  |  | $(3,1)_{2/3}$ | $t^{\prime}_{L}$ | $1/3$ | $0$ | $1$ | $1/3$ | $3$
|  |  | $(1,1)_{-1}$ | $\tau^{\prime}_{L}$ | $0$ | $-1$ | $-1$ | $-1$ | $3$
|  |  | $(1,1)_{0}$ | $\nu^{\prime}_{L}$ | $0$ | $0$ | $1$ | $-1$ | $3$
Table 5:  Field decomposition for the gauge multiplet $R={\bf 78}$. Fields
with orbifold parities $(+,+)$ have a zero mode in $W^{\alpha}_{78}$, while
$(-,-)$ entails a zero mode in $\Phi_{78}$.