aainstitutetext: Department of Physics, Oklahoma State University, Stillwater,
OK, 74078, USAbbinstitutetext: Jožef Stefan Institute, Jamova cesta 39, 1000
Ljubljana, Sloveniaccinstitutetext: Institute of Particle and Nuclear Physics,
Faculty of Mathematics and Physics, Charles University,
V Holešovičkách 2, 180 00 Prague 8, Czech Republic

# Trinification from $\mathbf{E_{6}}$ symmetry breaking

K.S. Babu b Borut Bajc c Vasja Susič<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

In the context of $\mathrm{E}_{6}$ Grand Unified Theories (GUTs), an
intriguing possibility for symmetry breaking to the Standard Model (SM) group
involves an intermediate stage characterized by either
$\mathrm{SU}(3)\times\mathrm{SU}(3)\times\mathrm{SU}(3)$ (trinification) or
$\mathrm{SU}(6)\times\mathrm{SU}(2)$. The more common choices of
$\mathrm{SU(5)}$ and $\mathrm{SO}(10)$ GUT symmetry groups do not offer such
breaking chains. We argue that the presence of a real (rank $2$ tensor)
representation $\mathbf{650}$ of $\mathrm{E}_{6}$ in the scalar sector is the
minimal and likely only reasonable possibility to obtain one of the novel
intermediate stages. We analyze the renormalizable scalar potential of a
single copy of the $\mathbf{650}$ and find vacuum solutions that support
regularly embedded subgroups
$\mathrm{SU}(3)\times\mathrm{SU}(3)\times\mathrm{SU}(3)$,
$\mathrm{SU}(6)\times\mathrm{SU}(2)$, and
$\mathrm{SO}(10)\times\mathrm{U}(1)$, as well as specially embedded subgroups
$\mathrm{F}_{4}$ and $\mathrm{SU}(3)\times\mathrm{G}_{2}$ that do not contain
the SM gauge symmetry. We show that for a suitable choice of parameters, each
of the regular cases can be obtained as the lowest among the analyzed minima
in the potential.

##  1 Introduction

An intriguing possibility of enlarging the gauge interactions of the Standard
Model (SM) is to embed them together in a Grand Unified Theory (GUT)
Georgi:1974sy . For such a GUT to be viable and consistent, its symmetry group
must be a simple Lie group, which has the SM group
$\mathrm{SU}(3)\times\mathrm{SU}(2)\times\mathrm{U}(1)$ as its subgroup, and
must admit complex representations (so as to produce a chiral theory). Only
the groups $\mathrm{SU}(N+3)$, $\mathrm{SO}(4N+2)$ and $\mathrm{E}_{6}$
satisfy these criteria, where $N\geq 2$ and the minimal cases in the unitary
and orthogonal families are the familiar $\mathrm{SU}(5)$ and
$\mathrm{SO}(10)$ groups. The focus of this paper is the third possibility,
viz., $\mathrm{E}_{6}$.

The exceptional Lie group $\mathrm{E}_{6}$ contains $\mathrm{SU}(5)$ and
$\mathrm{SO}(10)$ as its subgroups and is thus the largest of the usual GUT
symmetry possibilities, as recognized long ago Gursey:1975ki . Unlike
$\mathrm{SU}(5)$ and $\mathrm{SO}(10)$, the group $\mathrm{E}_{6}$ also has
the trinification group
$\mathrm{SU}(3)\times\mathrm{SU}(3)\times\mathrm{SU}(3)$ and
$\mathrm{SU}(6)\times\mathrm{SU}(2)$ as its subgroups (we shall sometimes
denote them as $G_{333}$ and $G_{62}$ for simplicity), and thus allows
breaking routes to the SM not present in the other two (minimal) GUT symmetry
groups. Although $\mathrm{E}_{6}$ also has other phenomenologically peculiar
features, such as matter unification of each generation joined by $2$ sterile
neutrinos and vector-like exotics in the fundamental representation
$\mathbf{27}$, it is the exotic intermediate symmetries that $\mathrm{E}_{6}$
can break into that are of interest to us in this paper.

A crucial role in obtaining $G_{333}$ or $G_{62}$ at the intermediate stage is
played by the representation $\mathbf{650}$, since it is the simplest and
seemingly only realistic111 Consider an $\mathrm{E}_{6}$ 4D Yang-Mills theory
with a single representation of real scalars $\mathbf{R}$, which contains
$G_{333}$ or $G_{62}$ singlets. The one-loop RGE for the unified gauge
coupling is $dg/dt=\beta g^{3}/(16\pi^{2})$. The two simplest candidates for
$\mathbf{R}$ are the representations $\mathbf{650}$ and $\mathbf{2430}$,
yielding $\beta=-19$ and $\beta=+91$, respectively. The second case leads to a
fast Landau pole, hence the strong preference for $\mathbf{650}$ in model
building.  representation that contains a SM-singlet of either, see e.g.
Slansky:1981yr ; Feger:2019tvk . Any reasonable GUT model utilizing the
$G_{333}$ or $G_{62}$ intermediate symmetries will thus require at least one
scalar representation $\mathbf{650}$ in its symmetry-breaking sector.

In this paper, we determine the (non-supersymmetric) scalar potential of a
single representation $\mathbf{650}$ and study its minima. This effort can be
seen as a first step in analyzing the class of $\mathrm{E}_{6}$ models that
aspires to use an exotic breaking route to the SM. We limit ourselves to the
analysis of the first breaking stage, which should be followed by at least one
more before reaching the SM. Also, although other representations are required
for a realistic model,222We note here that adding scalar representations
$\mathbf{27}\oplus\mathbf{351}^{\prime}$ to the $\mathbf{650}$ seems
especially promising to us, since their addition is well suited to describing
a realistic Yukawa sector, that reduces to the $\mathbf{10}\oplus\mathbf{126}$
case in $\mathrm{SO}(10)$ Babu:1992ia ; Bajc:2005zf ; Joshipura:2011nn ;
Ohlsson:2019sja  in the limit of $\mathrm{SO}(10)$-spinors in the
$\mathrm{E}_{6}$ scalar representations not acquiring VEVs (deducible from
Babu:2015psa ).  we do not commit to any particular fixed scalar sector beyond
the presence of one copy of the $\mathbf{650}$.

Since the inclusions $\mathbf{75}\subset\mathbf{210}\subset\mathbf{650}$
describe the largest irrep subparts with respect to the subgroup chain
$\mathrm{SU}(5)\subset\mathrm{SO}(10)\subset\mathrm{E}_{6}$, the study in this
paper can be viewed as an $\mathrm{E}_{6}$ analogue to the vacuum analysis
studies of the $\mathbf{75}$ in $\mathrm{SU}(5)$ GUT Koh:1982ji ;
Hubsch:1984pg ; Hubsch:1984qi ; Cummins:1985vg  or $\mathbf{210}$ in
$\mathrm{SO}(10)$ Chang:1984qr ; Chang:1985zq ; Chang:1986be ; He:1989rb .
Within $\mathrm{E}_{6}$ GUT, the representation $\mathbf{650}$ has been
considered before Lonsdale , albeit with a partial set of invariants.

We organize the paper as follows: we present some group theory preliminaries
required for describing the representation $\mathbf{650}$ in Section 2, write
down and analyze the scalar potential using one copy of this representation in
Section 3, and then conclude in Section 4. We also provide a set of
appendices, to which further technical details have been relegated.

##  2 Group Theory preliminaries of $\mathrm{E}_{6}$

###  2.1 Generators and representations of $\mathrm{E}_{6}$

The exceptional group $\mathrm{E}_{6}$ is simple, contains the SM group
$\mathrm{SU}(3)_{C}\times\mathrm{SU}(2)_{L}\times\mathrm{U}(1)_{Y}\equiv
G_{321}$ and has complex representations, and is thus a suitable candidate for
the unification group in GUT.

The $\mathrm{E}_{6}$ group has $78$ generators and is of rank $6$, see e.g.
Slansky:1981yr ; Feger:2019tvk  for group theory resources. The most
convenient way of considering the generators of $\mathrm{E}_{6}$ is through
the language of the trinification subgroup
$\mathrm{SU}(3)_{C}\times\mathrm{SU}(3)_{L}\times\mathrm{SU}(3)_{R}$, where
the labels $C$, $L$, and $R$ for the $\mathrm{SU}(3)$ factors refer to color,
left and right, respectively. According to the trinification decomposition of
the $\mathrm{E}_{6}$ adjoint representation

$\displaystyle\mathbf{78}$
$\displaystyle=(\mathbf{8},\mathbf{1},\mathbf{1})\;\oplus\;(\mathbf{1},\mathbf{8},\mathbf{1})\;\oplus\;(\mathbf{1},\mathbf{1},\mathbf{8})\;\oplus\;(\mathbf{3},\mathbf{\bar{3}},\mathbf{\bar{3}})\;\oplus\;(\mathbf{\bar{3}},\mathbf{3},\mathbf{3}),$
(1)

the $\mathrm{E}_{6}$ generators can be organized in matching order of the
representations above as

$\displaystyle t_{C}^{A},\quad t_{L}^{A},\quad t_{R}^{A},\quad
t^{\alpha}{}_{aa^{\prime}},\quad\bar{t}_{\alpha}{}^{aa^{\prime}}.$ (2)

The indices $\alpha$, $a$ and $a^{\prime}$ are the indices of the $C$, $L$ and
$R$ subgroups, respectively; they run from $1$ to $3$, and are fundamental
(anti-fundamental) if they are upper (lower). We also use a single label for
the adjoint index $A$ running from $1$ to $8$, always in the upper position,
with the subscript of the generator indicating which $\mathrm{SU}(3)$ factor
the $A$ refers to. This notation is standard and has been used before
Kephart:1981gf ; Bajc:2013qra ; Babu:2015psa .

We also follow the same references in the use of tensor formalism for
irreducible representations of $\mathrm{E}_{6}$. For the (anti)-fundamental
representation $\mathbf{27}$ ($\overline{\mathbf{27}}$) of $\mathrm{E}_{6}$,
we use an upper (lower) index $i$ that runs from $1$ to $27$. All irreducible
representations of $\mathrm{E}_{6}$ with dimension smaller than $1000$ can be
accessed through the following two tensor products:

$\displaystyle\mathbf{27}\otimes\mathbf{27}$
$\displaystyle=\overline{\mathbf{27}}_{S}\oplus\mathbf{351}^{\prime}_{S}\oplus\mathbf{351}_{A},$
(3) $\displaystyle\mathbf{27}\otimes\overline{\mathbf{27}}$
$\displaystyle=\mathbf{1}\oplus\mathbf{78}\oplus\mathbf{650},$ (4)

where the subscripts $S$ and $A$ indicate the symmetric and anti-symmetric
parts of the product, respectively.

Table 1: List of non-trivial irreducible representations (up to conjugation) of $\mathrm{E}_{6}$ with $\mathrm{dim}<1000$. They are written as tensors with a generic label $X$ satisfying certain $\mathrm{E}_{6}$-invariant conditions. $D_{2}$ denotes the Dynkin index of the representation.  irrep | indices | conditions | $D_{2}$ | self-conjugate?
---|---|---|---|---
$\phantom{0}\mathbf{27}$ | $X^{i}$ | / | $3$ | no
$\phantom{0}\mathbf{78}$ | $X^{i}{}_{j}$ | $X^{i}{}_{j}=X^{K}(t^{K})^{i}{}_{j}$ | $12$ | yes
$\mathbf{351}$ | $X^{ij}$ | $X^{ij}=-X^{ji}$ | $75$ | no
$\mathbf{351}^{\prime}$ | $X^{ij}$ |  $X^{ij}=X^{ji}$, $d_{ijk}\,X^{jk}=0$ | $84$ | no
$\mathbf{650}$ | $X^{i}{}_{j}$ |  $\mathbf{X}=\mathbf{X}^{\dagger}$, $X^{i}{}_{i}=0$, $X^{i}{}_{j}\,(t^{K})^{j}{}_{i}=0$ | $150$ | yes

Given these decompositions, and making use of the $\mathrm{E}_{6}$ completely-
symmetric invariant tensors $d_{ijk}$ and $d^{ijk}$, the irreducible
representations in question can be formally written as tensors with up to $2$
indices, as shown in Table 1. The $\mathrm{E}_{6}$ generators there have been
generically labeled as $t^{K}$, where the adjoint index $K$ runs from $1$ to
$78$.

Some further technical details, e.g. the commutation relations of
$\mathrm{E}_{6}$ generators and the definition of the invariant $d$-tensor,
are provided in Appendix A.1.

###  2.2 Maximal Subgroups of $\mathrm{E}_{6}$

Since $\mathrm{E}_{6}$ is a relatively large group, the possible breaking
chains to the SM group $G_{321}$ become very elaborate, much more so than e.g.
in $\mathrm{SO}(10)$ Chang:1984qr ; Deshpande:1992au ; Deshpande:1992em . In
fact, since it is a subgroup of $\mathrm{E}_{6}$, all $\mathrm{SO}(10)$
breaking chains become subchains of the larger $\mathrm{E}_{6}$ breaking
patterns.

Our primary interest in this paper are the new breaking chains to the SM not
available in $\mathrm{SO}(10)$ GUT, in particular breaking through $G_{333}$
and $G_{62}$. Both are examples of maximal subgroups of $\mathrm{E}_{6}$, and
we shall therefore limit the investigation to such cases — a limitation that
we shall briefly revisit at the end of this section. The scope of maximal
$\mathrm{E}_{6}$-subgroups with potential phenomenological relevance is then
rounded out by $\mathrm{SO}(10)\times\mathrm{U}(1)$, which is covered in this
work as well.

The list of maximal subgroups of $\mathrm{E}_{6}$ is provided in Table 2
(based on Feger:2019tvk ; Yamatsu:2015npn ). For each subgroup in the table,
we list whether it requires a regular (R) or special (S) embedding, the
decomposition of the adjoint representation $\mathbf{78}$ of $\mathrm{E}_{6}$
into this subgroups’ irreducible representations, the number of subgroup
singlets found when decomposing the $\mathbf{650}$ of $\mathrm{E}_{6}$, and
whether $G$ contains the SM group $G_{321}$ as a subgroup. In the
decomposition of the $\mathbf{78}$, the parts which belong to the adjoint of
the subgroup $G$ are underlined; in a spontaneous symmetry breaking scenario
to that subgroup the gauge bosons that are not underlined thus become massive.

Table 2: List of maximal subgroups $G$ of $\mathrm{E}_{6}$, along with embedding type (regular/special), decomposition of the $\mathbf{E}_{6}$ adjoint into irreducible representations of $G$ (with the adjoint representation(s) of $G$ underlined), and number $\\#$ of $G$-singlets in the $\mathbf{650}$. The last column indicates whether the SM group is a subgroup of $G$. We consider the first $5$ cases to be the “relevant” subgroups.  subgroup $G$ |  | type | decomposition of $\mathbf{78}$ | $\\#_{\mathbf{1}_{G}\in\mathbf{650}}$ | $G_{321}\stackrel{{\scriptstyle?}}{{\subset}}G$
---|---|---|---|---|---
$\mathrm{SU}(3)\times\mathrm{SU}(3)\times\mathrm{SU}(3)$ | $\equiv G_{333}$ | R | $\underline{(\mathbf{8},{\mathbf{1},\mathbf{1}})}\oplus\underline{(\mathbf{1},{\mathbf{8},\mathbf{1}})}\oplus\underline{(\mathbf{1},{\mathbf{1},\mathbf{8}})}\oplus$ | 2 | yes
|  |  | $\oplus(\mathbf{3},{\mathbf{\bar{3}},\mathbf{\bar{3}}})\oplus(\mathbf{\bar{3}},{\mathbf{3},\mathbf{3}})$ |  |
$\mathrm{SU}(6)\times\mathrm{SU}(2)$ | $\equiv G_{62}$ | R | $\underline{(\mathbf{35},\mathbf{1})}\oplus\underline{(\mathbf{1},\mathbf{3})}\oplus(\mathbf{20},\mathbf{2})$ | 1 | yes
$\mathrm{SO}(10)\times\mathrm{U}(1)$ | $\equiv G_{10,1}$ | R | $\underline{(\mathbf{45},0)}\oplus\underline{(\mathbf{1},0)}\oplus(\mathbf{16},+3)\oplus(\mathbf{\overline{16}},-3)$ | 1 | yes
$\mathrm{F}_{4}$ |  | S | $\underline{\mathbf{56}}\oplus\mathbf{26}$ | 1 | no
$\mathrm{SU}(3)\times\mathrm{G}_{2}$ |  | S | $\underline{(\mathbf{8},\mathbf{1})}\oplus\underline{(\mathbf{1},\mathbf{14})}\oplus(\mathbf{8},\mathbf{7})$ | 1 | no
$\mathrm{G}_{2}$ |  | S | $\underline{\mathbf{14}}\oplus\mathbf{64}$ | 0 | no
$\mathrm{SU}(3)$ |  | S | $\underline{\mathbf{8}}\oplus\mathbf{35}\oplus\mathbf{\overline{35}}$ | 0 | no
$\mathrm{Sp}(8)$ |  | S | $\underline{\mathbf{36}}\oplus\mathbf{42}$ | 0 | no

We observe from Table 2 that the SM group can be found in $G$ for the first
three cases, and hence these cases are of phenomenological relevance.
Incidentally, the first three cases are also exactly those with a regular
embedding of $G$ into $\mathrm{E}_{6}$, i.e. the embedding is rank-preserving.

The cases that may be obtained as the residual symmetry after first stage
breaking by the vacuum expectation value (VEV) of the representation
$\mathbf{650}$, however, are those which contain at least one singlet in this
representation. Table 2 shows these are the regular cases, as well as the
special embedding cases of $\mathrm{F}_{4}$ and
$\mathrm{SU}(3)\times\mathrm{G}_{2}$, which involve the exceptional groups
$\mathrm{F}_{4}$ and $\mathrm{G}_{2}$. It is these maximal cases with singlets
in the $\mathbf{650}$ that we focus on in our vacuum analysis, and we shall
refer to them as the “relevant” subgroups.

The “relevant” subgroups $G$ can be specified concretely by listing their
generators as a linear combinations of those from Eq. (2). The regular cases
can be defined by subsets of $\mathrm{E}_{6}$ generators

$\displaystyle\mathrm{SU}(3)^{3}:$ $\displaystyle\quad\\{t^{A}_{C},\
t^{A}_{L},\ t^{A}_{R}\\},$ (5)
$\displaystyle\mathrm{SU}(6)\times\mathrm{SU}(2):$
$\displaystyle\quad\\{t^{A}_{C},\ t^{A}_{L},\ t^{1,2,3,8}_{R},\
t^{\alpha}{}_{a3},\ \bar{t}_{\alpha}{}^{a3}\\},$ (6)
$\displaystyle\mathrm{SO}(10)\times\mathrm{U}(1):$
$\displaystyle\quad\\{t^{A}_{C},t^{1,2,3,8}_{L},t^{1,2,3,8}_{R},t^{\alpha}{}_{bb^{\prime}},\bar{t}_{\alpha}{}^{bb^{\prime}};\
(bb^{\prime})=(11),(12),(21),(22),(33)\\},$ (7)

while a more elaborate construction is necessary for those with a special
embedding, see Appendix A for technical details. In Eqs. (5)–(7) the index $A$
runs unrestricted from $1$ to $8$ and indices $a$ and $a^{\prime}$ from $1$ to
$3$, while specific index values are listed with commas in cases when only
some generators of that type are present, e.g. $t_{L}^{1,2,3,8}$. Along with
the already defined abbreviations $G_{333}$ and $G_{62}$, we shall sometimes
shorten $\mathrm{SO}(10)\times\mathrm{U}(1)$ to $G_{10,1}$, as indicated in
Table 2.

We finish this section on subgroups with a few noteworthy remarks:

  * •

The subgroup specifications of Eqs. (5)–(7) are only one possible embedding of
each into $\mathrm{E}_{6}$. Given a subalgebra embedding
$\mathfrak{g}\hookrightarrow\mathfrak{e}_{6}$, we can obtain an equivalent
embedding by conjugation, i.e. $A\mathfrak{g}A^{-1}$ also constitutes an
embedding of $\mathfrak{g}$ if we conjugate with any group element
$A\in\mathrm{E}_{6}$. This shows there are always an infinite number of ways
to embed $G$ into $\mathrm{E}_{6}$. Each case of $G$ has, however, only one
embedding (of its Lie algebra) into $\mathrm{E}_{6}$ up to conjugacy, cf. e.g.
lists of maximal algebras in Feger:2019tvk . The choice of one representative
for each case suffices for our vacuum analysis.

  * •

A related question concerns the embedding of the SM into $\mathrm{E}_{6}$.
Given a fixed embedding of SM, conjugating the embedding of $G$ by an element
$A\in\mathrm{E}_{6}$ may or may not change the SM embedding. From this point
of view, although all embeddings of $G$ into $\mathrm{E}_{6}$ are equivalent,
they may differ by how the SM group is embedded into $G$ (for example, how the
hypercharge is embedded). The classification of how SM embeds into the regular
subgroups $G$ will not be relevant for us in this paper, since it has no
bearing on the properties of the minima in the first stage of symmetry
breaking, and as such this question is beyond the scope of this study, but
will instead be presented in a follow-up.

  * •

In the case of breaking chains proceeding through trinification $G_{333}$,
there is an additional discrete symmetry that plays a role, as we shall see
explicitly in Section 3. We denote this symmetry by $D_{3}$. It essentially
permutes the three group factors of $G_{333}$ (and is hence isomorphic to the
permutation group $S_{3}$), and is generated by three parities that exchange
pairs of factors, dubbed as LR, CL and CR parity. One of these parities from
$D_{3}$ survives the breaking to $G_{333}$ via the $\mathbf{650}$. Note that
this phenomenon is analogous to $D$-parity in the $\mathrm{SO}(10)$ context
Kibble:1982dd ; Chang:1983fu ; Chang:1984uy , and in fact the $\mathrm{E}_{6}$
LR parity is equivalent to the $\mathrm{SO}(10)$ $D$-parity. We relegate the
technical details and further discussion of the discrete group $D_{3}$ to
Appendix B.

  * •

Finally, we elaborate upon our limitation of analyzing vacua only for the
“relevant” subgroups of Table 2. If one uses a single irreducible
representation for spontaneous symmetry breaking, as we do in the first stage
breaking, Michel’s conjecture states that the resulting symmetry is a maximal
little group of the representation, see Michel:1971th ; Slansky:1981yr  and
references therein. This suggests that large subgroups (such as maximal
subgroups) are indeed of greatest interest in the analysis. Namely, vacua
breaking to subgroups of the “relevant” ones would be in violation of Michel.
The cases allowed by Michel that we do not analyze are the maximal subgroups
of the last three cases in Table 2: all of these indeed contain singlet(s) in
the $\mathbf{650}$, and they are maximal little groups if they are not
subgroups of the “relevant” cases.
Although a complete classification of all breaking solutions of a single
$\mathbf{650}$ of $\mathrm{E}_{6}$ is beyond the scope of this paper, the
arguments above suggest the “relevant” cases may indeed be the only pertinent
ones, at least phenomenologically.

##  3 Analysis of scalar potential with one $\mathbf{650}$ of $\mathrm{E}_{6}$

###  3.1 The scalar potential

As seen in Eq. (4), the representation $\mathbf{650}$ is found in the tensor
product $\mathbf{27}\otimes\mathbf{\overline{27}}$. It can thus be implemented
as a $27\times 27$ matrix $\mathbf{X}$, written as $X^{i}{}_{j}$ in tensor
notation, with some additional constraints placed on it, cf. Table 1.

The most general renormalizable potential with the representation
$\mathbf{650}\equiv\mathbf{X}$ can be written as

$\displaystyle\begin{split}V_{650}(\mathbf{X})&=\phantom{+}M^{2}\cdot\mathrm{Tr}(\mathbf{X}^{2})\\\
&\quad+\;m_{1}\cdot\mathrm{Tr}(\mathbf{X}^{3})+m_{2}\cdot
X^{i}{}_{l}\,X^{j}{}_{m}\,X^{k}{}_{n}\;d^{lmn}d_{ijk}\\\
&\quad+\;\lambda_{1}\cdot(\mathrm{Tr}(\mathbf{X}^{2}))^{2}\;+\;\lambda_{2}\cdot\mathrm{Tr}(\mathbf{X}^{4})\;+\;\lambda_{3}\cdot(\mathbf{X}^{2})^{k}{}_{i}\;(\mathbf{X}^{2})^{l}{}_{j}\;D^{ij}{}_{kl}\\\
&\quad+\;\lambda_{4}\cdot
X^{i}{}_{i^{\prime}}\,X^{j}{}_{j^{\prime}}\,X^{k}{}_{k^{\prime}}\,X^{l}{}_{l^{\prime}}\;D^{i^{\prime}j^{\prime}}{}_{kl}\;D^{k^{\prime}l^{\prime}}{}_{ij}\;+\;\lambda_{5}\cdot
X^{i}{}_{l}\,X^{j}{}_{m}\,(\mathbf{X}^{2})^{k}{}_{n}\;d^{lmn}d_{ijk},\\\
\end{split}$ (8)

where the bold-typed $\mathbf{X}$ indicates its matrix form, $X^{i}{}_{j}$ the
components of this matrix, while $d_{ijk}$ is the primitive completely-
symmetric invariant tensor in $\mathrm{E}_{6}$ and $D^{ij}{}_{kl}$ is a
composite defined by

$\displaystyle D^{ij}{}_{kl}$ $\displaystyle:=d^{ijm}\,d_{klm}.$ (9)

A few more technical details for the $d$-tensors are given in Appendix A.1.
All indices involved in the above expression run from $1$ to $27$, and all
parameters $M^{2}$, $m_{k}$ and $\lambda_{l}$ are real.

Notice that there are $2$ cubic and $5$ quartic invariants in Eq. (8). This
result is non-trivial, since there can be many different ways in which the
$\mathrm{E}_{6}$ indices between the tensor $X^{i}{}_{j}$ and the invariant
tensors $d$ can be contracted, and furthermore the different contractions may
be related to one another. This difficulty can be solved if one knows the
number of independent invariants in advance, for which we used the symmetrised
tensor power function of the computer algebra program LiE Leeuwen1992LiE .
Having access to the explicit form of the tensor $d$ then allows for
constructing that many same-order invariants and checking their linear
independence explicitly.

To analyze the stationarity conditions of this potential, we need to consider
only the singlet directions in $\mathbf{X}$ with respect to the “relevant”
subgroups, as discussed in Section 2.2. We thus digress to specifying these
directions below.

For the regular cases in Table 2, a singlet of $G$ will also be a singlet
under the SM group $G_{321}$. The singlet directions of regular $G$ are thus
most conveniently viewed as directions in the sub-space of SM-singlets. The
representation $\mathbf{650}$ contains $11$ real singlets of the SM group,
which we label by

$\displaystyle\\{\,s,\quad a,\quad z,\quad x_{1},\quad x_{2},\quad X_{3},\quad
y_{1},\quad y_{2},\quad Y_{3}\,\\},$ (10)

where small letters are used for real VEVs and capital letters for complex
valued VEVs containing $2$ real degrees of freedom. They are partially
specified by listing their $G_{333}$ origin in Table 3, while their explicit
definition in terms of entries $X^{i}{}_{j}$ is given in Appendix C.

As seen from the $G_{333}$ decomposition in Table 3, the representation
$\mathbf{650}$ has two $G_{333}$ singlets labeled by $s$ and $a$, standing for
symmetric and anti-symmetric under left-right (LR) parity. This parity
exchanges the $\mathrm{SU}(3)_{L}$ and $\mathrm{SU}(3)_{R}$ subgroups, see
Appendix B for further elaboration. In the decomposition to $G_{10,1}$ and
$G_{62}$, however, there is only one singlet state in the $\mathbf{650}$. We
use a generic label $\tilde{s}$ for this state, and it can be specified in
each case as a linear combination of the SM-singlets listed in Eq. (10).
Explicitly, for the subgroups defined in Eqs. (6)–(7), the computed directions
are given in Table 4.

The $G$-singlet directions for the special embedding cases of $\mathrm{F}_{4}$
and $\mathrm{SU}(3)\times\mathrm{G}_{2}$ do not live in the SM-singlet
subspace, since these groups don’t contain the SM group. We still label the
VEV-direction in the $\mathbf{650}$ that retains these symmetries by
$\tilde{s}$, but we relegate their explicit definition to Appendix C.

Note that the VEVs in the $G$-singlet direction(s) are always normalized such
that the following hold:

$\displaystyle\langle\mathrm{Tr}(\mathbf{X}^{2})\rangle$
$\displaystyle=\tfrac{1}{2}\,(s^{2}+a^{2}),$
$\displaystyle\langle\mathrm{Tr}(\mathbf{X}^{2})\rangle$
$\displaystyle=\tfrac{1}{2}\,\tilde{s}^{2}.$ (11)

The former applies to $G_{333}$, and the latter to all other cases with only
one $G$-singlet in the $\mathbf{650}$.

Table 3: The $G_{333}$-locations of 11 SM real singlets from the $\mathbf{650}$ of $\mathrm{E}_{6}$. The two states $s$ and $a$ have a well-defined LR parity. Explicit definitions of states can be found in Appendix C.  $G_{333}$ irrep | LR parity | SM singlets inside
---|---|---
$(\mathbf{1},\mathbf{1},\mathbf{1})$ | $+1$ | $s$
$(\mathbf{1},\mathbf{1},\mathbf{1})$ | $-1$ | $a$
$(\mathbf{1},\mathbf{1},\mathbf{8})$ |  |  $x_{1}$, $x_{2}$, $\mathrm{Re}(X_{3})$, $\mathrm{Im}(X_{3})$
$(\mathbf{1},\mathbf{8},\mathbf{1})$ |  | $z$
$(\mathbf{1},\mathbf{8},\mathbf{8})$ |  |  $y_{1}$, $y_{2}$, $\mathrm{Re}(Y_{3})$, $\mathrm{Im}(Y_{3})$
Table 4: The VEV alignment of $\tilde{s}$ in the $\mathbf{650}$ which breaks $E_{6}$ to the regular subgroups $G$ defined in Eqs. (5)–(7). The $G_{333}$ case has $2$ possible independent directions $s$ and $a$ for the VEV. The special cases $\mathrm{F}_{4}$ and $\mathrm{SU}(3)\times\mathrm{G}_{2}$ are considered separately in Appendix C.  subgroup $G$ | VEV alignment
---|---
$G_{333}$ |  $s$, $a$
$G_{62}$ | $(s,a,x_{1},x_{2})=\tfrac{1}{2\sqrt{30}}\,(2\sqrt{3},6,-3\sqrt{2},-3\sqrt{6})\;\tilde{s}$
$G_{10,1}$ | $(s,z,x_{1},x_{2},y_{1},y_{2})=\tfrac{1}{\sqrt{80}}\,(2\sqrt{2},2\sqrt{3},\sqrt{3},3,-2\sqrt{3},6)\;\tilde{s}$

Returning to the analysis of the scalar potential $V_{650}$ in Eq. (8), the
stationarity conditions can be expressed only in terms of the defined singlet
directions for each of the relevant cases of Table 2. The $G$-non-singlets, on
the other hand, do not need to be considered since they have a vanishing VEV —
we refer to the potential with all non-singlet zero-VEVs inserted as the
restricted potential. It turns out all cases lead to a similar expression for
the restricted potential that takes the general form

$\displaystyle V(s,a)$
$\displaystyle=\tfrac{1}{2}M^{2}\;(s^{2}+a^{2})-\tfrac{1}{3}m\;s(s^{2}-3a^{2})+\tfrac{1}{4}\lambda\;(s^{2}+a^{2})^{2},$
(12)

where the parameter $M^{2}$ is the same as in $V_{650}$, while $m$ and
$\lambda$ are determined from the original parameters in Eq. (8) via

$\displaystyle m$ $\displaystyle:=\sum_{k=1}^{2}\alpha_{k}\,m_{k},$
$\displaystyle\lambda$ $\displaystyle:=\sum_{l=1}^{5}\beta_{l}\,\lambda_{l},$
(13)

where the numbers $\alpha_{k}$ and $\beta_{l}$ are subgroup dependent and are
given in Table 5. For all cases other than trinification, when only one
singlet is present, one obtains the restricted pontential by setting
$s=\tilde{s}$ and $a=0$.

The last column of Table 5 also gives explicit expressions for the masses of
those gauge bosons which correspond to broken symmetry generators, i.e. those
representations of $G$ not underlined in the decomposition of the
$\mathbf{78}$ in Table 2. These results come from the computation of all $78$
gauge boson masses in each case. They thus offer an independent check that we
correctly identified the singlets of $G$ in the representation $\mathbf{650}$,
since we obtain the correct counting of degeneracies in gauge boson masses.

We analyze the restricted potential of Eq. (12) and obtain minima candidates
for the potential in Eq. (8) in the next subsection.

Table 5: Coefficients $\alpha_{k}$ and $\beta_{l}$ needed to compute the parameters of the potential in Eq. (12) using Eq. (13). The last column gives masses acquired by the gauge bosons once the VEVs are engaged.  subgroups | $\alpha_{k}$ | $\beta_{l}$ | value of $M^{2}_{GUT}$
---|---|---|---
$G_{333}$ | $\tfrac{1}{4\sqrt{3}}(1,10)$ | $\tfrac{1}{18}(18,1,7,34,4)$ | $M^{2}_{(\mathbf{3},\mathbf{\bar{3}},\mathbf{\bar{3}})}=\tfrac{1}{3}\,g^{2}(s^{2}+a^{2})$
$G_{62}$ | $\tfrac{1}{4\sqrt{30}}(1,-44)$ | $\tfrac{1}{180}(180,7,67,652,-8)$ | $M^{2}_{(\mathbf{20},\mathbf{2})}=\tfrac{9}{20}\,g^{2}\tilde{s}^{2}$
$G_{10,1}$ | $\tfrac{1}{8\sqrt{30}}(29,20)$ | $\tfrac{1}{720}(720,397,97,520,40)$ | $M^{2}_{\mathbf{16}(-3)}=\tfrac{9}{16}\,g^{2}\tilde{s}^{2}$
$\mathrm{F_{4}}$ | $\tfrac{1}{\sqrt{78}}(25,34)$ | $\tfrac{1}{234}(234,217,295,394,292)$ | $M^{2}_{\mathbf{26}}=\tfrac{9}{13}\,g^{2}\tilde{s}^{2}$
$\mathrm{SU}(3)\times\mathrm{G}_{2}$ | $\tfrac{1}{\sqrt{42}}(5,23)$ | $\tfrac{1}{126}(126,13,55,181,43)$ | $M^{2}_{(\mathbf{8},\mathbf{7})}=\tfrac{9}{28}\,g^{2}\tilde{s}^{2}$

###  3.2 Analysis of the vacua

####  3.2.1 Solution to stationarity conditions

We start by analyzing the most general form of the restricted potential in Eq.
(12). Since the VEVs $s$ and $a$ are real, the potential is a polynomial
function mapping $\mathbb{R}^{2}\to\mathbb{R}$. For the potential to be
bounded from below, we demand $\lambda>0$.

First, we notice a three-fold symmetry in the restricted potential. Suppose
that we perform a rotation in the $(s,a)$ plane by an angle $\varphi$:

$\displaystyle\begin{pmatrix}s\\\ a\\\ \end{pmatrix}$ $\displaystyle\mapsto
R(\varphi)\begin{pmatrix}s\\\ a\\\ \end{pmatrix},$ $\displaystyle R(\varphi)$
$\displaystyle:=\begin{pmatrix}\cos\varphi&-\sin\varphi\\\
\sin\varphi&\phantom{-}\cos\varphi\\\ \end{pmatrix}.$ (14)

Inserting this transformation into the restricted potential, we get

$\displaystyle V_{650}(s,a)$
$\displaystyle\mapsto\tfrac{1}{2}M^{2}\;(s^{2}+a^{2})-\tfrac{1}{3}m\;\left(s(s^{2}-3a^{2})\cos(3\varphi)+a(a^{2}-3s^{2})\sin(3\varphi)\right)+\tfrac{1}{4}\lambda\;(s^{2}+a^{2})^{2}.$
(15)

Angles that satisfy $3\varphi=2\pi k$ for integer $k$ clearly yield back the
original restricted potential of Eq. (12). The potential $V$ thus exhibits a
$\mathbb{Z}_{3}$ symmetry in the $s$-$a$ plane, generated by $R(2\pi/3)$.
Furthermore, the flip $a\mapsto-a$ generates a $\mathbb{Z}_{2}$ parity
transformation, which also leaves the restricted potential invariant. All
together these transformations form an $S_{3}$ permutation symmetry of the
restricted potential.

The full two-singlet expression of the restricted potential is relevant only
for the $G_{333}$ trinification case, and in this context the $S_{3}$
transformations correspond to the discrete group $D_{3}$ from Section 2.2. To
reiterate, it is generated by the $\mathbb{Z}^{LR}_{2}$, $\mathbb{Z}^{CL}_{2}$
and $\mathbb{Z}^{CR}_{2}$ parities defined in Appendix B. The flip
$a\mapsto-a$ directly corresponds to $\mathbb{Z}^{LR}_{2}$ parity, in
accordance with Table 3.

We now proceed by analyzing the restricted potential directly: the
stationarity conditions

$\displaystyle\tfrac{\partial}{\partial s}V=\tfrac{\partial}{\partial a}V$
$\displaystyle=0$ (16)

yield $7$ formal solutions $(s_{i},a_{i})$, with $i=0,\ldots 6$ as follows:

$\displaystyle(s_{0},a_{0})$ $\displaystyle=(0,0),$ (17)
$\displaystyle(s_{1,2},a_{1,2})$
$\displaystyle=(\tfrac{m\pm\sqrt{m^{2}-4\lambda\,M^{2}}}{2\lambda},0),$ (18)

while the remaining $4$ solutions are the $(s_{j},a_{j})$ solutions for
$j=1,2$ rotated by either $R(2\pi/3)$ or $R(4\pi/3)$ due to the
$\mathbb{Z}_{3}$ symmetry of the restricted potential. When the formal
expressions for the solution yield real values, i.e. when the discriminant
$\mathcal{D}$ is positive

$\displaystyle\mathcal{D}$ $\displaystyle:=m^{2}-4\lambda\,M^{2}>0,$ (19)

non-trivial minima can exist.

Intuitively, demanding $\lambda>0$ and destabilizing the trivial solution by
$M^{2}<0$, the discriminant is positive for all $m\in\mathbb{R}$. Since at
least one of the points must be a local minimum in such a case, combining
$\mathbb{Z}_{3}$ with the Poincare-Hopf index theorem results in having $3$
minima and $3$ saddle points.

The pattern of minima is confirmed in the general case by the Hessian matrix
of $V$ with the inserted $(s_{1,2},a_{1,2})$ solutions:

$\displaystyle\partial^{2}V\Big{|}_{(s,a)=(s_{1,2},a_{1,2})}$
$\displaystyle=\frac{m\pm\sqrt{\mathcal{D}}}{2\lambda}\;\mathrm{diag}\left(\pm\sqrt{\mathcal{D}},3m\right).$
(20)

For $m>0$, together with the non-trivial minima and boundedness conditions
$\lambda,\mathcal{D}>0$, we therefore see that $(s_{i},a_{i})$ is a (global)
minimum of the restricted potential $V$ for $i=1$ (the point with $s>0$), and
a saddle point for $i=2$. For $m<0$, the role of minima and saddle points are
exchanged.333Note that the sign of $\mathbf{650}$ can be redefined it
necessary, so that one can take $m>0$ without loss of generality. The
parameter $m$ is, however, a derived parameter dependent on $G$, so the
redefinition is possible only for an analysis limited to vacua of a single
$G$, while $m>0$ may not be achieved for all $G$ simultaneously. Also, if
$M^{2}=0$, the saddle points merge with $P_{0}$ for any sign of $m$.

Figure 1: The contour plot in the $(s,a)$-plane for the restricted potential
$V$ of Eq. (12) taking $M^{2}=-1$, $m=0.3$ and $\lambda=1$ (using arbitrary
mass units). The stationary points $P_{i}=(s_{i},a_{i})$ are shown in red.

The potential $V$ is visualized in the contour plot of the restricted
potential in Figure 1. Specific parameter values
$(M^{2},m,\lambda)=(-1,0.3,1)$ were chosen. The stationary points are
indicated as red dots and denoted by $P_{i}=(s_{i},a_{i})$ for $i=0,\ldots,6$.
The plot clearly exhibits the three-fold $\mathbb{Z}_{3}$ rotational symmetry,
and it is symmetric under the $a\mapsto-a$ reflection. The point $P_{0}$ is a
local maximum, $P_{1,3,5}$ are global minima, and $P_{2,4,6}$ are saddle
points.

Suppose we define two new rotated sets of coordinates beside the pair $(s,a)$:

$\displaystyle(s^{\prime},a^{\prime})^{T}$
$\displaystyle:=R(2\pi/3)\,(s,a)^{T},$
$\displaystyle(s^{\prime\prime},a^{\prime\prime})^{T}$
$\displaystyle:=R(4\pi/3)\,(s,a)^{T}.$ (21)

We see that the points $P_{2}$, $P_{0}$ and $P_{1}$ lie on the $a=0$ line.
Similarly, the points $P_{4,0,3}$ lie on the $a^{\prime}=0$ line and
$P_{6,0,5}$ lie on the $a^{\prime\prime}=0$ line due to rotational symmetry.
The implication for trinification is that each of the $3$ minima preserves a
different $\mathbb{Z}_{2}$ parity: $\\{P_{1},P_{3},P_{5}\\}$ preserve the
$\\{$LR,CL,CR$\\}$ parities corresponding to
$\\{a=0,a^{\prime}=0,a^{\prime\prime}=0\\}$, respectively. In particular, any
trinification vacuum preserves a particular $\mathbb{Z}_{2}$ parity remnant of
the discrete symmetry $D_{3}$ of the restricted potential.

For all cases of subgroups $G$ other than trinification, when there is only
one singlet $\tilde{s}$, the stationarity condition is a cubic equation of
$\tilde{s}$ and therefore has $3$ formal solutions. These consist of setting
$s=\tilde{s}$ and $a=0$ in Eqs. (17)–(18), with the second derivative at
$\tilde{s}_{1,2}$ equal to the upper-left entry in the matrix of Eq. (20).

  * •

In the regime $M^{2}<0$ and $\lambda>0$, we always have $\mathcal{D}>m^{2}$,
and thus the solution $\tilde{s}_{0}=0$ is a local maximum and
$\tilde{s}_{1,2}$ are two local minima. For $m>0$, the $+$ solution
$\tilde{s}_{1}$ is the global minimum of $V$. This can be visualized in Figure
1 if we restrict to the dashed line $a=0$: $P_{1}$ and $P_{2}$ are local
minima along this line, with $P_{1}$ being also the global minimum.

  * •

In the alternative regime $M^{2},\lambda>0$, the trivial solution is a local
minimum and non-trivial solutions exist only for $\mathcal{D}>0$; in such a
case we have $|m|>\sqrt{\mathcal{D}}$, and so for $m>0$ the solution
$\tilde{s}_{1}$ is a local minimum, while $\tilde{s}_{2}$ is a local maximum.

  * •

For $M^{2}=0$, the trivial minimum and local maximum from the $M^{2}>0$ case
merge into a saddle point, and one local minimum remains ($s_{1}$ for $m>0$).

To summarize, for all symmetries $G$ there is at least one corresponding
(local) minimum of the restricted potential when $\lambda,\mathcal{D}>0$. For
$G_{333}$ there are $3$ degenerate minima, and for all other cases of $G$ in
Table 5 we have either one ($M^{2}\geq 0$) or two ($M^{2}<0$) local minima. In
the latter case, their difference in depth is controlled by the parameter $m$
of the cubic term.

We stress that these results hold for the restricted potential. In the full
potential $V_{650}$ of Eq. (8), one also has to consider the fields in non-
singlet directions. By construction, these were taken with zero VEVs and the
local minima of the restricted potential $V$ are stationary points of the full
potential $V_{650}$. Their ultimate fate, i.e. whether they become local
minima or saddle points of $V_{650}$, is addressed in Section 3.2.2.

####  3.2.2 The scalar spectrum in different vacua

Armed with the solutions for stationarity points for different cases of
subgroup $G$, the obvious next step is to determine the conditions under which
these are (local) minima of the potential $V$ in Eq. (8) rather than merely
saddle points. In other words, we are seeking conditions under which the
potential is stabilized in all $650$ field directions. This is easily
addressed by considering the masses of all the states in the $\mathbf{650}$,
which can be grouped according to the residual symmetry $G$.

We provide the masses of representations in different vacua in a series of
tables. The masses for the $G_{333}$ vacuum are listed in Table 6, for the
$G_{62}$ case in Table 7, and for the $G_{10,1}$ case in Table 8. In all
tables we also specify for each representation $R$ whether the fields inside
are complex or real; the reader can verify in each case that the total number
of real degrees of freedom adds up to $650$. Since the $\mathrm{F}_{4}$ and
$\mathrm{SU}(3)\times\mathrm{G}_{2}$ vacua are not of phenomenological
relevance, we omit the computation of associated masses for these cases.

The results in the tables were obtained by differentiating the full potential
of Eq. (8) with respect to two appropriately-transforming fields $\phi_{a}$
and $\phi_{b}$ (components of the in principle complex representation $R$ of
$G$) from $\mathbf{X}\sim\mathbf{650}$, and inserting the vacuum condition,
i.e. $(m^{2})^{a}{}_{b}=\partial^{a\ast}\partial_{b}V_{650}|$. In order to
obtain simpler expressions, we solved the stationarity conditions for $M^{2}$
instead of the VEV $s$ (or $\tilde{s}$). Note that for a fixed parameter point
$(M^{2},m_{k},\lambda_{l})$, the VEVs take different values in different
tables. Explicitly, $s$ in Tables 6 and $\tilde{s}$ in Tables 7 and 8 take the
value of the solution in Eq. (18) that corresponds to a local minimum of the
restricted potential $V$ (identified by $\partial^{2}V$ via Eq. (20)), where
the parameters $m$ and $\lambda$ are $G$-dependent and are determined from Eq.
(13) and Table 5. For trinification masses in Table 6, we chose the LR-
symmetric vacuum with $a=0$ out of the three discrete possibilities.

Table 6: Masses of irreducible representations in the $\mathrm{SU}(3)_{C}\times\mathrm{SU}(3)_{L}\times\mathrm{SU}(3)_{R}$ vacuum with LR parity. While $a=0$ has already been inserted, $s$ is a local minimum of $V$ determined from Eqs. (18)–(20), where $m$ and $\lambda$ are computed from Eq. (13) and Table 5.  irreps $R$ of $G_{333}$ | $\mathbb{R}/\mathbb{C}$ | masses-square $m^{2}_{R}$
---|---|---
$(\mathbf{1},\mathbf{1},\mathbf{1})_{1}$ | $\mathbb{R}$ | $\tfrac{1}{36}s\left(4s(18\lambda_{1}+\lambda_{2}+7\lambda_{3}+34\lambda_{4}+4\lambda_{5})-3\sqrt{3}(m_{1}+10m_{2})\right)$
$(\mathbf{1},\mathbf{1},\mathbf{1})_{2}$ | $\mathbb{R}$ | $\tfrac{1}{4}\sqrt{3}s(m_{1}+10m_{2})$
$(\mathbf{8},\mathbf{1},\mathbf{1})$ | $\mathbb{R}$ | $\tfrac{1}{4}s\left(\sqrt{3}m_{1}-2\sqrt{3}m_{2}+2s(4\lambda_{4}-\lambda_{5})\right)$
$(\mathbf{1},\mathbf{8},\mathbf{1})$, $(\mathbf{1},\mathbf{1},\mathbf{8})$ | $\mathbb{R}$ | $\tfrac{1}{12}s\left(18\sqrt{3}m_{2}+s(\lambda_{2}-5\lambda_{3}-20\lambda_{4}-5\lambda_{5})\right)$
$(\mathbf{3},\mathbf{\bar{3}},\mathbf{\bar{3}})_{1}$ | $\mathbb{C}$ | $\tfrac{1}{18}s\left(3\sqrt{3}(m_{1}+10m_{2})-2s(\lambda_{3}+2(8\lambda_{4}+\lambda_{5}))\right)$
$(\mathbf{3},\mathbf{\bar{3}},\mathbf{\bar{3}})_{2}$ | $\mathbb{C}$ | $0$
$(\mathbf{3},\mathbf{6},\mathbf{\bar{3}})$, $(\mathbf{3},\mathbf{\bar{3}},\mathbf{6})$ | $\mathbb{C}$ | $s\left(\sqrt{3}m_{2}-\tfrac{1}{6}s(12\lambda_{4}+\lambda_{5})\right)$
$(\mathbf{6},\mathbf{3},\mathbf{3})$ | $\mathbb{C}$ | $\tfrac{1}{12}s\left(3\sqrt{3}(m_{1}+2m_{2})-4s(4\lambda_{4}+\lambda_{5})\right)$
$(\mathbf{8},\mathbf{8},\mathbf{1})$, $(\mathbf{8},\mathbf{1},\mathbf{8})$ | $\mathbb{R}$ | $\tfrac{1}{4}s\left(\sqrt{3}(m_{1}+4m_{2})-s(8\lambda_{4}+\lambda_{5})\right)$
$(\mathbf{1},\mathbf{8},\mathbf{8})$ | $\mathbb{R}$ | $\tfrac{1}{12}s\left(2s(\lambda_{2}+\lambda_{3}-8\lambda_{4}+\lambda_{5})-3\sqrt{3}(m_{1}-2m_{2})\right)$
Table 7: Masses of irreducible representations in the $\mathrm{SU}(6)\times\mathrm{SU}(2)$ vacuum, where $\tilde{s}$ is a local minimum of $V$ determined from Eqs. (18)–(20), where $m$ and $\lambda$ are computed from Eq. (13) and Table 5.  irrep $R$ of $G_{62}$ | $\mathbb{R}/\mathbb{C}$ | mass-square $m^{2}_{R}$
---|---|---
$(\mathbf{1},\mathbf{1})$ | $\mathbb{R}$ | $\tfrac{1}{360}\tilde{s}\,\left(4\tilde{s}\,(180\lambda_{1}+7\lambda_{2}+67\lambda_{3}+652\lambda_{4}-8\lambda_{5})-3\sqrt{30}\,(m_{1}-44m_{2})\right)$
$(\mathbf{35},\mathbf{1})$ | $\mathbb{R}$ | $\tfrac{1}{120}\tilde{s}\,\left(2\tilde{s}\,(5\lambda_{2}-19\lambda_{3}-208\lambda_{4}+2\lambda_{5})-3\sqrt{30}\,(m_{1}+28m_{2})\right)$
$(\mathbf{20},\mathbf{2})$ | $\mathbb{R}$ | $0$
$(\mathbf{35},\mathbf{3})$ | $\mathbb{R}$ | $\tfrac{1}{40}\tilde{s}\,\left(4\tilde{s}\,(\lambda_{2}+\lambda_{3}-32\lambda_{4}-2\lambda_{5})-3\sqrt{30}\,(m_{1}+4m_{2})\right)$
$(\mathbf{70},\mathbf{2})$ | $\mathbb{C}$ | $-\tfrac{3}{40}\tilde{s}\,\left(6\sqrt{30}\,m_{2}+\tilde{s}\,(52\lambda_{4}-\lambda_{5})\right)$
$(\mathbf{189},\mathbf{1})$ | $\mathbb{R}$ | $\tfrac{1}{40}\tilde{s}\,\left(3\sqrt{30}\,(m_{1}-4m_{2})+2\tilde{s}\,(\lambda_{2}+\lambda_{3}-64\lambda_{4}+6\lambda_{5})\right)$
Table 8: Masses of irreducible representations in the $\mathrm{SO}(10)\times\mathrm{U}(1)$ vacuum, where $\tilde{s}$ is a local minimum of $V$ determined from Eqs. (18)–(20), where $m$ and $\lambda$ are computed from Eq. (13) and Table 5.  irrep $R$ of $G_{10,1}$ | $\mathbb{R}/\mathbb{C}$ | mass-square $m^{2}_{R}$
---|---|---
$(\mathbf{1},+0)$ | $\mathbb{R}$ | $\tfrac{1}{720}\tilde{s}\left(-3\sqrt{30}(29m_{1}+20m_{2})+2\tilde{s}(720\lambda_{1}+397\lambda_{2}+97\lambda_{3}+520\lambda_{4}+40\lambda_{5})\right)$
$(\mathbf{10},+6)$ | $\mathbb{C}$ | $\tfrac{1}{240}\tilde{s}\left(3\sqrt{30}(4m_{2}-5m_{1})+\tilde{s}(65\lambda_{2}+185\lambda_{3}-112\lambda_{4}+26\lambda_{5})\right)$
$(\mathbf{16},-3)$ | $\mathbb{C}$ | $0$
$(\mathbf{45},+0)$ | $\mathbb{R}$ | $\tfrac{1}{80}\tilde{s}\left(9\sqrt{30}(m_{1}+4m_{2})-6\tilde{s}(7\lambda_{2}-5\lambda_{3}-8\lambda_{4}+4\lambda_{5})\right)$
$(\mathbf{54},+0)$ | $\mathbb{R}$ | $\tfrac{1}{240}\tilde{s}\left(21\sqrt{30}m_{1}-60\sqrt{30}m_{2}+\tilde{s}(-127\lambda_{2}+173\lambda_{3}+200\lambda_{4}+140\lambda_{5})\right)$
$(\mathbf{144},-3)$ | $\mathbb{C}$ | $\tfrac{1}{96}\tilde{s}\left(12\sqrt{30}(m_{1}+m_{2})-\tilde{s}(52\lambda_{2}-32\lambda_{3}+76\lambda_{4}+\lambda_{5})\right)$
$(\mathbf{210},+0)$ | $\mathbb{R}$ | $\tfrac{1}{240}\tilde{s}\left(3\sqrt{30}(13m_{1}+4m_{2})-4\tilde{s}(31\lambda_{2}+\lambda_{3}+7(4\lambda_{4}+\lambda_{5}))\right)$

Considering the labeling in the tables: when there is more than one copy of
the representation $R$ of subgroup $G$ in the $\mathbf{650}$, we distinguish
the eigenstates by labeling them with a numeric index. This happens only for
group $G_{333}$ in Table 6.

For a given parameter point $(M^{2},m_{k},\lambda_{l})$, the VEV solution
associated to the symmetry $G$ is a local minimum of the potential $V_{650}$
if all the masses are non-tachyonic for that case.444Note that in a realistic
model, other scalar representations would be needed beside the $\mathbf{650}$.
The newly introduced states would also need to be non-tachyonic, but since
their mass expressions would necessarily involve only new parameters, that
analysis is of no relevance to the parameters in $V_{650}$ considered here
regardless of the scalar sector extension. We analyze the non-tachyonic
regions of parameter space numerically in Section 3.3, but conclude the
discussion here with some observations on the analytic form of mass
expressions:

  * •

A vanishing mass expression in Tables 6–8 corresponds to the would-be
Goldstone states, as confirmed by the lists of broken generators denoted by
non-underlined terms in the decomposition of $\mathbf{78}$ found in Table 2.
Note that this adjoint $\mathbf{78}$ is taken as a real representation, so the
complex representations in the decomposition have their degrees of freedom
listed twice — both in the original form and as a complex conjugate. In
particular, this remark applies for the
$(\mathbf{3},\mathbf{\bar{3}},\mathbf{\bar{3}})$ of $G_{333}$ and the
$(\mathbf{16},-3)$ of $G_{10,1}$.

The correct would-be Goldstone structure is a non-trivial consistency check of
our vacuum and mass computation.

  * •

The masses-square have a structure consistent with power-counting of mass
dimensions. The only dimensionful quantities in the theory are $m_{1}$,
$m_{2}$ and $M^{2}$, with the the latter being absorbed into the VEV $s$ or
$\tilde{s}$ in the mass expressions. The mass-square expressions can thus be
formed only from $7$ linearly independent terms: $m_{k}\tilde{s}$ ($k=1,2$)
arising from cubic terms in the scalar potential, and
$\lambda_{l}\tilde{s}^{2}$ (for $l=1,\ldots,5$) arising from quartic terms.
For the $G_{333}$ case, the same applies if we replace $\tilde{s}$ with $s$ in
the argumentation.

The explicit forms in the tables confirm this analytic structure. Furthermore,
the masses within each table are linearly independent; the exception is the
case $G_{333}$, where $9$ different mass-square expressions span a
$7$-dimensional linear space, implying $2$ independent mass relations.
Specifically, the $2$ mass sum rules in the trinification case can be written
as

$\displaystyle
2\,m^{2}_{(\mathbf{8},\mathbf{1},\mathbf{1})}+8\,m^{2}_{(\mathbf{8},\mathbf{8},\mathbf{1})}$
$\displaystyle=m^{2}_{(\mathbf{1},\mathbf{1},\mathbf{1})_{2}}+9\,m^{2}_{(\mathbf{6},\mathbf{3},\mathbf{3})},$
(22) $\displaystyle
m^{2}_{(\mathbf{1},\mathbf{8},\mathbf{8})}+9\,m^{2}_{(\mathbf{3},\mathbf{\bar{3}},\mathbf{\bar{3}})_{1}}+3\,m^{2}_{(\mathbf{6},\mathbf{3},\mathbf{3})}$
$\displaystyle=2\,m^{2}_{(\mathbf{1},\mathbf{1},\mathbf{1})_{2}}+2\,m^{2}_{(\mathbf{1},\mathbf{8},\mathbf{1})}+3\,m^{2}_{(\mathbf{3},\mathbf{6},\mathbf{\bar{3}})}+6\,m^{2}_{(\mathbf{8},\mathbf{8},\mathbf{1})}.$
(23)

  * •

All $3$ vacua in the trinification case $G_{333}$ are equally suitable, i.e.
have the same potential value. We chose the LR-symmetric case with $a=0$,
which then manifests in a LR-symmetric scalar spectrum, as seen from mass
degeneracies in Table 6. The spectra of the other two vacua consist of the
same masses reshuffled among the representations, in accordance with the
action of the parity that transforms the LR-symmetric vacuum to the desired
one.

The set of $3$ solutions in the trinification case also indicates a possible
formation of domain walls during the first stage of symmetry breaking. Also,
$\mathbb{Z}_{2}$-strings are the other type of possible topological defect
arising due to preserved parity, as has been pointed out for the $D$-parity
trinification case in Ref. Chakrabortty:2019fov . We don’t purse here the
question of topological defect further; they are not problematic for cosmology
if the scale of inflation is lower than the breaking scale.

  * •

Curiously, the mass of the second trinification singlet
$(\mathbf{1},\mathbf{1},\mathbf{1})_{2}$ in Table 6 is a linear combination of
the massive parameters $m_{k}$, and hence vanishes if one takes the limit
$m_{k}\to 0$.

This can be understood directly from the restricted potential $V(s,a)$. Taking
$m_{k}=0$ removes the cubic terms, and the remaining quadratic and quartic
terms are functions only of the combination $s^{2}+a^{2}$. In this limit the
restricted potential $V$ thus becomes rotationally symmetric in the $(s,a)$
plane, and the vacuum manifold of $V$ becomes a circle, i.e. $1$-dimensional,
on which all $P_{i}$ for $i$ from $1$ to $6$ lie. The second $G_{333}$-singlet
mass eigenstate can thus be interpreted as a pseudo-Goldstone boson associated
to the explicit breaking of the $\mathrm{SO}(2)$ rotational symmetry in the
$(s,a)$-plane by cubic terms with parameters $m_{k}$. Note, however, that in
the $m_{k}\to 0$ limit only the restricted potential $V$ in the
$G_{333}$-singlet plane becomes rotationally symmetric, and not the full
potential $V_{650}$. This is not surprising, since the aforementioned
$\mathrm{SO}(2)$ symmetry belongs to a $\mathrm{SO}(650)$ group of
transformations of the real representation $\mathbf{650}$, but not to the
smaller $\mathrm{E}_{6}$. In particular, the $M^{2}$ and $\lambda_{1}$ terms
are symmetric under this $\mathrm{SO}(2)$, while $\lambda_{l}$ for $l=2,3,4,5$
explicitly break it in the full space and preserve it only in the
$(s,a)$-plane.

###  3.3 Candidates for global minima

As a final step in our vacuum analysis, we (partially) address the question of
the global minimum of the potential $V_{650}$ in Eq. (8). This is achieved by
comparing the value of the potential $V_{650}$ of the solutions to vacua
corresponding to the five “relevant” cases of Table 2 that we found in Section
3.2, and we refer to the deepest among them as the “global” minimum.

We now proceed as follows: the procedure for determining the “global” minimum
for a given parameter point and the associated caveats are discussed in
Section 3.3.1, while concrete numerical results are then given in Section
3.3.2.

####  3.3.1 Procedure and limitations

First, let us specify the procedure by which a parameter point
$(M^{2},m^{2}_{k},\lambda_{l})$ is assessed, in part summarizing the results
obtained from local analyses. We perform the following computational steps:

  1. 1.

The potential $V_{650}$ must be bounded from below. We test this only in the
singlet directions; in particular, we confirm that
$\sum_{i}\beta_{i}\lambda_{i}>0$ for all cases of $\vec{\beta}$ in Table 5. If
this criterion is not passed, we do not consider the parameter point as
viable.

  2. 2.

All VEV solutions for the five “relevant” breaking possibilities in Table 2
are determined from Eq. (18), where $m$ and $\lambda$ are determined from Eq.
(13) and Table 5.

  3. 3.

Each VEV solution, together with its associated values of $m$ and $\lambda$,
is inserted back into $V$ of Eq. (12) to determine the corresponding values of
the potential in the candidate points. Additionally, the potential value for
the stationary point with a vanishing VEV $\langle\mathbf{650}\rangle=0$,
corresponding to the unbroken $\mathrm{E}_{6}$ phase, is considered.

  4. 4.

For the regular cases of Table 2, which contain the SM group and are thus the
physically relevant ones, we compute the masses for each of their VEV
solutions using Tables 6–8 and check for possible tachyonicity. We thus
unambiguously determine which cases correspond to local minima.

  5. 5.

The case with the lowest value of the potential that is also a determined to
be a local minimum is then considered the sole candidate for the “global”
minimum. It is considered a viable “global” minimum if it also passes the
following two physically motivated constraints:

     * (i)

The cubic couplings $m_{1}$ and $m_{2}$ should not be too large compared to
the heaviest scalar mass (computed in the “global” minimum candidate only),
otherwise the scattering amplitudes become non-unitary at low energies, see
e.g. Staub:2018vux .

     * (ii)

A similar limitation applies to the gauge boson masses in the “global” minimum
candidate: they should not be much larger than the heaviest scalar state lest
the tree-level analysis becomes threatened by large one-loop corrections to
the potential Coleman:1973jx .

For both criteria, the “much heavier” threshold is taken to be a factor $10$
at the level of masses-square, and is explicitly implemented as the following
simple criteria:555In practice, the criteria of (24) are unlikely to be
violated for a random parameter point; their exact formulation is thus not
crucial, and we essentially use them as a very simple way of confirming that
the breaking phenomenon occurs at roughly a single energy scale, which is not
destabilized by any of the massive parameters or VEVs at the quantum level.

$\displaystyle\max\\{m_{1}^{2},m_{2}^{2}\\}$ $\displaystyle\leq
10\,\max_{R}\,m^{2}_{R},$ $\displaystyle m^{2}_{\text{gauge}}$
$\displaystyle\leq 10\,\max_{R}\,m^{2}_{R},$ (24)

where $R$ goes over the irreducible scalar representations of $G$ (the
residual symmetry in the “global” minimum candidate) present in the
$\mathbf{650}$, and $m^{2}_{\text{gauge}}$ labels the gauge boson mass from
Table 5 when breaking into $G$ with the unified gauge coupling value taken as
$g\simeq 0.5$.

The above procedure allows to asses any point in parameter space and determine
its “global” minimum, which in turn serves as our best estimate for the actual
global minimum of the scalar potential. The procedure, however, does have some
caveats:

  * •

Local minima that are lower than the “global” one may in principle exist,
since we did not fully classify all VEV solutions in the $650$-dimensional
space of scalars. As discussed in Section 2.2, however, the only other
plausible minima consistent with Michel’s conjecture (assuming discrete
remnants of $\mathrm{E}_{6}$ are not present) lead to subgroups of the last
three cases of Table 2. Indeed, the representation $\mathbf{650}$ contains
singlets under all maximal subgroups of the three cases; we list all these
candidates in Table 9. Among them, the candidates consistent with Michel are
those who are not simultaneously subgroups of the “relevant” cases (up to
conjugation).

We do not proceeded further with this analysis, but one can argue that these
small groups are not favored to produce a local minimum in most of the
parameter space: under small subgroups of $\mathrm{E}_{6}$, the $\mathbf{650}$
decomposes into a large number of irreducible representations, leading to a
large set of non-tachyonicity conditions that would all need to simultaneously
hold.

  * •

The “global” analysis searches for the global minimum of the potential among
the local minima. Consider though the well-known counterexample of a “runaway”
potential in two dimensions:

$\displaystyle V(x,y)$ $\displaystyle=(M^{2}-xy)^{2}+m^{2}\,x^{2},$ (25)

where $m,M>0$. Clearly $V\geq 0$. The only stationary point is $(0,0)$ and it
is a saddle point, the value $V=0$ cannot be reached by any $(x,y)$, while for
any $\epsilon>0$ the point
$(x_{\epsilon},y_{\epsilon})=(m\sqrt{\epsilon},\frac{M^{2}}{m\sqrt{\epsilon}})$
gives the potential value $V(x_{\epsilon},y_{\epsilon})=\epsilon\,m^{4}$,
which is arbitrarily close to zero for a small enough $\epsilon$. This
potential is bounded from below, but does not have an absolute minimum at any
point.

The existence of the global minimum for a continuous potential is guaranteed
only in a compact domain, and the key feature of the counterexample is that
the potential does not diverge at infinity in all directions. Such a pathology
though is unlikely to arise in our $V_{650}$ case for the generic parameter
point, since already the quartic invariant associated to the parameter
$\lambda_{1}$ in Eq. (8) diverges at infinity for all directions of
$\mathbf{X}$.

Table 9: The maximal subgroups of groups $G$, which are in turn maximal subgroups of $\mathrm{E}_{6}$. We listed only those $G$ which don’t already have a $G$-invariant state in the $\mathbf{650}$.  $G$ | maximal subgroups of $G$
---|---
$G_{2}$ |  $\mathrm{SU}(3)$, $\mathrm{SU}(2)\times\mathrm{SU}(2)$, $\mathrm{SU}(2)$
$\mathrm{SU}(3)$ |  $\mathrm{SU}(2)\times\mathrm{U}(1)$, $\mathrm{SU}(2)$
$\mathrm{Sp}(8)$ |  $\mathrm{SU}(4)\times\mathrm{U}(1)$, $\mathrm{SU}(2)\times\mathrm{Sp}(6)$, $\mathrm{Sp}(4)\times\mathrm{Sp}(4)$, $\mathrm{SU}(2)$, $\mathrm{SU}(2)^{3}$
  * •

Boundedness from below was checked only in the $5$ main VEV directions, i.e.
$\lambda>0$ for all cases in Table 5. This is only a necessary condition that
the full potential $V_{650}$ is bounded from below.

Formulating the most general sufficient conditions for boundedness from below
in the full $650$-dimensional space is a difficult task. Although boundedness
can be checked in some special cases, e.g. in the case of an adjoint of
$\mathrm{SU}(5)$ MacKenzie:1982fi , we are not aware of a general procedure.
We do make the observation, however, that the invariants with coefficients
$\lambda_{1}$ and $\lambda_{2}$ in Eq. (8) give only positive directions for
quartic terms, since $\mathbf{X}$ is a Hermitian matrix. Therefore,
qualitatively, one sufficient condition for boundedness from below is to have
$\lambda_{1},\lambda_{2}>0$ dominating over the other quartic terms, i.e.
$\lambda_{1,2}\gg|\lambda_{3,4,5}|$.

Quantitatively, the argument can be pushed a bit further by noting that the
other quartic invariants involve the invariant tensors $d$ and $D$, which
consist only of numeric values $\pm 1$ or $0$, see Appendix A.1. It thus seems
likely that a condition in the form

$\displaystyle\lambda_{1}+\lambda_{2}>C\,\left(|\lambda_{3}|+|\lambda_{4}|+|\lambda_{5}|\right)$
(26)

would be sufficient for boundedness from below for some unknown positive
constant $C\sim\mathcal{O}(1)$.

  * •

The entire analysis is performed only at tree-level. Quantum corrections to
the effective scalar potential may be large, especially considering the large
number of fields that can run in the loops. This consideration may
considerably shrink the range of couplings in which the computation is
amenable to perturbative methods, see e.g. Jarkovska:2021jvw ;
Jarkovska:2023zwv  for a recent example from $\mathrm{SO}(10)$ GUT.

We do not pursue here this issue further, and simply limit the region for
dimensionless parameters to $|\lambda_{l}|\leq 1$.

Keeping in mind the limitations discussed above, the analysis of the
“relevant” cases is nevertheless expected to yield the best candidate for the
global minimum.

####  3.3.2 Results for “global” minima

We now investigate the parameter space $(M^{2},m_{k},\lambda_{l})$ for
“global” minima with a symmetry among the “relevant” cases in accordance with
the considerations of Section 3.3.1. The parameters are those present in
$V_{650}$ of Eq. (8).

Table 10: A list of benchmark points. We provide for each a label, the parameter values $(M^{2},m_{k},\lambda_{l})$, the existence of local minima for regular cases ($\checkmark$ for yes and $-$ for no), and the symmetry in the “global” minimum.  point | $M^{2}$ | $m_{1}$ | $m_{2}$ | $\lambda_{1}\phantom{0}$ | $\lambda_{2}$ | $\lambda_{3}$ | $\lambda_{4}$ | $\lambda_{5}\phantom{0}$ | $G_{10,1}$ | $G_{62}$ | $G_{333}$ | “global” minimum
---|---|---|---|---|---|---|---|---|---|---|---|---
A | $-1$ | $1.0$ | $0.0$ | $0.2\phantom{0}$ | $0.0$ | $0.2$ | $0.0$ | $0.0\phantom{0}$ | $\checkmark$ | $-$ | $-$ | $\mathrm{SO}(10)\times\mathrm{U}(1)$
B | $-1$ | $0.0$ | $1.0$ | $0.2\phantom{0}$ | $0.0$ | $0.0$ | $0.0$ | $0.0\phantom{0}$ | $-$ | $\checkmark$ | $-$ | $G_{62}$
C | $-1$ | $0.5$ | $1.0$ | $0.15$ | $0.2$ | $0.2$ | $0.2$ | $0.15$ | $-$ | $-$ | $\checkmark$ | $G_{333}$
D | $-1$ | $1.0$ | $1.0$ | $0.1\phantom{0}$ | $0.0$ | $0.0$ | $0.0$ | $0.0\phantom{0}$ | $-$ | $\checkmark$ | $-$ | $\mathrm{F}_{4}$
E | $-1$ | $1.0$ | $1.0$ | $0.5\phantom{0}$ | $0.6$ | $0.0$ | $0.2$ | $0.0\phantom{0}$ | $\checkmark$ | $\checkmark$ | $-$ | $\mathrm{SU}(3)\times\mathrm{G}_{2}$
F | $-1$ | $0.6$ | $0.7$ | $0.3\phantom{0}$ | $0.0$ | $0.3$ | $0.2$ | $0.4\phantom{0}$ | $\checkmark$ | $\checkmark$ | $\checkmark$ | $\mathrm{SO}(10)\times\mathrm{U}(1)$

First, we provide a list of benchmark points in Table 10 that serves as an
existence proof for various scenarios. The points were carefully selected so
that the following remarks can be made:

  * •

Notice that points A–E cover all the symmetry cases for a global minimum we
considered from Table 2.

  * •

Points A–C correspond to situations, where the global minimum breaks to one of
the regular subgroups. For each of these there are no local minima of the
potential corresponding to the other two regular cases, since they are
destabilized by tachyonicity and the associated stationary points are saddle
points of the full potential $V_{650}$.

  * •

Case F is an example where a local minimum exists for all $3$ types of regular
cases. The $\mathrm{SO}(10)\times\mathrm{U}(1)$ minimum is the deepest among
them.

  * •

Points D and E break into one of the special cases, where only the depth of
the minimum was checked and not the masses. Although local minima to regular
cases exist for these points, they are metastable.

Second, we perform a search for points where the “global” minimum is in one of
the regular scenarios of Table 2, i.e. one of the phenomenologically
interesting cases. Although a local minimum with a long enough vacuum lifetime
would already be phenomenologically viable, the demand for a “global” minimum
ensures exclusivity of each parameter point and thus yields a phase diagram in
the $8$-dimensional space $(M^{2},m_{k},\lambda_{l})$.

We limit our search to the parameter region

$\displaystyle|M^{2}|$ $\displaystyle\leq 1,$ $\displaystyle|m_{k}|$
$\displaystyle\leq 1,$ $\displaystyle|\lambda_{l}|$ $\displaystyle\leq 1,$
$\displaystyle m_{1}$ $\displaystyle\geq 0.$ (27)

The limitation on the dimensionless couplings $\lambda_{l}$ is essentially a
rough perturbativity bound, cf. Section 3.3.1, while $m_{1}$ can be made non-
negative by redefining the sign of $\mathbf{X}$ in $V_{650}$ if necessary.

Note that the “global” analysis does not change qualitatively under mass
rescaling, i.e. under $m_{k}\mapsto\eta\,m_{k}$ and
$M^{2}\mapsto\eta^{2}\,M^{2}$, so there is a one parameter redundancy in the
space $(M^{2},m_{k},\lambda_{l})$. When $M^{2}\neq 0$, one could in principle
rescale the dimensionful parameters so that $M^{2}=\pm 1$ and eliminate the
redundancy, but a separate analysis is then required for each discrete case.
Since this obscures the smooth transition in quantitative results when $M^{2}$
changes sign, we rather perform the analysis in the full parameter space. In
this context, the limits on $M^{2}$ and $m_{k}$ in Eq. (27) are not important,
provided they define a region with equal opportunity for either $M^{2}$ or
$m_{k}$ to dominate.

For each regular case, we scan for a dataset of $10^{4}$ viable parameter
points. The search was performed by first finding a starting generation of
points in each case, and then obtaining new ones by use of a stochastic
variant of the differential evolution algorithm diffevolalg  (version
“DE/rand/1” with a random choice of $F\in(0.5,2)$ for each point).

To present the results visually, the points must be projected from the 8D
space $(M^{2},m_{k},\lambda_{l})$ down to 1D or 2D for plotting. Colors encode
the different datasets: red for trinification $\mathrm{SU}(3)^{3}$ (left bar),
blue for $\mathrm{SU}(6)\times\mathrm{SU}(2)$ (middle bar), and green for
$\mathrm{SO}(10)\times\mathrm{U}(1)$ (right bar).

Figure 2: The $1$\- and $2$-$\sigma$ HDI ranges (with decreasing opacity) for
the massive parameters $m_{k}$ and the dimensionless couplings $\lambda_{k}$.
The colors denote the different cases of the resulting global minimum, as
shown in the legend.

We show the 1D highest density intervals (HDIs) for separate input parameters
in Figure 2. Due to mass-scale redundancy, we show only dimensionless ratios
of mass parameters. Also, $m_{1}/|M|>0$ in accordance with Eq. (27). The
vertical bars indicate the $1$\- and $2$-$\sigma$ HDIs in decreasing opacity.

Additionally, it is also instructive to show 2D correlation plots for specific
pairs of inputs. Figure 3 shows scatter plots in the
$(m_{2}/m_{1})$-$\lambda_{4}$ and $\lambda_{3}$-$\lambda_{4}$ planes. These
choices of pairs best discriminate between the various regular cases.

Figure 3: Correlation scatter plots in the planes
$(m_{2}/m_{1})$-$\lambda_{4}$ (left panel) and $\lambda_{3}$-$\lambda_{4}$
(right panel). The colors denote different cases, see the legends.

The results from Figures 2 and 3 show the (projected) regions, where parameter
values most often lead to a given type of “global” minimum. One can make the
following observations:

  * •

“Global” minima with $G_{333}$ or $G_{62}$ symmetry prefer large ratios
$|m_{2}/m_{1}|$, while $G_{10,1}$ prefers small such ratios, large positive
values of $\lambda_{3}$ and negative values of $\lambda_{2}$.

  * •

The parameter $\lambda_{4}$ is best for discriminating “globally” between the
$G_{333}$ and $G_{62}$ cases.

  * •

There is a high preference for $\lambda_{1}>0$ in all cases, and $\lambda_{3}$
and $\lambda_{4}$ also prefer positive ranges. This is likely due to the
conditions for boundedness from below of the full potential $V_{650}$, cf.
Section 3.3.1.

In summary, we see from Table 10 that “global” minima for all “relevant” cases
exist somewhere in parameter space, while Figures 2 and 3 reveal for each
regular case the preferred (projected) regions of parameters.

##  4 Conclusions

Unlike the smaller GUT symmetries $\mathrm{SU}(5)$ or $\mathrm{SO}(10)$, the
group $\mathrm{E}_{6}$ offers some unique opportunities in GUT model building,
namely a breaking chain to the SM group through $\mathrm{SU}(3)^{3}$
(trinification) or $\mathrm{SU}(6)\times\mathrm{SU}(2)$. The real
representation $\mathbf{650}$ of $\mathrm{E}_{6}$ is the minimal and only
realistic way to implement these breaking chains, since the next-largest
representation of $\mathrm{E}_{6}$ with $G_{333}$\- or $G_{62}$-singlets is of
dimension $2430$. A Yang-Mill theory with only a real scalar $\mathbf{650}$ is
still asymptotically free, while the presence of $\mathbf{2430}$ leads
inextricably to a Landau pole within an order of magnitude.

We investigated in this paper the most general renormalizable scalar potential
with a single copy of the representation $\mathbf{650}$, which consists of one
quadratic, two cubic and five quartic linearly independent invariants. We
obtained explicit VEV solutions that break $\mathrm{E}_{6}$ to the following
cases:

  * (i)

the regular maximal subgroups $\mathrm{SU}(3)^{3}$,
$\mathrm{SU}(6)\times\mathrm{SU}(2)$ and $\mathrm{SO}(10)\times\mathrm{U}(1)$,

  * (ii)

the special maximal subgroups $\mathrm{F}_{4}$ and
$\mathrm{SU}(3)\times\mathrm{G}_{2}$.

Although this may not be a complete classification of solutions, we argued
that the above list consists of the most important cases, at least in most of
parameter space.

Phenomenologically, the important cases are those with a regular maximal
subgroup, since these still retain the SM group as a subgroup. For these
vacuum solutions we computed the scalar spectrum, thus studying when they
correspond to a local minimum of the full scalar potential. In a (partial)
global analysis, we then scanned the parameter space for points which support
each of these regular cases as the lowest minimum (among those analyzed);
indeed, non-empty regions were identified for all of them, effectively
producing a phase diagram withing the $8$-dimensional parameter space.

An aspect of the trinification solution worth emphasizing is that it also
preserves a discrete part of $\mathrm{E}_{6}$, analogous to $D$-parity in
$\mathrm{SO}(10)$: either LR, CR or CL parity remain unbroken, where
intuitively these parities exchange the factors of
$\mathrm{SU}(3)_{C}\times\mathrm{SU}(3)_{L}\times\mathrm{SU}(3)_{R}$. The
imprint of the remaining parity is manifest in the spectrum, and it may lead
to topological defects, in particular domain walls and strings.

Finally, the representation $\mathbf{650}$ has a an intriguing outlook in
$\mathrm{E}_{6}$ model building, especially in the non-supersymmetric case
where the breaking is expected to occur in multiple stages. This paper
addressed the possibilities for the first stage in $\mathrm{E}_{6}$ breaking
without further regard to additional phenomenological considerations or a
concrete model setup; it serves as a piece in the wider effort of
$\mathrm{E}_{6}$ GUT model building, including our future work.

## Acknowledgments

The work of KSB is supported in part by the U.S. Department of Energy under
grant number DE-SC0016013. BB acknowledges the financial support from the
Slovenian Research Agency (research core funding No. P1-0035 and in part
research grant J1-4389). VS acknowledges financial support from the Grant
Agency of the Czech Republic (GAČR) via Contract No. 20-17490S, as well as
from the Charles University Research Center Grant No. UNCE/SCI/013. We thank
Raymond Volkas for discussion and pointing out Ref. Lonsdale .

##  Appendix A The special embeddings of $\mathrm{F}_{4}$ and
$\mathrm{SU}(3)\times\mathrm{G}_{2}$ into $\mathrm{E}_{6}$

We provide in this Appendix some technical details on dealing with the Lie
algebra of $\mathrm{E}_{6}$, as well as the special embeddings of
$\mathrm{F}_{4}$ and $\mathrm{SU}(3)\times\mathrm{G}_{2}$.

Note that a special embedding $H\subset G$ breaks the rank of the larger group
$G$. Suppose we have a basis of generators in $G$, such that every generator
has well-defined quantum numbers of $G$ (this is the basis of lowering and
raising operators). Suppose we have an analogous basis of generators in $H$,
such that each of those have well-defined quantum numbers in $H$. Due to the
rank-breaking property of the maximal embedding, the basis of $H$ cannot be
simply a subset of the generators of $G$. Instead, some generators of $H$ will
necessarily have to be linear combinations of those in $G$ — a feature we
shall indeed see explicitly in the embeddings of $\mathrm{F}_{4}$ and
$\mathrm{SU}(3)\times\mathrm{G}_{2}$. This makes the analysis of special
embeddings more complicated, hence its relegation to the appendix.

The most convenient treatment of the exceptional groups $\mathrm{E}_{6}$,
$\mathrm{F}_{4}$ and $\mathrm{G}_{2}$ is perhaps by making use of their
maximal regular subgroups $\mathrm{SU}(3)^{n}$, see Table 11. We describe each
of these cases in turn, and specify explicit embeddings in the following
subsections.

Table 11: A comparison of exceptional groups $\mathrm{E}_{6}$, $\mathrm{F}_{4}$ and $\mathrm{G_{2}}$, and the branching rules for their adjoints to a maximal $\mathrm{SU}(3)^{n}$ in each case, which facilitates a convenient language for their description.  $G$ | adjoint | decomposition | maximal $\mathrm{SU}(3)^{n}\subset G$
---|---|---|---
$\mathrm{E}_{6}$ |  $\mathbf{78}$ | $\mathbf{8}_{C}+\mathbf{8}_{L}+\mathbf{8}_{R}+(\mathbf{3},\mathbf{\bar{3}},\mathbf{\bar{3}})+(\mathbf{\bar{3}},\mathbf{3},\mathbf{3})$ | $\mathrm{SU}(3)^{3}\quad\equiv\quad\mathrm{SU}(3)_{C}\times\mathrm{SU}(3)_{L}\times\mathrm{SU}(3)_{R}$
$\mathrm{F}_{4}$ |  $\mathbf{52}$ | $\mathbf{8}_{C}+\mathbf{8}_{LR}+(\mathbf{3},\mathbf{\bar{6}})+(\mathbf{\bar{3}},\mathbf{6})$ | $\mathrm{SU}(3)^{2}\quad\equiv\quad\mathrm{SU}(3)_{C}\times\mathrm{SU}(3)_{LR}$
$\mathrm{G}_{2}$ |  $\mathbf{14}$ | $\mathbf{8}+\mathbf{3}+\mathbf{\bar{3}}$ | $\mathrm{SU}(3)^{1}$

###  A.1 The algebra of $\mathrm{E}_{6}$

As already stated in Section 2.1, the $78$ generators of $\mathrm{E}_{6}$ can
be labeled by

$\displaystyle(T_{C})^{A},\quad(T_{L})^{A},\quad(T_{R})^{A},\quad
t^{\alpha}{}_{aa^{\prime}},\quad\bar{t}_{\alpha}{}^{aa^{\prime}},$ (28)

where $A$ is the adjoint $\mathrm{SU}(3)$ index (for the appropriate factor),
while $\\{\alpha,a,a^{\prime}\\}$ are fundamental for
$\mathrm{SU}(3)_{\\{C,L,R\\}}$. The most general element in the Lie algebra
$\mathfrak{e}_{6}$ is thus written as

$\displaystyle\alpha_{C}{}^{A}\,T_{C}^{A}+\alpha_{L}{}^{A}\,T_{L}^{A}+\alpha_{R}{}^{A}\,T_{R}^{A}+\beta_{\alpha}{}^{aa^{\prime}}\,t^{\alpha}{}_{aa^{\prime}}+\beta^{\ast\,\alpha}{}_{aa^{\prime}}\,\bar{t}_{\alpha}{}^{aa^{\prime}},$
(29)

where the $24$ coefficients $\alpha_{C}{}^{A}$, $\alpha_{L}{}^{A}$,
$\alpha_{R}{}^{A}$ are real, while the $27$ coefficients
$\beta_{\alpha}{}^{aa^{\prime}}$ are complex. For quick reference by “sector”,
we shall sometimes suppress the indices and simply refer to the generators as
$\\{T_{C},T_{L},T_{R},t,\bar{t}\\}$.

The explicit $\mathrm{E}_{6}$ commutation relations are then written as

$\displaystyle\big{[}T_{C}^{A},T_{R}^{B}\big{]}=\big{[}T_{R}^{A},T_{L}^{B}\big{]}=\big{[}T_{L}^{A},T_{C}^{B}\big{]}$
$\displaystyle=0,$ (30) $\displaystyle\big{[}T_{C}^{A},T_{C}^{B}\big{]}$
$\displaystyle=if^{ABD}\;T_{C}^{D},$ (31)
$\displaystyle\big{[}T_{L}^{A},T_{L}^{B}\big{]}$
$\displaystyle=if^{ABD}\;T_{L}^{D},$ (32)
$\displaystyle\big{[}T_{R}^{A},T_{R}^{B}\big{]}$
$\displaystyle=if^{ABD}\;T_{R}^{D},$ (33)
$\displaystyle\big{[}T_{C}^{A},{t}^{\alpha}{}_{aa^{\prime}}\big{]}$
$\displaystyle=-\tfrac{1}{2}(\lambda^{A})^{\alpha}{}_{\beta}\;{t}^{\beta}{}_{aa^{\prime}},$
(34) $\displaystyle\big{[}T_{L}^{A},{t}^{\alpha}{}_{aa^{\prime}}\big{]}$
$\displaystyle=\phantom{-}\tfrac{1}{2}(\lambda^{A})^{b}{}_{a}\;{t}^{\alpha}{}_{ba^{\prime}},$
(35) $\displaystyle\big{[}T_{R}^{A},{t}^{\alpha}{}_{aa^{\prime}}\big{]}$
$\displaystyle=\phantom{-}\tfrac{1}{2}(\lambda^{A})^{b^{\prime}}{}_{a^{\prime}}\;{t}^{\alpha}{}_{ab^{\prime}},$
(36) $\displaystyle\big{[}T_{C}^{A},\bar{t}_{\alpha}{}^{aa^{\prime}}\big{]}$
$\displaystyle=\phantom{-}\tfrac{1}{2}(\lambda^{A})^{\beta}{}_{\alpha}\;\bar{t}_{\beta}{}^{aa^{\prime}},$
(37) $\displaystyle\big{[}T_{L}^{A},\bar{t}_{\alpha}{}^{aa^{\prime}}\big{]}$
$\displaystyle=-\tfrac{1}{2}(\lambda^{A})^{a}{}_{b}\;\bar{t}_{\alpha}{}^{ba^{\prime}},$
(38) $\displaystyle\big{[}T_{R}^{A},\bar{t}_{\alpha}{}^{aa^{\prime}}\big{]}$
$\displaystyle=-\tfrac{1}{2}(\lambda^{A})^{a^{\prime}}{}_{b^{\prime}}\;\bar{t}_{\alpha}{}^{ab^{\prime}},$
(39)
$\displaystyle\big{[}{t}^{\alpha}{}_{aa^{\prime}},{t}^{\beta}{}_{bb^{\prime}}\big{]}$
$\displaystyle=-\varepsilon^{\alpha\beta\gamma}\;\varepsilon_{abc}\;\varepsilon_{a^{\prime}b^{\prime}c^{\prime}}\;\bar{t}_{\gamma}{}^{cc^{\prime}},$
(40)
$\displaystyle\big{[}\bar{t}_{\alpha}{}^{aa^{\prime}},\bar{t}_{\beta}{}^{bb^{\prime}}\big{]}$
$\displaystyle=\phantom{-}\varepsilon_{\alpha\beta\gamma}\;\varepsilon^{abc}\;\varepsilon^{a^{\prime}b^{\prime}c^{\prime}}\;{t}^{\gamma}{}_{cc^{\prime}},$
(41)
$\displaystyle\big{[}\bar{t}_{\alpha}{}^{aa^{\prime}},{t}^{\beta}{}_{bb^{\prime}}\big{]}$
$\displaystyle=(\lambda^{A})^{\beta}{}_{\alpha}\;\delta^{a}{}_{b}\;\delta^{a^{\prime}}{}_{b^{\prime}}\;T_{C}^{A}-\delta^{\beta}{}_{\alpha}\;(\lambda^{A})^{a}{}_{b}\;\delta^{a^{\prime}}{}_{b^{\prime}}\;T_{L}^{A}-\delta^{\beta}{}_{\alpha}\;\delta^{a}{}_{b}\;(\lambda^{A})^{a^{\prime}}{}_{b^{\prime}}\;T_{R}^{A}.$
(42)

These relation are a modified version of those from Kephart:1981gf . We use a
conjugate embedding for the C $\mathrm{SU}(3)$-factor relative to those of L
and R in $G_{333}$, in accordance with Eq. (1). Our preferred embedding is
phenomenologically more convenient and is also the more common one adopted in
the literature Slansky:1981yr ; Bajc:2013qra ; Babu:2015psa ; Feger:2012bs ;
Feger:2019tvk , while Kephart:1981gf  implements a more symmetric embedding
with the last two terms in Eq. (1) replaced by
$(\mathbf{3},\mathbf{3},\mathbf{3})\oplus(\mathbf{\bar{3}},\mathbf{\bar{3}},\mathbf{\bar{3}})$.

Note that the only objects necessary to define $\mathrm{E}_{6}$ commutation
relations are $\mathrm{SU}(3)$-related:

  1. 1.

The Gell-Mann matrices $\lambda^{A}$.

  2. 2.

The $\mathrm{SU}(3)$ structure constants $f^{ABC}$:

$\displaystyle[\tfrac{1}{2}\lambda^{A},\tfrac{1}{2}\lambda^{B}]$
$\displaystyle=if^{ABC}\,\tfrac{1}{2}\lambda^{C}.$ (43)

  3. 3.

The $\mathrm{SU}(3)$ invariant tensors $\varepsilon_{abc}$,
$\varepsilon^{abc}$, $\delta^{a}{}_{b}$ (the completely anti-symmetric Levi-
Civita and Kronecker delta tensors in three dimensions).

Schematically, the commutation relations by sector can be understood most
easily from Table 12.

Table 12: The schematic table of the $\mathrm{E}_{6}$ commutation relations by sector.  $[.,.]$ | $T_{C}$ | $T_{L}$ | $T_{R}$ | $t$ | $\bar{t}$
---|---|---|---|---|---
$T_{C}$ | $T_{C}$ | 0 | 0 | $t$ | $\bar{t}$
$T_{L}$ | 0 | $T_{L}$ | 0 | $t$ | $\bar{t}$
$T_{R}$ | 0 | 0 | $T_{R}$ | $t$ | $\bar{t}$
$t$ | $t$ | $t$ | $t$ | $\bar{t}$ | $T_{C,L,R}$
$\bar{t}$ | $\bar{t}$ | $\bar{t}$ | $\bar{t}$ | $T_{C,L,R}$ | $t$

For completeness, we now provide also the action of the $\mathrm{E}_{6}$
generators on the fundamental representation $\mathbf{27}$. The representation
$\mathbf{27}$ decomposes under the $G_{333}$ subgroup (in our chosen
embedding) as

$\displaystyle\mathbf{27}$
$\displaystyle=(3,3,1)\oplus(1,\bar{3},3)\oplus(\bar{3},1,\bar{3})\equiv
L^{\alpha a}\oplus M_{a}{}^{a^{\prime}}\oplus N_{a^{\prime}\alpha}.$ (44)

In this $(L^{\alpha a},M_{a}{}^{a^{\prime}},N_{a^{\prime}\alpha})$ notation
for the $\mathbf{27}$, the generators act via

$\displaystyle T_{C}^{A}\;\begin{pmatrix}L^{\alpha
a},&M_{a}{}^{a^{\prime}},&N_{a^{\prime}\alpha}\\\ \end{pmatrix}$
$\displaystyle=\begin{pmatrix}\tfrac{1}{2}(\lambda_{A})^{\alpha}{}_{\beta}\;L^{\beta
a},&0,&-\tfrac{1}{2}(\lambda^{\ast}_{A})_{\alpha}{}^{\beta}\;N_{a^{\prime}\beta}\\\
\end{pmatrix},$ (45) $\displaystyle T_{L}^{A}\;\begin{pmatrix}L^{\alpha
a},&M_{a}{}^{a^{\prime}},&N_{a^{\prime}\alpha}\\\ \end{pmatrix}$
$\displaystyle=\begin{pmatrix}\tfrac{1}{2}(\lambda_{A})^{a}{}_{b}\;L^{\alpha
b},&-\tfrac{1}{2}(\lambda^{\ast}_{A})_{a}{}^{b}\;M_{b}{}^{a^{\prime}},&0\\\
\end{pmatrix},$ (46) $\displaystyle T_{R}^{A}\;\begin{pmatrix}L^{\alpha
a},&M_{a}{}^{a^{\prime}},&N_{a^{\prime}\alpha}\\\ \end{pmatrix}$
$\displaystyle=\begin{pmatrix}0,&\tfrac{1}{2}(\lambda_{A})^{a^{\prime}}{}_{b^{\prime}}\;M_{a}{}^{b^{\prime}},&-\tfrac{1}{2}(\lambda^{\ast}_{A})_{a^{\prime}}{}^{b^{\prime}}N_{b^{\prime}\alpha}\\\
\end{pmatrix},$ (47) $\displaystyle
t^{\alpha}{}_{aa^{\prime}}\;\begin{pmatrix}L^{\alpha
a},&M_{a}{}^{a^{\prime}},&N_{a^{\prime}\alpha}\\\ \end{pmatrix}$
$\displaystyle=\begin{pmatrix}\varepsilon^{\alpha\beta\gamma}\;\delta^{b}{}_{a}\;N_{a^{\prime}\gamma},&-\varepsilon_{abc}\;\delta^{b^{\prime}}{}_{a^{\prime}}\;L^{\alpha
c},&-\varepsilon_{a^{\prime}b^{\prime}c^{\prime}}\;\delta^{\alpha}{}_{\beta}\;M_{a}{}^{c^{\prime}}\\\
\end{pmatrix},$ (48)
$\displaystyle\bar{t}_{\alpha}{}^{aa^{\prime}}\;\begin{pmatrix}L^{\alpha
a},&M_{a}{}^{a^{\prime}},&N_{a^{\prime}\alpha}\\\ \end{pmatrix}$
$\displaystyle=\begin{pmatrix}\varepsilon^{abc}\;\delta^{\beta}{}_{\alpha}\;M_{c}{}^{a^{\prime}},&\varepsilon^{a^{\prime}b^{\prime}c^{\prime}}\;\delta^{a}{}_{b}\;N_{c^{\prime}\alpha},&-\varepsilon_{\alpha\beta\gamma}\;\delta^{a^{\prime}}{}_{b^{\prime}}\;L^{\gamma
a}\\\ \end{pmatrix}.$ (49)

The generator action has again been taken from Kephart:1981gf  and modified in
accordance with our preferred $G_{333}$ embedding, cf. discussion below Eq.
(42). Note that Eqs. (45)–(49) allow for an explicit construction of
$\mathrm{E}_{6}$ generator matrices in the fundamental representation
$\mathbf{27}$. These indeed satisfy the commutation relations of Eqs.
(30)–(42). The explicit form of the generators provides the computational
machinery behind the explicit results in this paper.

Finally, the group $\mathrm{E}_{6}$ has its own primitive invariant tensors:
$\delta^{i}{}_{j}$, $d_{ijk}$ and $d^{ijk}$, where $i,j,k$ are fundamental
$\mathrm{E}_{6}$ indices and run from $1$ to $27$. Their components can be
extracted from the relation

$\displaystyle\tfrac{1}{3!}\,d_{ijk}\;\mathbf{27}^{i}\,\mathbf{27}^{j}\,\mathbf{27}^{k}$
$\displaystyle=-\mbox{det}\,\mathbf{L}+\mbox{det}\,\mathbf{M}-\mbox{det}\,\mathbf{N}-\mathrm{Tr}(\mathbf{LMN}),$
(50)

where the $\mathbf{27}^{i}$ was written in terms of matrices
$(\mathbf{L},\mathbf{M},\mathbf{N})$ as in Eq. (44). The tensors $d_{ijk}$ and
$d^{ijk}$ have the same numerical components, are completely symmetric, and in
a suitable basis all their entries are either $\pm 1$ or $0$, see Bajc:2013qra
. In that basis they have the normalization

$\displaystyle d^{ikl}d_{jkl}$ $\displaystyle=10\,\delta^{i}{}_{j}.$ (51)

###  A.2 The algebra of $\mathrm{F}_{4}$ and its embedding into
$\mathrm{E}_{6}$

The group $\mathrm{F}_{4}$ has $52$ generators. Making use of their
decomposition into irreducible representations of the
$\mathrm{SU}(3)_{C}\times\mathrm{SU}(3)_{LR}$ maximal subgroup, see Table 11,
the $\mathrm{F}_{4}$ generators can be labeled as follows:

$\displaystyle(\Phi_{C})^{A},\quad(\Phi_{LR})^{A},\quad\phi^{\alpha}{}_{ab},\quad\bar{\phi}_{\alpha}{}^{ab},$
(52)

where symmetry is imposed in $(a,b)$ (so that it forms a $6=(3\times 3)_{s}$
of $\mathrm{SU}(3)_{LR}$). As usual, $A$ is the adjoint index of
$\mathrm{SU}(3)$, and $\\{\alpha,a\\}$ are fundamental for
$\mathrm{SU}(3)_{C,LR}$. The subscripts $C$ and $LR$ in the maximal subgroup
$\mathrm{SU}(3)^{2}$ already anticipate a particular embedding into
$\mathrm{E}_{6}$. We abbreviate the labels of $\mathrm{F}_{4}$ generators by
sector as $\\{\Phi_{C},\Phi_{LR},\phi,\bar{\phi}\\}$.

One possible embedding is to define the $\mathrm{F}_{4}$ generators in terms
of $\mathrm{E}_{6}$ generators as follows:

$\displaystyle\Phi_{C}^{A}$ $\displaystyle:=T_{C}^{A},$ (53)
$\displaystyle\Phi_{LR}^{A}$ $\displaystyle:=T_{L}^{A}+T_{R}^{A},$ (54)
$\displaystyle\phi^{\alpha}{}_{ab}$
$\displaystyle:=t^{\alpha}{}_{ab}+t^{\alpha}{}_{ba},$ (55)
$\displaystyle\bar{\phi}_{\alpha}{}^{ab}$
$\displaystyle:=\bar{t}_{\alpha}{}^{ab}+\bar{t}_{\alpha}{}^{ba}.$ (56)

Notice that the set of $\mathrm{F}_{4}$ generators is obtained from
$\mathrm{E}_{6}$ generators by symmetrizing with respect to LR parity from
Appendix B. This is a particularly simple embedding, in the context of which
the $\mathfrak{f}_{4}$ Lie algebra can be understood as the maximal subalgebra
of $\mathfrak{e}_{6}$ that consists of LR-symmetric elements.

Using the commutation relations of $\mathrm{E}_{6}$ from Eqs. (30)–(42), we
get the following consistent set of $\mathrm{F}_{4}$ commutation relations:

$\displaystyle\big{[}\Phi_{C}^{A},\Phi_{LR}^{B}\big{]}$ $\displaystyle=0,$
(57) $\displaystyle\big{[}\Phi_{C}^{A},\Phi_{C}^{B}\big{]}$
$\displaystyle=if^{ABC}\;\Phi_{C}^{C},$ (58)
$\displaystyle\big{[}\Phi_{LR}^{A},\Phi_{LR}^{B}\big{]}$
$\displaystyle=if^{ABC}\;\Phi_{LR}^{C},$ (59)
$\displaystyle\big{[}\Phi_{C}^{A},\phi^{\alpha}{}_{ab}\big{]}$
$\displaystyle=-\tfrac{1}{2}(\lambda^{A})^{\alpha}{}_{\beta}\;\phi^{\beta}{}_{ab},$
(60) $\displaystyle\big{[}\Phi_{LR}^{A},\phi^{\alpha}{}_{ab}\big{]}$
$\displaystyle=\phantom{-}\tfrac{1}{2}\left((\lambda^{A})^{c}{}_{a}\;\phi^{\alpha}{}_{cb}+(\lambda^{A})^{c}{}_{b}\;\phi^{\alpha}{}_{ac}\right),$
(61) $\displaystyle\big{[}\Phi_{C}^{A},\bar{\phi}_{\alpha}{}^{ab}\big{]}$
$\displaystyle=\phantom{-}\tfrac{1}{2}(\lambda^{A})^{\beta}{}_{\alpha}\;\bar{\phi}_{\beta}{}^{ab},$
(62) $\displaystyle\big{[}\Phi_{LR}^{A},\bar{\phi}_{\alpha}{}^{ab}\big{]}$
$\displaystyle=-\tfrac{1}{2}\left((\lambda^{A})^{a}{}_{c}\;\bar{\phi}_{\alpha}{}^{cb}+(\lambda^{A})^{b}{}_{c}\;\bar{\phi}_{\alpha}{}^{ac}\right),$
(63) $\displaystyle\big{[}\phi^{\alpha}{}_{ab},\phi^{\beta}{}_{cd}\big{]}$
$\displaystyle=-\varepsilon^{\alpha\beta\gamma}\;(\varepsilon_{ace}\,\varepsilon_{bdf}+\varepsilon_{bce}\,\varepsilon_{adf})\;\bar{\phi}_{\gamma}{}^{ef},$
(64)
$\displaystyle\big{[}\bar{\phi}_{\alpha}{}^{ab},\bar{\phi}_{\beta}{}^{cd}\big{]}$
$\displaystyle=+\varepsilon_{\alpha\beta\gamma}\;(\varepsilon^{ace}\,\varepsilon^{bdf}+\varepsilon^{bce}\,\varepsilon^{adf})\;\phi^{\gamma}{}_{ef},$
(65)
$\displaystyle\big{[}\bar{\phi}_{\alpha}{}^{ab},\phi^{\beta}{}_{cd}\big{]}$
$\displaystyle=2\,(\lambda^{A})^{\beta}{}_{\alpha}\;(\delta^{a}{}_{c}\;\delta^{b}{}_{d}+\delta^{a}{}_{d}\;\delta^{b}{}_{c})\;\Phi_{C}^{A}-$
$\displaystyle\quad-\delta^{\beta}{}_{\alpha}\;\left((\lambda^{A})^{a}{}_{c}\,\delta^{b}{}_{d}+(\lambda^{A})^{a}{}_{d}\,\delta^{b}{}_{c}+(\lambda^{A})^{b}{}_{d}\,\delta^{a}{}_{c}+(\lambda^{A})^{b}{}_{c}\,\delta^{a}{}_{d}\right)\;\Phi_{LR}^{A}.$
(66)

The commutation relations are consistent with the generator
$\phi^{\alpha}{}_{ab}$ being symmetric in the indices $a$ and $b$, as they
should be.

As in $\mathrm{E}_{6}$, the commutation relations of $\mathrm{F}_{4}$ were
possible to write down by using only $\mathrm{SU}(3)$-related objects: the
Gell-Mann matrices $\lambda^{A}$, the Levi-Civita tensors $\varepsilon_{abc}$
and the $\mathrm{SU}(3)$ structure constants $f^{ABC}$. This is the advantage
offered by considering $\mathrm{F}_{4}$ in the language of its
$\mathrm{SU}(3)^{2}$ maximal subalgebra.

Schematically, the $\mathrm{F}_{4}$ commutation relations by sector are
displayed in Table 13. The generators from different sectors do not
necessarily have a consistent normalization.

Table 13: The schematic table of the $\mathrm{F}_{4}$ commutation relations by sector.  $[.,.]$ | $\Phi_{C}$ | $\Phi_{LR}$ | $\phi$ | $\bar{\phi}$
---|---|---|---|---
$\Phi_{C}$ | $\Phi_{C}$ | 0 | $\phi$ | $\bar{\phi}$
$\Phi_{LR}$ | 0 | $\Phi_{LR}$ | $\phi$ | $\bar{\phi}$
$\phi$ | $\phi$ | $\phi$ | $\bar{\phi}$ | $\Phi_{C,LR}$
$\bar{\phi}$ | $\bar{\phi}$ | $\bar{\phi}$ | $\Phi_{C,LR}$ | $\phi$

###  A.3 The algebra of $\mathrm{SU}(3)\times\mathrm{G}_{2}$ and its embedding
into $\mathrm{E}_{6}$

We label the $8$ generators of $\mathrm{SU}(3)$ by

$\displaystyle T^{A}$ (67)

and the $14$ generators of $G_{2}$ by

$\displaystyle G^{A},\quad g^{a},\quad\bar{g}_{a}.$ (68)

We used the label $A$ for the adjoint index of $\mathrm{SU}(3)$, and the
(upper) $a$ as the fundamental index of $\mathrm{SU}(3)$. Furthermore, we made
use of the maximal $\mathrm{SU}(3)$ subgroup of $\mathrm{G}_{2}$ to decompose
the generators of $\mathrm{G}_{2}$, see Table 11. In the abbreviated notation
by sector, the generators of $\mathrm{SU}(3)\times\mathrm{G}_{2}$ are written
simply as $\\{T,G,g,\bar{g}\\}$.

One possible embedding of $\mathrm{SU}(3)\times\mathrm{G}_{2}$ into
$\mathrm{E}_{6}$ is to define its generators as follows:

$\displaystyle T^{12+}$ $\displaystyle=T_{L}^{45+}+t^{2}{}_{23},$
$\displaystyle T^{45+}$ $\displaystyle=t^{2}{}_{12}-t^{2}{}_{21},$
$\displaystyle T^{67+}$ $\displaystyle=T_{R}^{45+}+t^{2}{}_{32},$ (69)
$\displaystyle T^{12-}$ $\displaystyle=T_{L}^{45-}+\bar{t}_{2}{}^{23},$
$\displaystyle T^{45-}$ $\displaystyle=\bar{t}_{2}{}^{12}-\bar{t}_{2}{}^{21},$
$\displaystyle T^{67-}$ $\displaystyle=T_{R}^{45-}+\bar{t}_{2}{}^{32},$ (70)
$\displaystyle T^{3}$
$\displaystyle=\langle\tfrac{1}{2},-\tfrac{1}{2\sqrt{3}},0,\tfrac{2}{\sqrt{3}},0,-\tfrac{1}{\sqrt{3}}\rangle,$
(71) $\displaystyle T^{8}$
$\displaystyle=\langle\tfrac{\sqrt{3}}{2},-\tfrac{1}{2},0,0,0,1\rangle,$ (72)
$\displaystyle G^{12+}$ $\displaystyle=t^{1}{}_{22},$ $\displaystyle G^{45+}$
$\displaystyle=t^{3}{}_{22},$ $\displaystyle G^{67+}$
$\displaystyle=T_{C}^{45+},$ (73) $\displaystyle G^{12-}$
$\displaystyle=\bar{t}_{1}{}^{22},$ $\displaystyle G^{45-}$
$\displaystyle=\bar{t}_{3}{}^{22},$ $\displaystyle G^{67-}$
$\displaystyle=T_{C}^{45-},$ (74) $\displaystyle G^{3}$
$\displaystyle=\langle-\tfrac{1}{2},-\tfrac{1}{2\sqrt{3}},-\tfrac{1}{2},\tfrac{1}{2\sqrt{3}},-\tfrac{1}{2},\tfrac{1}{2\sqrt{3}}\rangle$
(75) $\displaystyle G^{8}$
$\displaystyle=\langle\tfrac{1}{2\sqrt{3}},\tfrac{5}{6},-\tfrac{1}{2\sqrt{3}},\tfrac{1}{6},-\tfrac{1}{2\sqrt{3}},\tfrac{1}{6}\rangle,$
(76) $\displaystyle g^{1}$
$\displaystyle=t_{L}^{12+}+t_{R}^{12+}+t^{2}{}_{33},$ $\displaystyle g^{2}$
$\displaystyle=t^{1}{}_{12}+t^{1}{}_{21}+\bar{t}_{3}{}^{11},$ $\displaystyle
g^{3}$ $\displaystyle=t^{3}{}_{12}+t^{3}{}_{21}-\bar{t}_{1}{}^{11},$ (77)
$\displaystyle\bar{g}_{1}$
$\displaystyle=t_{L}^{12-}+t_{R}^{12-}+\bar{t}_{2}{}^{33},$
$\displaystyle\bar{g}_{2}$
$\displaystyle=\bar{t}_{1}{}^{12}+\bar{t}_{1}{}^{21}+t^{3}{}_{11},$
$\displaystyle\bar{g}_{3}$
$\displaystyle=\bar{t}_{3}{}^{12}+\bar{t}_{3}{}^{21}-t^{1}{}_{11}.$ (78)

Above, we made use of the following “vector notation” for the diagonal
generators:

$\displaystyle\langle\alpha_{1},\alpha_{2},\alpha_{3},\alpha_{4},\alpha_{5},\alpha_{6}\rangle$
$\displaystyle:=\alpha_{1}\;T_{C}^{3}+\alpha_{2}\;T_{C}^{8}+\alpha_{3}\;T_{L}^{3}+\alpha_{4}\;T_{L}^{8}+\alpha_{5}\;T_{R}^{3}+\alpha_{6}\;T_{R}^{8}.$
(79)

The diagonal generators $\\{T^{3},T^{8},G^{3},G^{8}\\}$ form a set of
orthogonal states in the sense of the Killing form
$K(\mathbf{X},\mathbf{Y})=\mathrm{Tr}(\mathrm{ad}_{\mathbf{X}}\mathrm{ad}_{\mathbf{Y}})$,
although they are not normalized in the same way.

Another observation is that the generators of the factor $\mathrm{G}_{2}$ are
symmetric under LR parity, and hence the group $\mathrm{G}_{2}$ defined here
is a subgroup of the group $\mathrm{F}_{4}$, whose embedding is defined in
Appendix A.2. The generators $T$ of the $\mathrm{SU}(3)$ factor, however, are
not LR-parity symmetric, ensuring that $\mathrm{SU}(3)\times\mathrm{G}_{2}$ is
not a subgroup of $\mathrm{F}_{4}$ and can thus be a maximal subgroup of
$\mathrm{E}_{6}$.

Inserting the commutation relations of $\mathrm{E}_{6}$ from Eqs. (30)–(42),
it is possible to work out the following consistent set of
$\mathrm{SU}(3)\times\mathrm{G}_{2}$ commutation relations:

$\displaystyle[T^{A},G^{A}]$
$\displaystyle=[T^{A},g^{a}]=[T^{A},\bar{g}_{a}]=0,$ (80)
$\displaystyle[T^{A},T^{B}]$ $\displaystyle=if^{ABC}\;T^{C},$ (81)
$\displaystyle[G^{A},G^{B}]$ $\displaystyle=if^{ABC}\;G^{C},$ (82)
$\displaystyle[G^{A},g^{a}]$
$\displaystyle=-\tfrac{1}{2}\,(\lambda^{A})^{a}{}_{b}\;g^{b},$ (83)
$\displaystyle[G^{A},\bar{g}_{a}]$
$\displaystyle=+\tfrac{1}{2}\,(\lambda^{A})^{b}{}_{a}\;\bar{g}_{b}$ (84)
$\displaystyle[g^{a},g^{b}]$
$\displaystyle=-2\,\varepsilon^{abc}\;\bar{g}_{c},$ (85)
$\displaystyle[\bar{g}_{a},\bar{g}_{b}]$
$\displaystyle=+2\,\varepsilon_{abc}\;g^{c},$ (86)
$\displaystyle[\bar{g}_{a},g^{b}]$
$\displaystyle=3\,(\lambda^{A})^{b}{}_{a}\,G^{A}.$ (87)

Note that the above commutation relations use $T^{A}$ and $G^{A}$ in the
Hermitian basis, while the generator definitions of Eq. (69)–(78) were given
in a complex basis of raising and lowering operators. The conversion between
them can be done via the relations

$\displaystyle T^{AB\pm}$ $\displaystyle=T^{A}\pm iT^{B},$ $\displaystyle
G^{AB\pm}$ $\displaystyle=G^{A}\pm iG^{B}.$ (88)

Analogous to $\mathrm{E}_{6}$ and $\mathrm{F}_{4}$, the language of the
maximal subgroup $\mathrm{SU}(3)$ in $\mathrm{G}_{2}$ again allowed us to
write down the $\mathrm{G}_{2}$ commutation relations only with
$\mathrm{SU}(3)$-related objects. In fact, the use of
$\mathrm{SU}(3)$-covariant indices structurally already restricts the form of
the commutation relations up to overall numeric coefficients. The commutation
relations also have a straightforward interpretation in this $\mathrm{SU}(3)$
language, for example the relation for $[G^{A},g^{a}]$ simply states that
$g^{a}$ transform as a triplet under the $\mathrm{SU}(3)$ subgroup defined by
$G^{A}$.

For better intuition, we again write the commutation relations of
$\mathrm{G}_{2}$ by sector in Table 14. The generators from different sectors
do not necessarily have a consistent normalization.

Table 14: The schematic table of the $\mathrm{G}_{2}$ commutation relations by sector.  $[.,.]$ | $G$ | $g$ | $\bar{g}$
---|---|---|---
$G$ | $G$ | $g$ | $\bar{g}$
$g$ | $g$ | $\bar{g}$ | $G$
$\bar{g}$ | $\bar{g}$ | $G$ | $g$

##  Appendix B Discrete symmetries in $\mathrm{E}_{6}$

One could consider reshuffling the labels for color (C), left (L) and right
(R) of the $\mathrm{SU}(3)$ factors in trinification $G_{333}$. Such
transformations form a permutation group of $3$ objects, which we denoted by
$D_{3}$ in Section 2.2. The group $D_{3}$ is generated by 3 parities, which
exchange pairs of $\mathrm{SU}(3)$ factors: $\mathbb{Z}_{2}^{LR}$,
$\mathbb{Z}_{2}^{CL}$ and $\mathbb{Z}_{2}^{CR}$, also dubbed LR, CL and CR
parity.

The permutation group $D_{3}$ is a discrete subgroup of $\mathrm{E}_{6}$. This
can be seen by explicitly constructing the parities; it turns out we can
define them by

$\displaystyle P_{LR}$ $\displaystyle:=e^{i\pi A_{LR}}\,e^{i\pi B_{LR}},$
$\displaystyle P_{CL}$ $\displaystyle:=e^{i\pi A_{CL}}\,e^{i\pi B_{CL}},$
$\displaystyle P_{CR}$ $\displaystyle:=e^{i\pi A_{CR}}\,e^{i\pi B_{CR}},$ (89)

where the $A$\- and $B$-matrices are all $\mathfrak{e}_{6}$ algebra elements:

$\displaystyle A_{LR}$
$\displaystyle:=\tfrac{1}{2i}(t^{1}{}_{22}-\bar{t}_{1}{}^{22})+\tfrac{1}{2i}(t^{1}{}_{11}-\bar{t}_{1}{}^{11}),$
$\displaystyle B_{LR}$
$\displaystyle:=\tfrac{1}{2i}(t^{1}{}_{33}-\bar{t}_{1}{}^{33})-t^{7}_{C},$
(90) $\displaystyle A_{CL}$
$\displaystyle:=\tfrac{1}{2i}(t^{2}{}_{21}-\bar{t}_{2}{}^{21})+\tfrac{1}{2i}(t^{1}{}_{11}-\bar{t}_{1}{}^{11}),$
$\displaystyle B_{CL}$
$\displaystyle:=\tfrac{1}{2i}(t^{3}{}_{31}-\bar{t}_{3}{}^{31})-t^{7}_{R},$
(91) $\displaystyle A_{CR}$
$\displaystyle:=\tfrac{1}{2i}(t^{2}{}_{12}-\bar{t}_{2}{}^{12})+\tfrac{1}{2i}(t^{1}{}_{11}-\bar{t}_{1}{}^{11}),$
$\displaystyle B_{CR}$
$\displaystyle:=\tfrac{1}{2i}(t^{3}{}_{13}-\bar{t}_{3}{}^{13})-t^{7}_{L}.$
(92)

The actions of the parities on the $\mathbf{27}$ can then be conveniently
specified in the language of the $(\mathbf{L},\mathbf{M},\mathbf{N})$ matrices
from Eq. (44). Using the parity definitions of Eqs. (89)–(92) and the action
of $\mathrm{E}_{6}$ generators on the $\mathbf{27}$ from Eqs. (45)–(49), we
obtain

$\displaystyle P_{LR}:$
$\displaystyle\quad(\mathbf{L},\mathbf{M},\mathbf{N})\mapsto(\mathbf{N}^{T},\mathbf{M}^{T},\mathbf{L}^{T}),$
(93) $\displaystyle P_{CL}:$
$\displaystyle\quad(\mathbf{L},\mathbf{M},\mathbf{N})\mapsto(\mathbf{L}^{T},-\mathbf{N}^{T},-\mathbf{M}^{T}),$
(94) $\displaystyle P_{CR}:$
$\displaystyle\quad(\mathbf{L},\mathbf{M},\mathbf{N})\mapsto(-\mathbf{M}^{T},-\mathbf{L}^{T},\mathbf{N}^{T}).$
(95)

The parities thus act as specific permutations of the states in the
representation $\mathbf{27}$ (up to minus signs). The presence of minuses is
due to our use of the phenomenologically more convenient embedding of
trinification, see discussion below Eq. (42).

Armed with the prescription of Eqs. (93)–(95), we can easily confirm that the
defined transformations are parities in the mathematical sense:

$\displaystyle P_{LR}^{2}$ $\displaystyle=P_{CL}^{2}=P_{CR}^{2}=\mathbb{1}.$
(96)

Furthermore, we define the composite transformation $P_{CLR}\equiv
P_{LR}P_{CR}$; the group elements of $D_{3}$ can now be written as

$\displaystyle D_{3}$
$\displaystyle=\\{\mathbb{1},P_{CLR},P_{CLR}^{2},P_{LR},P_{CL},P_{RC}\\}.$
(97)

It can be checked explicitly that the set closes under multiplication, and the
group structure is indeed that of a permutation group $S_{3}$ of $3$ elements.
The element $P_{CLR}$ generates a $\mathbb{Z}_{3}$ cyclic subgroup, and can be
understood intuitively to cyclically permute $C\mapsto L\mapsto R\mapsto C$.
In the context of Section 3.2.1 and Eq. (14), we can identify $P_{CLR}$ with
clockwise rotations of $3$-fold symmetry: $P_{CLR}=R(4\pi/3)$ and
$P_{CLR}^{2}=R(2\pi/3)$.

The action of the parities can be extended to any representation of
$\mathrm{E}_{6}$ by tensoriality. We now consider how the parities act on the
adjoint $\mathbf{78}$, which in tensor form has one upper and one lower index,
cf. Table 1. The action of a parity $P$ on a generator $\mathbf{T}$ (written
as a matrix) is then given explicitly by

$\displaystyle\hat{P}(\mathbf{T})$ $\displaystyle=\mathbf{P\,T\,P}^{T},$ (98)

where the matrix $\mathbf{P}$ is constructed via Eqs. (93)–(95). This gives
the explicit results of Table 15. The expressions confirm the intuition behind
what these parities do, i.e. they exchange $\mathrm{SU}(3)$ factors (up to
taking the generator in the conjugate representation).

Table 15: The action of the $3$ parities $\mathbb{Z}_{2}^{LR}$, $\mathbb{Z}_{2}^{CL}$ on the basis elements of the adjoint representation $\mathbf{78}$ of $\mathrm{E}_{6}$ (when the adjoint representation is considered as the span of generators in the fundamental representation).  map $P$ | $\hat{P}(t^{A}_{C})$ | $\hat{P}(t^{A}_{L})$ | $\hat{P}(t^{A}_{R})$ | $\hat{P}(t^{\alpha}{}_{aa^{\prime}})$ | $\hat{P}(\bar{t}_{\alpha}{}^{aa^{\prime}})$
---|---|---|---|---|---
$P_{LR}$ | $-(t^{A}_{C})^{T}$ | $-(t^{A}_{R})^{T}$ | $-(t^{A}_{L})^{T}$ | $-(t^{\alpha}{}_{a^{\prime}a})^{T}$ | $-(\bar{t}_{\alpha}{}^{a^{\prime}a})^{T}$
$P_{CL}$ | $t^{A}_{L}$ | $t^{A}_{C}$ | $-(t^{A}_{R})^{T}$ | $-(t^{a}{}_{\alpha a^{\prime}})^{T}$ | $-(\bar{t}_{a}{}^{\alpha a^{\prime}})^{T}$
$P_{CR}$ | $t^{A}_{R}$ | $-(t^{A}_{L})^{T}$ | $t^{A}_{C}$ | $-(t^{a^{\prime}}{}_{a\alpha})^{T}$ | $-(\bar{t}_{a^{\prime}}{}^{a\alpha})^{T}$

To summarize, we saw explicitly that Eqs. (89)–(92) indeed define parities
forming the discrete $D_{3}$ subgroup of $\mathrm{E}_{6}$ with all intuitive
properties of exchanging the $\mathrm{SU}(3)$ factor labels of the $G_{333}$
subgroup.

The importance of the $D_{3}$ group is obvious from the vacuum analysis when
breaking along the $G_{333}$ direction. In particular, it appears as a
symmetry of the restricted potential of $G_{333}$-singlets in Eq. (12) and is
visually manifest in Figure 1. Furthermore, the $G_{333}$ vacua preserve a
$\mathbb{Z}_{2}$ remnant (one of the parities), cf. Section 3.2.1. Placing
ourselves in e.g. the LR symmetric vacuum, the full preserved symmetry is in
fact $G_{333}\rtimes\mathbb{Z}_{2}^{LR}$, as confirmed also by the LR
symmetric spectrum in Table 6. We emphasize that the product with the discrete
group is semi-direct, since elements of $D_{3}$ do not commute with those of
$G_{333}$, i.e. they act on $G_{333}$ elements non-trivially in accordance
with Table 15.

Finally, we observe that given the definition of $P_{LR}$ in Eqs. (89) and
(90), LR parity is also part of the $\mathrm{SO}(10)$ subgroup of
$\mathrm{E}_{6}$ that is specified (together with an Abelian factor) in Eq.
(7). LR parity can thus be identified (up to phase conventions) as $D$-parity
Kibble:1982dd ; Chang:1983fu ; Chang:1984uy ; Chang:1984qr  in the
$\mathrm{SO}(10)$ context, and has also been referred to as such in the
context of trinification originating from $\mathrm{E}_{6}$
Chakrabortty:2019fov ; Dash:2020jlc . Following these references, $D$-parity
is defined as

$\displaystyle P_{D}$ $\displaystyle=e^{i\pi T_{67}}e^{i\pi
T_{23}}=-\Sigma_{67}\Sigma_{23},$ (99)

where $T_{ij}\equiv\Sigma_{ij}/2=[\Gamma_{i},\Gamma_{j}]/(4i)$ are the
$\mathrm{SO}(10)$ generators in the spinorial representation666Throughout the
paper, we refer to groups with orthogonal algebras simply as $\mathrm{SO}(n)$,
but strictly speaking we in fact always have the simply-connected variants
$\mathrm{Spin}(n)$ in mind. The reader should take this into account when
topological considerations or representation theory are discussed. (the full
$\mathbf{16}\oplus\mathbf{\overline{16}}$), with $\Gamma_{i}$ the gamma
matrices of the $10$-dimensional Clifford algebra
$\\{\Gamma_{i},\Gamma_{j}\\}=2\delta_{ij}$, where indices $i,j$ run from $1$
to $10$. Note that the definition assumes the embedding of the Pati-Salam
group
$G_{422}\equiv\mathrm{SU}(4)_{C}\times\mathrm{SU}(2)_{L}\times\mathrm{SU}(2)_{R}$
into $\mathrm{SO}(10)$ in a convention, where the indices $1\leq i,j\leq 6$
refer to the $\mathrm{SO}(6)\cong\mathrm{SU}(4)_{C}$ factor. The definition of
$P_{D}$ can then be extended to all other $\mathrm{SO}(10)$ representations by
tensoriality. Observe that the last equality in Eq. (99), i.e. the evaluation
of the exponentials, is a simplification possible in $\mathrm{SO}(10)$ due to
the Clifford algebra relation, while no such simplification is possible in the
case of $\mathrm{E}_{6}$ parities of Eq. (89).

Also, as a last clarification, we do not aspire to consider any of the
$\mathrm{E}_{6}$ parities as an extension of the space-time parity as done in
the left-right model Mohapatra:1974gc ; Mohapatra:1974hk ; Senjanovic:1975rk .

##  Appendix C Location of singlets in the $\mathbf{650}$

We first consider the fundamental representation $\mathbf{27}$, whose
decomposition into trinification representations is given in Eq. (44). In the
spirit of Appendix A in Bajc:2013qra , the field content of this
representation is even better elucidated in terms of its SM irreducible
components and the use of SM-fermion language:

$\displaystyle\mathbf{27}$ $\displaystyle=(Q\oplus L\oplus d^{c}\oplus
u^{c}\oplus e^{c})\oplus(d^{\prime}\oplus d^{\prime c}\oplus L^{\prime}\oplus
L^{\prime c})\oplus(\nu^{c}\oplus n),$ (100)

where these fields transform under the SM group as

$\displaystyle Q$ $\displaystyle\sim(\mathbf{3},\mathbf{2},+\tfrac{1}{6}),$
$\displaystyle d^{c},d^{\prime c}$
$\displaystyle\sim(\mathbf{\bar{3}},\mathbf{1},+\tfrac{1}{3}),$ $\displaystyle
L,L^{\prime}$ $\displaystyle\sim(\mathbf{1},\mathbf{2},-\tfrac{1}{2}),$
$\displaystyle e^{c}$ $\displaystyle\sim(\mathbf{1},\mathbf{1},+1),$
$\displaystyle u^{c}$
$\displaystyle\sim(\mathbf{\bar{3}},\mathbf{1},-\tfrac{2}{3}),$ $\displaystyle
d^{\prime}$ $\displaystyle\sim(\mathbf{3},\mathbf{1},-\tfrac{1}{3}),$
$\displaystyle L^{\prime c}$
$\displaystyle\sim(\mathbf{1},\mathbf{2},+\tfrac{1}{2}),$
$\displaystyle\nu^{c},n$ $\displaystyle\sim(\mathbf{1},\mathbf{1},0).$ (101)

The first grouping of representations (in terms of parentheses) in Eq. (100)
consists of all SM fermions in one generation. The second grouping consists of
vector-like down-type quarks $d^{\prime}\oplus d^{\prime c}$ and vector-like
lepton doublets $L^{\prime}\oplus L^{\prime c}$. The third grouping contains
SM singlets: $\nu^{c}$ is a right-handed neutrino in the subpart $\mathbf{16}$
of the $\mathrm{SO}(10)$ subgroup, and $n$ is a $\mathrm{SO}(10)$ singlet
(denoted by $s$ in Bajc:2013qra ). Writing in terms of
$\mathrm{SU}(3)_{C}\times\mathrm{U}(1)_{EM}$ components of the EW broken
phase, we use the notation

$\displaystyle Q$ $\displaystyle=(u,d),$ $\displaystyle L$
$\displaystyle=(\nu,e),$ $\displaystyle L^{\prime}$
$\displaystyle=(\nu^{\prime},e^{\prime})$ $\displaystyle L^{\prime c}$
$\displaystyle=(\nu^{\prime c},e^{\prime c}).$ (102)

We suppressed all color indices in the notation. Collecting all the states
together, the $\mathbf{27}$ is explicitly written in the following basis:

$\displaystyle\mathbf{e}_{i}=\\{u,\;d,\;d^{\prime},\;u^{c},\;d^{c},\;d^{\prime
c},\;e,\;e^{\prime},\;e^{c},\;e^{\prime c},\;\nu,\;\nu^{\prime},\;\nu^{\prime
c},\;\nu^{c},\;n\\}.$ (103)

For the sake of being fully explicit, we insert this basis into the matrices
$(L^{\alpha a},M_{a}{}^{a^{\prime}},N_{a^{\prime}\alpha})$ of Eq. (44) in the
following way:

$\displaystyle\mathbf{L}$
$\displaystyle=\begin{pmatrix}u_{1}&d_{1}&d^{\prime}_{1}\\\
u_{2}&d_{2}&d^{\prime}_{2}\\\ u_{3}&d_{3}&d^{\prime}_{3}\\\ \end{pmatrix},$
$\displaystyle\mathbf{M}$ $\displaystyle=\begin{pmatrix}\nu^{\prime
c}&e^{\prime}&e\\\ e^{\prime c}&-\nu^{\prime}&-\nu\\\ e^{c}&\nu^{c}&n\\\
\end{pmatrix},$ $\displaystyle\mathbf{N}$
$\displaystyle=\begin{pmatrix}u^{c}_{1}&u^{c}_{2}&u^{c}_{3}\\\
-d^{c}_{1}&-d^{c}_{2}&-d^{c}_{3}\\\ d^{\prime c}_{1}&d^{\prime
c}_{2}&d^{\prime c}_{3}\\\ \end{pmatrix}.$ (104)

The quarks were written with explicit color indices in their subscripts, and
the presence of minuses is part of our conventions for greater convenience.
The two minuses in $\mathbf{M}$ are present due to this representation
transforming as a $\mathbf{\bar{3}}$ under $\mathrm{SU}(3)_{L}$, and hence the
lepton doublets $L$ and $L^{\prime}$ are embedded as a $\mathbf{\bar{2}}$ of
$\mathrm{SU}(2)_{L}$; the minuses ensure the definitions align with those in
the SM. Similarly, the minuses in $\mathbf{N}$ indicate the
$\mathrm{SU}(2)_{R}$ quark doublet $(u^{c},d^{c})$ is embedded as a
$\mathbf{\bar{2}}$.

We are now ready to consider the representation $\mathbf{650}$. Since its
components $X^{i}{}_{j}$ are written with one upper and one lower fundamental
index, cf. Table 1, its basis is a $650$-dimensional subspace spanned by
$\mathbf{e}_{i}\otimes\mathbf{e}^{\ast j}\equiv\mathbf{e}_{i}\mathbf{e}^{\ast
j}$. It is in this “$\mathbf{e}\mathbf{e}^{\ast}$” basis that we specify some
the states of interest in the $\mathbf{650}\equiv\mathbf{X}$.

The SM singlets listed in Eq. (10) and Table 3 have the following specific
form in $\mathbf{X}$:

$\displaystyle s$
$\displaystyle\simeq\tfrac{1}{6\sqrt{3}}\big{(}(uu^{\ast}+dd^{\ast}+d^{\prime}d^{\prime\ast})$
$\displaystyle\qquad-2(\nu^{\prime c}\nu^{\prime
c\ast}+e^{\prime}e^{\prime\ast}+ee^{\ast}+e^{\prime c}e^{\prime
c\ast}+\nu^{\prime}\nu^{\prime\ast}+\nu\nu^{\ast}+e^{c}e^{c\ast}+\nu^{c}\nu^{c\ast}+nn^{\ast})$
$\displaystyle\qquad+(u^{c}u^{c\ast}+d^{c}d^{c\ast}+d^{\prime c}d^{\prime
c\ast})\big{)}$ (105) $\displaystyle a$
$\displaystyle\simeq\tfrac{1}{6}\big{(}(uu^{\ast}+dd^{\ast}+d^{\prime}d^{\prime\ast})-(u^{c}u^{c\ast}+d^{c}d^{c\ast}+d^{\prime
c}d^{\prime c\ast})\big{)}$ (106) $\displaystyle x_{1}$
$\displaystyle\simeq\tfrac{1}{6\sqrt{2}}\big{(}2(u^{c}u^{c\ast}+\nu^{\prime
c}\nu^{\prime c\ast}+e^{\prime c}e^{\prime c\ast}+e^{c}e^{c\ast})$
$\displaystyle\qquad-(d^{c}d^{c\ast}+d^{\prime c}d^{\prime
c\ast}+e^{\prime}e^{\prime\ast}+ee^{\ast}+\nu^{\prime}\nu^{\prime\ast}+\nu\nu^{\ast}+\nu^{c}\nu^{c\ast}+nn^{\ast})\big{)}$
(107) $\displaystyle x_{2}$
$\displaystyle\simeq\tfrac{1}{2\sqrt{6}}\big{(}(d^{c}d^{c\ast}+e^{\prime}e^{\prime\ast}+\nu^{\prime}\nu^{\prime\ast}+\nu^{c}\nu^{c\ast})-(d^{\prime
c}d^{\prime c\ast}+ee^{\ast}+\nu\nu^{\ast}+nn^{\ast})\big{)}$ (108)
$\displaystyle X_{3}$
$\displaystyle\simeq\tfrac{1}{2\sqrt{3}}(e^{\prime}e^{\ast}+\nu^{\prime}\nu^{\ast}+\nu^{c}n^{\ast}+d^{\prime
c}d^{c\ast})$ (109) $\displaystyle y_{1}$
$\displaystyle\simeq\tfrac{1}{6\sqrt{2}}\big{(}(2\nu^{\prime c}\nu^{\prime
c\ast}-e^{\prime}e^{\prime\ast}-ee^{\ast})+(2e^{\prime c}e^{\prime
c\ast}-\nu^{\prime}\nu^{\prime\ast}-\nu\nu^{\ast})-2(2e^{c}e^{c\ast}-\nu^{c}\nu^{c\ast}-nn^{\ast})\big{)}$
(110) $\displaystyle y_{2}$
$\displaystyle\simeq\tfrac{1}{2\sqrt{6}}\big{(}-(e^{\prime}e^{\prime\ast}-ee^{\ast})-(\nu^{\prime}\nu^{\prime\ast}-\nu\nu^{\ast})+2(\nu^{c}\nu^{c\ast}-nn^{\ast})\big{)}$
(111) $\displaystyle Y_{3}$
$\displaystyle\simeq\tfrac{1}{2\sqrt{3}}(e^{\prime}e^{\ast}+\nu^{\prime}\nu^{\ast}-2\nu^{c}n^{\ast})$
(112) $\displaystyle z$
$\displaystyle\simeq\tfrac{1}{6\sqrt{2}}\big{(}(uu^{\ast}+dd^{\ast}+\nu^{\prime
c}\nu^{\prime c\ast}+e^{\prime}e^{\prime\ast}+ee^{\ast}+e^{\prime c}e^{\prime
c\ast}+\nu^{\prime}\nu^{\prime\ast}+\nu\nu^{\ast})$
$\displaystyle\qquad-2(d^{\prime}d^{\prime\ast}+e^{c}e^{c\ast}+\nu^{c}\nu^{c\ast}+nn^{\ast})\big{)}.$
(113)

The conjugates $X_{3}^{\ast}$ and $Y_{3}^{\ast}$ are obtained by exchanging
the order in each term and complex conjugating each fermion label (equivalent
to Hermitian conjugation in $\mathbf{X}$). Also, the states $X_{3}$ and
$Y_{3}$ are the only SM-singlet states with off-diagonal entries in
$\mathbf{X}$. These states are normalized to $1/2$ for real fields and $1$ for
complex fields:

$\displaystyle X^{i}{}_{j}\,X^{\ast}{}_{i}{}^{j}$
$\displaystyle=\tfrac{1}{2}(s^{2}+a^{2}+x_{1}^{2}+x_{2}^{2}+y_{1}^{2}+y_{2}^{2}+z^{2})+|X_{3}|^{2}+|Y_{3}|^{2}.$
(114)

As always, the overall sign for each real field and phase for a complex field
is conventional.

We emphasize again that color indices were suppressed in our notation; the
quark terms should thus be taken as a sum over all colors, e.g.
$uu^{\ast}\equiv u^{a}u^{\ast}{}_{a}$, where $a$ is a color index going from
$1$ to $3$.

Since these states in Eqs. (105)–(113) are SM singlets, they could
alternatively be written in the language of the EW unbroken phase with weak
indices suppressed, e.g.

$\displaystyle z$
$\displaystyle\simeq\tfrac{1}{6\sqrt{2}}\big{(}(QQ^{\ast}+LL^{\ast}+L^{\prime}L^{\prime\ast}+L^{\prime
c}L^{\prime
c\ast})-2(d^{\prime}d^{\prime\ast}+e^{c}e^{c\ast}+\nu^{c}\nu^{c\ast}+nn^{\ast})\big{)}.$
(115)

Finally, we specify the singlets of the exceptional cases. For the case of the
$\mathrm{F}_{4}$, with its embedding specified in Eqs. (53)–(56), the singlet
state $\tilde{s}$ has the direction

$\displaystyle\tilde{s}_{\mathrm{F}_{4}}$
$\displaystyle\simeq\frac{1}{6\,\sqrt{39}}\left(\sum_{i=1}^{27}\mathbf{e}_{i}\,\mathbf{e}^{\ast
i}-9\left(ss^{\ast}+\nu^{\prime}\nu^{\prime\ast}+\nu^{\prime c}\nu^{\prime
c\ast}\right)-9\left(\nu^{\prime
c}s^{\ast}-\nu^{\prime}s^{\ast}-\nu^{\prime}\nu^{\prime
c\ast}+h.c.\right)\right).$ (116)

For the other exceptional case $\mathrm{SU}(3)\times\mathrm{G}_{2}$ with the
embedding given in Eqs. (69)–(78), the singlet has the direction

$\displaystyle\tilde{s}_{\mathrm{SU}(3)\times\mathrm{G}_{2}}$
$\displaystyle\simeq\frac{1}{12\,\sqrt{21}}\;\bigg{(}4\,\sum_{i=1}^{27}\mathbf{e}_{i}\,\mathbf{e}^{\ast
i}-9\left(u_{2}u_{2}^{\ast}+u^{c}_{2}u^{c\ast}_{2}+ee^{\ast}+e^{c}e^{c\ast}+\nu^{\prime}\nu^{\prime\ast}+\nu^{\prime
c}\nu^{\prime c\ast}\right)-$
$\displaystyle\quad-18\left(ss^{\ast}+d^{\prime}_{2}d^{\prime\ast}_{2}+d^{\prime
c}_{2}d^{\prime c\ast}_{2}\right)-9\left(u_{2}e^{\ast}-\nu^{\prime}\nu^{\prime
c\ast}+u_{2}^{c}e^{c\ast}+h.c.\right)\bigg{)}.$ (117)

Notice that the singlets in the exceptional cases cannot be written in terms
of SM representations, i.e. $\tilde{s}_{\mathrm{F}_{4}}$ explicitly breaks
$\mathrm{SU}(2)_{L}$ while $\tilde{s}_{\mathrm{SU}(3)\times\mathrm{G}_{2}}$
breaks both $\mathrm{SU}(2)_{L}$ and $\mathrm{SU}(3)_{C}$. This is not
surprising, since the exceptional cases do not have a SM group as a subgroup,
and thus their singlets are not SM singlets.

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