aainstitutetext: Instituto de Física Corpuscular, Universidad de Valencia and
CSIC, Edificio Institutos Investigación, C/Catedrático José Beltrán 2, 46980
Paterna, Spainbbinstitutetext: Universidad Politécnica de Madrid, ETSIST,
C/Nikola Tesla s/n, 28031 Madrid, Spainccinstitutetext: Valencia College,
Osceola Science Department, Kissimmee, FL 34744, USAddinstitutetext: School of
Physics & Astronomy, University of Southampton, Southampton SO17 1BJ, UK

# Flavon vacuum alignment beyond SUSY

Claudia Hagedorn b M.L. López-Ibáñez c M. Jay Pérez a,d Moinul Hossain Rahat a
and Oscar Vives<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

In flavor models the vacuum alignment of flavons is typically achieved via the
$F$-terms of certain fields in the supersymmetric limit. We propose a method
for preserving such alignments, up to a rescaling of the vacuum expectation
values, even after supersymmetry (and the flavor symmetry) are softly broken,
facilitating the vacuum alignment in models which are non-supersymmetric at
low energies. Examples of models with different flavor groups, namely $A_{4}$,
$T_{7}$, $S_{4}$ and $\Delta(27)$, are discussed.

##  1 Introduction

The origin of the curious triplication of the fermionic representations of the
Standard Model (SM), and of their patterns of masses and mixing, remains an
open question. As symmetries have proven crucial in understanding and
organizing the gauge sector, a well-studied approach to this problem has been
to employ symmetries acting on flavor space in the quark and lepton sector [1,
2, 3]; for reviews see e.g. [4, 5, 6, 7, 8].

In many models with flavor symmetries, these are broken spontaneously and a
peculiar form of the vacuum is required in order to correctly describe the
fermion masses and/or mixing.111See e.g. [9] for models in which the flavor
symmetry is broken at the boundaries of an extra dimension. For this reason,
supersymmetric (SUSY) extensions of the SM are often considered in which gauge
singlets, flavons, are responsible for this breaking. If the latter occurs
while SUSY is still intact, the vacuum can be aligned via $F$-terms (of
further fields, often called driving fields), see e.g. [10, 11].222If the
employed fields transform under a new gauge symmetry, $D$-terms can also be
relevant for the vacuum alignment, see e.g. [12, 13, 14].

An important point is the sufficient segregation of different symmetry
breaking sectors such that the vacuum of flavons contributing to e.g. the
neutrino and the charged lepton sector, respectively, can be independently
(and in different directions) aligned. In general, further symmetries, also
called shaping symmetries, have to be invoked in order to achieve this aim.
However, such a procedure can usually not be applied in non-SUSY models in
which e.g. quartic interactions involving two different fields and their
complex conjugates are invariant.333For examples in which symmetries alone are
sufficient in order to appropriately align the vacuum, see [15, 16]. The study
of the orbit space of N-Higgs doublet potentials, see e.g. [17, 18, 19, 20,
21], allows to analyze especially highly symmetric potentials and their
minima. It is, thus, assumed that certain couplings are absent (or highly
suppressed), although their expected size is of order one.444In extra-
dimensional models, the flavons belonging to different symmetry breaking
sectors can be separated via their localization in the extra dimension and,
consequently, couplings between these are suppressed, see e.g. [22].

In this paper, we study how vacuum alignments that are achieved via $F$-terms
in the SUSY limit can be realized, up to a rescaling of the vacuum expectation
values (VEVs), in non-SUSY models. For this, we include certain soft SUSY (and
potentially also flavor symmetry) breaking terms in the potential. We compute
the expected size of the rescaling factor. Furthermore, we identify conditions
which flavor-symmetry violating soft SUSY breaking terms must fulfill in order
to maintain the direction of the aligned vacuum.

These conditions are similar to those obtained in [23], where the authors have
explored the vacuum alignment in multi-Higgs doublet potentials with a softly
broken discrete symmetry. In particular, they have shown that the vacuum
alignment, obtained in the symmetric potential, remains preserved, up to a
rescaling of the VEVs, as long as soft breaking terms have the aligned vacuum
as an eigenvector. We focus in this study on flavons which are triplets of a
(discrete) flavor symmetry and analyze certain potentials in general.
Furthermore, we present concrete examples with the flavor symmetries $A_{4}$,
$T_{7}$, $S_{4}$ and $\Delta(27)$. These have been widely used in the
literature, see e.g. [24, 25, 10, 22, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35,
36, 37, 38].

The paper is organized as follows. In section 2 we determine the conditions
under which a vacuum alignment found for a SUSY potential remains preserved
even after introducing certain soft SUSY (and flavor symmetry) breaking terms
for both flavons in real and complex representations. In section 3 we
exemplify these results in an $A_{4}$ model with one or two flavons. Section 4
contains further examples in which flavons transform as (complex) triplets of
$T_{7}$ and $\Delta(27)$ as well as a case with general soft masses for the
group $S_{4}$. We summarize in section 5. A summary of the relevant properties
of the different flavor groups considered, details of the minimization and the
effects of higher-order terms in the $A_{4}$ model are relegated to the
appendices.

##  2 Vacuum alignment from SUSY to non-SUSY potentials

In this section, we present the general idea and show that it is indeed
possible to maintain the vacuum alignment achieved in the SUSY limit upon
including soft SUSY breaking terms. We consider only cases with isolated
minima, and assume that the effects of the soft SUSY breaking terms are small,
given that the flavor symmetry is usually spontaneously broken at a high
energy scale where SUSY is still intact, while the scale of the soft SUSY
breaking terms is taken to be of the order of a (few) TeV.

The aim is to obtain the same vacuum alignment from the non-SUSY potential
(with certain soft SUSY breaking effects) as in the SUSY limit, up to a real
rescaling factor $\zeta$ that should be close to one, see Eq. (14).555The
following considerations can be generalized for complex $\zeta$. In the first
part of this section we focus on the size of $\zeta$ and the necessary
conditions on the soft SUSY breaking terms in order to maintain the direction
of the aligned vacuum, while in the latter part we comment on more general
changes in the vacuum alignment.

###  2.1 Employed superpotentials

We begin by specifying the framework that we consider in the following. First
of all, we assume that the fields responsible for the breaking of the flavor
symmetry are gauge singlets, commonly denoted as flavons. These we call $\phi$
throughout, potentially with subscripts that refer to a certain flavon
multiplet and its components. As is well-known [10, 11], the achievement of a
certain vacuum alignment can be facilitated by the introduction of a
continuous $R$-symmetry $U(1)_{R}$ and a further set of gauge singlet fields,
called driving fields. These are denoted by $\Phi$ and also potentially have
subscripts.

Assuming that the driving fields carry $R$-charge $2$, while the flavons have
no $R$-charge, the superpotential $W$ is at maximum linear in the driving
fields. In fact, the terms relevant for the derivation of the scalar potential
are all linear in the driving fields, while terms representing Yukawa-type
interactions do not contain these fields, since supermultiplets containing SM
fermions are assigned $R$-charge 1. In the current study, only the former part
of $W$ is of interest. Furthermore, we make the simplifying assumption that it
is enough to consider only renormalizable terms of $W$.666The impact of non-
renormalizable operators is expected to be small. If the vacuum alignment
necessitates such terms, one can always imagine to introduce further fields,
both flavons and driving fields, such that all non-renormalizable terms can
originate from renormalizable ones in a certain ultra-violet completion.

In order to present the idea, it is sufficient to focus on the situation of
one driving field and one flavon. Since we imagine that the flavon is
responsible for the generation of a certain flavor pattern (e.g. fermion
mixing), this field is supposed to transform in a non-trivial representation
of the (discrete) flavor symmetry $G_{f}$, usually as a two- or three-
dimensional irreducible representation. For concreteness, we take
$\phi\sim{\bf 3}$, where this representation can be either real or
complex.777In the case of even-dimensional irreducible representations, these
can also be pseudo-real and such instance can be discussed analogously. For
the driving field, we have in general two options: it can either be a
(trivial) singlet of $G_{f}$ or, like the flavon, in a non-trivial
representation. In order to determine its assignment we consider separately
the case in which $\phi$ is in a real representation and $\phi$ being complex
with respect to $G_{f}$. In either case, we want to ensure the appearance of
an explicit mass scale in the superpotential, either in the form of a mass
term $M^{2}\,\Phi$ or in the form of a dimensionful coupling
$M\,\Phi\,\phi$,888As we see in the example given in section 3, $M$ might also
originate from the spontaneous breaking of a further symmetry and, thus, be
set by another VEV. such that the scale of the VEV of the flavon $\phi$ is
determined by $M$.999In particular, we would like to avoid the VEV of any of
the flavons being related to a flat direction of the potential and,
consequently, its value being fixed only once additional terms, e.g. soft SUSY
breaking terms, are taken into account as well. At the same time, the
inclusion of a dimensionful parameter should ensure that none of the
components of the flavons and driving fields remains massless.

For $\phi$ being a real triplet, $\phi\sim{\bf 3}$, the relevant terms in the
superpotential are

$W_{s}=M^{2}\,\Phi+\lambda\,\Phi\,\phi^{2}$ (1)

with the driving field $\Phi\sim{\bf 1}$, i.e. $\Phi$ transforms as trivial
singlet of the flavor symmetry $G_{f}$. These two couplings necessarily exist,
since $\phi$ is a real triplet. Further types of terms cannot exist, unless
more fields are included in the considerations. A typical example of such a
potential can be found in section 3 for the flavor symmetry $A_{4}$.

The corresponding $F$-term (of the driving field $\Phi$) reads101010If not
stated otherwise, the repeated appearance of an index indicates that we sum
over this index.

$F_{s}\equiv-F_{\Phi}^{\star}=M^{2}+\lambda\,c_{ij}\,\phi_{i}\,\phi_{j}$ (2)

with the coefficients $c_{ij}$ taking into account the possible non-trivial
contraction of the indices of two triplets in order to obtain the trivial
singlet. We can simplify this expression by choosing a basis in which the
latter has the form of an ordinary scalar product, $\phi_{i}^{2}$. Thus, we
use

$F_{s}=M^{2}+\lambda\,\phi_{i}^{2}\;.$ (3)

Since we assume that the vacuum alignment of the flavons occurs in the SUSY
limit, the vanishing of this $F$-term partially aligns the flavon VEVs

$F_{s}|_{V_{S}}=M^{2}+\lambda\,\phi_{i}^{2}|_{V_{S}}=0\;,\;\;i.e.\;\;\phi_{i}^{2}|_{V_{S}}=-\frac{M^{2}}{\lambda}\;.$
(4)

Given that $\Phi$ is a singlet, only one constraint can be obtained on the
flavon VEVs and, consequently, further driving fields (and flavons) are
necessary in order to fix the VEVs of all components of the flavon
$\phi$.111111In the more general case in which flavons and driving fields can
also be charged under a new gauge symmetry, the vanishing of $D$-terms aligns
the vacuum as well. We note that in this case both parameters $M$ and
$\lambda$ can be made real by a suitable choice of the phases of the fields
$\Phi$ and $\phi$. So, the form of the potential $V_{S}$ itself is given by

$V_{S}=F_{s}\,F_{s}^{\star}=M^{4}+M^{2}\,\lambda\,(\phi_{i}^{2}+(\phi_{i}^{\star})^{2})+V_{4}$
(5)

with $V_{4}$ containing the quartic terms. In the SUSY limit, we find,
consequently, for the first derivative of $V_{S}$ with respect to
$\phi_{j}^{\star}$

$\frac{\partial
V_{S}}{\partial\phi_{j}^{\star}}\Big{|}_{V_{S}}=0=2\,M^{2}\,\lambda\,\phi_{j}^{\star}|_{V_{S}}+\frac{\partial
V_{4}}{\partial\phi_{j}^{\star}}\Big{|}_{V_{S}}\;.$ (6)

Another possible form of $W$, with $\phi$ still being a real triplet, is

$W_{t}=M\,\Phi\,\phi+\lambda\,\Phi\,\phi^{2}$ (7)

with the driving field $\Phi$ transforming in the same way as $\phi$, namely
as a $\Phi\sim{\bf 3}$. While the first coupling is guaranteed to exist, the
second one requires that the product of two triplets contains the triplet
itself in its symmetric part. Note that we assume here for simplicity that the
multiplicity of the triplet in the product of two triplets is one (which is
true for many discrete groups). One such example is the group $S_{4}$, see
[6], section 4.3 as well as appendix A.4.

Likewise, we can study the case of $\phi$ being a complex triplet. Then, the
driving field necessarily also has to be a triplet. For $\phi\sim{\bf 3}$,
$\Phi$ has to transform as a ${\bf\bar{3}}$ such that we can write down the
first term in $W_{t}$ in Eq. (7). As is common, we use a basis in which the
representation matrices of the complex conjugate representation are the
complex conjugate of those of the representation. The second term in Eq. (7)
requires the product of two triplets to contain a triplet in its symmetric
part. Again, we assume that its multiplicity is one, such that a unique term
of the form $\Phi\,\phi^{2}$ exists. The group $T_{7}$ is an example of a
flavor symmetry that leads to a superpotential which is of the form of $W_{t}$
and we discuss it explicitly in section 4.1. The case of having two
independent cubic terms, $\Phi\,\phi^{2}$, is obtained with $\Delta(27)$, for
which we also present an example of in section 4.2. We note that a driving
field that is a singlet can usually not be employed, unless further flavons
are present in the setup, since the product of a complex triplet with itself
does not contain the trivial singlet ${\bf 1}$.121212One may wonder whether
one could use a driving field transforming as a non-trivial singlet, but then
the first term in Eq. (1) becomes forbidden, as long as it is assumed that the
flavor symmetry is unbroken at the level of the superpotential. A further
option is to introduce flavons in reducible representations of $G_{f}$ that
are composed of a triplet and its complex conjugate so that the product of two
of these reducible representations contains a trivial singlet.

In this case, the $F$-terms, one for each component of the driving field
$\Phi$, read

$F_{t,k}\equiv-
F_{\Phi_{k}}^{\star}=M\,\phi_{k}+\lambda\,c_{k,ij}\,\phi_{i}\,\phi_{j}$ (8)

with the coefficients $c_{k,ij}$ representing the relevant combination of
components of the fields $\Phi$ and $\phi$ leading to a trivial
singlet.131313We separate the index $k$ from $i$ and $j$, since the former
refers to the index of the driving field, while the latter denote the index of
the flavons. Note that the coefficients $c_{k,ij}$ are symmetric in the second
and third indices $i$ and $j$, i.e. $c_{k,ij}=c_{k,ji}$. Again, the vacuum of
the flavons is aligned in the SUSY limit, i.e. the $F$-terms have to vanish

$F_{t,k}|_{V_{S}}=M\,\phi_{k}|_{V_{S}}+\lambda\,c_{k,ij}\,\phi_{i}|_{V_{S}}\,\phi_{j}|_{V_{S}}=0\;.$
(9)

We note that it is in general possible to make the parameters $M$ and
$\lambda$ real. The corresponding part of the potential reads

$V_{S}=F_{t,k}\,F_{t,k}^{\star}\;.$ (10)

We remark here that while we present the idea assuming one pair of driving
field $\Phi$ and flavon $\phi$ at a time, some of the considered examples, see
sections 3 and 4, contain more driving fields and flavons in order to ensure
that the VEVs of all components of the flavon can be aligned, as in realistic
models usually at least two of these are present. Additionally, the
consideration of more than one flavon permits one to study further constraints
arising from the requirement that the VEVs in the non-SUSY minimum should be
rescaled by the same factor $\zeta$. The situation of several flavons may also
be reduced to the instance with one flavon only when it is possible to
integrate out all except one.

Finally, the vacuum values of the driving fields are aligned with the help of
the $F$-terms of the flavons and, due to the linearity of the superpotentials
$W_{s}$ and $W_{t}$ in the driving fields, can always be set to zero by
assuming a vanishing VEV for the driving fields. We assume such a vacuum
structure in the following. Furthermore, we consider all components of the
driving fields to be very heavy and, thus, irrelevant at low energies,
allowing us to restrict our focus to the flavon potential only.

###  2.2 Adding soft SUSY breaking terms

In order to break SUSY we add certain soft SUSY breaking terms. We remain
agnostic about the origin of these terms and assume that these can have the
structure we invoke in the following in order to preserve the vacuum alignment
obtained in the SUSY limit. In the case of several flavons, they should also
lead to the same rescaling for all flavon VEVs. For this reason, we do not add
all possible types of soft SUSY breaking terms (e.g. A-terms and B-terms) that
are compatible with the flavor symmetry, but only those that are necessary in
order to ensure that the masses of the corresponding SUSY partners can be
lifted. Consequently, we add only a universal soft mass term for the
flavon141414We can also introduce soft SUSY breaking terms for the driving
fields. For the soft mass parameters being positive, the corresponding vacuum
remains zero. In this case, driving fields do not play a role in the current
study.

$V_{\mathrm{soft},\,G_{f}}=m^{2}\,\phi^{\star}_{k}\,\phi_{k}\;.$ (11)

We note that this term is also always invariant under $G_{f}$. Furthermore,
the soft mass parameter $m^{2}$ is expected to be of the order of a (few) TeV
and, thus, in general much smaller than $M$, $m^{2}\ll M^{2}$.

In certain instances, it might be of advantage for the soft SUSY breaking
terms, in our case the soft masses, to explicitly but softly break the flavor
symmetry. Then, the most general form is

$V_{\mathrm{soft},\,gen}=m_{kl}^{2}\,\phi^{\star}_{k}\,\phi_{l}\;$ (12)

with $m_{kl}^{2}$ being a hermitean matrix, $m_{kl}^{2}=(m_{lk}^{2})^{\star}$,
such that $V_{\mathrm{soft},\,gen}$ itself is real. In general such arbitrary
soft masses do not preserve the vacuum alignment achieved in the SUSY limit,
but have to fulfill certain conditions, similar to those found in the context
of soft breaking mass terms in multi-Higgs doublet potentials [23], as we
detail below.

While we present the idea also for general soft masses, we mainly concentrate
on the case of flavor-symmetry preserving soft masses in the examples, found
in sections 3 and 4.

###  2.3 Non-SUSY potentials and their vacuum

With the information given in the preceding subsections, we can write down the
potential $V$ at low energies

$V=V_{S}+V_{\mathrm{soft}}$ (13)

where the potential $V_{S}$ can be found in Eqs. (5) and (10), respectively,
and $V_{\mathrm{soft}}$ is either of the form as in Eq. (11) or (12), and
study its vacuum. In particular, we can show that the vacuum alignment
obtained in the SUSY limit, called $\phi|_{V_{S}}$, is not altered, when
taking into account the soft SUSY masses, up to a possible rescaling with a
real factor $\zeta$ close to one, i.e. the VEV of the non-SUSY potential,
denoted by $\phi|_{V}$, is given by

$\phi|_{V}=\zeta\,\phi|_{V_{S}}\;\;\;\mbox{with}\;\;\;\zeta\approx 1\;.$ (14)

Furthermore, for general soft masses we derive constraints on their form,
needed to preserve Eq. (14).

In the case of the driving field $\Phi$ being a singlet, we note that for the
vacuum aligned in the SUSY limit the following holds

$\lambda\,\phi_{i}^{2}|_{V_{S}}=-M^{2}$ (15)

because of Eq. (4). Similarly, we obtain for $\Phi$ being a triplet that the
vacuum alignment in the SUSY limit has to fulfill

$\lambda\,c_{k,ij}\,\phi_{i}|_{V_{S}}\,\phi_{j}|_{V_{S}}=-M\,\phi_{k}|_{V_{S}}$
(16)

arising from Eq. (9).

####  2.3.1 Case 1: Singlet driving field, universal soft mass $m^{2}$

In this case, the form of the potential $V$ is given by

$V=V_{S}+V_{\mathrm{soft},\,G_{f}}$ (17)

with $V_{S}$ as found in Eq. (5). We begin by supposing that the vacuum of $V$
is related to the vacuum in the SUSY limit by a simple rescaling $\zeta$, see
Eq. (14). The $F$-term $F_{s}$ then takes the form

$F_{s}|_{V}=M^{2}+\lambda\,\phi_{i}^{2}|_{V}=M^{2}+\lambda\,\zeta^{2}\,\phi_{i}^{2}|_{V_{S}}=M^{2}(1-\zeta^{2})$
(18)

using Eqs. (14) and (4). Treating the soft mass term in an analogous way, we
can write the potential $V$ in its presumed minimum as a function of $\zeta$

$V(\zeta)=V(\phi|_{V})=M^{4}(1-\zeta^{2})^{2}+m^{2}\,\zeta^{2}\,\phi^{\star}_{k}|_{V_{S}}\,\phi_{k}|_{V_{S}}\;.$
(19)

Computing the first derivative, we see that $\zeta$ can take three
(approximate) values (for $m^{2}\ll M^{2}$)

$\zeta=0\;\;,\;\;\zeta\approx-1+\frac{m^{2}}{4\,M^{4}}\,\phi^{\star}_{k}|_{V_{S}}\,\phi_{k}|_{V_{S}}\;\;,\;\;\zeta\approx
1-\frac{m^{2}}{4\,M^{4}}\,\phi^{\star}_{k}|_{V_{S}}\,\phi_{k}|_{V_{S}}\;.$
(20)

Plugging these possible values for the minimum into the potential, we see
that, while $V(\zeta=0)=M^{4}$, the other two potential extrema lead to

$V(\zeta)\approx m^{2}\,\phi^{\star}_{k}|_{V_{S}}\,\phi_{k}|_{V_{S}}\;.$ (21)

As we consider only situations in which the minima are isolated and $m^{2}\ll
M^{2}$, the only plausible solution is, indeed, $\zeta\approx 1$. The other
solution, $\zeta\approx-1$, is a consequence of the symmetry of the potential
in the flavon $\phi$. We can also analyze the first derivative of the
potential with respect to the flavon $\phi_{a}^{\star}$, which can be written
as

$\frac{\partial
V}{\partial\phi_{a}^{\star}}=2\,M^{2}\,\lambda\,\phi_{a}^{\star}+\frac{\partial
V_{4}}{\partial\phi_{a}^{\star}}+m^{2}\,\phi_{a}$ (22)

using the form in Eq. (5). From

$2\,M^{2}\,\lambda\,\phi_{a}^{\star}|_{V}+\frac{\partial
V_{4}}{\partial\phi_{a}^{\star}}\Big{|}_{V}+m^{2}\,\phi_{a}|_{V}=0$ (23)

and supposing that the rescaling of the vacuum holds, see Eq. (14), the vacuum
is real and using Eq. (6), we have

$\zeta\,\left(2\,(1-\zeta^{2})\,M^{2}\,\lambda+m^{2}\right)\,\phi_{a}|_{V_{S}}=0\;.$
(24)

This leads to the same result for $\zeta$ as in Eq. (20), if we take into
account that the vacuum in the SUSY limit is constrained by Eq. (4).

####  2.3.2 Case 2: Singlet driving field, general soft masses $m_{kl}^{2}$

With general soft masses, the potential $V$ reads

$V=V_{S}+V_{\mathrm{soft},\,gen}\;,$ (25)

where $V_{S}$ is given in Eq. (5). We proceed in a similar way as for case 1.
First, we assume again that the vacuum is only rescaled with a real parameter
$\zeta$, see Eq. (14). We consider the first derivative of the potential in
Eq. (25) with respect to $\phi_{a}^{\star}$

$\frac{\partial
V}{\partial\phi_{a}^{\star}}=2\,M^{2}\,\lambda\,\phi_{a}^{\star}+\frac{\partial
V_{4}}{\partial\phi_{a}^{\star}}+m_{ai}^{2}\,\phi_{i}\;.$ (26)

Using Eqs. (6) and (14) and $\zeta$ real, we obtain the following equation

$\zeta\,\Big{(}2\,(1-\zeta^{2})\,M^{2}\,\lambda\,\phi_{a}^{\star}|_{V_{S}}+m_{ai}^{2}\,\phi_{i}|_{V_{S}}\Big{)}=0\;.$
(27)

With this we can not only compute the size of the rescaling $\zeta$, but also
determine the form of the general soft masses that is compatible with this
minimum in the non-SUSY case. The latter becomes more obvious by noticing that
for a real vacuum, $\phi_{i}|_{V_{S}}=\phi_{i}^{\star}|_{V_{S}}$, the equation
can be written as (discarding the solution $\zeta=0$)

$m_{ai}^{2}\,\phi_{i}|_{V_{S}}=-2\,(1-\zeta^{2})\,M^{2}\,\lambda\,\phi_{a}|_{V_{S}}\;,$
(28)

meaning that the vacuum in the SUSY limit should be an eigenvector of the soft
mass matrix $m_{kl}^{2}$ with eigenvalue $-2\,(1-\zeta^{2})\,M^{2}\,\lambda$
in order to be compatible with the simple rescaling of the vacuum. This is
similar to the findings of [23] obtained for multi-Higgs doublet potentials.

####  2.3.3 Case 3: Triplet driving field, universal soft mass $m^{2}$

Here, we have as potential $V$

$V=V_{S}+V_{\mathrm{soft},\,G_{f}}$ (29)

with $V_{S}$ found in Eq. (10). We consider also in this case the value of the
potential as a function of the parameter $\zeta$, assuming that Eq. (14)
holds. We find

$V(\zeta)=V(\phi|_{V})=V_{S}|_{V}+V_{\mathrm{soft},\,G_{f}}|_{V}=\zeta^{2}\,\left((1-\zeta)^{2}\,M^{2}+m^{2}\right)\,\phi^{\star}_{k}|_{V_{S}}\,\phi_{k}|_{V_{S}}\;,$
(30)

since

$\displaystyle F_{t,k}|_{V}$ $\displaystyle=$ $\displaystyle
M\,\phi_{k}|_{V}+\lambda\,c_{k,ij}\,\phi_{i}|_{V}\,\phi_{j}|_{V}$ (31)
$\displaystyle=$
$\displaystyle\zeta\,M\,\phi_{k}|_{V_{S}}+\lambda\,\zeta^{2}\,c_{k,ij}\,\phi_{i}|_{V_{S}}\,\phi_{j}|_{V_{S}}$
$\displaystyle=$ $\displaystyle\zeta\,(1-\zeta)\,M\,\phi_{k}|_{V_{S}}\;,$

see Eqs. (9) and (16). Extremizing $V$ in Eq. (30), we obtain as (approximate)
solutions (for $m^{2}\ll M^{2}$)

$\zeta=0\;\;,\;\;\zeta\approx\frac{1}{2}+\frac{m^{2}}{M^{2}}\;\;,\;\;\zeta\approx
1-\frac{m^{2}}{M^{2}}\;.$ (32)

The values of the potential at these points are $V(\zeta=0)=0$ and

$V\left(\zeta\approx\frac{1}{2}+\frac{m^{2}}{M^{2}}\right)\approx\frac{1}{16}\,M^{2}\,\phi^{\star}_{k}|_{V_{S}}\,\phi_{k}|_{V_{S}}\;\;,\;\;V\left(\zeta\approx
1-\frac{m^{2}}{M^{2}}\right)\approx
m^{2}\,\phi^{\star}_{k}|_{V_{S}}\,\phi_{k}|_{V_{S}}\;.$ (33)

Inspecting these, only the last one can correspond to a minimum that arises
from a small perturbation of the one obtained in the SUSY limit.

The first derivative of $V$ with respect to the field $\phi_{a}^{\star}$ at
the minimum of this potential is given by

$\left.\frac{\partial
V}{\partial\phi_{a}^{\star}}\right|_{V}=0=F_{t,k}|_{V}\,\left.\frac{\partial
F_{t,k}^{\star}}{\partial\phi_{a}^{\star}}\right|_{V}+m^{2}\,\phi_{a}|_{V}\;.$
(34)

Expanding the $F$-terms around the minimum obtained in the SUSY limit, i.e.

$F_{t,k}|_{V}\approx\left.\frac{\partial
F_{t,k}}{\partial\phi_{b}}\right|_{V_{S}}\,(\phi_{b}|_{V}-\phi_{b}|_{V_{S}})\;\;,\left.\frac{\partial
F_{t,k}^{\star}}{\partial\phi_{a}^{\star}}\right|_{V}\approx\left.\frac{\partial
F_{t,k}^{\star}}{\partial\phi_{a}^{\star}}\right|_{V_{S}}\;,$ (35)

and assuming the rescaling, see Eq. (14), we get

$(\zeta-1)\,\left.\frac{\partial
F_{t,k}^{\star}}{\partial\phi_{a}^{\star}}\right|_{V_{S}}\,\left.\frac{\partial
F_{t,k}}{\partial\phi_{b}}\right|_{V_{S}}\,\phi_{b}|_{V_{S}}+\zeta\,m^{2}\,\phi_{a}|_{V_{S}}\approx
0\;.$ (36)

We identify the first expression as mass matrix of the flavons in the SUSY
limit

${\cal M}^{2}_{ab}\equiv\left.\frac{\partial
F_{t,k}^{\star}}{\partial\phi_{a}^{\star}}\right|_{V_{S}}\,\left.\frac{\partial
F_{t,k}}{\partial\phi_{b}}\right|_{V_{S}}\;.$ (37)

Using the expectation that $\zeta$ is close to one, up to corrections of order
$m^{2}/M^{2}$, we arrive at

$(1-\zeta)\,{\cal M}^{2}_{ab}\,\phi_{b}|_{V_{S}}\approx
m^{2}\,\phi_{a}|_{V_{S}}\;.$ (38)

This equation has, indeed, as solution $(1-\zeta)$ of order $m^{2}/M^{2}$,
since the eigenvalues of the mass matrix ${\cal M}^{2}_{ab}$ are of order
$M^{2}$.

####  2.3.4 Case 4: Triplet driving field, general soft masses $m_{kl}^{2}$

For the last case, we begin with the potential $V$ being of the form

$V=V_{S}+V_{\mathrm{soft},\,gen}$ (39)

with $V_{S}$ taken from Eq. (10). The first derivative of $V$ with respect to
the field $\phi_{a}^{\star}$ at the minimum of this potential is of the form

$\left.\frac{\partial
V}{\partial\phi_{a}^{\star}}\right|_{V}=0=F_{t,k}|_{V}\,\left.\frac{\partial
F_{t,k}^{\star}}{\partial\phi_{a}^{\star}}\right|_{V}+m_{al}^{2}\,\phi_{l}|_{V}\;.$
(40)

Expanding the $F$-terms around the minimum obtained in the SUSY limit as in
Eq. (35) and employing the rescaling, see Eq. (14), we have

$(\zeta-1)\,{\cal
M}^{2}_{ab}\,\phi_{b}|_{V_{S}}+\zeta\,m_{al}^{2}\,\phi_{l}|_{V_{S}}\approx 0$
(41)

with the mass matrix ${\cal M}^{2}_{ab}$ given in Eq. (37). Neglecting higher
orders in $m^{2}/M^{2}$, we find

$(1-\zeta)\,{\cal M}^{2}_{ab}\,\phi_{b}|_{V_{S}}\approx
m_{al}^{2}\,\phi_{l}|_{V_{S}}\;.$ (42)

Now, if the vacuum $\phi_{b}|_{V_{S}}$ is an eigenvector of the mass matrix
${\cal M}^{2}_{ab}$, we see that this condition simplifies and the form of the
general soft masses is constrained to also have this vacuum as eigenvector to
a certain eigenvalue. This result is equivalent to the one obtained in Eq.
(28).

If the vacuum $\phi_{b}|_{V_{S}}$ and the coefficients $c_{k,ij}$ are real and
the latter fulfill $c_{k,ij}=c_{i,kj}=c_{j,ik}$, it is, indeed,
straightforward to show that the vacuum $\phi_{b}|_{V_{S}}$ is an eigenvector
of the mass matrix ${\cal M}^{2}_{ab}$. Consider $\frac{\partial
F_{t,k}}{\partial\phi_{b}}$ which can be computed from Eq. (8) as

$\frac{\partial
F_{t,k}}{\partial\phi_{b}}=M\,\delta_{kb}+2\,\lambda\,c_{k,ib}\,\phi_{i}$ (43)

using the symmetry of the coefficients $c_{k,ij}$. Thus, we have in the vacuum
in the SUSY limit

$\left.\frac{\partial
F_{t,k}}{\partial\phi_{b}}\right|_{V_{S}}\,\phi_{b}|_{V_{S}}=M\,\phi_{k}|_{V_{S}}+2\,(-M\phi_{k}|_{V_{S}})=-M\,\phi_{k}|_{V_{S}}\;,$
(44)

taking into account Eq. (16). For real $\phi_{b}|_{V_{S}}$ and $c_{k,ij}$ and
remembering that $M$ and $\lambda$ can be made real without loss of
generality, we have

${\cal M}^{2}_{ab}\,\phi_{b}|_{V_{S}}=\left.\frac{\partial
F_{t,k}^{\star}}{\partial\phi_{a}^{\star}}\right|_{V_{S}}\,(-M\,\phi_{k}|_{V_{S}})=-M^{2}\,\phi_{a}|_{V_{S}}-2\,M\,\lambda\,c_{k,ia}\,\phi_{i}|_{V_{S}}\,\phi_{k}|_{V_{S}}\;.$
(45)

This can be further simplified, if $c_{k,ij}=c_{i,kj}=c_{j,ik}$ holds, and we
again use Eq. (16)

${\cal
M}^{2}_{ab}\,\phi_{b}|_{V_{S}}=-M^{2}\,\phi_{a}|_{V_{S}}-2\,M\,(-M\,\phi_{a}|_{V_{S}})=M^{2}\,\phi_{a}|_{V_{S}}\;.$
(46)

In this case, the condition in Eq. (42) reduces to

$(1-\zeta)\,M^{2}\,\phi_{a}|_{V_{S}}\approx m_{al}^{2}\,\phi_{l}|_{V_{S}}\;,$
(47)

meaning the general soft mass matrix should have $\phi|_{V_{S}}$ as
eigenvector to the eigenvalue $(1-\zeta)\,M^{2}$.

###  2.4 Comment on other corrections to vacuum alignment

In the preceding part, we have focused on corrections to the vacuum alignment
in the form of a real rescaling $\zeta$. One may wonder whether corrections
orthogonal to the direction of the vacuum alignment, obtained in the SUSY
limit, could lead to a deeper minimum of the non-SUSY potential than the one
due to the rescaled vacuum. We can write the vacuum in the form

$\phi|_{V}=\zeta\,\phi|_{V_{S}}+\alpha\,\phi_{\perp,1}+\beta\,\phi_{\perp,2}$
(48)

with $\alpha$ and $\beta$ being (real) coefficients and $\phi_{\perp,1}$ and
$\phi_{\perp,2}$ being vectors orthogonal to the vacuum $\phi|_{V_{S}}$ and
orthogonal to each other. Plugging this form of $\phi|_{V}$ into the $F$-terms
of the driving fields leads to a positive contribution to the potential $V$
for $\alpha$ and/or $\beta$ non-zero. Furthermore, for universal soft masses
this form of $\phi|_{V}$ gives for non-zero $\alpha$ and/or $\beta$ rise to a
larger value than for $\alpha=\beta=0$ (corresponding to the rescaled vacuum).
Thus, having any of the parameters $\alpha$ and $\beta$ non-zero does not lead
to smaller values of the potential than the rescaled vacuum. For general soft
masses one can require these to be of the form

$m_{kl}^{2}=c_{0}\,\phi|_{V_{S}}\,\phi|_{V_{S}}^{\dagger}+c_{1}\,\phi_{\perp,1}\,\phi_{\perp,1}^{\dagger}+c_{2}\,\phi_{\perp,2}\,\phi_{\perp,2}^{\dagger}$
(49)

such that $\phi|_{V_{S}}$, $\phi_{\perp,1}$ and $\phi_{\perp,2}$ are
eigenvectors of the general soft masses with (real) eigenvalues proportional
to $c_{0}$, $c_{1}$ and $c_{2}$, respectively. These can be chosen
appropriately such that the rescaled vacuum leads to the smallest value of the
potential $V$.

##  3 Vacuum alignment in $A_{4}$ model

In this section we discuss in detail the case of flavons that transform as
(real) triplets under the flavor symmetry $A_{4}$. We first illustrate the
example of a single flavon and then generalize to the case of more flavons.

###  3.1 One flavon triplet

Suppose $\phi_{a}^{T}\equiv(a_{1},a_{2},a_{3})$ transforms as triplet of
$A_{4}$, see Tab. 1, and that in the SUSY limit, only one of the components of
$\phi_{a}$ attains a non-zero VEV. This alignment is achieved with the help of
the driving fields $\Phi_{a}$ and $\Phi_{d}$, $\Phi_{a}\sim{\bf 1}$ and
$\Phi_{d}\sim{\bf 3}$ under $A_{4}$. The relevant superpotential is of the
form

$\displaystyle W_{a}$ $\displaystyle=$
$\displaystyle\lambda_{a}\,\Phi_{a}\,\left(\,\phi_{a}^{2}-x^{2}\,\right)\>+\>\lambda_{d}\,\Phi_{d}\,\phi_{a}^{2}\,.$
(50)

We note that in addition to the flavor symmetry $A_{4}$ and the $R$-symmetry
$U(1)_{R}$ we use a $Z_{N}$ shaping symmetry in order to restrict the number
of allowed terms. The first term of the superpotential is $SO(3)$-invariant.
The term $\lambda_{a}\,\Phi_{a}\,x^{2}$, which is not invariant under the
$Z_{N}$ shaping symmetry, is assumed to be generated by the interaction with
other fields that get a VEV at a higher scale. We also assume that the
couplings $\lambda_{a}$ and $\lambda_{d}$ as well as the parameter $x$ are
real.

####  3.1.1 Vacuum alignment in SUSY limit

In the SUSY limit, the vacuum alignment of the flavon $\phi_{a}$ is determined
by the vanishing of the $F$-terms of the driving fields. These are given by

$\displaystyle\frac{\partial W_{a}}{\partial\Phi_{a}}$ $\displaystyle=$
$\displaystyle-
F_{a^{0}}^{\star}~{}=~{}\lambda_{a}\,\left(\,a_{1}^{2}+a_{2}^{2}+a_{3}^{2}-x^{2}\,\right),$
(51) $\displaystyle\frac{\partial W_{a}}{\partial\Phi_{d_{i}}}$
$\displaystyle=$ $\displaystyle-
F_{d_{i}^{0}}^{\star}~{}=~{}\lambda_{d}\,a_{i+1}\,a_{i+2}\,.$ (52)

In these equations, the subscripts related to the components of the triplets
are understood to be cyclic with values from 1 to 3.

Fields | $\phi_{a}$ | $\Phi_{a}$ | $\Phi_{d}$
---|---|---|---
$A_{4}$ | 3 | 1 | 3
$U(1)_{R}$ | 0 | 2 | 2
$Z_{N}$ | 1 | $N-2$ | $N-2$
Table 1: Charge assignment for the flavon and driving fields of the $A_{4}$
model with a single flavon.

Setting Eqs. (51) and (52) to zero fixes both the direction and size of the
VEV of $\phi_{a}$. In particular, Eq. (52) establishes that
$\langle{\phi_{a}}\rangle$ must have only one component different from zero,
and Eq. (51) sets its size. Without loss of generality, we choose

$\displaystyle\langle\phi_{a}\rangle^{T}$ $\displaystyle=$
$\displaystyle(x,\,0,\,0)\;.$ (53)

The freedom to choose the position of the non-zero component reflects the fact
that the minimum of the potential is symmetric under $A_{4}$, i.e. the
solution shown in Eq. (53) can be transformed by any element of the group
(which changes the position of the non-zero component) and still be a
solution. This is, similarly, true for the sign of the flavon VEV.

The corresponding potential $V_{S}$ can be written as

$\displaystyle V_{S}$ $\displaystyle=\lambda_{a}^{2}\
\left|a_{1}^{2}+a_{2}^{2}+a_{3}^{2}-x^{2}\right|^{2}+\lambda_{d}^{2}\left(|a_{1}|^{2}|a_{2}|^{2}+|a_{2}|^{2}|a_{3}|^{2}+|a_{3}|^{2}|a_{1}|^{2}\right)\;.$
(54)

In general, the extrema of the potential are derived from the vanishing of the
first derivatives with respect to the components $a_{i}$ of the flavon
$\phi_{a}$

$\displaystyle\left.\frac{\partial V_{S}}{\partial a_{i}}\right|_{\rm ext}$
$\displaystyle=$ $\displaystyle
2\,\lambda_{a}^{2}\,a_{i}\left((a_{1}^{\star})^{2}+(a_{2}^{\star})^{2}+(a_{3}^{\star})^{2}-x^{2}\right)+{\lambda_{d}^{2}\,a_{i}^{\star}\,\left(\left|a_{i+1}\right|^{2}+\left|a_{i+2}\right|^{2}\right)}=0\;.\quad$
(55)

It is easy to check that these conditions are consistent with the vacuum
alignment in Eq. (53) for the minimum derived from the vanishing of the
$F$-terms. The value of the potential $V_{S}$ is zero at this minimum, as
expected.

The next step is to break SUSY, but maintain this vacuum alignment.

####  3.1.2 Vacuum alignment including soft SUSY breaking

As discussed in section 2, the vacuum alignment derived in the SUSY limit can
be preserved, if the soft mass matrix fulfills certain conditions. In the
$A_{4}$ model, we introduce a universal soft mass for the flavon $\phi_{a}$ so
that the potential $V$ becomes

$V~{}=~{}V_{S}\>+\>\mu_{a}^{2}\,a_{i}^{\star}\,a_{i}$ (56)

with $V_{S}$ given in Eq. (54). Since the added soft mass is universal, we
expect that the alignment achieved in the SUSY limit remains preserved.

Applying the results of section 2, we assume that the aligned vacuum is only
rescaled by a real parameter $\zeta$ and can compute the value of the
potential $V$ as function of $\zeta$

$V(\zeta)~{}=~{}V(\phi_{a}|_{V})~{}=~{}\lambda_{a}^{2}~{}x^{4}(1-\zeta^{2})^{2}\>+\>\zeta^{2}\mu_{a}^{2}\,x^{2}\;.$
(57)

Minimizing the potential with respect to $\zeta$, we obtain

$0~{}=~{}2\,\zeta\;\left(2\,(\zeta^{2}-1)\,\lambda_{a}^{2}~{}x^{4}\>+\>\mu_{a}^{2}\,x^{2}\right)\;.$
(58)

Therefore, we have (discarding $\zeta=0$)

$\zeta^{2}~{}=~{}1-\frac{\mu_{a}^{2}\,}{2\,\lambda_{a}^{2}~{}x^{2}}$ (59)

which is consistent with the expectation that $\zeta$ is of order one up to
corrections suppressed by $m^{2}/M^{2}$.

This result can be compared to an explicit calculation which shows that the
addition of the universal soft mass to the potential, indeed, preserves the
vacuum alignment of the flavon. The minimization conditions are similar to the
previous ones in Eq. (55) except for the new contribution proportional to the
soft mass parameter $\mu_{a}^{2}$

$\displaystyle\left.\frac{\partial V}{\partial a_{i}}\right|_{\rm ext}$
$\displaystyle=$ $\displaystyle\left.\frac{\partial V_{S}}{\partial
a_{i}}\right|_{\rm ext}~{}+~{}\mu_{a}^{2}\,a_{i}^{\star}~{}=~{}0\;.$ (60)

As we have the freedom to choose the direction of the VEV, setting
$a_{2}=a_{3}=0$, the solution of Eq. (60) yields as global minimum of the
potential

$\displaystyle a_{1}^{2}$
$\displaystyle=x^{2}\left(1-\frac{\mu_{a}^{2}}{2\,\lambda_{a}^{2}\,x^{2}}\right),\quad
a_{2}=a_{3}=0,$ (61)

see the detailed discussion in appendix B. The alignment of the flavon remains
the same, but its magnitude is rescaled by an amount proportional to the soft
mass parameter $\mu_{a}^{2}$. This is consistent with Eq. (59).

###  3.2 Two flavon triplets

We now present the case of an $A_{4}$ model with two flavon triplets whose
vacua are aligned orthogonally in the SUSY limit. In addition to the flavon
and driving fields discussed in this section so far, we introduce a second
flavon $\phi_{b}^{T}\equiv(b_{1},b_{2},b_{3})$ and driving fields $\Phi_{b}$,
$\Phi_{c}$ and $\Phi_{e}$ that allow to fix the vacuum alignment of the
flavons. We use the following superpotential

$\displaystyle W_{ab}$ $\displaystyle=$
$\displaystyle\lambda_{a}\,\Phi_{a}\,\left(\,\phi_{a}^{2}-x^{2}\,\right)\>+\>\lambda_{b}\,\Phi_{b}\,\left(\,\phi_{b}^{2}-y^{2}\right)\>+\>\lambda_{c}\,\Phi_{c}\,\phi_{a}\,\phi_{b}$
(62)
$\displaystyle+\lambda_{d}\,\Phi_{d}\,\phi_{a}^{2}\>+\>\lambda_{e}\,\Phi_{e}\,\phi_{b}^{2}\;.$

The charge assignment of the flavons and driving fields is listed in Tab. 2.
Note that we have to use a further shaping symmetry, $Z_{M}$. The term
$\lambda_{b}\,\Phi_{b}\,y^{2}$ is not $Z_{M}$-invariant and is assumed to
arise in a similar way as the term $\lambda_{a}\,\Phi_{a}\,x^{2}$. Like the
couplings $\lambda_{a}$ and $\lambda_{d}$ as well as the parameter $x$, we
take $\lambda_{b}$, $\lambda_{c}$, $\lambda_{e}$ and $y$ to be real.

Fields | $\phi_{a}$ | $\phi_{b}$ | $\Phi_{a}$ | $\Phi_{b}$ | $\Phi_{c}$ | $\Phi_{d}$ | $\Phi_{e}$
---|---|---|---|---|---|---|---
$A_{4}$ | 3 | 3 | 1 | 1 | 1 | 3 | 3
$U(1)_{R}$ | 0 | 0 | 2 | 2 | 2 | 2 | 2
$Z_{N}$ | 1 | 0 | $N-2$ | 0 | $N-1$ | $N-2$ | 0
$Z_{M}$ | 0 | 1 | 0 | $M-2$ | $M-1$ | 0 | $M-2$
Table 2: Charge assignment for the flavons and driving fields of the $A_{4}$
model with two flavons.

The vacuum alignment in the SUSY limit is obtained from the vanishing of the
$F$-terms. In addition to Eqs. (51) and (52), we have for the $F$-terms

$\displaystyle\frac{\partial W_{ab}}{\partial\Phi_{b}}$ $\displaystyle=$
$\displaystyle-
F_{b^{0}}^{\star}~{}=~{}\lambda_{b}\,\left(\,b_{1}^{2}+b_{2}^{2}+b_{3}^{2}-y^{2}\,\right)\,,$
(63) $\displaystyle\frac{\partial W_{ab}}{\partial\Phi_{c}}$ $\displaystyle=$
$\displaystyle-
F_{c^{0}}^{\star}~{}=~{}\lambda_{c}\,\left(a_{1}\,b_{1}+a_{2}\,b_{2}+a_{3}\,b_{3}\right)\,,$
(64) $\displaystyle\frac{\partial W_{ab}}{\partial\Phi_{e_{i}}}$
$\displaystyle=$ $\displaystyle-
F_{e_{i}^{0}}^{\star}~{}=~{}\lambda_{e}\,b_{i+1}\,b_{i+2}\;$ (65)

with cyclic indices assumed. As before, Eqs. (63)-(65) determine the size of
the VEV of $\phi_{b}$ and imply that only one of the components of this flavon
can get a non-zero VEV. In particular, Eq. (64) is used to ensure the
orthogonality of the vacua of $\phi_{a}$ and $\phi_{b}$. Keeping the alignment
of the VEV of $\phi_{a}$ as given in Eq. (53), we choose

$\displaystyle\langle\phi_{b}\rangle^{T}=(0,\,y,\,0).$ (66)

The corresponding potential reads

$\displaystyle V_{S}$ $\displaystyle=\lambda_{a}^{2}\
\left|a_{1}^{2}+a_{2}^{2}+a_{3}^{2}-x^{2}\right|^{2}+\lambda_{b}^{2}\
\left|b_{1}^{2}+b_{2}^{2}+b_{3}^{2}-y^{2}\right|^{2}+\lambda_{c}^{2}\
\left|a_{1}b_{1}+a_{2}b_{2}+a_{3}b_{3}\right|^{2}$
$\displaystyle+\lambda_{d}^{2}\left(|a_{1}|^{2}|a_{2}|^{2}+|a_{2}|^{2}|a_{3}|^{2}+|a_{3}|^{2}|a_{1}|^{2}\right)+\lambda_{e}^{2}\left(|b_{1}|^{2}|b_{2}|^{2}+|b_{2}|^{2}|b_{3}|^{2}+|b_{3}|^{2}|b_{1}|^{2}\right).$
(67)

Extremizing the potential with respect to the components of the flavons, we
get

$\displaystyle\left.\frac{\partial V_{S}}{\partial a_{i}}\right|_{\rm ext}$
$\displaystyle=$ $\displaystyle
2\,\lambda_{a}^{2}\,a_{i}\left((a_{1}^{\star})^{2}+(a_{2}^{\star})^{2}+(a_{3}^{\star})^{2}-x^{2}\right)+\lambda_{c}^{2}\,b_{i}\left(a_{1}^{\star}b_{1}^{\star}+a_{2}^{\star}b_{2}^{\star}+a_{3}^{\star}b_{3}^{\star}\right)$
(68) $\displaystyle+$
$\displaystyle{\lambda_{d}^{2}\,a_{i}^{\star}\left(\left|a_{i+1}\right|^{2}+\left|a_{i+2}\right|^{2}\right)}=0\,,\quad$
$\displaystyle\left.\frac{\partial V_{S}}{\partial b_{i}}\right|_{\rm ext}$
$\displaystyle=$ $\displaystyle
2\,\lambda_{b}^{2}\,b_{i}\left((b_{1}^{\star})^{2}+(b_{2}^{\star})^{2}+(b_{3}^{\star})^{2}-y^{2}\right)+\lambda_{c}^{2}\,a_{i}\left(a_{1}^{\star}b_{1}^{\star}+a_{2}^{\star}b_{2}^{\star}+a_{3}^{\star}b_{3}^{\star}\right)$
(69) $\displaystyle+$
$\displaystyle\lambda_{e}^{2}\,b_{i}^{\star}\,\left(\left|b_{i+1}\right|^{2}+\left|b_{i+2}\right|^{2}\right)~{}=~{}0\;.\quad$

These constraints are satisfied by the vacuum alignment in Eqs. (53) and (66).

As before, we use universal soft mass terms for the flavons. These are of the
form $\mu_{a}^{2}\,a_{i}^{\star}\,a_{i}$ and
$\mu_{b}^{2}\,b_{i}^{\star}\,b_{i}$. As a consequence, the vacuum alignment
obtained in the SUSY limit remains preserved. In the following, we assume a
single real rescaling $\zeta$ for both vacua and calculate the condition that
results for the soft mass parameters $\mu_{a}^{2}$ and $\mu_{b}^{2}$.

The relevant potential $V$ as function of $\zeta$ is given by

$V(\zeta)~{}=~{}V(\phi_{a},\phi_{b}|_{V})~{}=~{}\lambda_{a}^{2}~{}x^{4}(1-\zeta^{2})^{2}\>+\>\lambda_{b}^{2}~{}y^{4}(1-\zeta^{2})^{2}\>+\>\zeta^{2}\mu_{a}^{2}\,x^{2}\>+\>\zeta^{2}\mu_{b}^{2}\,y^{2}.$
(70)

Minimizing the potential and discarding $\zeta=0$, we obtain

$\zeta^{2}~{}=~{}1-\frac{\mu_{a}^{2}\,x^{2}\>+\>\mu_{b}^{2}\,y^{2}}{2\left(\lambda_{a}^{2}~{}x^{4}\>+\>\lambda_{b}^{2}~{}y^{4}\right)}\,.$
(71)

Nevertheless, we can proceed as in the case of one flavon only and extremize
the potential $V$ with respect to the flavons

$\displaystyle\left.\frac{\partial V}{\partial\phi_{i}}\right|_{\rm ext}$
$\displaystyle=$ $\displaystyle\left.\frac{\partial
V_{S}}{\partial\phi_{i}}\right|_{\rm
ext}~{}+~{}\mu_{i}^{2}\,\phi_{i}^{\star}~{}=~{}0$ (72)

for $i=a,b$. As before, we keep $a_{2}=a_{3}=0$. Then, the solution of Eq.
(72) yields as global minimum of the potential

$\displaystyle a_{1}^{2}$
$\displaystyle=x^{2}\,\left(1-\frac{\mu_{a}^{2}}{2\,\lambda_{a}^{2}\,x^{2}}\right)\,,\quad
a_{2}=a_{3}=0\;,$ (73) $\displaystyle b_{1}$ $\displaystyle=0\,,\quad
b_{2}^{2}=y^{2}\,\left(1-\frac{\mu_{b}^{2}}{2\,\lambda_{b}^{2}\,y^{2}}\right)\,,\quad
b_{3}=0\;,$ (74)

as shown in appendix B. We, thus, have to impose the condition
$\;\mu_{a}^{2}/(\lambda_{a}^{2}\,x^{2})=\mu_{b}^{2}/(\lambda_{b}^{2}\,y^{2})\;$
such that

$\zeta^{2}~{}=~{}1-\frac{\mu_{a}^{2}}{2\,\lambda_{a}^{2}~{}x^{2}}~{}=~{}1-\frac{\mu_{b}^{2}}{2\,\lambda_{b}^{2}~{}y^{2}}\,.$
(75)

This result is consistent with the one in Eq. (71) for the assumed relation
among the soft mass parameters.

###  3.3 Effect of higher-order terms on SUSY potential

The purpose of the $Z_{n}$ shaping symmetries is not only to constrain the
number of terms in the superpotential at the renormalizable level, but also to
control the effect of higher-order terms on the vacuum alignment achieved with
the help of the $F$-terms of the driving fields. These higher-order terms
contain in general more than two flavons (and always one driving field in
order to keep the $R$-symmetry intact) and are thus non-renormalizable. They
are taken to be suppressed by the cutoff scale $\Lambda$ which is assumed to
be larger than all scales present in the superpotential, especially larger
than the scales $x$ and $y$.

In order to show that the effect of higher-order terms can be kept well under
control, we study the flavon combinations $\phi_{a}^{n}$, $\phi_{b}^{m}$ and
$\phi_{a}^{n}\,\phi_{b}^{m}$ for $n$ and $m$ integer and $n+m$ larger than
two, their VEVs following from the VEVs of the flavons $\phi_{a}$ and
$\phi_{b}$, see Eqs. (53) and (66), and how they couple to the different
driving fields $\Phi_{a}$, …, $\Phi_{e}$. Indeed, we can see that only small
shifts in the size of the VEVs of $\phi_{a}$ and $\phi_{b}$, $x$ and $y$, are
induced, as long as we assume that the indices of the $Z_{n}$ shaping
symmetries, $N$ and $M$, are both even. The smallest possible viable choice is
$N=4$ and $M=4$, since in the case of $N$ and/or $M$ being two, the driving
fields $\Phi_{a}$ and $\Phi_{d}$ and/or $\Phi_{b}$ and $\Phi_{e}$ would turn
out to be uncharged under the shaping symmetries, see Tab. 2, and, thus,
further terms would become allowed. Choosing values for $N$ and/or $M$ larger
than four is also possible and leads to a larger suppression of the shift in
the size of the VEVs of the flavons. We note that $N$ and $M$ may also take on
distinct values.

A detailed discussion of the flavon combinations, their VEVs and their impact
on the $F$-terms of the different driving fields can be found in appendix C.

###  3.4 Comment on generalization to three flavon triplets

The example of two flavons whose VEVs are aligned orthogonally can be
generalized to three flavon triplets with orthogonal alignment by adding
another flavon, suitably chosen driving fields and a further shaping symmetry
$Z_{P}$. All statements made regarding the vacuum alignment in the SUSY limit
and including soft SUSY breaking terms, as well as those regarding the impact
of higher-order terms can be adapted straightforwardly to the case of three
flavon triplets.

##  4 Vacuum alignment for other groups

In the previous example for the group $A_{4}$, it has been shown that the
explicit minimization of the scalar potential renders the same results as
those derived in section 2.

In the following, further examples are presented based on the groups $T_{7}$,
$\Delta(27)$ and $S_{4}$. These cover different situations including the case
of real and complex triplet representations as well as universal and general
soft masses.

###  4.1 $T_{7}$ model

We consider first the simple case with two fields, a flavon and a driving
field. Since the group $T_{7}$ only contains complex triplet representations,
the flavon, $\phi^{T}\equiv(a_{1},a_{2},a_{3})$, and the driving field,
$\Phi^{T}\equiv(a^{0}_{1},a^{0}_{2},a^{0}_{3})$, transform as a triplet and
antitriplet, respectively. More information about the group $T_{7}$ can be
found in appendix A.2 and references therein.

For our discussion, the relevant terms of the superpotential are

$\displaystyle W$ $\displaystyle=$ $\displaystyle
M\,\Phi\,\phi+\lambda\,\Phi\,\phi^{2}$ (76) $\displaystyle=$ $\displaystyle
M\,\left(a^{0}_{1}\,a_{1}+a^{0}_{2}\,a_{2}+a^{0}_{3}\,a_{3}\right)+\lambda\,\left(a^{0}_{1}\,a_{3}^{2}+a^{0}_{2}\,a_{1}^{2}+a^{0}_{3}\,a_{2}^{2}\right)\;.$

Differentiating with respect to the components of the driving field, the
vanishing of their $F$-terms gives

$\displaystyle\frac{\partial W}{\partial a^{0}_{i}}$ $\displaystyle=$
$\displaystyle-
F_{i}^{\star}~{}=~{}M\,a_{i}\>+\>\lambda\,a_{i+2}^{2}~{}=~{}0\;,$ (77)

where again cyclic indices are understood. Apart from the trivial solution,
the alignment arising from Eq. (77) is

$\left.\phi\right|_{V_{S}}~{}=~{}-\frac{M}{\lambda}\,\begin{pmatrix}\omega_{7}^{n}\\\\[2.5pt]
\omega_{7}^{2\,n}\\\\[2.5pt]
\omega_{7}^{4\,n}\end{pmatrix}\quad\text{with}\quad\omega_{7}=e^{\frac{2\,\pi\,i}{7}},~{}~{}n=0,\dots,6\;.$
(78)

Including universal soft masses, see Eq. (11), and taking the vacuum of the
non-SUSY potential $V$ to be rescaled by $\zeta$ compared to the vacuum in Eq.
(78), $V$ as function of $\zeta$ reads

$V(\zeta)~{}=~{}V\left(\left.\phi\right|_{V}\right)~{}=~{}\zeta^{2}\,\left((1-\zeta)^{2}\,M^{2}+m^{2}\right)\frac{3\,M^{2}}{\lambda^{2}}\,,$
(79)

compare Eq. (30). Assuming $m^{2}\ll M^{2}$, the possible extrema correspond
to

$\zeta_{0}=0\,,\qquad\zeta_{1}\approx\frac{1}{2}+\frac{m^{2}}{M^{2}}\,,\qquad\zeta_{2}\approx
1-\frac{m^{2}}{M^{2}}\,,$ (80)

where $\zeta_{0}$ is the trivial minimum, $\zeta_{1}$ corresponds to a local
maximum and $\zeta_{2}$ is the shifted minimum, consistent with Eq. (32).

###  4.2 $\Delta(27)$ model

This example aims to show that the results of section 2 can also be applied to
the case of more than one flavon. This model has two pairs of flavons and
driving fields. The flavons, $\phi_{a}^{T}\equiv(a_{1},\,a_{2},\,a_{3})$ and
$\phi_{b}^{T}\equiv(b_{1},b_{2},b_{3})$, transform as triplet and antitriplet
of $\Delta(27)$, respectively. The corresponding driving fields,
$\Phi_{a}^{T}\equiv(a_{1}^{0},\,a_{2}^{0},\,a_{3}^{0})$ and
$\Phi_{b}^{T}\equiv(b_{1}^{0},b_{2}^{0},b_{3}^{0})$, are instead
$\Phi_{a}\sim{\bf\bar{3}}$ and $\Phi_{b}\sim{\bf 3}$.

The superpotential at the renormalizable level is given by

$\displaystyle W$ $\displaystyle=$ $\displaystyle
M_{a}\,\Phi_{a}\,\phi_{a}\>+\>M_{b}\,\Phi_{b}\,\phi_{b}$ (81) $\displaystyle+$
$\displaystyle\Phi_{a}\,\left(\lambda_{1}\,\left.\phi_{b}^{2}\right|_{{\bf
3}_{1}}+\lambda_{2}\,\left.\phi_{b}^{2}\right|_{{\bf
3}_{2}}\right)\>+\>\Phi_{b}\,\left(\lambda_{3}\,\left.\phi_{a}^{2}\right|_{{\bf\bar{3}}_{1}}+\lambda_{4}\,\left.\phi_{a}^{2}\right|_{{\bf\bar{3}}_{2}}\right)\;.$

We note that there are two independent cubic terms of the form
$\Phi_{a}\,\phi_{b}^{2}$ and $\Phi_{b}\,\phi_{a}^{2}$, respectively. The
explicit form of the resulting triplets (antitriplets) from the contractions
$\phi_{b}^{2}$ ($\phi_{a}^{2}$) can be found in appendix A.3. It can be
checked that the conditions arising from the vanishing of the $F$-terms of the
driving fields are

$\displaystyle\frac{\partial W}{\partial a^{0}_{i}}$ $\displaystyle=$
$\displaystyle-
F_{a^{0}_{i}}^{\star}~{}=~{}M_{a}\,a_{i}\>+\>\lambda_{1}\,b_{i}^{2}\>+\>\lambda_{2}\,b_{i+1}\,b_{i+2}~{}=~{}0\,,$
(82) $\displaystyle\frac{\partial W}{\partial b^{0}_{i}}$ $\displaystyle=$
$\displaystyle-
F_{b^{0}_{i}}^{\star}~{}=~{}M_{b}\,b_{i}\>+\>\lambda_{3}\,a_{i}^{2}\>+\>\lambda_{4}\,a_{i+1}\,a_{i+2}~{}=~{}0\,,$
(83)

where again cyclic indices are understood. Two types of alignment are

$\left.\phi_{a}\,,\,\phi_{b}\right|_{V_{S_{1}}}\propto\begin{pmatrix}1\\\ 0\\\
0\end{pmatrix}\quad\text{with}\quad\langle
a_{1}\rangle=-\omega^{n}\,\frac{M_{a}^{1/3}M_{b}^{2/3}}{\lambda_{1}^{1/3}\lambda_{3}^{2/3}}\,,\quad\langle
b_{1}\rangle=\frac{M_{a}M_{b}}{\lambda_{1}\lambda_{3}}\,\langle
a_{1}\rangle^{-1}$ (84)

and

$\displaystyle\left.\phi_{a}\,,\,\phi_{b}\right|_{V_{S_{2}}}\propto\begin{pmatrix}1\\\
1\\\ 1\end{pmatrix}\quad\text{with}\quad$ $\displaystyle\langle
a_{1}\rangle=-\omega^{n}\,\frac{M_{a}^{1/3}M_{b}^{2/3}}{\left(\lambda_{1}+\lambda_{2}\right)^{1/3}\left(\lambda_{3}+\lambda_{4}\right)^{2/3}}\,,$
(85) $\displaystyle\langle
b_{1}\rangle=\frac{M_{a}M_{b}}{\left(\lambda_{1}+\lambda_{2}\right)\left(\lambda_{3}+\lambda_{4}\right)}\,\langle
a_{1}\rangle^{-1}\;,$ (86)

where $n=0,1,2$ and $\omega=e^{\frac{2\,\pi\,i}{3}}$. Including universal soft
masses, see Eq. (11), we can analyze the minima of the non-SUSY potential $V$.
As in the case of the $A_{4}$ model with two flavon triplets, compare section
3.2, the soft mass parameters are chosen such that the vacuum of the flavons
$\phi_{a}$ and $\phi_{b}$ is rescaled by the same factor $\zeta.$ If we take
the first type of alignment as an example, this requires, at first order in
$m^{2}/M^{2}$,

$\displaystyle\frac{m_{b}^{2}}{m_{a}^{2}}\approx\left(\frac{M_{b}\,\lambda_{1}}{M_{a}\,\lambda_{3}}\right)^{2/3}~{}\frac{2\,M_{a}^{4/3}\,\lambda_{1}^{2/3}-M_{b}^{4/3}\,\lambda_{3}^{2/3}}{2\,M_{b}^{4/3}\,\lambda_{3}^{2/3}-M_{a}^{4/3}\,\lambda_{1}^{2/3}}+{\cal{O}}\left(\frac{m_{a}^{2}}{M_{a,b}^{2}}\right)$
(87)

for $M_{a}^{4/3}\lambda_{1}^{2/3}\leq 2M_{b}^{4/3}\lambda_{3}^{2/3}\,$. The
rescaling $\zeta$ then reads

$\zeta\approx
1-\frac{\lambda_{1}^{2/3}\,m_{a}^{2}}{2\,M_{a}^{2/3}\,M_{b}^{4/3}\,\lambda_{3}^{2/3}-M_{a}^{2}\,\lambda_{1}^{2/3}}+{\cal{O}}\left(\frac{m_{a}^{4}}{M_{a,b}^{4}}\right)\;.$
(88)

Furthermore, the value of the potential is given by

$\displaystyle V\left(\left.\phi_{a},\phi_{b}\right|_{V_{1}}\right)$
$\displaystyle\approx$
$\displaystyle\frac{M_{a}^{2/3}\,M_{b}^{4/3}\,m_{a}^{2}}{\lambda_{3}^{4/3}\,\lambda_{1}^{2/3}}~{}\frac{M_{a}^{4/3}\,\lambda_{1}^{2/3}+M_{b}^{4/3}\,\lambda_{3}^{2/3}}{2\,M_{b}^{4/3}\,\lambda_{3}^{2/3}-M_{a}^{4/3}\,\lambda_{1}^{2/3}}+{\mathcal{O}}(m_{a}^{4})\;.$
(89)

An analogous analysis can be performed for the second type of alignment in
order to obtain the relation among the soft mass parameters, the shifted
minimum and the value of the potential $V$.

###  4.3 $S_{4}$ model

In this example we study the constraints on general soft masses arising from
requiring that they do not alter the vacuum alignment. The group $S_{4}$
contains two real triplets, $3$ and $3^{\prime}$. Further details and the
relevant multiplication rule can be found in appendix A.4. We consider one
flavon, $\phi^{T}\equiv(a_{1},a_{2},a_{3})$, and one driving field,
$\Phi^{T}\equiv(a_{1}^{0},a_{2}^{0},a_{3}^{0})$, both transforming as $3$.

The renormalizable terms in the superpotential are

$\displaystyle W$ $\displaystyle=$ $\displaystyle
M\>\Phi\,\phi~{}+~{}\frac{\lambda}{2}\>\Phi\,\phi^{2}$ (90) $\displaystyle=$
$\displaystyle
M\>\left(a^{0}_{1}\,a_{1}+a^{0}_{2}\,a_{2}+a^{0}_{3}\,a_{3}\right)~{}+~{}\lambda\>\left(a^{0}_{1}\,a_{2}\,a_{3}+a^{0}_{2}\,a_{1}\,a_{3}+a^{0}_{3}\,a_{1}\,a_{2}\right)\;.$

Setting the $F$-terms of the components of the driving field to zero leads to

$\frac{\partial W}{\partial
a^{0}_{i}}=-F_{i}^{\star}~{}=~{}M\,a_{i}\>+\>\lambda\,a_{i+1}\,a_{i+2}~{}=~{}0\;,$
(91)

with cyclic indices being understood. Eq. (91) is compatible with the
following alignments (discarding the trivial vacuum)

$\left.\phi\right|_{V_{S}}=-\frac{M}{\lambda}\begin{pmatrix}~{}1~{}\\\\[2.5pt]
1\\\\[2.5pt] 1\end{pmatrix}\,,~{}\begin{pmatrix}~{}1\\\\[2.5pt]
-1~{}\\\\[2.5pt] -1\end{pmatrix}\,,~{}\begin{pmatrix}-1~{}\\\\[2.5pt]
~{}1\\\\[2.5pt] -1\end{pmatrix}\,,~{}\begin{pmatrix}-1~{}\\\\[2.5pt]
-1\\\\[2.5pt] ~{}1\end{pmatrix}\;.$ (92)

For universal soft masses, see Eq. (11), and rescaling the SUSY vacuum by
$\zeta$, we obtain the potential $V$ as function of $\zeta$

$V\left(\zeta\right)~{}=~{}V\left(\left.\phi\right|_{V}\right)~{}=~{}\zeta^{2}\left((1-\zeta)^{2}\,M^{2}+m^{2}\right)\frac{3\,M^{2}}{\lambda^{2}}\;,$
(93)

compare Eq. (30). As expected, the solutions for $\zeta$ are those in Eq.
(32).

For general soft masses, see Eq. (12), we instead have to check Eq. (42) in
order to determine their form compatible with a rescaling only of the vacuum
as well as the value of $\zeta$. We exemplify this for the first alignment
mentioned in Eq. (92) and compute the form of the mass matrix ${\cal
M}_{ab}^{2}$, see Eq. (37),

${\cal M}^{2}=M^{2}\begin{pmatrix}~{}3~{}&-1&-1\\\ -1&~{}3~{}&-1\\\
-1&-1&~{}3~{}\end{pmatrix}\;.$ (94)

As we can see, the first alignment in Eq. (92) is, indeed, an eigenvector of
this matrix with eigenvalue $M^{2}$. From Eq. (42), this alignment also has to
correspond to an eigenvector of the general soft mass matrix $m_{kl}^{2}$
whose form (assuming, for simplicity, that $m_{kl}^{2}$ is real) is then given
by

$m^{2}=\begin{pmatrix}m_{11}^{2}&m_{12}^{2}&m_{12}^{2}+m_{22}^{2}-m_{33}^{2}\\\
m_{12}^{2}&m_{22}^{2}&~{}m_{11}^{2}+m_{12}^{2}-m_{33}^{2}~{}\\\
~{}m_{12}^{2}+m_{22}^{2}-m_{33}^{2}~{}&~{}m_{11}^{2}+m_{12}^{2}-m_{33}^{2}~{}&m_{33}^{2}\end{pmatrix}\;.$
(95)

For $\zeta$ we get, at first order in $m^{2}/M^{2}$,

$\zeta\approx
1-\frac{m_{11}^{2}}{M^{2}}-2\,\frac{m_{12}^{2}}{M^{2}}-\frac{m_{22}^{2}}{M^{2}}+\frac{m_{33}^{2}}{M^{2}}.$
(96)

Similar results are obtained for the other three alignments, mentioned in Eq.
(92).

##  5 Summary

In this work, we demonstrate a way to implement vacuum alignments, obtained in
SUSY theories, in non-SUSY models with (discrete) flavor symmetries. As is
well-studied, the vacua of gauge singlets, flavons, can be aligned in specific
directions in SUSY theories with the help of the $F$-terms of driving fields.
It is often desirable to apply such an alignment mechanism in non-SUSY models
in which it is notoriously difficult to obtain the correct vacuum alignment
without suppressing some couplings by hand. This can be achieved by adding
certain soft SUSY (and potentially flavor symmetry) breaking masses to the
SUSY potential. The only effect of these terms is to rescale the aligned
vacuum by a factor close to one, up to corrections of the order of the soft
mass parameters which are small compared to the mass scales in the
superpotential.

In the case of general soft masses, we have identified conditions that must be
fulfilled in order to maintain the alignment, up to rescaling. These are
similar to those found for mass terms softly breaking the flavor symmetry in
multi-Higgs doublet potentials [23]. Beyond the general case we have also
discussed examples with the well-known flavor symmetries $A_{4}$, $T_{7}$,
$\Delta(27)$ and $S_{4}$. For concreteness, we have assumed the flavons to be
triplets of the flavor symmetry. In section 3, we have investigated the vacuum
alignment in $A_{4}$ models with one or two flavons and shown that the
alignment realized in the SUSY limit is only rescaled after the inclusion of
universal soft masses. Furthermore, we have presented examples with $T_{7}$,
$\Delta(27)$ and $S_{4}$ in order to study concrete cases with flavons in
complex three-dimensional representations as well as with a potential with
general soft masses in section 4.

## Acknowledgments

CH would like to thank Matteo Bertolini, Andrea Romanino, Michael A. Schmidt
and Marco Serone for discussions. MJP would like to thank the theory
department at the Universidad de Valencia for the opportunity to visit, during
which part of this work was completed. MHR acknowledges financial support from
the STFC Consolidated Grant ST/T000775/1. OV and MHR acknowledge financial
support from the Spanish AEI-MICINN
PID2020-113334GB-I00/AEI/10.13039/501100011033 and Generalitat Valenciana
project CIPROM/2021/054. The work of CH is supported by the Spanish MINECO
through the Ramón y Cajal program RYC2018-024529-I, by the national grant
PID2020-113644GB-I00, by the Generalitat Valenciana through PROMETEO/2021/083
as well as by the European Union’s Horizon 2020 research and innovation
program under the Marie Skłodowska-Curie grant agreement No. 860881
(HIDDe$\nu$ network) and under the Marie Skłodowska-Curie Staff Exchange grant
agreement No 101086085 (ASYMMETRY).

##  Appendix A Group theory details

In this appendix, we summarize information about the employed flavor
symmetries $A_{4}$, $T_{7}$, $\Delta(27)$ and $S_{4}$.

###  A.1 Group theory of $A_{4}$

The group $A_{4}$ has twelve distinct elements and four irreducible
representations. Apart from the trivial singlet ${\bf 1}$ it has two complex
conjugated singlets, ${\bf 1^{\prime}}$ and ${\bf 1^{\prime\prime}}$, and one
real triplet ${\bf 3}$. This group can be generated by two generators, $s$ and
$t$, that satisfy $s^{2}=(s\,t)^{3}=t^{3}=e$, where $e$ is the neutral
element. For the three-dimensional representation ${\bf 3}$ these elements can
be chosen as the $3\times 3$ matrices

$S=\begin{pmatrix}1&0&0\\\ 0&-1&0\\\ 0&0&-1\end{pmatrix}\hskip 28.45274pt{\rm
and}\hskip 28.45274ptT=\begin{pmatrix}0&0&1\\\ 1&0&0\\\ 0&1&0\end{pmatrix}.$
(97)

The multiplication rule of two triplets is

$\displaystyle\left({\bf 3}_{a}\otimes{\bf 3}_{b}\right)$ $\displaystyle=$
$\displaystyle\left(a_{1}\,b_{1}+a_{2}\,b_{2}+a_{3}\,b_{3}\right)_{\bf 1}$
(98) $\displaystyle+$
$\displaystyle\left(a_{1}\,b_{1}+\omega^{2}\,a_{2}\,b_{2}+\omega\,a_{3}\,b_{3}\right)_{\bf
1^{\prime}}$ $\displaystyle+$
$\displaystyle\left(a_{1}\,b_{1}+\omega\,a_{2}\,b_{2}+\omega^{2}\,a_{3}\,b_{3}\right)_{\bf
1^{\prime\prime}}$ $\displaystyle+$
$\displaystyle\left(a_{2}\,b_{3},\,a_{3}\,b_{1},\,a_{1}\,b_{2}\right)_{\bf
3}+\left(a_{3}\,b_{2},\,a_{1}\,b_{3},\,a_{2}\,b_{1}\right)_{\bf 3}$

with $\omega=e^{\frac{2\,\pi\,i}{3}}$. Further details can be found in e.g.
[24].

###  A.2 Group theory of $T_{7}$

The group $T_{7}$ has 21 distinct elements. It contains five irreducible
representations: three singlets, ${\bf 1}$ and the complex conjugated pair
${\bf 1^{\prime}}$ and ${\bf 1^{\prime\prime}}$, and two complex conjugated
triplets ${\bf 3}$ and ${\bf\bar{3}}$. This group can be generated by two
generators, $s$ and $t$, which fulfill $s^{7}=t^{3}=e$ and $s\,t=t\,s^{4}$.
For the three-dimensional representation ${\bf 3}$, the generators can be
chosen as

$S=\begin{pmatrix}\omega_{7}&0&0\\\ 0&\omega_{7}^{2}&0\\\
0&0&\omega_{7}^{4}\end{pmatrix}\hskip 28.45274pt{\rm and}\hskip
28.45274ptT=\begin{pmatrix}0&1&0\\\ 0&0&1\\\ 1&0&0\end{pmatrix}$ (99)

where $\omega_{7}=e^{\frac{2\,\pi\,i}{7}}$. The relevant product rules for
$T_{7}$ are

$\displaystyle\left({\bf 3}_{a}\otimes{\bf 3}_{b}\right)$ $\displaystyle=$
$\displaystyle(a_{3}\,b_{3},\,a_{1}\,b_{1},\,a_{2}\,b_{2})_{\bf 3}$
$\displaystyle+$
$\displaystyle(a_{2}\,b_{3},\,a_{3}\,b_{1},\,a_{1}\,b_{2})_{\bf\bar{3}}$
$\displaystyle+$
$\displaystyle(a_{3}\,b_{2},\,a_{1}\,b_{3},\,a_{2}\,b_{1})_{\bf\bar{3}}\;,$

$\displaystyle\left({\bf 3}_{a}\otimes{\bf\bar{3}}_{b}\right)$
$\displaystyle=$ $\displaystyle(a_{1}\,b_{1}+a_{2}\,b_{2}+a_{3}\,b_{3})_{\bf
1}$ $\displaystyle+$
$\displaystyle(a_{1}\,b_{1}+\omega^{2}\,a_{2}\,b_{2}+\omega\,a_{3}\,b_{3})_{\bf
1^{\prime}}$ $\displaystyle+$
$\displaystyle(a_{1}\,b_{1}+\omega\,a_{2}\,b_{2}+\omega^{2}\,a_{3}\,b_{3})_{\bf
1^{\prime\prime}}$ $\displaystyle+$
$\displaystyle(a_{2}\,b_{1},\,a_{3}\,b_{2},\,a_{1}\,b_{3})_{\bf 3}$
$\displaystyle+$
$\displaystyle(a_{1}\,b_{2},\,a_{2}\,b_{3},\,a_{3}\,b_{1})_{\bf\bar{3}}\;.$

with $\omega=e^{\frac{2\,\pi\,i}{3}}$. Further information can, for example,
be found in [27].

###  A.3 Group theory of $\Delta(27)$

The group $\Delta(27)$ contains 27 different elements. It has nine irreducible
singlets, ${\bf 1_{1}},\dots,{\bf 1_{9}}$, that correspond to the trivial
singlet ${\bf 1_{1}}$ and four pairs of complex conjugated singlets, as well
as two complex conjugated triplets, ${\bf 3}$ and ${\bf\bar{3}}$. This group
can be described in terms of two generators, $s$ and $t$, that satisfy
$s^{3}=t^{3}=(s\,t)^{3}=(s^{2}\,t)^{3}=e$. In the representation ${\bf 3}$
these generators are represented by the matrices

$S=\begin{pmatrix}1&0&0\\\ 0&\omega&0\\\ 0&0&\omega^{2}\end{pmatrix}\hskip
28.45274pt{\rm and}\hskip 28.45274ptT=\begin{pmatrix}0&1&0\\\ 0&0&1\\\
1&0&0\end{pmatrix}$ (101)

with $\omega=e^{\frac{2\,\pi\,i}{3}}$. The relevant product rule for
$\Delta(27)$ reads

$\left({\bf 3}_{a}\otimes{\bf
3}_{b}\right)=(a_{1}\,b_{1},\,a_{2}\,b_{2},\,a_{3}\,b_{3})_{{\bf\bar{3}}_{1}}\\\
+(a_{2}\,b_{3},\,a_{3}\,b_{1},\,a_{1}\,b_{2})_{{\bf\bar{3}}_{2}}\\\
+(a_{3}\,b_{2},\,a_{1}\,b_{3},\,a_{2}\,b_{1})_{{\bf\bar{3}}_{3}}\;.$ (102)

For more information see e.g. [39].

###  A.4 Group theory of $S_{4}$

The group $S_{4}$ has 24 different elements and five real irreducible
representations: two singlets, ${\bf 1}$ and ${\bf 1^{\prime}}$, one doublet,
${\bf 2}$, and two triplets, ${\bf 3}$ and ${\bf 3^{\prime}}$. The group can
be defined by two generators, $a$ and $b$, fulfilling: $a^{4}=e=b^{3}$,
$a\,b^{2}\,a=b$ and $a\,b\,a=b\,a^{2}\,b$. In particular, for the
representation $\bf 3$, the following pair of real matrices can be chosen

$A=\begin{pmatrix}-1&0&~{}0\\\ 0&0&-1\\\
0&~{}1~{}&~{}0\end{pmatrix}\quad\text{and}\quad
B=\begin{pmatrix}~{}0~{}&0&~{}1~{}\\\ 1&0&0\\\ 0&1&0\end{pmatrix}\;.$ (103)

The relevant product rule for the triplet $\bf 3$ is

$\displaystyle\left({\bf 3}_{a}\otimes{\bf 3}_{b}\right)$ $\displaystyle=$
$\displaystyle(a_{1}\,b_{1}+a_{2}\,b_{2}+a_{3}\,b_{3})_{\bf 1}$ (104)
$\displaystyle+$
$\displaystyle\left(\frac{a_{2}\,b_{2}-a_{3}\,b_{3}}{\sqrt{2}},\,\frac{-2\,a_{1}\,b_{1}+a_{2}\,b_{2}+a_{3}\,b_{3}}{\sqrt{6}}\right)_{\bf
2}$ $\displaystyle+$
$\displaystyle(a_{2}\,b_{3}+a_{3}\,b_{2},\,a_{1}\,b_{3}+a_{3}\,b_{1},\,a_{1}\,b_{2}+a_{2}\,b_{1})_{\bf
3}$ $\displaystyle+$
$\displaystyle(a_{3}\,b_{2}-a_{2}\,b_{3},\,a_{1}\,b_{3}-a_{3}\,b_{1},\,a_{2}\,b_{1}-a_{1}\,b_{2})_{\bf
3^{\prime}}\;.$

For more information on $S_{4}$ see e.g. [31].

##  Appendix B Minimum of $A_{4}$ potential including soft SUSY breaking

###  B.1 One flavon triplet

First we analyze the extremization conditions in Eq. (60) for the $A_{4}$
potential including soft SUSY breaking masses and derive the form of the
shifted minimum. Writing Eq. (60) in terms of the components of $\phi_{a}$, we
get

$\displaystyle
2\,\lambda_{a}^{2}\,a_{1}\,\left((a_{1}^{*})^{2}-x^{2}\right)+\mu_{a}^{2}\,a_{1}^{*}=0\,,$
(105)

which yields
$a_{1}^{*}=a_{1}\,(2\,\lambda_{a}^{2}\,x^{2})/(2\,\lambda_{a}^{2}\,|a_{1}|^{2}+\mu_{a}^{2})$,
implying that $a_{1}$ and $a_{1}^{*}$ have the same phase and must be real. In
that case

$\displaystyle a_{1}^{2}$
$\displaystyle=x^{2}-\frac{\mu_{a}^{2}}{2\,\lambda_{a}^{2}},\quad
a_{2}=a_{3}=0\;.$ (106)

We can check that this shifted vacuum leads to a minimum of the potential by
examining its Hessian, expressed as a $3\times 3$ symmetric matrix
$\mathcal{H}={\partial^{2}V}/{\partial a_{i}\partial a_{j}}\,$. The principal
minors $\mathcal{H}_{i}$ of this Hessian are defined as the determinants of
the $i\times i$ upper left submatrices. According to Sylvester’s criterion,
all principal minors of $\mathcal{H}$ must be positive at the point where the
potential has a local minimum [40]. Explicitly calculating them, we get

$\mathcal{H}_{1}=8\,\lambda_{a}^{2}\,\left(x^{2}-\frac{\mu_{a}^{2}}{2\,\lambda_{a}^{2}}\right),\quad\mathcal{H}_{2}=2\,\lambda_{d}^{2}\,\mathcal{H}_{1}\,\left(x^{2}-\frac{\mu_{a}^{2}}{2\,\lambda_{a}^{2}}\right),\quad\mathcal{H}_{3}=2\,\lambda_{d}^{2}\,\mathcal{H}_{2}\,\left(x^{2}-\frac{\mu_{a}^{2}}{2\,\lambda_{a}^{2}}\right)\;.$
(107)

These are positive, implying that the vacuum in Eq. (106) yields a local
minimum of the potential.

It remains to verify that this minimum is also a global minimum of the
potential including soft SUSY breaking masses. The value of the potential at
the local minimum is given by

$\displaystyle V_{\rm
min}=\mu_{a}^{2}\,\left(x^{2}-\frac{\mu_{a}^{2}}{4\,\lambda_{a}^{2}}\right)\;.$
(108)

We would like to check whether or not a different choice of shifted vacuum
could yield a value of the potential smaller than this. Suppose the addition
of soft SUSY breaking masses causes a (real) shift $x^{2}\rightarrow
x^{2}+\delta_{a}$. The value of the potential then reads

$\displaystyle V_{\rm
min}(\delta_{a})=\mu_{a}^{2}\,\left(x^{2}+\delta_{a}+\frac{\lambda_{a}^{2}\,\delta_{a}^{2}}{\mu_{a}^{2}}\right)$
(109)

with $\delta_{a}$ unknown. Extremizing the potential with respect to this
quantity, $\partial V_{\rm min}/\partial\delta_{a}=0$, yields
$\delta_{a}=-{\mu_{a}^{2}}/{(2\,\lambda_{a}^{2})}\;.$ Hence, the minimum in
Eq. (106) is the global one of the potential.

###  B.2 Two flavon triplets

We move on to the case of two flavon triplets. Writing the extremization
conditions in Eq. (72) in terms of the components of $\phi_{a}$ and
$\phi_{b}$, we find for $a_{2}=a_{3}=0$

$\displaystyle
2\,\lambda_{a}^{2}\,a_{1}\,\left((a_{1}^{*})^{2}-x^{2}\right)+\lambda_{c}^{2}\,a_{1}^{*}\,|b_{1}|^{2}+\mu_{a}^{2}\,a_{1}^{*}=0\,,$
(110) $\displaystyle\lambda_{c}^{2}\,a_{1}^{*}\,b_{1}^{*}\,b_{2}=0\,,$ (111)
$\displaystyle\lambda_{c}^{2}\,a_{1}^{*}\,b_{1}^{*}\,b_{3}=0\,,$ (112)
$\displaystyle
2\,\lambda_{b}^{2}\,b_{1}\,\left((b_{1}^{*})^{2}+(b_{2}^{*})^{2}+(b_{3}^{*})^{2}-y^{2}\right)+\lambda_{c}^{2}\,|a_{1}|^{2}\,b_{1}^{*}+\lambda_{e}^{2}\,b_{1}^{*}\,\left(|b_{2}|^{2}+|b_{3}|^{2}\right)+\mu_{b}^{2}\,b_{1}^{*}=0\,,$
(113) $\displaystyle
2\,\lambda_{b}^{2}\,b_{2}\,\left((b_{1}^{*})^{2}+(b_{2}^{*})^{2}+(b_{3}^{*})^{2}-y^{2}\right)+\lambda_{e}^{2}\,b_{2}^{*}\,\left(|b_{1}|^{2}+|b_{3}|^{2}\right)+\mu_{b}^{2}\,b_{2}^{*}=0\,,$
(114) $\displaystyle
2\,\lambda_{b}^{2}\,b_{3}\,\left((b_{1}^{*})^{2}+(b_{2}^{*})^{2}+(b_{3}^{*})^{2}-y^{2}\right)+\lambda_{e}^{2}\,b_{3}^{*}\,\left(|b_{1}|^{2}+|b_{2}|^{2}\right)+\mu_{b}^{2}\,b_{3}^{*}=0\;.$
(115)

From Eqs. (111) and (112), we get two cases, (I) $b_{1}=0$ and (II)
$b_{2}=b_{3}=0$.

##### Case (I)

Eq. (110) leads to $a_{1}^{*}=\
a_{1}\,(2\,\lambda_{a}^{2}\,x^{2})/(2\,\lambda_{a}^{2}\,|a_{1}|^{2}+\mu_{a}^{2})$,
which implies that $a_{1}$ must be real and is given by

$\displaystyle a_{1}^{2}$
$\displaystyle=x^{2}-\frac{\mu_{a}^{2}}{2\,\lambda_{a}^{2}}\;.$ (116)

Furthermore, setting $b_{1}=0$, Eqs. (114) and (115) are solved for $b_{2}=0$
and $b_{3}=0$, respectively. However, we are not interested in the trivial
solution, $b_{1}=b_{2}=b_{3}=0$. We, therefore, consider the following two
cases.

###### Case (I.a)

If we assume $b_{3}=0$, Eq. (114) implies that $b_{2}$ must be real, and we
arrive at

$\displaystyle b_{1}=0,\quad b_{2}^{2}$
$\displaystyle=y^{2}-\frac{\mu_{b}^{2}}{2\,\lambda_{b}^{2}}\,,\quad
b_{3}=0\;.$ (117)

We note that an analogous result can be obtained for $b_{2}=0$ instead.

###### Case (I.b)

If $b_{2}\neq 0$ and $b_{3}\neq 0$, Eqs. (114) and (115) yield

$\displaystyle(b_{2}^{*})^{2}+(b_{3}^{*})^{2}-y^{2}$
$\displaystyle=-\frac{b_{2}^{*}}{b_{2}}\,\left(\frac{\lambda_{e}^{2}\,|b_{3}|^{2}+\mu_{b}^{2}}{2\,\lambda_{b}^{2}}\right)=-\frac{b_{3}^{*}}{b_{3}}\,\left(\frac{\lambda_{e}^{2}\,|b_{2}|^{2}+\mu_{b}^{2}}{2\,\lambda_{b}^{2}}\right)\,,$
(118)

implying that $b_{2}$ and $b_{3}$ must have the same phase. Then, each term on
the left-hand side of Eq. (114) or (115) has to have the phase of $b_{2}^{*}$,
telling us that $b_{2}$ and $b_{3}$ are real. Consequently, Eq. (118) leads to
$b_{1}=0,b_{2}^{2}=b_{3}^{2}={(2\,\lambda_{b}^{2}\,y^{2}-\mu_{b}^{2})}/{(4\,\lambda_{b}^{2}+\lambda_{e}^{2})}\,$.

The potential for case (I.a) and (I.b) can be evaluated to be

$\displaystyle V_{\rm I.a}$
$\displaystyle=\mu_{a}^{2}\,\left(x^{2}-\frac{\mu_{a}^{2}}{4\,\lambda_{a}^{2}}\right)+\mu_{b}^{2}\,\left(y^{2}-\frac{\mu_{b}^{2}}{4\,\lambda_{b}^{2}}\right)\,,\quad
V_{\rm I.b}=V_{\rm
I.a}+\frac{\lambda_{e}^{2}\,(2\,\lambda_{b}^{2}\,y^{2}-\mu_{b}^{2})^{2}}{4\,\lambda_{b}^{2}\,(4\,\lambda_{b}^{2}+\lambda_{e}^{2})}\;.$
(119)

Clearly, case (I.b) cannot yield the global minimum.

##### Case (II)

For $b_{2}=b_{3}=0$, Eqs. (110) and (113) imply that both $a_{1}$ and $b_{1}$
are real and we find

$\displaystyle a_{1}^{2}$
$\displaystyle=\frac{2\lambda_{b}^{2}\,\mu_{a}^{2}-4\,\lambda_{a}^{2}\,\lambda_{b}^{2}\,x^{2}+2\,\lambda_{b}^{2}\,\lambda_{c}^{2}\,y^{2}-\lambda_{c}^{2}\,\mu_{b}^{2}}{\lambda_{c}^{4}-4\,\lambda_{a}^{2}\,\lambda_{b}^{2}},\quad
b_{1}^{2}=\frac{2\,\lambda_{a}^{2}\,\mu_{b}^{2}-4\,\lambda_{a}^{2}\,\lambda_{b}^{2}\,y^{2}+2\,\lambda_{a}^{2}\,\lambda_{c}^{2}\,x^{2}-\lambda_{c}^{2}\,\mu_{a}^{2}}{\lambda_{c}^{4}-4\,\lambda_{a}^{2}\,\lambda_{b}^{2}}\;.$
(120)

The value of the potential at this point is

$\displaystyle V_{\rm II}$ $\displaystyle=V_{\rm
I.a}+\lambda_{a}^{2}\,\left[a_{1}^{2}-\left(x^{2}-\frac{\mu_{a}^{2}}{2\,\lambda_{a}^{2}}\right)\right]^{2}+\lambda_{b}^{2}\,\left[b_{1}^{2}-\left(y^{2}-\frac{\mu_{b}^{2}}{2\,\lambda_{b}^{2}}\right)\right]^{2}+\lambda_{c}^{2}\,a_{1}^{2}\,b_{1}^{2}\,,$
(121)

which, again, cannot be the global minimum.

We conclude that the potential is minimized for case (I.a), implying that the
universal soft masses do not change the vacuum alignment, but simply induce a
shift in the vacuum proportional to the soft mass parameters. One may verify
that this leads to a local minimum of the potential from the Hessian, which is
now a $6\times 6$ symmetric matrix
$\mathcal{H}={\partial^{2}V}/{\partial\varphi_{i}\partial\varphi_{j}}$, where
$\varphi^{T}\equiv(a_{1},a_{2},a_{3},b_{1},b_{2},b_{3})$. The principal minors
of the Hessian are given by

$\displaystyle\mathcal{H}_{1}$
$\displaystyle=8\,\lambda_{a}^{2}\,\left(x^{2}-\frac{\mu_{a}^{2}}{2\,\lambda_{a}^{2}}\right)\,,\quad\mathcal{H}_{2}=2\,\mathcal{H}_{1}\,\left[\lambda_{d}^{2}\left(x^{2}-\frac{\mu_{a}^{2}}{2\,\lambda_{a}^{2}}\right)+\lambda_{c}^{2}\,\left(y^{2}-\frac{\mu_{b}^{2}}{2\,\lambda_{b}^{2}}\right)\right]\,,$
$\displaystyle\mathcal{H}_{3}$
$\displaystyle=2\,\lambda_{d}^{2}\,\mathcal{H}_{2}\,\left(x^{2}-\frac{\mu_{a}^{2}}{2\,\lambda_{a}^{2}}\right)\,,$
$\displaystyle\mathcal{H}_{4}$
$\displaystyle=\frac{\lambda_{d}^{2}}{\lambda_{a}^{2}}\mathcal{H}_{1}^{2}\left[\lambda_{c}^{2}\lambda_{d}^{2}\left(x^{2}-\frac{\mu_{a}^{2}}{2\lambda_{a}^{2}}\right)^{2}+\lambda_{d}^{2}\lambda_{e}^{2}\left(x^{2}-\frac{\mu_{a}^{2}}{2\lambda_{a}^{2}}\right)\left(y^{2}-\frac{\mu_{b}^{2}}{2\lambda_{b}^{2}}\right)+\lambda_{c}^{2}\lambda_{e}^{2}\left(y^{2}-\frac{\mu_{b}^{2}}{2\lambda_{b}^{2}}\right)^{2}\right],$
$\displaystyle\mathcal{H}_{5}$
$\displaystyle=8\,\lambda_{b}^{2}\,\mathcal{H}_{4}\,\left(y^{2}-\frac{\mu_{b}^{2}}{2\,\lambda_{b}^{2}}\right)\,,\quad\mathcal{H}_{6}=2\,\lambda_{e}^{2}\,\mathcal{H}_{5}\,\left(y^{2}-\frac{\mu_{b}^{2}}{2\,\lambda_{b}^{2}}\right)\,,$
(122)

which are all positive, confirming that the shifted vacua in Eqs. (116) and
(117) lead to a local minimum with the value of the potential given by $V_{\rm
I.a}$ in Eq. (119). As before, we can further check whether or not it is the
global minimum assuming that $x^{2}\rightarrow x^{2}+\delta_{a}$ and
$y^{2}\rightarrow y^{2}+\delta_{b}$ could give rise to a value of the
potential smaller than $V_{\rm I.a}$. We find

$\displaystyle V_{\rm
I.a}\,(\delta_{a},\delta_{b})=\mu_{a}^{2}\,\left(x^{2}+\delta_{a}+\frac{\lambda_{a}^{2}\,\delta_{a}^{2}}{\mu_{a}^{2}}\right)+\mu_{b}^{2}\,\left(y^{2}+\delta_{b}+\frac{\lambda_{b}^{2}\,\delta_{b}^{2}}{\mu_{b}^{2}}\right)\;.$
(123)

Extremizing $V_{\rm I.a}\,(\delta_{a},\delta_{b})$, we get
$\delta_{a}=-{\mu_{a}^{2}}/{(2\,\lambda_{a}^{2})}$ and
$\delta_{b}=-{\mu_{b}^{2}}/{(2\,\lambda_{b}^{2})}$, leading to the shifted
vacua found in Eqs. (116) and (117).

##  Appendix C Higher-order terms in SUSY $A_{4}$ potential and shaping
symmetries

In this appendix, we present the form and possible impact of the higher-order
terms on the SUSY $A_{4}$ potential and how the choice of the indices of the
$Z_{n}$ shaping symmetries can reduce their effect to only a (small) shift in
the size of the VEVs of the flavons. In the following, we refer to the case of
two flavon triplets. Clearly, these results can be directly applied to the
case of one flavon triplet only and also generalized to three flavon triplets.

We study the combinations $\phi_{a}^{n}$, $\phi_{b}^{m}$ and
$\phi_{a}^{n}\,\phi_{b}^{m}$ with $n$ and $m$ integer and $n+m$ larger than
two.

Assuming the vacuum alignment given in Eq. (53), for the combination
$\phi_{a}^{n}$ with $n$ even only the covariants that transform as singlets
have a non-zero VEV, while for $n$ odd only the covariant that transforms as
triplet has a non-vanishing VEV, whose form is proportional to
$\langle\phi_{a}\rangle^{T}$. Similarly, we find for $\phi_{b}^{m}$ with the
vacuum alignment shown in Eq. (66) that the covariants with a non-zero VEV are
either singlets for $m$ even or the triplet for $m$ odd, with its VEV being
proportional to $\langle\phi_{b}\rangle^{T}$.

For the flavon combinations $\phi_{a}^{n}\,\phi_{b}^{m}$ with $n$ and $m$ both
integer and larger than zero, we can classify the covariants acquiring a non-
vanishing VEV for the vacuum alignment in Eqs. (53) and (66) according to
whether $n+m$, $n$ and $m$ are even or odd. In particular, we have for $n+m$
odd with $n$ even (odd) and $m$ odd (even) that only the covariant which
transforms as triplet has a non-zero VEV that is proportional to
$\langle\phi_{b}\rangle^{T}$ ($\langle\phi_{a}\rangle^{T}$). For $n+m$ even
(and equal or larger than four) with both $n$ and $m$ also even only the
covariants that are singlets acquire a VEV, while for $n+m$ even with both $n$
and $m$ odd only the triplet has a non-zero VEV, whose form is proportional to
$\langle\phi_{a}\,\phi_{b}\rangle^{T}$, i.e. it is proportional to $(0,0,1)$.

In the next step, we check for each flavon combination whether or not it can
be coupled in a $Z_{n}$\- and $A_{4}$-invariant way to one (or more) of the
driving fields $\Phi_{a}$, …, $\Phi_{e}$ as well as if such a coupling can
give rise to a non-zero contribution to the $F$-terms of the driving fields
and, thus, has an impact on the vacuum alignment of the flavons.

Before doing so, we emphasize that the driving fields $\Phi_{a}$ and
$\Phi_{b}$ are responsible for the size of the VEVs of the flavons and not for
their alignment. Thus, any contribution from higher-order terms to the
$F$-terms of $\Phi_{a}$ and $\Phi_{b}$ can only lead to a shift in the flavon
VEVs and, thus, is acceptable, as long as this shift is small compared to the
scales $x$ and $y$. The size of such shifts is determined by the choice of the
indices of the $Z_{n}$ shaping symmetries. On the contrary, the driving fields
$\Phi_{c}$, $\Phi_{d}$ and $\Phi_{e}$ are responsible for the alignment of the
flavon VEVs and, consequently, the impact of higher-order terms on their
$F$-terms should be suppressed/absent due to the choice of the shaping
symmetries.

Furthermore, we remind that in order to form an invariant under $Z_{N}\times
Z_{M}$ flavon combinations coupling to $\Phi_{a}$ and $\Phi_{d}$ have to have
the charges $(2,0)$, while flavon combinations coupling to $\Phi_{b}$ and
$\Phi_{e}$ should carry the charges $(0,2)$. Eventually, those coupling in an
invariant manner to the driving field $\Phi_{c}$ must have the $Z_{n}$ charges
$(1,1)$. The invariance under $A_{4}$ requires the flavon combinations
coupling to $\Phi_{a}$, $\Phi_{b}$ and $\Phi_{c}$ to be a trivial singlet,
whereas the ones coupling to $\Phi_{d}$ and $\Phi_{e}$ have to transform as
triplet, see Tab. 2.

Clearly, the flavon combinations $\phi_{a}^{n}$ with $n$ integer cannot couple
in an invariant way to the driving fields $\Phi_{b}$, $\Phi_{c}$ and
$\Phi_{e}$. An invariant coupling is instead possible to the driving fields
$\Phi_{a}$ and $\Phi_{d}$, in case $n$ equals $2+\alpha\,N$ with $\alpha$
integer and larger than zero.151515The choice $\alpha=0$ leads to
renormalizable terms that are included in the superpotential. As mentioned,
for $2+\alpha\,N$ being even the covariants transforming as singlets acquire a
non-zero VEV and, thus, a contribution to the $F$-term of $\Phi_{a}$ in
general arises. Similarly, for $2+\alpha\,N$ odd the covariant being the
triplet gets a non-vanishing VEV which affects the $F$-terms of $\Phi_{d}$ and
as a consequence the vacuum alignment of the flavon $\phi_{a}$. This can be
easily avoided, if $N$ is even. Furthermore, $N$ should be larger than two,
since otherwise the driving fields $\Phi_{a}$ and $\Phi_{d}$ would be neutral
and additional couplings become allowed.

Similarly, combinations of the form $\phi_{b}^{m}$ with $m$ integer cannot
form an invariant with the driving fields $\Phi_{a}$, $\Phi_{c}$ and
$\Phi_{d}$. For $m$ being $2+\beta\,M$ with $\beta$ integer and larger than
zero, $\phi_{b}^{m}$ can be coupled in an invariant way to $\Phi_{b}$ and
$\Phi_{e}$. With the same arguments as above it follows that also $M$ should
be even and larger than two.

The impact of the flavon combinations $\phi_{a}^{n}\,\phi_{b}^{m}$ with $n$
and $m$ integer and $n+m$ larger than two remains to be analyzed. These can,
depending on the values of $n$ and $m$, couple to all driving fields. Indeed,
$\phi_{a}^{2+\alpha\,N}\phi_{b}^{\beta\,M}$ for $\alpha$ and $\beta$ integer
(and at least one of them larger than zero) can couple to $\Phi_{a}$ and
$\Phi_{d}$. Since $N$ and $M$ are already chosen to be even, both exponents
are even and, hence, also their sum such that the covariants that are singlets
get a non-zero VEV, meaning that only a shift in the size of the VEV of the
flavon $\phi_{a}$ can be induced by these combinations. Likewise, the flavon
combinations $\phi_{a}^{\alpha\,N}\phi_{b}^{2+\beta\,M}$ with $\alpha$ and
$\beta$ integer can couple to $\Phi_{b}$ and $\Phi_{e}$. Given $N$ and $M$
even, however, these can only impact the size of the VEV of the flavon
$\phi_{b}$. Eventually, we see that
$\phi_{a}^{1+\alpha\,N}\,\phi_{b}^{1+\beta\,M}$ with $\alpha$ and $\beta$
integer161616At least one of these should be non-zero, as otherwise the
resulting term is renormalizable and included in the superpotential. can
couple to $\Phi_{c}$. Nevertheless, these combinations cannot lead to a
contribution to the $F$-term of $\Phi_{c}$, since both exponents are odd,
while their sum, $2+\alpha\,N+\beta\,M$, is even for $N$ and $M$ both even
and, consequently, only the triplet gets a non-vanishing VEV, while $\Phi_{c}$
is a singlet of $A_{4}$.

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