DESY-22-082
IFT-UAM/CSIC-20-144
FTUAM-20-21
IPPP/22/31

Discrete Goldstone Bosons

Victor Enguita-Viletaa, Belen Gavelaa, Rachel Houtzb, Pablo Quilezc,d,

aDepartamento de Fisica Teorica, Universidad Autonoma de Madrid,
and IFT-UAM/CSIC, Cantoblanco, 28049, Madrid, Spain
bInstitute for Particle Physics Phenomenology, Durham University,
Durham DH1 3LE, U.K.
cDepartment of Physics, University of California, San Diego, 9500 Gilman
Drive, La Jolla, CA 92093, USA
dDeutsches Elektronen-Synchrotron DESY, Notkestr. 85, 22607 Hamburg, Germany

###### Abstract

Exact discrete symmetries, if non-linearly realized, can reduce the
ultraviolet sensitivity of a given theory. The scalars stemming from
spontaneous symmetry breaking are massive without breaking the discrete
symmetry, and those masses are protected from divergent quadratic corrections.
This is in contrast to non-linearly realized continuous symmetries, for which
the masses of pseudo-Goldstone bosons require an explicit breaking mechanism.
The symmetry-protected masses and potentials of those discrete Goldstone
bosons offer promising physics avenues, both theoretically and in view of the
blooming experimental search for ALPs. We develop this theoretical setup using
invariant theory and focusing on the maximally natural minima of the
potential. For these, we show that typically a subgroup of the ultraviolet
discrete symmetry remains explicit in the spectrum, i.e. realized “à la
Wigner”; this subgroup can be either abelian or non-abelian. This suggests
tell-tale experimental signals for those minima: at least two (three)
degenerate scalars produced simultaneously if abelian (non-abelian), while the
specific ratios of multi-scalar amplitudes provide a hint of the full
ultraviolet discrete symmetry. Examples of exact ultraviolet $A_{4}$ and
$A_{5}$ symmetries are explored in substantial detail.

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###### Contents

  1. 1 Introduction
  2. 2 Non-linearly realized discrete symmetries
  3. 3 Scalars in a triplet of $A_{4}$
    1. 3.1 The invariants
    2. 3.2 The potential
      1. 3.2.1 Embedding in a continuous group: $SO(3)$
      2. 3.2.2 Understanding the nature of quantum quadratic protection
      3. 3.2.3 Maximally natural extrema of an invariant potential
      4. 3.2.4 MaNa extrema for the triplet of $A_{4}$
    3. 3.3 Phenomenology of $A_{4}$ dGBs
      1. 3.3.1 Degenerate spectrum
      2. 3.3.2 Simultaneous production
      3. 3.3.3 Experimental signals from interactions with derivative couplings
      4. 3.3.4 Counting the number of dGBs produced in experiment
      5. 3.3.5 Inferrence of the underlying discrete symmetry
    4. 3.4 A related symmetry: $S_{4}$
  4. 4 Scalars in a triplet of $A_{5}$
    1. 4.1 Invariants and potential
      1. 4.1.1 Embedding in a continuous group: $SO(3)$.
      2. 4.1.2  MaNa extrema
    2. 4.2 Phenomenological signals with an $A_{5}$ triplet
      1. 4.2.1 Generalization to low-energy $Z_{N}$ symmetric spectra
  5. 5 Scalars in a quadruplet of $A_{5}$ : non-abelian symmetry of the spectrum
    1. 5.1 Invariants and Potential
      1. 5.1.1 Embedding in a continuous group: $SO(4)$
      2. 5.1.2 MaNa extrema
    2. 5.2 Phenomenological signals from an $A_{5}$ quadruplet
  6. 6 Conclusions
  7. A Brief summary of $S_{4}$
    1. A.1 $S_{4}$ in its $\boldsymbol{3}$ representation
    2. A.2 The $\boldsymbol{3^{\prime}}$ of $S_{4}$
    3. A.3 Phenomenological signals from a triplet of $S_{4}$

##  1 Introduction

How is it possible to obtain scalar particles which are naturally much lighter
than the overall scale of the theory? This question is at the heart of the
electroweak hierarchy problem and of other open fine-tuning issues in particle
physics. We address it here using exact hidden (aka spontaneously broken) non-
linearly realized discrete symmetries.

Outstanding tensions and conundra of the Standard Model of Particle Physics
(SM) have often instead been confronted using exact hidden continuous
symmetries. The delicate issue is then that, for the theory to be realistic,
the resulting massless Nambu-Goldstone bosons (GBs) must somehow acquire a
small mass: they must become pseudo Nambu-Goldstone bosons (pGBs). Examples of
the latter include the axion that may solve the strong CP problem [1, 2, 3],
the Majoron if the neutrino masses are dynamical Majorona ones [4], theories
of extra-dimensions in which the Wilson loop around a compact dimension acts
like an axion in four-dimensions [5, 6], or the plethora of U(1) pGBs which
often appear in string-inspired phenomenological models [7, 8]. Furthermore,
the Higgs itself may be a pGB of some strong dynamics at very high energies
[9, 10, 11].

Small masses of pGBs in continuous –non-linearly realized– symmetries must
arise from an arbitrary explicit breaking of the continuous symmetry. Those
masses are therefore necessarily proportional to the parameters of the
explicit breaking mechanism, and in this sense the construction may be
technically natural if those parameters are small. In practice, this often
leads to fine-tunings and new hierarchy problems, spoiling the naturalness and
beauty of the solutions. Tensions arise because, in most constructions, data
require the pGBs masses to be extremely small compared to the overall scale of
the theory.

Exact but non-linearly realized discrete symmetries have the potential to
substantially ameliorate and restabilize that situation. The point is that
discrete symmetries, when non-linearly realized, can render trivial the only
possible invariant quadratic of the scalar fields involved in the non-linear
constraint. The low-energy theory is then protected from being quadratically
sensitive to very high –ultraviolet(UV)– scales, even in the presence of
marginal couplings. Nevertheless, the discrete symmetry does admit scalar
invariants of higher dimension. In other words, it permits non-zero masses for
the scalars which implement the non-linear scenario, without breaking the
discrete symmetry. We will denote these massive scalar fields “discrete Nambu-
Goldstone bosons” (dGBs). This name underscores the relation of these scalars
with the pGBs of the continuous group(s) in which the discrete group can be
embedded, to be developed below. The relevant point is that while the GBs of
exact continuous symmetries are exactly massless, the dGBs get masses even if
the discrete symmetry is exact.

Furthermore, the discrete invariance has the potential to strongly separate
the dGB masses from the high-energy scale of the theory. A key ingredient to
assess the smallness of dGB masses is the dimensionality of the first scalar
invariant in their –discrete-symmetry invariant– potential. We will argue that
(in a large class of UV completions of the mechanism) the higher its
dimension, the smaller their coefficients are expected to be, and thus the
smaller the dGB masses.

It is illustrative to consider the discrete symmetry groups that can be
embedded in a given continuous group and their fate under explicit breaking.
Under explicit breaking of the continuous group, either all its discrete
subgroups undergo as well explicit breaking (as in the customary constructions
of pGBs) or alternatively some discrete subgroup may remain untouched. It is
in this second option that that preserved exact ultraviolet (UV) discrete
symmetry provides the enhanced UV protection. Furthermore, it can also provide
the strong suppression of the pGBs masses, that is, of the dGB masses, as we
will develop. From this point of view, the discrete symmetries under
discussion are not introduced ad hoc as an extra ingredient, but are part of a
continuous symmetry. Simply, the potential that breaks explicitly the
continuous group is required to be invariant under one of its discrete
subgroups, instead of being the most general one. The higher the mass
dimension of the first term in the potential which breaks the continuous
symmetry but is allowed by the discrete invariance, the lighter the pGB can
be. A global continuous symmetry $G$ may then be viewed as an approximate
symmetry induced by an exact discrete invariance $D$ (which may be a subgroup
of $G$).111This view matches well with the perspective that no global symmetry
is ultimately fundamental, as they are expected to be broken and destabilized
at least by gravity [12, 13, 14, 15], while exact discrete ones become more
easily harmless, for instance via gauging. It is in this context that the
concept of dGB’s makes sense (i.e. they are a subset of the set of possible
pGB’s). This may also have a bearing on the analysis of an immediate
consequence of spontaneously broken discrete symmetries: the existence of
topological defects, in particular domain walls which typically have important
cosmological consequences.222The case of spontaneously broken $Z_{2}$ symmetry
has been extensively studied in the literature [16, 17, 18], and non abelian
domain walls have been analyzed in Refs. [19, 20, 21, 22, 23]).

The fact that spontaneously broken discrete symmetries can improve the UV
convergence of theories with pGBs was first argued in Ref. [24]. The case of
abelian discrete $Z_{N}$ groups (for which $U(1)$ is the obvious approximate
continuous symmetry) has been explored in Refs. [24, 25, 26], where it was
shown that a lighter-than-usual QCD axion is a valid solution to the strong CP
problem. Here we develop instead the much richer case of non-abelian UV
discrete symmetries, for which a first attempt appeared in Ref. [27]. The
latter considered a scalar triplet of $A_{4}$, showing that no low-energy
physical effects can stem from the $A_{4}$ invariant quadratic in the scalar
fields, and instead the first relevant invariant is cubic. Our analysis of
this particular scenario is different and provides novel results: we will show
that explicit symmetries may survive in the low-energy spectrum and
interactions. Next, the bulk of the paper explores various new realizations of
$A_{4}$, $A_{5}$ and other discrete symmetries. One general question to also
be addressed here is how to naturally obtain very high dimensionality for the
first discrete-symmetry scalar invariants, that is, how to obtain even further
suppressed dGB masses.

As theoretical tool we will use Effective Field Theory (EFT) and invariant
analysis rather than commit to specific UV models. The basis of the game is to
find the minima of the discrete symmetry invariants that can be built out of
the scalar representation(s) at hand. The most general potential will be an
arbitrary function of all such possible invariants. We focus here on the
natural minima of the potential, that is, the minima whose locations are less
or not at all dependent of the values of the parameters of the potential [28,
29]. Nevertheless, for illustrative purposes we will also show in detail the
cancellation of UV divergences among different diagrams, within two specific
UV complete models.

In addition, the identification of which abelian or non-abelian discrete
symmetries remain exact at low-energies (i.e. realized à la Wigner in the
spectrum) for the maximally natural minima will be shown to have a major
bearing on the possible tell-tale experimental signals of the mechanism.
Specifically, the degeneracies of the pGB spectra and the predicted ratios of
multi-scalar amplitudes will be analyzed.

The structure of the paper can be easily inferred from the Table of Contents.

##  2 Non-linearly realized discrete symmetries

We will analyze in this work scalar fields which belong to irreducible real
representations of discrete symmetries. Consider a generic scalar field $\Phi$
in an irreducible $m$-dimensional real representation of the discrete symmetry
group $D$, $\Phi\equiv(\phi_{1},\phi_{2}.......\phi_{m})$. A non-linearity
constraint can be expressed as the requirement that, at low energies, the
fields satisfy the quadratic restriction

$\Phi^{T}\Phi=\phi_{1}^{2}+\phi_{2}^{2}+\ldots+\phi_{m}^{2}=f^{2}\,,$ (2.1)

where $f$ is a constant with mass dimension one. Eq. (2.1) reduces by one the
number of independent degrees of freedom of the low energy theory. These are
$m-1$ spin zero particles, which would be the massless GBs of spontaneously
broken continuous symmetries, and are the dGBs of discrete symmetries, which
will be shown to be massive while the discrete symmetry remains exact.

Let us consider the model-independent tool of EFT. The dGB scale is defined by
$f$ in Eq. (2.1), and its associated scale $\Lambda\sim 4\pi f$ will weigh
down effective operators of mass dimension larger than four, for instance all
those containing momentum insertions other than kinetic energy terms. Naive
dimensional analysis (NDA) [30, 31, 32] will be used to formulate the EFT. To
this end, the non-linear constraint above can be expressed in terms of a
dimensionless functional $U$, which is a function of the $m-1$ low-energy
degrees of freedom,

$U\equiv\frac{\Phi}{f}\,,\qquad\text{with}\quad U^{T}U=1\,,$ (2.2)

and one can apply the CCWZ construction [33, 34] to parametrize the dGB inside
$U$. This function is the real equivalent of the customary pion field
parametrization of non-linearly realized continuous symmetries.333For
instance, in spontaneously global $SU(N)$ the function
$U=\exp{i\vec{\pi_{i}}\lambda_{i}/f}$ where $\pi$ denote the pion fields,
$\lambda_{i}$ the group generators and $f$ de pGB scale, is customarily used,
i.e. $U^{\dagger}U=1$. A possible choice for $U$ reads

$f\,U\equiv\Phi(\pi_{1},\dots,\pi_{m-1})=\left(\begin{array}[]{c}\phi_{1}\\\
\phi_{2}\\\ \vdots\\\
\phi_{m}\end{array}\right)=\exp\left[\frac{1}{f}\left(\begin{array}[]{ccccc}0&\dots&0&\pi_{1}\\\
\vdots&\vdots&\vdots&\vdots\\\ 0&\dots&0&\pi_{m-1}\\\
-\pi_{1}&\dots&-\pi_{m-1}&0\end{array}\right)\right]\left(\begin{array}[]{c}0\\\
0\\\ \vdots\\\ f\end{array}\right)\,,$ (2.3)

where $\pi_{1}\dots\pi_{m-1}$ denote the $m-1$ physical dGB degrees of
freedom. Whatever the parametrization, the important point is that we focus on
the dynamics of the dGB degrees of freedom including the impact of the
possible terms in their symmetry-invariant scalar potential. The dGB
Lagrangian can be parametrized as

$\displaystyle\mathcal{L}_{\text{dGB}}$
$\displaystyle=f^{2}\Lambda^{2}\,\tilde{\mathcal{L}}\left(\frac{\partial}{\Lambda},U,c_{\alpha},L_{\alpha}\right)\,=\,\frac{\Lambda^{4}}{(4\pi)^{2}}\,\tilde{\mathcal{L}}\left(\frac{\partial}{\Lambda},U,c_{\alpha},L_{\alpha}\right)\,,$
(2.4)

where $\Lambda=4\pi f$ has been assumed,444Strictly speaking $\Lambda\leq 4\pi
f$ [30]. the tilde signals here and in what follows dimensionless functions,
and the compact notation recalls that derivative terms are suppressed by
powers of $\Lambda$ in NDA, while $U$ does not scale. The dimensionless
$c_{\alpha}$ and $L_{\alpha}$ coefficients will weigh down respectively the
invariants in the dGB potential and terms involving derivatives, i.e.

$\mathcal{L}_{\text{dGB}}=\frac{f^{2}}{4}\,\left[\partial^{\mu}U^{T}\partial_{\mu}U\right]-V_{\text{dGB}}+{\mathcal{L}}_{\not{\partial}}\,,$
(2.5)

where $V_{\text{dGB}}$ denotes the potential

$V_{\text{dGB}}\equiv{f^{2}}\Lambda^{2}\,\widetilde{V}(U,c_{\alpha})=\frac{\Lambda^{4}}{(4\pi)^{2}}\,\widetilde{V}(U,c_{\alpha})\,,$
(2.6)

with $\widetilde{V}(U,c_{\alpha})$ a dimensionless function. All terms
containing $\partial_{\mu}U$ components –except the kinetic term– are encoded
in ${\mathcal{L}}_{\not{\partial}}$,

${\mathcal{L}}_{\not{\partial}}=L_{1}\,\left[\partial_{\mu}U^{T}\partial^{\mu}U\right]^{2}+L_{2}\,\left[\partial_{\mu}U^{T}\partial_{\nu}U\right]\,\left[\partial^{\mu}U^{T}\partial^{\nu}U\right]+\cdots$
(2.7)

where dots stand for both higher order terms built up purely of $U$
derivatives, and mixed terms made out of both $U$ derivatives and non-
derivative insertions of $U$ (e.g. $\prod_{{\scriptscriptstyle
i<j<k}}U_{i}\partial_{\mu}U_{j}\partial^{\mu}U_{k}$). For the purely
derivative terms, NDA suggests coefficients
$\hat{L}_{\alpha}\sim{\mathcal{O}}(1)$ or slightly smaller, related to the
customary $L_{i}$ above by powers of $4\pi$, e.g.
$\hat{L}_{1}\equiv(4\pi)^{2}L_{1},\,\hat{L}_{2}\equiv(4\pi)^{2}L_{2}$.555 In
NDA the first operator in Eq. (2.7) would be written as
$\hat{L}_{1}\,\Lambda^{4}/{(4\pi)^{2}}\,\left[(\partial
U^{T}/{\Lambda})(\partial
U/\Lambda)\right]^{2}=\hat{L}_{1}/{(4\pi)^{2}}\,\left[\partial U^{T}\partial
U\right]^{2}$, from which the relations among the coefficients follow. The
value $L_{1}\sim 10^{-2}-10^{-3}$ measured from data is thus consistent with
$\hat{L}_{1}\sim 1$, as expected because NDA is designed precisely so that the
the $4\pi$ factors from radiative corrections are consistent with NDA
coefficients of order one. The coefficients of mixed scalar-derivative terms
in ${\mathcal{L}}_{\not{\partial}}$, as well as those in the dGB potential
$c_{\alpha}$, are instead expected to be strongly suppressed because of the
constraints stemming from the UV discrete symmetry, though, as argued further
below.

The analysis of the terms involving derivatives will be deferred to subsequent
work (except for qualitative comments in the phenomenological analysis further
below). We will focus next instead in the complete scalar potential invariant
under the UV discrete symmetry, as it sources the dGB masses. Several
representative discrete groups and representations will be considered.

##  3 Scalars in a triplet of $A_{4}$

In the example in Ref. [27], a triplet of $A_{4}$ was studied,
$\Phi\equiv(\phi_{1},\phi_{2},\phi_{3})^{\intercal}$. This case is analyzed it
in detail in this section, where we will also expand on the analysis tools
used throughout this paper. Details on the group theory considerations and
subsequent analysis are provided in an ancillary Mathematica file.

###  3.1 The invariants

For a given representation $\Phi$ of the UV discrete symmetry, the most
general potential $V(\Phi)$ can be expressed as a function of all possible
invariants built out of that representation. Invariant theory of finite groups
simplifies our task by noting that, for the discrete groups we will study, the
invariants form a polynomial ring [35, 36]. The number of independent
invariants must match the number of independent degrees of freedom of the
representation. For an $A_{4}$ triplet of real scalars, the potential can be
indeed constructed as a function of the following three invariants:

$\displaystyle\mathcal{I}_{2}=$
$\displaystyle\,\sum\phi_{i}^{2}=\phi_{1}^{2}+\phi_{2}^{2}+\phi_{3}^{2}\,,\hfill$
(3.1) $\displaystyle\mathcal{I}_{3}=$
$\displaystyle\,\prod_{{\scriptscriptstyle
i<j<k}}\phi_{i}\phi_{j}\phi_{k}=\phi_{1}\phi_{2}\phi_{3}\,,\hfill$ (3.2)
$\displaystyle\mathcal{I}_{4}=$
$\displaystyle\,\sum\phi_{i}^{4}=\phi_{1}^{4}+\phi_{2}^{4}+\phi_{3}^{4}\,\,,\hfill$
(3.3)

which are often called _primary invariants_ , plus a fourth –secondary–
invariant of mass-dimension six which is a non-polynomial function of these
three. The equation which relates secondary and primary invariants is often
referred to in the literature as a _syzygy_ [37, 38], which in this case reads

$-4\,\mathcal{I}_{6}^{2}=-2\,\mathcal{I}_{4}^{3}+5\,\mathcal{I}_{4}^{2}\,\mathcal{I}_{2}^{2}-4\,\mathcal{I}_{4}\,\mathcal{I}_{2}^{4}+36\,\mathcal{I}_{4}\,\mathcal{I}_{3}^{2}\,\mathcal{I}_{2}+\mathcal{I}_{2}^{6}-20\,\mathcal{I}_{3}^{2}\,\mathcal{I}_{2}^{3}+108\,\mathcal{I}_{3}^{4}\,.$
(3.4)

This sixth-order invariant coincides in fact with the determinant of the
Jacobian for the three primary invariants above (up to a trivial
normalization),

$\operatorname{det}\mathcal{J}=8\,\mathcal{I}_{6}(\phi)=(\phi_{1}^{2}-\phi_{2}^{2})(\phi_{1}^{2}-\phi_{3}^{2})(\phi_{2}^{2}-\phi_{3}^{2})\,,$
(3.5)

where

$\mathcal{J}\equiv\frac{\partial(\mathcal{I}_{2},\mathcal{I}_{3},\mathcal{I}_{4})}{\partial(\phi_{1},\phi_{2},\phi_{3})}\,.$
(3.6)

It follows from the above that the understanding of the behavior of
$\mathcal{I}_{2}$, $\mathcal{I}_{3}$ and $\mathcal{I}_{4}$ will provide
essential information for the characterization of the general potential
$V(\phi_{1},\phi_{2},\phi_{3})$.

At energies $E<\Lambda$, the non-linear constraint in Eq. (2.1) implies

$\mathcal{I}_{2}|_{(|\Phi|^{2}=f^{2})}=f^{2}\,U^{T}U=\phi_{1}^{2}+\phi_{2}^{2}+\phi_{3}^{2}=f^{2}\,,$
(3.7)

a constraint which is crucial for the UV (in)sensitivity of the low-energy
effective theory to quadratic divergences: the only possible quadratic scalar
invariant becomes trivial at low energies. In other words, no quadratic
correction to the dGB masses can arise at any order from physics dynamics at
scales higher than $\Lambda$.666Other non-linearity constraints that also
preserve the $A_{4}$ symmetry are possible [39]. Alternative choices include
$\phi_{1}^{2}+\exp{(i\omega})\phi_{2}^{2}+\exp{(i2\omega)}\phi_{3}^{2}=0$,
where $\omega$ is a phase, or
$\phi_{1}^{4}+\phi_{2}^{4}+\phi_{3}^{4}=\text{cte}$. Those choices would not
provide protection from quantum quadratic divergences, though. We thank F.
Feruglio for this comment.

###  3.2 The potential

The most general analytic potential can be expressed as a polynomial of the
primary and secondary invariants. Taking into account that $\mathcal{I}_{2}$
does not contribute to the potential for the dGB, and that even powers of the
secondary invariant $\mathcal{I}_{6}$ can be expressed in terms of the others
via the syzigy in Eq. 3.4, the most general analytic potential for the dGB
reads

$\displaystyle V_{\rm
dGB}={f^{2}}\Lambda^{2}\,\sum_{a,b,c}^{\infty}\hat{c}_{abc}\,\left[\frac{\mathcal{I}_{3}}{f^{3}}\right]^{a}\,\left[\frac{\mathcal{I}_{4}}{f^{4}}\right]^{b}\,\left[\frac{\mathcal{I}_{6}}{f^{6}}\right]^{c}\quad\text{
with }a,b\in\mathbb{N}\text{ and }c=0,1\,,$ (3.8)

where the $\hat{c}_{abc}$ coefficients are defined in NDA,777Recall that in
NDA, the weight of fields and couplings is: an overall factor of the
Lagrangian $\Lambda^{2}f^{2}=\Lambda^{4}/(4\pi)^{2}$, and scalings
$\partial/\Lambda$, $4\pi\phi/\Lambda$ for a generic scalar field, $4\pi
A/\Lambda$ for a gauge field $A$, $4\pi\Psi/(\Lambda)^{3/2}$ for a fermion
field. and where the $\mathcal{I}_{n}/f^{n}$ dependence stems from the
$U$-dependence in Eq. (2.6). Note that in this case the complete
$V_{\text{dGB}}$ potential -i.e. including all possible invariants of any
order– is a function888This function may be non-polynomial if any
$\hat{c}_{a,b,1}\neq 0$, i.e. if Eq. 3.8 includes the secondary invariant. of
only two non-trivial invariants ($\mathcal{I}_{3}$ and $\mathcal{I}_{4}$)
after the non-linearity constraint and the syzygy are taken into account,
which is consistent with the fact that a real triplet scalar sources two dGB
degrees of freedom.

The main results in this paper apply to that complete potential, that is, they
do not rely on any expansion or truncation of the potential and we will focus
on its most natural minima. From this perspective, the reader can skip the
illustrative discussion that comes next and go directly to Sec. 3.2.3 if
wished.

It can be argued, though, that the lower dimension terms are expected to
dominate the dGB effective potential of spontaneously broken discrete
symmetries [27], as we proceed to discuss next. For illustrative purposes, let
us expand Eq. (3.8) as follows

$V_{\text{dGB}}={f^{2}}\Lambda^{2}\,\big{[}\hat{c}_{3}\,\frac{\mathcal{I}_{3}}{f^{3}}+\hat{c}_{4}\,\frac{\mathcal{I}_{4}}{f^{4}}+\hat{c}_{6}\frac{\mathcal{I}_{6}}{f^{6}}+\hat{c}_{7}\frac{\mathcal{I}_{7}}{f^{7}}\cdots\big{]}\,,$
(3.9)

which can be rewritten in a more customary notation as

$V_{\text{dGB}}=c_{3}\Lambda\,{\mathcal{I}_{3}}+c_{4}\,\mathcal{I}_{4}+c_{6}\,\frac{\mathcal{I}_{6}}{\Lambda^{2}}+c_{7}\,\frac{\mathcal{I}_{7}}{\Lambda^{4}}+\cdots\,,$
(3.10)

where $c_{n}\equiv(4\pi)^{n-2}\hat{c}_{n}$. In these equations any
$\mathcal{I}_{n>6}$ can be expressed in terms of lower dimensional ones, e.g.
$\mathcal{I}_{7}=\mathcal{I}_{3}\,\mathcal{I}_{4}$, and dots indicate terms
with higher dimension invariants. The relation with Eq. (3.8) is given by
$\hat{c}_{3}=\hat{c}_{100}$, $\hat{c}_{4}=\hat{c}_{010}$,
$\hat{c}_{6}=\hat{c}_{011}$, $\hat{c}_{7}=\hat{c}_{110}$, etc.

Were all $\hat{c}_{i}$ coefficients of the same order, all terms in the series
would contribute on equal grounds to the dGB potential. In particular, dGB
masses will be shown to stem from all terms in the potential which do not
contain $\mathcal{I}_{6}$. For generic $\hat{c}_{i}\sim\mathcal{O}(1)$ or
slightly smaller, the resulting pion masses would be $\sim\Lambda$ (instead of
much smaller as expected for pGBs). This is as expected for generic fields in
a theory with a given overall scale, unless a symmetry protects the size of
the mass terms, as we will analyze next for the case under study.

Indeed, while NDA is designed so that the coefficients on the EFT are expected
to be ${\cal{O}}(1)$ or slightly smaller, this is not yet the case for the
$\hat{c}_{\alpha}$ defined above, though, because a stronger source of
suppression can be expected in the presence of an exact discrete invariance.
Its enforcement in the couplings of $\Phi$ to the BSM fields (which are
dynamic at energy scales above $\Lambda$) typically leads to a particular
structure of the EFT couplings at lower energies, for instance a power-like
dependence on those BSM couplings. The situation is somewhat alike to that for
little Higgs models [40, 41, 42, 43] or clockwork constructions [44, 45, 46,
47, 48, 49]: a chain of interactions is required to enforce invariant
operators so that, while the individual couplings of the BSM theory may be
slightly smaller than one, the effective operator coefficient is a power of
them, ensuring strong exponential suppressions. A similar situation can be
expected in the presence of discrete symmetries. Indeed, the enforcement of
the UV exact discrete symmetry in the couplings of $\Phi$ to the BSM fields
results in a typical dependence of the form

$\hat{c}_{n}\,\sim\,\epsilon^{n}\,,$ (3.11)

where $\epsilon$ is a small quantity $\epsilon<1$. For instance, in the
scenario with an UV abelian discrete symmetry explored in Refs. [24, 25] it
was identified $\epsilon\sim(m_{u}/m_{d})$, where $m_{u}$ and $m_{d}$ denote
respectively the up and down quark masses, and the dGB mass exhibited a
$\sim(m_{u}/m_{d})^{n}$ suppression.

Such $n$-dependent exponential suppression via a small parameter has also been
shown to appear in specific UV completions of non-abelian discrete invariance,
see Ref. [27]. In the latter, a real scalar triplet of $A_{4}$ is considered,
$\Phi$, with Yukawa couplings $y$ to exotic heavy fermions with mass
$M\geq\Lambda$: $n$ is then the number of exotic fermion exchanges among
$\Phi$ fields needed to obtain an effective scalar operator invariant under
the discrete symmetry, and in consequence the effective operator coefficients
obey $\hat{c}_{n}\sim y^{n}$. More in detail, the NDA analysis of that case
indicates a potential of the form

$V=\frac{M^{4}}{(4\pi)^{2}}\sum_{n}\left(\hat{y}\frac{4\pi\Phi}{M}\right)^{n}=\frac{M^{4}}{(4\pi)^{2}}\sum_{n}\left(y\frac{\Phi}{M}\right)^{n}=\Lambda^{2}f^{2}\sum_{n}y^{n}\left(\frac{\Lambda}{M}\right)^{n-4}\left(\frac{\Phi}{\Lambda}\right)^{n}\,,$
(3.12)

with $y\equiv 4\pi\hat{y}$. It follows that in this example $\hat{c}_{n}$ in
Eq. (3.9) is given by

$\hat{c}_{n}\sim y^{n}\,\left(\frac{\Lambda}{M}\right)^{n-4}\,,$ (3.13)

which shows that all operator coefficients are suppressed by powers of the
Yukawa coupling as $y^{n}$ as far as $y<1$, and the operators with $n\geq 4$
are additionally suppressed by the ratio of scales whenever $\Lambda<M$, the
larger $n$ the stronger the suppression. For the particular case of scenarios
which admit cubic scalar invariants –as in the case studied in this section–
it also follows that $y<4\pi\left({\Lambda}/{M}\right)^{1/3}$ is necessary and
sufficient to obtain dGB masses $m_{\text{dGB}}$ smaller than the EFT scale,
i.e. $m_{\text{dGB}}^{2}<\Lambda^{2}$, as desired. This requirement does not
apply in other scenarios to be discussed below, where the UV discrete symmetry
will forbid cubic scalar invariants. The above is only one specific example of
how the (hierarchical) suppression of the coefficients of the EFT can arise
when an exact discrete symmetry is present. The suppression by the scale that
generates the invariant operators, $M$, can also be of interest in cases
explored in later sections, in which the leading invariant is a non-
renormalizable operator. If $M$ is even mildly larger than $\Lambda$, then
$\hat{c}_{n}$ naturally inherits a large suppression without relying on $y<1$.
Whether the scale $M$ is separated from or coincides with $\Lambda$ is a
detail of the UV model building, however, and from here on we will keep our
discussion more general.

An important question is that of the quantum stability of small $\hat{c}_{n}$
coefficients. They are indeed necessarily stable if assumed small, because in
the limit in which they are taken to zero the Lagrangian exhibits a larger
–continuous– symmetry, see Sec. 3.2.1.

In summary, a hierarchical ordering with $n$ of the coefficients of the
invariant scalar operators may be naturally at play in the presence of a UV
discrete symmetry which remains exact at all scales –albeit spontaneously
broken. The consequence is that the lowest order scalar invariants should
provide a good approximation to the whole EFT potential. This property also
underlies the interest of scenarios in which the lowest-order non-trivial
scalar invariant corresponds to very large $n$, for which an even stronger
suppression of the dGB masses can be expected. A simple scenario of this type
will be presented in the next section.

####  3.2.1 Embedding in a continuous group: $SO(3)$

It is illuminating to consider the embedding of the discrete groups under
consideration in continuous ones. In a nutshell, the results in this paper
could be presented for pedagogical reasons as a study of spontaneously broken
continuous symmetries supplemented with an explicit breaking potential which
is invariant under some of the discrete subgroups of the continuous symmetry,
instead of being the customary most general one.

From this point of view, all terms in the dGB potential preserve the discrete
symmetry but break explicitly the continuous symmetry of the embedding group.
That is, all coefficients $\hat{c}_{\alpha}$ in $V_{\text{dGB}}$ (Eq. (3.9))
are parameters that break that continuous symmetry.999Analogous considerations
apply to the coefficients of the mixed derivative-non-derivative operators.
From this perspective, it is natural that they are smaller than one, because
they herald the explicit breaking of a continuous global symmetry, alike to
the usual breaking parameter of chiral Lagrangians which is proportional to
the quark masses: $m_{q}/\Lambda<1$. In consequence, the putative GBs acquire
small masses. The crucial advantage of these dGBs with respect to generic pGBs
is that, even if there are marginal operators breaking the continuous symmetry
in the UV, their contribution to the low-energy dGB potential will amount to
contributions only to $c_{n>2}$ coefficients: no quadratic divergences can
arise through radiative contributions. This will hold as long as the first
invariant that breaks the continuous symmetry while respecting the discrete
one has dimension larger than two, which is precisely what happens in the UV
discrete symmetry scenarios studied here. In this context, whenever the dGB
potential is solely generated via these suppressed radiative corrections, one
can argue that the exact discrete symmetry is inducing the existence of an
approximate continuous one. In other words, the restrictions that discrete
symmetries impose on the allowed operators of the Lagrangian induce the
existence of approximate continuous symmetries.

The smallest continous group that can contain $A_{4}$ as subgroup with a
scalar triplet is $SO(3)$, see Table 1, with $\Phi$ being thus a triplet of
both groups. The only possible pattern of spontaneous symmetry breaking (SSB)
of $SO(3)$ with scalar fields in the $\boldsymbol{3}$ representation is
$SO(3)\to SO(2)$.

$\displaystyle\begin{array}[]{|c|c|c|c|c|c|c|}\hline\cr\vrule\lx@intercol\hfil\text{Finite
subgroups of $SO(3)$}\hfil\lx@intercol\vrule\lx@intercol\\\
\hline\cr\hline\cr&Z_{2}&Z_{N>2}&D_{2n}&A_{4}&S_{4}&A_{5}\\\
\hline\cr\hbox{\multirowsetup\begin{sideways}\text{Irreducible
reps.}\end{sideways}}&1&1&1&1&1&1\\\ &\ 1^{\prime}&2&2&\ 1^{\prime}&\
1^{\prime}&3\\\ &&&&\ \ 1^{\prime\prime}&\ \ 1^{\prime\prime}&\ 3^{\prime}\\\
&&&&3&2&4\\\ &&&&&3&5\\\ &&&&&\ 3^{\prime}&\\\ \hline\cr\end{array}$ Table 1:
Finite subgroups of $SO(3)$ and their irreducible representations [50].

This would source two GB degrees of freedom below the SSB scale, two “pions”.
The only possible quadratic term invariant under the $A_{4}$ discrete
symmetry, $\mathcal{I}_{2}$, is also $SO(3)$ invariant and it will yield no
contribution to the pion masses. In contrast,
$\mathcal{I}_{3},\mathcal{I}_{4},\mathcal{I}_{6}...$ are invariants of $A_{4}$
only: they break explicitly the continuous $SO(3)$ symmetry. That is, by
requiring that the potential that breaks explicitly $SO(3)$ is invariant under
one of its discrete subgroups, the quadratic sensitivity to high scales is
absent, as advertized, while the pions of the continuous theory acquire a mass
from operators with dimension three [27] (with coupling $c_{3}$) and larger.

A pertinent question is whether all finite groups allow an embedding in a
continuous group. Indeed, any finite group $D$ is isomorphic to a subgroup of
the symmetric group of $m$ elements, $S_{m}$, where $m$ is precisely the order
of $D$ [51]. Therefore, $D$ is also a subgroup of the continuous Lie groups
GL${}_{m}(\mathds{R})$ or GL${}_{m}(\mathds{C})$ and can in fact be realized
as a subgroup of $O(m)$, and thus also as a subgroup of $SO(2m)$. There are,
on the other hand, cases in which the smallest embedding continuous group
turns out to be smaller. A familiar example is that of the alternating group
of $4$ elements, $A_{4}$, which is obviously a subgroup of the permutation
group $S_{4}$: the smallest continuous group which contains $A_{4}$ is then
$SO(3)$. In the presence of spontaneous symmetry breaking, both the discrete
symmetry and its continuous embedding group are spontaneously broken. Our
analysis, however, does not rely on a specific choice of continuous group
embedding.

####  3.2.2 Understanding the nature of quantum quadratic protection

In order to better grasp the origin of the absence of quadratic sensitivity of
dGB, it is interesting to consider specific examples of UV complete models
where the cancellation of the quadratic divergences becomes explicit.

#### A UV model with exotic fermions

Let us consider a model with a fundamental scalar triplet of $A_{4}$, $\Phi$,
and a triplet of fermions,
$\mathbf{\Psi}\equiv(\Psi_{1},\Psi_{2},\Psi_{3})^{\intercal}$ (which was first
studied in Ref. [27]). There are two possible $A_{4}$-symmetric Yukawa
interactions among the scalars and the fermions101010See Ref. [52] for details
on how to construct $A_{4}$ singlets which can be written as:

$\begin{split}\mathcal{L}_{\text{int
}}&=\left[y_{G}\left(\begin{array}[]{c}{[\overline{\Psi}_{2},\Psi_{3}]}\\\
{[\overline{\Psi}_{3},\Psi_{1}]}\\\
{[\overline{\Psi_{1}},\Psi_{2}]}\end{array}\right)+y_{\not{G}}\left(\begin{array}[]{l}\\{\overline{\Psi}_{2},\Psi_{3}\\}\\\
\\{\overline{\Psi}_{3},\Psi_{1}\\}\\\
\\{\overline{\Psi}_{1},\Psi_{2}\\}\end{array}\right)\right]^{\top}\cdot\,\Phi=\,\\\
&=y_{G}\left(\left[\overline{\Psi}_{2},\Psi_{3}\right]\phi_{1}+\left[\overline{\Psi}_{1},\Psi_{3}\right]\phi_{2}+\left[\overline{\Psi}_{1},\Psi_{2}\right]\phi_{3}\right)\\\
&\hskip
130.08731pt+y_{\not{G}}\left(\\{\overline{\Psi}_{2},\Psi_{3}\\}\phi_{1}+\\{\overline{\Psi}_{1},\Psi_{3}\\}\phi_{2}+\\{\overline{\Psi}_{1},\Psi_{2}\\}\phi_{3}\right)\,,\end{split}$
(3.14)

which for small Yukawa couplings presents an approximate continuous $G\equiv
SO(3)$ symmetry. All interactions in this Lagrangian have mass-dimension four
with a dimensionless coefficient, i.e. they are marginal couplings, and one
could naively assume that these two Yukawa couplings would generate UV
sensitive pion masses. Nevertheless:

  * •

The term proportional to $y_{{G}}$ preserves $SO(3)$, and thus cannot source a
potential for the GBs of the theory due to the Goldstone theorem. In other
words, any scalar potential would need to be a function of the $SO(3)$ primary
invariant $\mathcal{I}_{2}=\phi^{T}\phi$ which does not depend on the pions
and this leads to a vanishing pion potential.

  * •

The term proportional to $y_{\not{G}}$ explicitly breaks $SO(3)$ but is
invariant under $A_{4}$. This term does source a potential and masses for the
pions of the theory, but it does not lead to quadratic sensitivity to scales
heavier than that of the dGB Lagrangian.

$+$

Figure 1: Diagrams a priori  sourcing quadratic divergences, with $y=y_{{G}}$
in the $SO(3)$ symmetric case in Eq. (3.15), and $y=y_{\not{G}}$ in the
SO(3)-breaking but $A_{4}$-preserving case in Eq. (3.16). Only the diagram on
the left for $\pi_{1}$ is present instead for the explicit breaking in Eq.
(3.18) with $y=y_{\not{A}_{4}}$.

In order to illustrate in detail the last statement, let us compute explicitly
the possible loops induced by $y_{{G}}$ and $y_{\not{G}}$ and obtain the
cancellation of the divergences in both cases. Expanding to second order in
the pion fields,
111111$\phi_{1}\simeq\pi_{1},\phi_{2}\simeq\pi_{2},\phi_{3}\simeq
f\big{[}1-\frac{1}{2}{(\pi_{1}^{2}+\pi_{2}^{2})}/{f^{2}}\big{]}$. the
$SO(3)$-symmetric Yukawa interactions read

$\mathcal{L}_{\text{int, G
}}=y_{{G}}\,\pi_{1}\left(\overline{\Psi}_{2}\Psi_{3}-\overline{\Psi}_{3}\Psi_{2}\right)+y_{{G}}\,\pi_{2}\left(\overline{\Psi}_{3}\Psi_{1}-\overline{\Psi}_{1}\Psi_{3}\right)\\\
+y_{{G}}\,f\left(1-\frac{1}{2}\frac{\pi_{1}^{2}+\pi_{2}^{2}}{f^{2}}\right)\left(\overline{\Psi}_{1}\Psi_{2}-\overline{\Psi}_{2}\Psi_{1}\right)+\mathcal{O}(\pi^{3})\,,$
(3.15)

and a very similar expression holds for the $SO(3)$-breaking but
$A_{4}$-preserving Yukawa couplings:

$\mathcal{L}_{\text{int. }{\not{G}}}=\quad
y_{\not{G}}\Big{[}\pi_{1}\left(\overline{\Psi}_{2}\Psi_{3}+\overline{\Psi}_{3}\Psi_{2}\right)+\pi_{2}\left(\overline{\Psi}_{3}\Psi_{1}+\overline{\Psi}_{1}\Psi_{3}\right)\\\
+f\left(1-\frac{1}{2}\frac{\pi_{1}^{2}+\pi_{2}^{2}}{f^{2}}\right)\left(\overline{\Psi}_{1}\Psi_{2}+\overline{\Psi}_{2}\Psi_{1}\right)\Big{]}+\mathcal{O}(\pi^{3})\,.$
(3.16)

The relevant Feynman diagrams for both Yukawa Lagrangians are depicted in Fig.
1. We can now compute the _a priori_ divergent loop contributions to the mass
of the pions: these are guaranteed by symmetry to vanish for the $y_{{G}}$
contributions, but they turn out to vanish for the $y_{\not{G}}$ contributions
as well. Indeed, in both cases, the correlation between the
$\mathcal{O}\left(\pi_{i}\Psi\Psi\right)$ interaction and the
$\mathcal{O}\left(\pi_{i}^{2}\Psi\Psi\right)$ interaction implies that the
quadratic divergences stemming from those two Feynman diagrams combine as

$\displaystyle\delta
m_{\pi_{1,2}}^{2}\propto\frac{1}{2}y\Lambda^{2}-\frac{y}{2f}\,yf\Lambda^{2}=0\,,$
(3.17)

where $y$ is either $y_{{G}}$ or $y_{\not{G}}$. In the case of the
($SO(3)$-breaking ) $y_{\not{G}}$ terms, the cancellation of quadratic
divergences is a consequence of the unbroken discrete $A_{4}$ symmetry. This
property can be traced back to the fact that the first $A_{4}$-invariant that
breaks the continuous symmetry is of dimension three and thus requires the
insertion of at least three Yukawa couplings 121212If the fermions have a bare
mass term $m_{\psi}$, the mass of the dGBs is $m_{\pi_{1,2}}^{2}\propto
y_{\not{G}}^{3}\,m_{\psi}f$. Otherwise an enhanced $A_{4}\times Z_{2}$
symmetry arises for which the first $SO(3)$-breaking invariant has dimension
four and thus $m_{\pi_{1,2}}\propto y_{\not{G}}^{4}f^{2}$ [27]. , i.e.
$m_{\pi_{1,2}}^{2}\propto y_{\not{G}}^{3}$ while all contributions $\propto
y_{\not{G}}^{2}$ must cancel out.

The conclusion is that, in order for the quadratic divergences to cancel, one
does not need to impose the full $SO(3)$ symmetry, but instead it is enough to
impose ultraviolet $A_{4}$ invariance. In the latter case, other mass terms
survive for the dGBs, though. This is in contrast with generic pGB with
arbitrary $SO(3)$-breaking terms. For instance, an explicit $SO(3)$ breaking
which also breaks the $A_{4}$ subgroup is given by

$\mathcal{L}_{\not{A}_{4}}=y_{\not{A}_{4}}\phi_{1}\,\overline{\Psi}_{1}\Psi_{1}=y_{\not{A}_{4}}\,\pi_{1}\overline{\Psi}_{1}\Psi_{1}+\mathcal{O}(\pi^{3})\,,$
(3.18)

in which case only the UV divergent diagram on the left of Fig. 1 is present
and only for $\pi_{1}$, and thus a divergent contribution to the pion mass
does remain,

$\displaystyle\delta
m_{\pi_{1}}^{2}\propto\frac{1}{2}y_{\not{A}_{4}}^{2}\Lambda^{2}\neq 0\,.$
(3.19)

#### A UV model with heavy scalars

The example above with exotic fermions is not unique. We consider next a BSM
scalar sector invariant under an exact ultraviolet $A_{4}$ symmetry and
inducing effective $\phi^{n}$ couplings. An example of this kind where one can
see at play the mechanism of cancellation of quadratic sensitivity to high
scales involves the interaction of the $A_{4}$ triplet $\phi$ with another
scalar triplet $S$ which does not take a vacuum expectation value (vev) and
has a large mass $m_{S}\gg f$. Let’s assume that these scalars interact
through the following $A_{4}$-symmetric and $SO(3)$-violating quartic
couplings,131313We have repeated the exercise for all possible $A_{4}$
invariant couplings of the scalars $S$ and confirmed that the same pattern of
cancellations holds.

$\displaystyle\mathcal{L}_{\text{int
}\not{G}}=\frac{\lambda_{\not{G}}}{4}\left(\phi_{1}^{2}S_{1}^{2}+\phi_{2}^{2}S_{2}^{2}+\phi_{3}^{2}S_{3}^{2}\right)\,.$
(3.20)

Assuming that the theory has a high energy cut-off $\Lambda_{UV}$, one can
easily check that the quadratically divergent diagrams in Fig. 2 arrange
themselves in a way that they only contribute to the $\mathcal{I}_{2}$
invariant, leaving the dGB masses insensitive to the UV scale, as illustrated
in Fig. 2.

$+$

$+$

$\qquad\propto\quad\lambda_{\not{G}}\,\mathcal{I}_{2}\Lambda_{UV}^{2}$

Figure 2: Insensitivity of the dGB masses to the cutoff in the model with
heavy scalars.

The dGBs do not remain massless though, and a mass term
$\propto\lambda_{\not{G}}^{2}$ is generated via loops through the invariant
$\mathcal{I}_{4}$, see Fig. 3.

$+$

$+$

$\qquad\propto\quad\lambda_{\not{G}}^{2}\,\mathcal{I}_{4}$

Figure 3: Finite contribution to the dGB mass through the quartic invariant.

An even more explicit manifestation of the cancellation arises when one
computes the interaction of the scalars $S_{i}$ with the dGB themselves.
Expanding again at second order in the dGBs, Eq. (3.20) results in

$\displaystyle\mathcal{L}_{\text{int
}\not{G}}\simeq\lambda_{\not{G}}\left[\pi_{1}^{2}S_{1}^{2}+\pi_{2}^{2}S_{2}^{2}+f^{2}S_{3}^{2}-\left(\pi_{1}^{2}+\pi_{2}^{2}\right)S_{3}^{2}\right]+\mathcal{O}(\pi_{i}^{3})\,.$
(3.21)

The computation of the corresponding loops, depicted in Fig. 4, shows that in
this case the cancellation arises due to a correlation among the quartic
interactions with $S_{1,2}$ ensured by the $A_{4}$ symmetry.

$+$

$\qquad\propto\quad\lambda_{\not{G}}\Lambda_{UV}^{2}-\lambda_{\not{G}}\Lambda_{UV}^{2}=0$

Figure 4: Cancellation of the UV divergences resulting from diagrams that
could have contributed to the dGB masses in the model with heavy scalars, Eq.
(3.21).

####  3.2.3 Maximally natural extrema of an invariant potential

The description and discussion in this subsection will apply as well to the
other discrete representations and groups to be explored later on.

In order to study the properties of the dGB and identify the physical states,
it is necessary to study the minima of the potential. Looking at the most
general potential in Eqs. (3.8) or (3.9), this may seem a colossal task given
that the coefficients are arbitrary. Nevertheless, one can study and classify
the natural extrema, those that are less or not at all dependent on a specific
combination of coefficients in the potential, but rather arise as a
consequence of the symmetry. A subset of the natural extrema we explore below
will not depend at all on the specific form or tuning of coefficients in the
potential, and these we will call maximally natural (MaNa) extrema. The MaNa
extrema are of particular interest for the analysis of the minima of a physics
potential. These points are guaranteed to be (absolute or local) extrema of
the complete potential since they are extrema of all the primary and secondary
invariants.

Consider a generic potential $V(x)$ for a set of $n$ independent scalar fields
$x_{i}$. If the potential preserves a symmetry, it can be expressed as a
function of just $n$ independent functions $I_{i}(x)$ invariant under the
symmetry: the primary invariants $I_{i}(x)$.141414The inversion of the
relation of the invariants in terms of the variables is unique. One may then
identify the inverse relation $x_{i}=x_{i}(I_{j})$ and express any new
invariant $I^{\prime}$ in terms of the independent set $\\{I_{j}\\}$;
$I^{\prime}=I^{\prime}(x_{i})=I^{\prime}(x_{i}(I_{j}))$. The most general
potential will be a function of all such possible invariants,
$V(x)=V[I_{i}(x)]$. It follows that all the extrema of the potential (and
among them the true vacuum) can be obtained imposing

$\frac{\partial V}{\partial x_{j}}=\sum_{i}\frac{\partial V}{\partial
I_{i}}\frac{\partial I_{i}}{\partial x_{j}}=\sum_{j}\frac{\partial V}{\partial
I_{i}}\,J_{ij}=0\,.$ (3.22)

Figure 5: Manifold defined by the $\mathcal{I}_{3}(\Phi)$ and
$\mathcal{I}_{4}(\Phi)$ invariants of $A_{4}$, with $\Phi$ a triplet
fulfilling $\Phi^{T}\Phi=\phi_{1}^{2}+\phi_{2}^{2}+\phi_{3}^{2}=f^{2}$. The
little group which remains invariant at each natural extrema is indicated.

This equation can be seen as the Jacobian matrix of the change of variables
from the fields to the invariants, $J_{ji}\equiv{\partial I_{i}}/{\partial
x_{j}}$ times the vector ${\partial V}/{\partial I_{i}}$, and allows one to
identify two kinds of extrema of a potential:

  1. 1)

Model-dependent extrema: If the rank of the Jacobian is maximal, then the
extremal points necessarily correspond to a vanishing vector ${\partial
V}/{\partial I_{i}}=0$. In consequence, the extrema will depend on the
specific parameters of the combination of invariants that build the potential.
This is the case, for example, of the Higgs potential of the SM.

  2. 2)

Natural extrema: Those points that are extrema of the invariants
$\mathcal{I}_{i}$ themselves and therefore correspond to

$\mbox{det}\,[J_{ji}]=0\,,$ (3.23)

that is, the rank of the Jacobian is less than maximal. They are called
natural extrema as they are set by the symmetry (that determines the
invariants) and therefore depend less on the specific coefficients of the
potential. In general terms, the reduction of the rank implies the appearance
of symmetries left explicit, i.e. unbroken in the spectrum.

The essential point of the analysis of the natural extrema is that the space
of the physical variables $x$ has no boundary, while the manifold spanned by
$I_{i}(x)$ does have boundaries. Fig. 5 illustrates the manifold and its
boundaries for the particular case of a non-linearly realized $A_{4}$ symmetry
with a scalar triplet. Extrema of $V$ in the bulk of the manifold correspond
to case 1) above and are always model-dependent extrema, while extrema on the
boundaries are natural extrema [29]: they fall in category 2) above. The
boundaries are characterized by the rank of the Jacobian matrix being less
than maximum and are thus described by $n-1$ dimensional manifolds (e.g.
surfaces, for $n=3$), each characterized by a different little group (i.e. the
set of transformations that leave the corresponding extremum invariant). Such
manifolds meet along $n-2$ dimensional manifolds (e.g. lines) which in turn
meet along even lower dimensionality manifolds (e.g. singular points), etc.
Each of these boundaries corresponds to a particular little group, with again
the smaller the rank of the Jacobian, the larger the little group.

Indeed, those points whose little group is a maximal subgroup151515 Given a
group, a subgroup is maximal if the only subgroup that contains it is the
complete group itself, i.e. a subgroup that can be included only in the full
group. of $D$ are expected to be the most natural minima [28]. Extrema of $V$
on a given boundary [29] are dubbed natural precisely because they are more
natural than the generic extrema in the interior of the manifold. The former
require the vanishing of only Rank$(J)=n-k$ derivatives of $V$ with respect to
the invariants given that, on the boundary, the Jacobian $J$ has $k$ vanishing
eigenvectors (orthogonal to the boundary). Note as well that the derivatives
of the invariants must vanish on the boundaries in the directions orthogonal
to them, which forces the existence of a singular point wherever $n$
boundaries intersect. Therefore one should always find an extremum at such
intersections, which is where all derivatives vanish. The natural minima at
the boundary intersections, that is, the zero-dimensional vertices of the
boundary, are what we have called the MaNa minima. The MaNa minima are
maximally natural in that their position (the vevs) do not rely on the values
of the parameters of the potential.

Each point $P$ in the manifold of the invariants corresponds in general to
several possible vacuum expectation values of the vectors $\Phi$, which we
will denote $\Phi_{P}$: they constitute an orbit. Each element of the exact UV
discrete invariance group $D$ either leaves the vector invariant (and
therefore belongs to its little group) or transforms the vector into another
element of the orbit. In consequence, the following counting rule applies:

$\displaystyle\text{order}(L)\times\\#\Phi_{P}=\text{order}(D)\,,$ (3.24)

where $\\#\Phi_{P}$ is the number of MaNa extrema for the representation
studied, and $L$ denotes the little group at $\Phi_{P}$.

In this work, we focus on the exploration of the MaNa minima for each of the
–spontaneoulsy broken– discrete groups to be considered.

####  3.2.4 MaNa extrema for the triplet of $A_{4}$

In this scenario, considering the three degrees of freedom at high energies,
$\Phi\equiv(\phi_{1},\phi_{2},\phi_{3})^{\intercal}$, the invariant manifold
would be three-dimensional. This is reduced to a two-dimensional manifold when
the non-linear constraint Eq. (3.7) is taken into account, which is the setup
of interest to explore the dGB potential. This invariant manifold is that
spanned by the set of two invariants $\\{\mathcal{I}_{3},\mathcal{I}_{4}\\}$
which source all possible terms in the dGB potential. This is depicted in Fig.
5: the points inside the bulk of the manifold correspond to the rank two
Jacobian, the boundaries are lines, and the invariant little groups at the
MaNa extrema points –the vertices $A$, $B$ and $C$– are indicated.

The coordinates of the MaNa points in field space are shown in Table 2,
together with the value of the invariants and their exact –explicit–
invariances. Note that the coordinate values enforce the non-linear constraint
in Eq. (3.7), as required. It is easy to check that Eq. (3.24) is satisfied:
$2\times 6\,\text{(for A)}=3\times 4\,\text{(for B)}=3\times 4\,\text{(for
C)}=12$, and the number of extrema is $6+4+4=14$. The three MaNa points $A$,
$B$ and $C$ are related by $A_{4}$ transformations. The discrete symmetries
that may remain explicit are typically a maximal subgroup of the UV discrete
group $A_{4}$: this is the case for $Z_{3}$ identified for the extrema $B$ and
$C$ in Table 2, which are both MaNa minima161616Depending on the sign of the
coefficient $\hat{c}_{i}$, these extrema are either maxima or minima., while
$A$ is a saddle point and its little group is $Z_{2}$ (which is not a maximal
subgroup of $A_{4}$).

The discrete symmetry is reflected in the geometrical locations of the MaNa
extrema in field space. For instance, the $Z_{3}$-symmetric minima arrange
themselves in the shape of two tetrahedrons, as depicted in Fig. 6.

Figure 6: Geometrical distribution in field space of the two $Z_{3}$-symmetric
MaNa extrema of the dGB potential for a scalar triplet of $A_{4}$. Points B of
Table 2 are depicted in green and points C are in red.
$\displaystyle\begin{array}[]{|c||cc|ccc|c|c|}\lx@intercol\hfil\text{{ MaNa
extrema for a triplet of} $\boldsymbol{A_{4}}$}\hfil\lx@intercol\\\
\hline\cr\text{Point}&\mathcal{I}_{3}&\,\mathcal{I}_{4}&\phi_{1}&\phi_{2}&\phi_{3}&\text{Little
group}&\text{Nature}\\\\[1.0pt] \hline\cr\hline\cr\hbox{\multirowsetup
A}&\hbox{\multirowsetup 0}&\hbox{\multirowsetup 1}&0&0&\pm
1&\hbox{\multirowsetup\text{$Z_{2}$}}&\hbox{\multirowsetup\text{Saddles}}\\\
&&&0&\pm 1&0&&\\\ &&&\pm 1&0&0&&\\\
\hline\cr\rule{0.0pt}{9.90276pt}\hbox{\multirowsetup
B}&\hbox{\multirowsetup$\frac{1}{3\sqrt{3}}$}&\hbox{\multirowsetup$\frac{1}{3}$}&\frac{1}{\sqrt{3}}&\frac{1}{\sqrt{3}}&\frac{1}{\sqrt{3}}&\hbox{\multirowsetup\text{$Z_{3}$}}&\hbox{\multirowsetup\text{Minima}}\\\
&&&-\frac{1}{\sqrt{3}}&\frac{1}{\sqrt{3}}&-\frac{1}{\sqrt{3}}&&\\\
&&&\pm\frac{1}{\sqrt{3}}&-\frac{1}{\sqrt{3}}&\mp\frac{1}{\sqrt{3}}&&\\\\[3.44444pt]
\hline\cr\rule{0.0pt}{9.90276pt}\hbox{\multirowsetup
C}&\hbox{\multirowsetup$-\frac{1}{3\sqrt{3}}$}&\hbox{\multirowsetup$\frac{1}{3}$}&-\frac{1}{\sqrt{3}}&-\frac{1}{\sqrt{3}}&-\frac{1}{\sqrt{3}}&\hbox{\multirowsetup$Z_{3}$}&\hbox{\multirowsetup\text{Minima}}\\\
&&&\frac{1}{\sqrt{3}}&-\frac{1}{\sqrt{3}}&\frac{1}{\sqrt{3}}&&\\\
&&&\pm\frac{1}{\sqrt{3}}&\frac{1}{\sqrt{3}}&\mp\frac{1}{\sqrt{3}}&&\\\\[3.44444pt]
\hline\cr\end{array}$ (3.36) Table 2: Location and symmetries of the MaNa
extrema of the manifold spanned by the invariants that build up the dGB
potential, for the case of a triplet scalar of $A_{4}$. The values of the
invariants and the locations of the extrema are given in units of the dGB
scale $f$. The location in field space of the MaNa minima is depicted in Fig.
6. The manifold is illustrated in Fig. 5.

##### Expansion in terms of low-energy degrees of freedom.

The parametrization of the real scalar field $\Phi$ in terms of physical dGBs
in Eq. (2.3) spans in this case two low-energy fields $\pi_{i=1,2}$,

$\Phi(\pi_{1},\pi_{2})=\left(\begin{array}[]{c}\phi_{1}\\\ \phi_{2}\\\
\phi_{3}\end{array}\right)=\exp\left[\frac{1}{f}\left(\begin{array}[]{cccc}0&0&\pi_{1}\\\
0&0&\pi_{2}\\\
-\pi_{1}&-\pi_{2}&0\end{array}\right)\right]\left(\begin{array}[]{c}0\\\ 0\\\
f\end{array}\right)\,.$ (3.37)

The expansion of the low-energy dGB Lagrangian in Eq. (2.5) around the MaNa
extrema in Table 2, in terms of these physical pions fields (dGBs), must be
done carefully. The point is that the minima in that table are rotated with
respect to the direction around which the pions are defined in Eq. (5.8). In
order to maintain the latter parametrization consistently, it is necessary to
apply a three-dimensional real rotation $R\in SO(3)$ that aligns each MaNa
point with the $z$-direction in field-component space, which corresponds to
the unbroken generator. The invariants can then be simply redefined as

$\mathcal{I}^{\prime}_{i}(\Phi)=\mathcal{I}_{i}(R^{-1}\Phi)\,,$ (3.38)

where $\Phi$ is as defined by Eq. (5.8); any MaNa extremum $\Phi_{s}$ is
therefore transformed into $\Phi^{\prime}_{s}=R\,\Phi_{s}=(0,0,f)^{\top}$. The
absence of this step has prevented Ref. [27] to identify the surviving
explicit symmetry of the spectrum, that we unveil here to be $Z_{3}$.171717If
one keeps the parametrization in Eq. (5.8) and expands around the vaccuum
expectation values (vevs) of the pion fields in Table 2,
$\langle\pi_{1}\rangle=\langle\pi_{2}\rangle=(f/\sqrt{2})\cos^{-1}\left(1/\sqrt{3}\right)$,
as done in Ref. [27], the resulting kinetic terms of the pions would not be
canonically normalized. Upon redefinition to canonically normalize the fields,
our results are recovered. Therefore, $Z_{3}$ is the symmetry that should
appear realized à la Wigner in the low-energy spectrum for the MaNa minima of
the potential, as discussed next in further detail.

The expansion around any of the $Z_{3}$ symmetric vevs (for instance
$\Phi_{C}=-\frac{1}{\sqrt{3}}(1,1,1)$) of the $A_{4}$ invariants for the
scalar triplet in terms of physical pion fields, parametrized as in Eq. (5.8),
yields

$\displaystyle\mathcal{I}_{3}$
$\displaystyle=\frac{f}{\sqrt{3}}\left[-\frac{f^{2}}{3}+\left(\pi_{1}^{2}+\pi_{2}^{2}\right)-\frac{1}{3\sqrt{2}f}\left(\pi_{1}^{3}-3\pi_{1}\pi_{2}^{2}\right)-\frac{17}{24f^{2}}\left(\pi_{1}^{2}+\pi_{2}^{2}\right)^{2}\,\right]+\cdots$
(3.39) $\displaystyle\mathcal{I}_{4}$
$\displaystyle=\frac{4f^{2}}{3}\left[\frac{f^{2}}{4}+\left(\pi_{1}^{2}+\pi_{2}^{2}\right)+\frac{1}{\sqrt{2}f}\left(\pi_{1}^{3}-3\pi_{1}\pi_{2}^{2}\right)-\frac{29}{24}\left(\pi_{1}^{2}+\pi_{2}^{2}\right)^{2}\,\right]\,+\cdots$
(3.40) $\displaystyle\mathcal{I}_{6}$
$\displaystyle=-\frac{4}{3}\sqrt{\frac{2}{3}}f^{3}\left(\pi_{2}^{3}-3\pi_{1}^{2}\pi_{2}\right)\,+\cdots$
(3.41)

where dots indicate terms of higher order in the pion fields, and the non-
linear constraint $(|\Phi|^{2}=f^{2}$) in Eq. (3.7) is automatically
implemented (here and all through the rest of the paper, the non-linear
constraint will be implicit whenever the dGB dynamics is analyzed). These
equations show that the dGB fields appear arranged in an irreducible
representation of the little group of the MaNa minima: a $\boldsymbol{2}$ of
$Z_{3}$. Indeed, two primary invariants exist for the latter which, in the
pion parametrization chosen read

$\displaystyle\mathcal{I}_{2}^{(\boldsymbol{2},Z_{3})}$
$\displaystyle=\pi_{1}^{2}+\pi_{2}^{2}\,,$ (3.42)
$\displaystyle\mathcal{I}_{3}^{(\boldsymbol{2},Z_{3})}$
$\displaystyle=\pi_{1}^{3}-3\pi_{1}\pi_{2}^{2}\,,$ (3.43)

and those are precisely the field combinations exhibited by Eqs.
(3.39)-(3.40). There is also a secondary $Z_{3}$ invariant (obtained via a
syzygy)

$\mathcal{I}_{3^{\prime}}^{(\boldsymbol{2},Z_{3})}=\pi_{2}^{3}-3\pi_{1}^{2}\pi_{2}\,,$
(3.44)

whose pion dependence equals that for the secondary $A_{4}$ invariant
$\mathcal{I}_{6}$ in Eq. (3.41). In fact, the role of the two cubic invariants
$\mathcal{I}_{3}^{(\boldsymbol{2},Z_{3})}$ and
$\mathcal{I}_{3^{\prime}}^{(\boldsymbol{2},Z_{3})}$ can be exchanged, or
substituted by combinations of them, by changing the pion parametrization in
Eq. (5.8). Note that we have denoted here the $Z_{3}$ invariants as

$\mathcal{I}_{n}^{(\boldsymbol{m},G)}$ (3.45)

where $n$ indicates the mass-dimension of the invariant, $G$ the invariance
group and $\boldsymbol{m}$ the field representation. This notation will be
implemented all through the rest of the paper, except for the triplet of
$A_{4}$, for which we will maintain the simplified notation above, i.e.
$\mathcal{I}_{2}\equiv\mathcal{I}_{2}^{(\boldsymbol{3},A_{4})}$,
$\mathcal{I}_{3}\equiv\mathcal{I}_{3}^{(\boldsymbol{3},A_{4})}$,
$\mathcal{I}_{4}\equiv\mathcal{I}_{4}^{(\boldsymbol{3},A_{4})}$ and
$\mathcal{I}_{6}\equiv\mathcal{I}_{6}^{(\boldsymbol{3},A_{4})}$.

###  3.3 Phenomenology of $A_{4}$ dGBs

We have shown that an unbroken $Z_{3}$ symmetry will be realized à la Wigner
in the dGB spectrum as long as the vevs of the scalar correspond to the MaNa
extrema. Some experimental consequences follow. We focus below in those
signals associated with the terms in the potential discussed above. The
detailed phenomenological analysis for the interaction terms in the Lagrangian
Eq. (2.5) which involve derivatives will be left for later work, although the
pattern of signals will have strong similarities to those from the potential
terms, as argued further below.181818 To keep our discussion general enough,
note that the phenomenological analysis presented here assumes that the global
minimum of the potential is one of the MaNa extrema identified above. It is
possible that a deeper global minimum arises in Nature which depends instead
on a precise combination of couplings, though, and the analysis would depend
in that case on the value of the potential parameters.

####  3.3.1 Degenerate spectrum

The first physical prediction stems from the pions being in an irreducible
doublet representation of $Z_{3}$: two degenerate dGBs are expected,

$\begin{split}m_{\pi_{1}}^{2}=m_{\pi_{2}}^{2}.\end{split}$ (3.46)

Degeneracy introduces freedom into the definition of the dGBs, though. They
can be redefined by an arbitrary rotation of angle $\beta$,

$\begin{split}\left\\{\begin{array}[]{c}\pi_{1}\\\
\pi_{2}\end{array}\right.\longrightarrow\left\\{\begin{array}[]{l}\hat{\pi}_{1}=\pi_{1}\cos\beta+\pi_{2}\sin\beta\\\
\hat{\pi}_{1}=\pi_{1}\sin\beta-\pi_{2}\cos\beta\,,\end{array}\right.\end{split}$
(3.47)

which in turn redefines the $Z_{3}$ invariants. For instance, for the cubic
ones,

$\displaystyle\\!\mathcal{I}_{3}^{(\boldsymbol{2},Z_{3})}$
$\displaystyle\\!\\!\\!\\!\\!\rightarrow
c_{3\beta}\,\mathcal{I}_{3}^{(\boldsymbol{2},Z_{3})}+s_{3\beta}\,\mathcal{I}_{3^{\prime}}^{(\boldsymbol{2},Z_{3})}$
$\displaystyle\\!\\!\\!\\!\\!=(\pi_{1}^{3}-3\pi_{1}\pi_{2}^{2})\cos
3\beta+(\pi_{2}^{3}-3\pi_{1}^{2}\pi_{2})\sin 3\beta\,,$ (3.48)
$\displaystyle\\!\mathcal{I}_{3^{\prime}}^{(\boldsymbol{2},Z_{3})}$
$\displaystyle\\!\\!\\!\\!\\!\rightarrow-
s_{3\beta}\,\mathcal{I}_{3}^{(\boldsymbol{2},Z_{3})}+c_{3\beta}\,\mathcal{I}_{3^{\prime}}^{(\boldsymbol{2},Z_{3})}$
$\displaystyle\\!\\!\\!\\!\\!=-(\pi_{1}^{3}-3\pi_{1}\pi_{2}^{2})\sin
3\beta+(\pi_{2}^{3}-3\pi_{1}^{2}\pi_{2})\cos 3\beta\,,$ (3.49)

where the short-hand notation $c_{3\beta}\equiv\cos(3\beta)$ and
$s_{3\beta}\equiv\sin(3\beta)$ has been used. This means that the cubic
interactions correspond to an arbitrary linear combination of the little group
invariants $\mathcal{I}_{3}^{(\boldsymbol{2},Z_{3})}$ and
$\mathcal{I}_{3^{\prime}}^{(\boldsymbol{2},Z_{3})}.$ The predictions for the
physical observables are reparametrization invariant, though, as expected and
illustrated further below.

####  3.3.2 Simultaneous production

The second prediction is that, unlike for usual ALPs, dGBs are not expected to
be produced alone. This is an unavoidable consequence of the fact that the
pion fields belong to non-trivial irreducible multiplets, as long as the SM
fields do not carry charges under the discrete symmetry. In any collision or
decay sourcing dGBs, at least two dGBs are then expected to be simultaneously
produced.

Consider for illustration a generic interaction Lagrangian between SM fields
which are singlets of the discrete symmetry –encoded in an operator
$\mathcal{O}^{SM}$– and the dGBs stemming from the $A_{4}$ invariant in the
scalar potential with lowest dimension, $\mathcal{I}_{3}$,

$\mathcal{L}^{\text{int}}\propto\frac{1}{M^{m}}\,\mathcal{O}^{SM}\mathcal{I}_{3}\,,$
(3.50)

where $m$ is some integer power which depends on the dimensionality of
$\mathcal{O}^{SM}$, and $M$ denotes here the high-energy scale of the UV
complete theory where both sectors would be connected. Some expected
topologies for dGB production from SM processes which follow from Eq. (3.39)
are illustrated in Fig. 7. The simultaneous emission of two or three
degenerate dGB are thus the simplest processes.

(a)

(b)

Figure 7: Production of dGBs from SM collisions for a triplet of non-linearly
realized $A_{4}$. The SM fields and interactions appear in black, the dGBs are
shown in red.

While the two dGBs are physically distinct fields, they are degenerate. The
latter means that they cannot be individually distinguished in experiments.
The dGB observables will be given by the sum of the probabilities for each
possible individual final dGB state. This is alike to QCD in which the three
colors of a quark cannot be experimentally separated even in the perturbative
regime, and thus the perturbative QCD cross sections typically require one to
compute the probabilities for each possible channel with given initial and
final colors, to average over color in the initial state, and to sum over all
possible final color configurations. Therefore, for the example in Eq. (3.50)
the probability will be proportional to

$\text{Prob}(\text{SM}\rightarrow
n\,\pi)\propto\,\sum_{i,j....l=1,2}|\mathcal{A}(\text{SM}\rightarrow\underbrace{\pi_{i}\pi_{j}....\pi_{l}}_{n})|^{2}\,,$
(3.51)

where $n$ denotes the number of dGBs produced (i.e. $n=2,3$ or $4$ for the
example in Eq. (3.50)). Upon this proviso, it is easy to check that the
physical observables are invariant under field reparametrizations, as they
should, i.e. be independent of the arbitrary parameter $\beta$ in Eqs. (3.47)
and (3.49). While this is trivial for the two-dGB processes in Fig. 7(a), the
3-dGB processes in Fig. 7(b) are shown to be $\beta$-invariant only upon
implementation of the sum in Eq. (3.51).

The third prediction depends on a particular structure of the relative
probabilities for multi-dGB channels, which may point to the full UV discrete
symmetry. For the real scalar triplet of $A_{4}$ and assuming that, to a good
approximation, Nature’s dGB interactions can be described by the lowest
dimension non-trivial operator, i.e. as in Eq. (3.50), it follows that the
cross sections $\sigma(\text{SM}\rightarrow n\pi)$ for 2-, 3-, and 4-dGB
production processes can be calculated using the interactions of Eq. (3.39)
and

$\displaystyle\sigma(A_{\text{SM}}B_{\text{SM}}\to n\pi)$
$\displaystyle=\displaystyle\int\frac{1}{2E_{A}2E_{B}|v_{A}-v_{B}|}\left|\mathcal{A}(A_{\text{SM}}B_{\text{SM}}\to
n\pi)\right|^{2}d\Pi_{n}\,,$ (3.52)

where $E_{A,B}$ and $v_{A,B}$ are the energies and velocities respectively of
the initial SM particles, and $d\Pi_{n}$ is the differential $n$-body phase
space. Interestingly, the expansion of the $\mathcal{I}_{3}$ invariant in
terms of the pions in Eq. 3.39 predicts certain specific relations among the
n-pion interactions. Neglecting momentum-dependent $\mathcal{O}^{SM}$ in Eq.
(3.50), the integral reduces to calculating the $n$-body final state phase
space, and we can write:

$\displaystyle\qquad\frac{\sigma(\text{SM}\rightarrow
2\pi)}{\sigma(\text{SM}\rightarrow
3\pi)}=2f^{2}\frac{\Pi_{2}}{\Pi_{3}}\,,\qquad\frac{\sigma(\text{SM}\rightarrow
3\pi)}{\sigma(\text{SM}\rightarrow
4\pi)}=\frac{36f^{2}}{19(17)^{2}}\frac{\Pi_{3}}{\Pi_{4}}\,,\qquad$ (3.53)

In the $m_{\pi}\to 0$ limit, we find:

$\displaystyle\qquad\frac{\sigma(\text{SM}\rightarrow
2\pi)}{\sigma(\text{SM}\rightarrow
3\pi)}=64\pi^{2}\frac{f^{2}}{E_{\text{CM}}^{2}}\,,\qquad\frac{\sigma(\text{SM}\rightarrow
3\pi)}{\sigma(\text{SM}\rightarrow
4\pi)}=\frac{6(24\pi)^{2}}{19(17)^{2}}\frac{f^{2}}{E_{\text{CM}}^{2}}\,.\qquad$
(3.54)

Information about the initial state cancels out when taking the above ratios,
and so Eq (3.54) holds for any number of particles in the initial state, and
also gives the ratios of decay rates $\Gamma(\text{SM}\to n\pi)$. These ratios
are specific to the $\mathcal{I}_{3}$ invariant under consideration, see Eq.
3.39, and thus may constitute an interesting tool to disentangle the full UV
discrete symmetry. Remarkably, the invariants of the potential provide a
handle to infer (at least partially) the UV discrete symmetry from the low
energy observables. This is an improvement over the case of nonlinearly
realized continuous symmetries, in which the full UV symmetry cannot be
extracted from the dynamics of pGBs below the SSB scale [53].

In this illustration, the SM sector is assumed to be a singlet of the unbroken
UV discrete symmetry $D$. Were some SM fields to carry charges under that
symmetry, single dGB production and other channels would open. The theoretical
construction would require one to embed SM fields in representations of the
discrete group, which is a major task left for future work.

####  3.3.3 Experimental signals from interactions with derivative couplings

Although we have focused above on the signals stemming from the terms in the
(discrete-symmetry invariant) potential, further signals are expected from the
terms in the dGB Lagrangian Eq. (2.5) containing $\Phi$ derivatives, such as
for instance the interaction term

$\mathcal{L}^{\text{int}}\propto\frac{1}{M^{m}}\,\mathcal{O}^{SM}\partial_{\mu}\Phi^{T}\partial^{\mu}\Phi\,.$
(3.55)

It should be not expected a priori  for these terms to have coefficients
suppressed due to the UV discrete symmetry, as they are also allowed by the
embedding continuous symmetry, see Eq. (2.7). Nevertheless, they will exhibit
the differentiating characteristics of its momentum dependence, which is a
tool to disentangle them from those stemming from scalar invariants discussed
in the previous subsection. They will share, though, the main characteristic,
i.e. to lead to the simultaneous production of at least two dGBs, with no
single dGB event, as far as the SM fields are uncharged under the discrete
symmetry. Analogous considerations hold for the interactions involving mixed
derivative/non-derivative dGB terms. We leave to future work a detailed
phenomenological analysis of the interaction terms involving dGB derivatives.
These comments apply as well to the rest of the UV discrete symmetry scenarios
to be discussed below.

####  3.3.4 Counting the number of dGBs produced in experiment

A typical signal of an ALP (or generic dark matter particle) is the
observation of events with missing energy in excess of the SM background,
assuming that neutral particle is stable or can escape the detector before
decaying. The question of how to count how many degenerate invisible particles
are simultaneously produced in a collision has been already addressed [54, 55]
in two cases: degenerate particles with mass negligible compared to the
collision energy, and degenerate but massive particles.

It was shown that the distribution of visible (SM) particles suffices to infer
the number of invisible particles ejected, without the need to reconstruct
specific characteristics of the invisible sector. The key is the end-point
behavior of certain observables, which has a strong power-law dependence on
the number of invisible particles in the event. This method has the strength
and clarity of a purely kinematic analysis, as it relies on the property that,
in the end point of observables for the visible particles, the invisible ones
must be produced either parallel to each other (if massless) or at rest.
Possible discriminators include the end point of the invariant mass and/or the
invariant transverse mass of the visible particles, among others. In
particular, the distributions for one versus more than one emitted invisible
particles differ widely.

This approach, designed to hunt for multicomponent dark matter, can be
translated to the setup of this work. In summary, the absence of events with
just one invisible (e.g. missing energy) track and the appearance instead of
multi-invisible tracks is the first experimental tell-tale signal consistent
with a BSM theory UV protected by the mechanism explored in this
paper.191919The case in which some SM particles would carry discrete symmetry
charges may lead to a different pattern.  This is the case whenever the non-
trivial irreducible representations of the theory are lighter than the
possible singlet ones, a pattern that holds through all the examples studied
in this work.

####  3.3.5 Inferrence of the underlying discrete symmetry

It may not be possible to infer the complete UV discrete symmetry from only
the low-energy part of the spectrum and interactions. The question is whether
one can identify the explicit à la Wigner symmetry remaining, which by
consistency would allow one to at least delineate the set of possible UV
invariances. More importantly, they can be the lighthouse signal of an UV
stable BSM theory with scalars.

Once the number of degenerate dGBs is experimentally established, it may be
possible to identify the final discrete symmetry of the spectrum because the
precise relative weights of the terms cubic in dGB fields, and of each set of
terms of higher order, is a trademark of the final explicit symmetry and
representation. In the case discussed of a doublet of $Z_{3}$, these are the
combinations in parenthesis in Eqs. (3.39)-(3.41).

Under the further assumption that the lowest-dimension invariant dominates the
potential, the relative weights of the two/three/four/… dGB amplitudes can
provide hints of the complete invariant at play, as illustrated above. The
point is that those ratios are a feature of the complete UV discrete symmetry
D, not of the explicitly realized one. That is, although there is not a one-
to-one correspondence between the amplitudes that follow from the first terms
of a leading invariant and a particular UV discrete symmetry, the
possibilities typically narrow down to a very reduced set of UV symmetries.
This type of analysis has been illustrated for the real triplet of $A_{4}$ in
Eqs. (3.54) above and will be further illuminated below, upon the comparison
with the signals expected from alternative UV discrete scenarios.

###  3.4 A related symmetry: $S_{4}$

The case of the non-abelian UV discrete symmetry $S_{4}$ is close to that of
$A_{4}$, as $S_{4}=A_{4}\times Z_{2}$. For a triplet of real scalars, there
will again be three primary invariants (corresponding to two for the dGB
potential, taking into account the non-linearity constraint). For $S_{4}$
there are two possible triplet representations, $\boldsymbol{3}$ and
$\boldsymbol{3^{\prime}}$. The interest of a triplet of $S_{4}$ as compared
with a triplet of $A_{4}$ is two-fold:

  * •

Further suppressed dGB masses for the $\boldsymbol{3^{\prime}}$
representation. In this case, the three primary invariants are of dimension
two, four and six, $\\{\mathcal{I}_{2},\mathcal{I}_{4},\mathcal{I}_{6}\\}$.
For the dGB potential, as $\mathcal{I}_{2}=f^{2}$ upon imposing the non-
linearity condition, the first operator contributing to the dGB masses is
$\mathcal{I}_{4}$, and furthermore the most general renormalizable potential
reduces to this invariant.

  * •

The predicted dGB spectrum can have non-abelian properties. Indeed, the
discrete symmetries exhibited by the MaNa minima (for both the
$\boldsymbol{3}$ and $\boldsymbol{3^{\prime}}$ cases), include (very simple)
non-abelian ones: e.g. $S_{3}=Z_{3}\times Z_{2}$.

The $S_{4}$ theory also allows us to showcase how different exact UV discrete
symmetries may be disentangled at low energies. While both $S_{4}$ and $A_{4}$
predict the same number of degenerate dGBs (two), the assumption that the
leading term of the dGB potential corresponds to the lowest dimensional
invariant leads to different ratios among the multi-dGB amplitudes stemming
from pure scalar invariants: for $S_{4}$ the dGB potential is expected to be
dominated by $\mathcal{I}_{4}$ and therefore the ratios depend on the
coefficients in Eq. 3.40 (the cubic invariant is absent) while for $A_{4}$
those ratios are fixed by $\mathcal{I}_{3}$, as studied in Eq. 3.54. For
instance, the following cross-section ratios hold for the triplet of $S_{4}$
developed around its $Z_{3}$-symmetric vacuum:

$\displaystyle\frac{\sigma\left(\text{SM}\rightarrow
2\pi\right)}{\sigma\left(\text{SM}\rightarrow
3\pi\right)}=2(8\pi)^{2}\frac{f^{2}}{E_{\text{CM}}^{2}}\,,\qquad\frac{\sigma\left(\text{SM}\rightarrow
3\pi\right)}{\sigma\left(\text{SM}\rightarrow
4\pi\right)}=\frac{27(8\pi)^{2}}{53371}\frac{f^{2}}{E_{\text{CM}}^{2}}\,,$
(3.56)

to be compared with the analogous ones in Eq. (3.54) for the triplet of
$A_{4}$.

We refer to App. A for the technical analysis and the detailed classification
of the $S_{4}$ MaNa extrema. The next two chapters will instead focus on
examples that demonstrate automatically very suppressed dGB masses or non-
abelian symmetric spectra.

##  4 Scalars in a triplet of $A_{5}$

This section analyzes the case of an exact (albeit non-linearly realized)
$A_{5}$ invariance with scalars in a $\boldsymbol{3}$ or a
$\boldsymbol{3}^{\prime}$ representation. It will be shown how this discrete
symmetry may further protect the dGB masses, as the first possible discrete
invariant contributing to the generic potential scalar is of higher dimension
than that for the $A_{4}$ (and $S_{4}$) triplets discussed in the previous
section.

###  4.1 Invariants and potential

As the high-energy (the low-energy) theory has three (two) dynamical scalar
degrees of freedom, we expect again three (two) primary invariants to describe
the scalar physics at those energies.

For a real scalar triplet, the most general scalar potential invariant under
the UV $A_{5}$ symmetry will be expressed as a function of all possible
$A_{5}$ invariants built out of that representation. The dimension and
quantity of the primary and secondary invariants follows from the Molien
generating function [56, 57].202020 For comprehensive reviews of this
formalism see Refs. [58, 59]. All other analytical invariants will be
polynomial combinations of the primary and secondary invariants. The Molien
functions for the $\boldsymbol{3}$ and $\boldsymbol{3}^{\prime}$ of $A_{5}$
are identical, and read

$\mathcal{F}_{A_{5}}(\boldsymbol{1},\boldsymbol{3},\lambda)=\frac{1+\lambda^{15}}{(1-\lambda^{2})(1-\lambda^{6})(1-\lambda^{10})}\,,$
(4.1)

which implies three primary invariants of order 2, 6, and 10, together with a
syzygy that allows one to construct a 15-th order secondary invariant in terms
of the primary invariants. For their explicit construction, one can profit
from the fact that $A_{4}$ is a subgroup of $A_{5}$ [52]. In consequence, the
three primary invariants for the $\boldsymbol{3}$ of $A_{5}$ can be written in
a compact way as combinations of the three primary $A_{4}$ invariants
($\mathcal{I}_{2}$, $\mathcal{I}_{3}$, $\mathcal{I}_{4}$) and the secondary
$A_{4}$ invariant ($\mathcal{I}_{6}$) given in Eqs. (3.1)-(3.4),

$\displaystyle\mathcal{I}_{2}^{(\boldsymbol{3},A_{5})}$
$\displaystyle=\mathcal{I}_{2}\,,$ (4.2)
$\displaystyle\mathcal{I}_{6}^{(\boldsymbol{3},A_{5})}$
$\displaystyle=22\,\mathcal{I}_{3}^{2}+\mathcal{I}_{2}\,\mathcal{I}_{4}-2\sqrt{5}\,\mathcal{I}_{6}\,,$
(4.3) $\displaystyle\mathcal{I}_{10}^{(\boldsymbol{3},A_{5})}$
$\displaystyle=\mathcal{I}_{2}^{2}\,\mathcal{I}_{4}+38\,\mathcal{I}_{3}^{2}\,\mathcal{I}_{4}-\frac{7}{11}\,\mathcal{I}_{2}^{3}\,\mathcal{I}_{4}-\frac{128}{11\sqrt{5}}\,\mathcal{I}_{2}^{2}\,\mathcal{I}_{6}+\frac{6}{\sqrt{5}}\,\mathcal{I}_{4}\,\mathcal{I}_{6}\,.$
(4.4)

The invariants for the $\boldsymbol{3^{\prime}}$ representation of $A_{5}$ are
exactly the same as for the $\boldsymbol{3}$.

Once again, the quadratic invariant $\mathcal{I}_{2}^{(\boldsymbol{3},A_{5})}$
will not contribute to the dGB potential, given that the non-linearity
constraint in Eq. (3.7) also holds here. This theory is therefore protected
from quadratic instabilities via quantum corrections induced by physical
scales in the theory heavier than $\Lambda$. Furthermore, the first invariant
which can induce a mass for the dGBs now has dimension six,

$V_{\text{dGB}}={f^{2}}\Lambda^{2}\,\big{[}\hat{c}_{6}\frac{\mathcal{I}_{6}}{f^{6}}+\hat{c}_{10}\frac{\mathcal{I}_{10}}{f^{10}}+\hat{c}_{12}\frac{\mathcal{I}_{6}^{2}}{f^{12}}+\hat{c}_{15}\frac{\mathcal{I}_{15}}{f^{15}}\cdots\big{]}\,,$
(4.5)

and thus the dGB masses are expected to be weighted down by a strong
$\epsilon^{6}$ suppression factor due to the UV discrete symmetry, see Eq.
(3.11) and the subsequent discussion. Therefore, the dGB masses are much more
suppressed than in the case of $A_{4}$ with a scalar triplet (which exhibited
an $\epsilon^{3}$ suppression).

Furthermore, note that in this case the physical impact of the complete
potential should be well approximated by just one invariant,
$\mathcal{I}_{6}$, given the strong hierarchy of the potential coefficients:
the impact of the other primary invariant,
$\mathcal{I}_{10}^{(\boldsymbol{3},A_{5})}$, the secondary invariant
$\mathcal{I}_{15}^{(\boldsymbol{3},A_{5})}$, and the rest of the terms, can be
disregarded here altogether given their very high suppression.

####  4.1.1 Embedding in a continuous group: $SO(3)$.

Alike to $A_{4}$, $A_{5}$ is a subgroup of $SO(3)$, for which two irreducible
triplet representations are possible, see Table 1. Again, the only possible
pattern of spontaneous symmetry breaking (SSB) of the continuous group with
those representations is $SO(3)\to SO(2)$, and consequently two physical pion
fields, $\pi_{1}$ and $\pi_{2}$, would survive at low energies, see Eqs. (3.7)
and (5.8).

The highly suppressed potential coefficients and their strong hierarchy imply
that an exact (and spontaneously broken) UV $A_{5}$ symmetry induces the
existence of an approximate continuous (spontaneously broken) $SO(3)$
invariance, which is explicitly broken to a very mild extent. This leads to
highly suppressed pGBs masses as compared with the overall effective scale
$\Lambda$.

As stated before, in what follows we do not rely on any particular choice of
continuous group among those in which the discrete group can be embedded.

####  4.1.2  MaNa extrema

There are a total of 62 MaNa extrema of the $A_{5}$ invariants described
above. They can be classified as:

  * •

12 stable extrema with $Z_{5}$ as their little group. They are the vertices of
a dodecahedron in Fig. 8(a).

  * •

20 stable extrema with $Z_{3}$ as little group. Geometrically, they are the
vertices of an icosahedron in Fig. 8(b).

  * •

30 saddle points with $Z_{2}$ as their little group. A total of 20 lie at the
midpoints of the edges of the icosahedron.

The counting constraint in Eq. (3.24) works out nicely, $60=12\times
5=20\times 3=30\times 2$, and also the number of extrema is consistent,
$62=12+20+30$. Table 3 shows the location in field space of some
representatives of the MaNa extrema, together with the value of the invariants
at those points and their exact (explicit) invariances.

(a)

(b)

Figure 8: Geometrical distribution in field space of the MaNa extrema which
are invariant under $Z_{5}$ (a) and $Z_{3}$ (b). The field values are given in
units of $f$.
$\displaystyle\begin{array}[]{|c||cc|ccc|c|c|}\lx@intercol\hfil\text{{MaNa
extrema for a triplet of} $\boldsymbol{A_{5}}$}\hfil\lx@intercol\\\
\hline\cr\hbox{\multirowsetup\text{Point}}&\hbox{\multirowsetup$\mathcal{I}_{6}$}&\,\hbox{\multirowsetup$\mathcal{I}_{10}$}&\phi_{1}&\phi_{2}&\phi_{3}&\hbox{\multirowsetup\text{Little
group}}&\hbox{\multirowsetup\text{Nature}}\\\
&&&\lx@intercol\hfil\textit{(Representatives)}\hfil\lx<EMAIL_ADDRESS>\hline\cr\hline\cr\hbox{\multirowsetup
A}&\hbox{\multirowsetup$\frac{1}{5}$}&\hbox{\multirowsetup$-\frac{472}{1375}$}&\pm\sqrt{\frac{5-\sqrt{5}}{10}}&0&\mp\sqrt{\frac{5+\sqrt{5}}{10}}&\hbox{\multirowsetup$Z_{5}$}&\hbox{\multirowsetup\text{Minima}}\\\\[3.44444pt]
\hline\cr\rule{0.0pt}{9.90276pt}\hbox{\multirowsetup
B}&\hbox{\multirowsetup$\frac{31}{27}$}&\hbox{\multirowsetup$\frac{328}{891}$}&\frac{1}{\sqrt{3}}&\frac{1}{\sqrt{3}}&\frac{1}{\sqrt{3}}&\hbox{\multirowsetup\text{$Z_{3}$}}&\hbox{\multirowsetup\text{Minima}}\\\\[3.44444pt]
&&&\sqrt{\frac{3+\sqrt{5}}{6}}&0&\sqrt{\frac{3-\sqrt{5}}{6}}&&\\\\[3.44444pt]
\hline\cr\rule{0.0pt}{15.0694pt}\hbox{\multirowsetup
C}&\hbox{\multirowsetup$1$}&\hbox{\multirowsetup$\frac{4}{11}$}&0&0&\pm
1&\hbox{\multirowsetup$Z_{2}$}&\hbox{\multirowsetup\text{Saddles}}\\\
&&&\frac{\left(-1+\sqrt{5}\right)}{4}&-\frac{1}{2}&\frac{\left(1+\sqrt{5}\right)}{4}&&\\\\[3.44444pt]
\hline\cr\end{array}$ Table 3: Location and symmetries of a few representative
MaNa extrema for the case of a triplet scalar of $A_{5}$. The values of the
invariants and the locations are normalized to $f=1$. The location in field
space of the ensemble of MaNa minima is depicted in Fig. 8. The manifold is
illustrated in Fig. 9.

The locations in field space of the ensemble of the MaNa extrema is depicted
in Fig. 8. The manifold defined by the two $A_{5}$ primary invariants relevant
for the potential is depicted in Fig. 9.

Figure 9: The manifold defined by the
$\mathcal{I}^{(\boldsymbol{3},A_{5})}_{6}(\Phi)$ and
$\mathcal{I}^{(\boldsymbol{3},A_{5})}_{10}(\Phi)$ invariants of the non-
linearly realized $A_{5}$ with a scalar triplet. The MaNa extrema correspond
to the singular points of this manifold: $A,B\text{ and }C$.

The set of $Z_{3}$-symmetric points and the set of $Z_{5}$-symmetric points
cannot be simultaneously minima or maxima. Depending on the sign of the
$\hat{c}_{i}$ coefficients, either one of the other are minima, while the
$Z_{2}$-symmetric extrema are always saddle points. It is interesting to note
that the larger the symmetry at a MaNa minima, the deeper the potential well
at that point.

The expansion of the invariant $\mathcal{I}_{6}^{(\boldsymbol{3},A_{5})}$ in
terms of dGBs around the $Z_{3}$-symmetric MaNa minimum
$\Phi_{B}\,=\,\frac{1}{\sqrt{3}}\left(1,1,1\right)^{\top}$ yields the
following contribution to the potential:

$\mathcal{I}_{6}^{(\boldsymbol{3},A_{5})}=\frac{32}{9}\,f^{4}\,\left[\frac{31}{96}f^{2}-\left(\pi_{1}^{2}+\pi_{2}^{2}\right)+\frac{10\sqrt{2}}{24f}\left(\pi_{1}^{3}-3\pi_{1}\pi_{2}^{2}\right)\right.\\\
\ \ \ \ \
\left.+\frac{\sqrt{30}}{4f}\left(\pi_{2}^{3}-3\pi_{1}^{2}\pi_{2}\right)+\frac{31}{12f^{2}}\left(\pi_{1}^{2}+\pi_{2}^{2}\right)^{2}\right]\,,$
(4.6)

which is a combination of the quadratic and the two cubic $Z_{3}$ invariants
for a doublet representation, see Eqs. (3.42)-(3.44). If the potential can be
approximated by this lowest invariant (with the appropriate sign $V_{\rm
dGB}\sim-\mathcal{I}_{6}^{(\boldsymbol{3},A_{5})}$), this $Z_{3}$-symmetric
point will correspond to a minimum.

In contrast, the expansion of $\mathcal{I}_{6}^{(\boldsymbol{3},A_{5})}$
around the $Z_{5}$-symmetric MaNa minimum $\Phi_{A}$ yields

$\begin{split}\\!\mathcal{I}_{6}^{(\boldsymbol{3},A_{5})}&=\frac{32}{5}f^{4}\left[\frac{f^{2}}{32}+\left(\pi_{1}^{2}+\pi_{2}^{2}\right)\\!-\\!\frac{31}{12f^{2}}\left(\pi_{1}^{2}+\pi_{2}^{2}\right)^{2}\\!-\\!\frac{1}{4f^{3}}\left(\pi_{1}^{5}-10\pi_{1}^{3}\pi_{2}^{2}+5\pi_{1}\pi_{2}^{4}\right)\right],\end{split}$
(4.7)

showing that the nature of this family of MaNa extrema is revealed by terms
which are of fifth order in the pion dependence, reflecting the 5-point
symmetry of the theory. Indeed, the Molien function for the irreducible
doublet representation of $Z_{5}$ reads

$\mathcal{F}_{Z_{5}}(\boldsymbol{1},\boldsymbol{2},\lambda)=\frac{1+\lambda^{5}}{(1-\lambda^{2})(1-\lambda^{5})}\,,$
(4.8)

which implies two primary invariants of mass dimension two and five,

$\displaystyle\mathcal{I}_{2}^{(\boldsymbol{2},Z_{5})}$
$\displaystyle=\pi_{1}^{2}+\pi_{2}^{2}$ (4.9)
$\displaystyle\mathcal{I}_{5}^{(\boldsymbol{2},Z_{5})}$
$\displaystyle=\pi_{1}^{5}-10\pi_{1}^{3}\pi_{2}^{2}+5\pi_{1}\pi_{2}^{4}\,,$
(4.10)

plus a secondary invariant also of dimension five,

$\displaystyle\mathcal{I}_{5}^{(\boldsymbol{2},Z_{5})}$
$\displaystyle=\pi_{2}^{5}-10\pi_{1}^{2}\pi_{2}^{3}+5\pi_{1}^{4}\pi_{2}\,.$
(4.11)

The expansions in terms of dGBs for
$\mathcal{I}_{6}^{(\boldsymbol{3^{\prime}},A_{5})}$ are identical to those for
$\mathcal{I}_{6}^{(\boldsymbol{3},A_{5})}$. Again, the physical results can be
easily proven to be independent of changes of parametrization of the dGB
fields, see Eq. (3.47), as they should be.

###  4.2 Phenomenological signals with an $A_{5}$ triplet

Given the $Z_{5}$ symmetry at the MaNa minimum with the largest little group,
the signals to be expected mirror those obtained earlier for the $A_{4}$ case,
see Sec. 3.3, that is:

  * •

Two degenerate dGBs, as befits the only irreducible representation of $Z_{N}$
with real fields and dimension larger than one (the $\boldsymbol{2}$). Their
masses are expected to be naturally more suppressed than those for the $A_{4}$
case, though, as compared with the overall UV scale, as explained above.

  * •

The simultaneous emission of two, four or five degenerate dGBs in SM initiated
processes is expected (as long as the SM fields are singlets of the UV
discrete symmetry). No events with only one dGB emitted are then expected, nor
with three (in contrast to the $A_{4}$ scenario).

The latter property can be illustrated with the generic interaction
Lagrangian,

$\mathcal{L}^{\text{int}}\propto\frac{1}{M^{m}}\,\mathcal{O}^{SM}\mathcal{I}_{6}^{(\boldsymbol{3},A_{5})}+\cdots\,,$
(4.12)

expanded around the dGB fields at the MaNa minimum with the largest little
group, which we showed above to exhibit a à la Wigner $Z_{5}$ symmetry. The
expansion around this minimum –see Eq. (4.7)– shows that no tree-level
production topology is expected with three dGBs emitted (unlike for the UV
$A_{4}$ case), and the first vertex with an odd number of dGBs involves five
dGBs. This is illustrated in Fig. 10. The considerations about
reparametrization invariance and the production probability in Eq. (3.51)
apply as well in this case.

(a)

(b)

(c)

Figure 10: Production of dGBs from SM collisions, for a triplet of non-
linearly realized $A_{5}$. While events with three dGBs are absent, those with
five dGBs are characteristic of this kind of dGBs whenever the
$Z_{5}$-symmetric MaNa minimum are the absolute minimum. The color code is
that of Fig. 7.

In consequence, this scenario yields a $Z_{5}$-symmetric spectrum resulting in
tell-tale relative probabilities for multi-dGB channels from the interaction
in Eq. (4.12), which are null for three dGB production. Using Eq. (3.51), we
find:

$\frac{\sigma(\text{SM}\rightarrow 2\pi)}{\sigma(\text{SM}\rightarrow
4\pi)}=\frac{18f^{4}}{19(31)^{2}}\frac{\Pi_{2}}{\Pi_{4}}\,,\qquad\frac{\sigma(\text{SM}\rightarrow
4\pi)}{\sigma(\text{SM}\rightarrow
5\pi)}=\frac{19(31)^{2}f^{2}}{3(45)^{2}}\frac{\Pi_{4}}{\Pi_{5}}\,.$ (4.13)

Evaluating the final state phase space gives explicitly:

$\frac{\sigma(\text{SM}\rightarrow 2\pi)}{\sigma(\text{SM}\rightarrow
4\pi)}=\frac{216(4\pi)^{4}}{19(31)^{2}}\frac{f^{4}}{E_{\text{CM}}^{4}}\,,\qquad\frac{\sigma(\text{SM}\rightarrow
4\pi)}{\sigma(\text{SM}\rightarrow
5\pi)}=\frac{19(31)^{2}(8\pi)^{2}}{(45)^{2}}\frac{f^{2}}{E_{\text{CM}}^{2}}\,.$
(4.14)

These are to be compared with Eq. (3.54) for the $Z_{5}$-symmetric spectrum
(stemming from a scalar triplet of a UV $A_{5}$ symmetry) developed in the
previous section.

Analogous remarks to those in Sect. 3.3.3 for the signals expected from
interaction terms containing $\partial_{\mu}U$ components apply in this case.

####  4.2.1 Generalization to low-energy $Z_{N}$ symmetric spectra

The examples above have been constrained to triplet (real) scalar
representations of simple UV discrete symmetries, non-linearly realized,212121
Which happen to be contained in $SO(3)$. which we showed lead to MaNa minima
with explicit $Z_{3}$ or $Z_{5}$ symmetry (or involving them). Larger
irreducible representations of various UV discrete symmetries could be
considered along the same lines, which may leave a low-energy spectra
invariant under some $Z_{N}$ symmetry.

The important point is that, whatever the value of $N$, the only irreducible
representations of $Z_{N}$ –aside from singlets– are doublets,222222 Rotations
of angle $2\pi/N$ in the plane $\mathds{R}^{2}$ are always a representation of
$Z_{N}$ with real fields. Such a rotation has no invariant subspace and thus
it is irreducible and of dimension two. This is not the case rotations in
$\mathds{R}^{n>2}$. In consequence, the only irreducible representations for
any $Z_{N}$ are doublets and singlets. whose lowest dimensional invariant has
dimension two. This implies that whenever the final preserved discrete
symmetry of the spectrum is abelian, i.e. $Z_{N}$, two degenerate dGBs (or
several sets of them) are to be expected, and no events with just one
invisible (e.g. missing energy) track should be found among the leading
signals, as far as the SM is a singlet of the discrete symmetry. Furthermore,
the characteristic tell-tale production topology for the absolute minimum is
that of $N$-point dGB interactions.

The experimental methods discussed in Sec. 3.3.4 to disentangle how many
invisible particles are ejected in a collision or decay are valid for any $N$.
The relative probabilities of events with different numbers of dGBs contain
information which a priori can allow one to identify the surviving discrete
symmetry of the low-energy spectrum and interactions, as exemplified above.
This in turn may delimitate by consistency the set of possible UV discrete
invariances which are ultimately responsible for the increased UV convergence
of the BSM theory.

A qualification is pertinent here for the case of large UV discrete groups.
The UV protection under study relies on a non-linear realization of the
discrete symmetry $D$ for an irreducible $m$-dimensional real representation
of the group, and this leads to $m-1$ dGBs at low energies. Large discrete
groups may not have irreducible representations of dimension smaller than
four, though (e.g. $m\geq 5$ for $A_{6}$). If the surviving explicit symmetry
of the spectrum is abelian, it follows then that the $m-1$ low-energy dGB
degrees of freedom will belong to a reducible representation. The latter is
expected to decompose as a combination of dGB doublets (for $m$ odd), or a
combination of doublets plus a singlet (for $m$ even). The examples that we
have analyzed suggest that the singlet may be typically heavier than the
doublets. In summary, the leading experimental signals (with SM fields
uncharged under $D$) are still characterized then by the absence of single ALP
emission, and the simultaneous emission of (sets of) two degenerate ALPs, as a
tell-tale of an abelian-symmetric spectrum.

A pertinent question is whether the prediction of simultaneous dGB production
could be broken in favor of emission of a single dGB instead, for instance if
the UV symmetry were to be classically exact but anomalous (assuming the SM
sector is uncharged under the discrete group).232323We thank Anson Hook for a
question prompting this consideration. This cannot happen, though, as the non-
abelian UV symmetries discussed are subgroups of continuous ones, which cannot
be anomalous because the non-abelian generators are traceless. This remark
only depends on the non-abelian character of the UV discrete symmetry and in
consequence applies as well to the next section.

##  5 Scalars in a quadruplet of $A_{5}$ : non-abelian symmetry of the
spectrum

The analysis in the two previous sections led to predict low-energy spectra
explicitly symmetric under abelian groups, e.g. $Z_{3}$ and $Z_{5}$ (except
for the case of $S_{3}$ from $S_{4}$ which led to low-energy consequences
similar to those for abelian little groups), as long as the scalar vevs
correspond to the MaNa minima. The physical dGBs to be detected are then
degenerate ones belonging to the largest irreducible representation of any
$Z_{N}$ group: the $\boldsymbol{2}$, while the interactions could allow one to
disentangle the precise abelian group, i.e. to identify $N$.

We address here whether the explicit (i.e. à la Wigner) discrete symmetry of
the spectrum can be non-abelian instead, and what would then be the
differentiating signals. A non-trivial case shown below to realize this
scenario is an EFT built out of a real scalar field in a $\boldsymbol{4}$ of
$A_{5}$.

###  5.1 Invariants and Potential

For four independent degrees of freedom at high-energies, four primary $A_{5}$
invariants are expected, with all other possible invariants being a function
of those. Indeed, the Molien function for a quadruplet of $A_{5}$ reads

$\mathcal{F}_{A_{5}}(\boldsymbol{1},\boldsymbol{4},\lambda)=\frac{1+\lambda^{10}}{(1-\lambda^{2})(1-\lambda^{3})(1-\lambda^{4})(1-\lambda^{5})}\,,$
(5.1)

which indicates four primary invariants with mass dimension two, three, four
and five. We choose the particular realization of the $\boldsymbol{4}$
representation of $A_{5}$, $\\{\phi_{1},\phi_{2},\phi_{3},\phi_{4}\\}$, to be
that for which one of its $A_{4}$ subgroups lives in the three first
components of the quadruplet, i.e. $\\{\phi_{1},\phi_{2},\phi_{3}\\}$ forms a
triplet of the $A_{4}$ subgroup. The $A_{5}$ invariants can then be written in
terms of polynomials constructed with the invariants $\mathcal{I}_{2}$,
$\mathcal{I}_{3}$ and $\mathcal{I}_{4}$ for the $\boldsymbol{3}$ of $A_{4}$,
see Eqs. (3.1)-(3.3), plus $\phi_{4}$,

$\displaystyle\mathcal{I}_{2}^{(\boldsymbol{4},A_{5})}$
$\displaystyle=\mathcal{I}_{2}+\phi_{4}^{2}\,,$ (5.2)
$\displaystyle\mathcal{I}_{3}^{(\boldsymbol{4},A_{5})}$
$\displaystyle=\mathcal{I}_{3}-\frac{\phi_{4}}{2\sqrt{5}}\mathcal{I}_{2}+\frac{\phi_{4}^{3}}{2\sqrt{5}}\,,$
(5.3) $\displaystyle\mathcal{I}_{4}^{(\boldsymbol{4},A_{5})}$
$\displaystyle=\mathcal{I}_{4}+\frac{12}{\sqrt{5}}\mathcal{I}_{3}\phi_{4}+\frac{12}{5}\mathcal{I}_{2}\phi_{4}^{2}+\frac{\phi_{4}^{4}}{5}\,,$
(5.4) $\displaystyle\mathcal{I}_{5}^{(\boldsymbol{4},A_{5})}$
$\displaystyle=\mathcal{I}_{4}\phi_{4}-\frac{1}{2}\mathcal{I}_{2}\phi_{4}-\frac{4}{\sqrt{5}}\mathcal{I}_{3}\phi_{4}^{2}-\frac{\phi_{4}^{3}}{5}\mathcal{I}_{2}+\frac{\phi_{4}^{5}}{50}\,.$
(5.5)

The non-linear constraint reads in this case,

$\mathcal{I}_{2}^{(\boldsymbol{4},A_{5})}=\Phi^{T}\Phi=\phi_{1}^{2}+\phi_{2}^{2}+\phi_{3}^{2}+\phi_{4}^{2}=f^{2}\,,$
(5.6)

and thus the first explicit breaking of the continuous symmetry will appear
through a cubic $A_{5}$ invariant. In other words, as in all previous
scenarios, the theory is protected from quadratic corrections at the quantum
level.

In the spirit of EFT, the generic potential can thus be written as

$V_{\text{dGB}}={f^{2}}\Lambda^{2}\,\left[\hat{c}_{3}\,\frac{\mathcal{I}_{3}}{f^{3}}+\hat{c}_{4}\,\frac{\mathcal{I}_{4}}{f^{4}}+\hat{c}_{5}\frac{\mathcal{I}_{5}}{f^{5}}+\hat{c}_{6}\frac{\mathcal{I}_{6}}{f^{6}}\cdots\right]\,,$
(5.7)

where $\mathcal{I}_{6}$ is formed out of primary invariants
$\mathcal{I}_{6}=\mathcal{I}_{3}^{2}$. The manifold of invariants spanned by
the set of primary invariants
$\\{\mathcal{I}_{3}^{(\boldsymbol{4},A_{5})},\mathcal{I}_{4}^{(\boldsymbol{4},A_{5})},\mathcal{I}_{5}^{(\boldsymbol{4},A_{5})}\\}$
is analyzed next, subject to the non-linearity constraint Eq. (5.6).

$\displaystyle\begin{array}[]{|c||ccc|cccc|c|c|}\lx@intercol\hfil\text{{ MaNa
extrema for a quadruplet of} $\boldsymbol{A_{5}}$}\hfil\lx@intercol\\\
\hline\cr\hbox{\multirowsetup\text{Point}}&\hbox{\multirowsetup$\mathcal{I}_{3}$}&\hbox{\multirowsetup$\mathcal{I}_{4}$}&\,\hbox{\multirowsetup$\mathcal{I}_{5}$}&\phi_{1}&\phi_{2}&\phi_{3}&\phi_{4}&\hbox{\multirowsetup\text{Little
group}}&\hbox{\multirowsetup\text{Nature}}\\\
&&&&\lx@intercol\hfil\textit{(Representatives)}\hfil\lx<EMAIL_ADDRESS>\hline\cr\hline\cr\hbox{\multirowsetup
A}&\hbox{\multirowsetup$\frac{1}{2\sqrt{5}}$}&\hbox{\multirowsetup$\frac{1}{5}$}&\hbox{\multirowsetup$\frac{1}{50}$}&0&0&0&1&\hbox{\multirowsetup$A_{4}$}&\hbox{\multirowsetup\text{Minima}}\\\
&&&&-\frac{\sqrt{5}}{4}&-\frac{\sqrt{5}}{4}&-\frac{\sqrt{5}}{4}&\frac{1}{4}&&\\\\[3.44444pt]
\cline{1-8}\cr\hbox{\multirowsetup
B}&\hbox{\multirowsetup$-\frac{1}{2\sqrt{5}}$}&\hbox{\multirowsetup$\frac{1}{5}$}&\hbox{\multirowsetup$-\frac{1}{50}$}&0&0&0&-1&&\\\
&&&&\frac{\sqrt{5}}{4}&\frac{\sqrt{5}}{4}&\frac{\sqrt{5}}{4}&-\frac{1}{4}&&\\\\[3.44444pt]
\hline\cr\rule{0.0pt}{12.91663pt}\hbox{\multirowsetup
C}&\hbox{\multirowsetup$\frac{1}{3\sqrt{30}}$}&\hbox{\multirowsetup$\frac{31}{30}$}&\hbox{\multirowsetup$-\frac{4}{25}\sqrt{\frac{2}{3}}$}&\sqrt{\frac{5}{24}}&\sqrt{\frac{5}{24}}&\sqrt{\frac{5}{24}}&\sqrt{\frac{3}{8}}&\hbox{\multirowsetup$S_{3}$}&\hbox{\multirowsetup\text{Saddles}}\\\\[3.44444pt]
&&&&-\sqrt{\frac{5}{6}}&0&0&-\frac{1}{\sqrt{6}}&&\\\\[3.44444pt]
\cline{1-8}\cr\rule{0.0pt}{14.2083pt}\hbox{\multirowsetup
D}&\hbox{\multirowsetup$-\frac{1}{3\sqrt{30}}$}&\hbox{\multirowsetup$\frac{31}{30}$}&\hbox{\multirowsetup$\frac{4}{25}\sqrt{\frac{2}{3}}$}&-\sqrt{\frac{5}{24}}&-\sqrt{\frac{5}{24}}&-\sqrt{\frac{5}{24}}&-\sqrt{\frac{3}{8}}&&\\\
&&&&\sqrt{\frac{5}{6}}&0&0&\frac{1}{\sqrt{6}}&&\\\\[3.44444pt]
\hline\cr\end{array}$ Table 4: Location and symmetries of some representative
MaNa extrema for the case of a real scalar fields in a quadruplet of $A_{5}$.
All dimensional entries are normalized to $f=1$. The complete manifold is
illustrated in Fig. 11.

####  5.1.1 Embedding in a continuous group: $SO(4)$

While the scenarios explored in previous sections with scalar triplets could
all be embedded in $SO(3)$, for a quadruplet of $A_{5}$ the smallest
continuous embedding corresponds to $SO(4)$. From the point of view of the
continuous symmetry, this theory is again insensitive to quadratic
divergences, while the first contributions to the pion masses stem from the
dimension three $A_{5}$-symmetric invariant.

####  5.1.2 MaNa extrema

There are a total of 30 MaNa extrema for the invariant potential built out of
the $\boldsymbol{4}$ representation of $A_{5}$:

  * •

10 $A_{4}$-symmetric MaNa extrema , and

  * •

20 MaNa extrema with $S_{3}$ as their little group,

i.e. all MaNa extrema are non-abelian as desired. The counting rule in Eq.
(3.24) works out nicely after one realizes that all the $A_{4}$\- symmetric
points do not belong to the same orbit, since transformations under $A_{5}$
are only able to relate among themselves half of them. The same applies to the
set of $S_{3}$-symmetric points. In consequence, there are four different
orbits in total which all comply neatly with the counting criterium.

Representatives of each MaNa extremum are shown in Table 4. The manifold
spanned by the three primary invariants of this scenario is depicted in Fig.
11.

Figure 11: Manifold defined by
$\mathcal{I}^{(\boldsymbol{3},A_{5})}_{3}(\Phi)$,
$\mathcal{I}^{(\boldsymbol{3},A_{5})}_{4}(\Phi)$ and
$\mathcal{I}^{(\boldsymbol{3},A_{5})}_{5}(\Phi)$ invariants of $A_{5}$ when
$\Phi$ is a $SO(4)$ triplet fulfiling
$\phi_{1}^{2}+\phi_{2}^{2}+\phi_{3}^{2}+\phi_{4}^{2}=f^{2}$.

##### Expansion in terms of low-energy degrees of freedom.

We will use again the standard pion parametrization in Eq. (2.3) to project
the results in Table 4 in terms of the three low energy dGBs,

$\Phi(\pi_{1},\pi_{2},\pi_{3})=\exp\left[\frac{1}{f}\left(\begin{array}[]{cccc}0&0&0&\pi_{1}\\\
0&0&0&\pi_{2}\\\ 0&0&0&\pi_{3}\\\
-\pi_{1}&-\pi_{2}&-\pi_{3}&0\end{array}\right)\right]\left(\begin{array}[]{c}0\\\
0\\\ 0\\\ f\end{array}\right)\,.$ (5.8)

In order to obtain physically meaningful results, one must rotate those points
among the MaNa extrema in Table 4 which do not point in the $(0,0,0,1)$
direction chosen for the pion parametrization, as described in Eq. (3.38)
albeit with a four-dimensional rotation in parameter space. Two types of
extrema appear, both exhibiting non-abelian invariances, described next.

#####  MaNa extrema with $A_{4}$ as their little group.

The minima in this class are stable. For them, the low-energy spectrum
exhibits and exact (à la Wigner) non-abelian $A_{4}$ symmetry. Consider for
illustration the representative MaNa extremum $\Phi_{B}=(0,0,0,-1)$. In terms
of the dGB degrees of freedom, the three primary invariants relevant for the
potential take the form

$\displaystyle\mathcal{I}_{3}^{(\boldsymbol{4},A_{5})}$
$\displaystyle=\frac{\sqrt{5}f}{4}\left[-\frac{2f^{2}}{5}+\left(\pi_{1}^{2}+\pi_{2}^{2}+\pi_{3}^{2}\right)-\frac{4}{\sqrt{5}f}\,\pi_{1}\pi_{2}\pi_{3}-\frac{41}{60f^{2}}\left(\pi_{1}^{2}+\pi_{2}^{2}+\pi_{3}^{2}\right)^{2}\right]\,,$
(5.9)
$\mathcal{I}_{4}^{(\boldsymbol{4},A_{5})}=2f^{2}\left[\frac{f^{2}}{10}+\left(\pi_{1}^{2}+\pi_{2}^{2}+\pi_{3}^{2}\right)+\frac{6}{\sqrt{5}f}\pi_{1}\pi_{2}\pi_{3}+\frac{1}{2f^{2}}\left(\pi_{1}^{4}+\pi_{2}^{4}+\pi_{3}^{4}\right)\right.\\\
\left.-\frac{43}{30f^{2}}\left(\pi_{1}^{2}+\pi_{2}^{2}+\pi_{3}^{2}\right)^{2}\right]\,,$
(5.10)
$\mathcal{I}_{5}^{(\boldsymbol{4},A_{5})}=\frac{f^{3}}{4}\left[\frac{2f^{2}}{25}+\left(\pi_{1}^{2}+\pi_{2}^{2}+\pi_{3}^{2}\right)+\frac{16}{\sqrt{5}f}\pi_{1}\pi_{2}\pi_{3}-\frac{4}{f^{2}}\left(\pi_{1}^{4}+\pi_{2}^{4}+\pi_{3}^{4}\right)\right.\\\
\left.+\frac{19}{60f^{2}}\left(\pi_{1}^{2}+\pi_{2}^{2}+\pi_{3}^{2}\right)^{2}\right]\,.$
(5.11)

If the dGB potential can be approximated by the lowest relevant invariant
(with the appropriate sign $V_{\rm
dGB}\sim-\mathcal{I}_{3}^{(\boldsymbol{4},A_{5})}$), the $A_{4}$-symmetric
point will correspond to a minimum. The expressions above show that the low
energy theory exhibits an exact $A_{4}$ symmetry for a $\boldsymbol{3}$
representation, i.e.

  * •

The low-energy spectrum consists of a triplet of degenerate scalars
$\\{\pi_{1},\pi_{2},\pi_{3}\\}$. This is in contrast with the spectrum of two
degenerate dGBs found whenever the exact symmetry of the low-energy theory was
a discrete abelian group of any order.

  * •

The invariants in terms of $\\{\pi_{1},\pi_{2},\pi_{3}\\}$ in Eqs.
(5.9)-(5.11) are combinations of those for a $\boldsymbol{3}$ representation
of $A_{4}$, see for instance Eqs. (3.1)-(3.3).

##### MaNa extrema with $S_{3}$ as their little group.

These are saddle points. It is worth it to remark, though, that the saddle
character stems exclusively from the cubic invariant,
$\mathcal{I}_{3}^{(\boldsymbol{4},A_{5})}$. Specifically, denoting by
$\Phi_{A,B}$ and $\Phi_{C,D}$ the extrema in Table 4 with little groups
respectively invariant under $A_{4}$ and $S_{3}$ , the correspondence is

$\displaystyle\left.\mathcal{I}_{3}^{(\boldsymbol{4},A_{5})}\right|_{\Phi_{A,B}}\longrightarrow\
\text{Minima}$
$\displaystyle\left.\mathcal{I}_{3}^{(\boldsymbol{4},A_{5})}\right|_{\Phi_{C,D}}\longrightarrow\
\text{Saddles}$

Therefore, if some constraint were to forbid $\mathcal{I}_{3}$, a stable dGB
theory constructed around the $S_{3}$-invariant extrema would be possible. In
that case, the low-energy spectrum would arrange itself in a
$\boldsymbol{1}+\boldsymbol{2}$ of $S_{3}$: the triplet degeneracy would be
lifted; a doublet and a singlet representation could be then expected. This
would open new avenues for model-building and lead to a very different
phenomenology. We do not pursue this path here, as extra symmetries would have
to be advocated to sustain such a construction. In their absence, the non-
abelian spectra suggest at least three degenerate dGBs to be detected, in
contrast with two degenerate dGBs for abelian symmetric spectra.

###  5.2 Phenomenological signals from an $A_{5}$ quadruplet

Once again, one finds the absence of quadratic sensitivity of the dGB
potential to ultraviolet scales, and dGB masses whose size can be further
separated from the scale of the effective theory $\Lambda$, as compared with
the expectations for generic pGBs from non-linearly realized and explicitly
broken symmetries. The essential low-energy property of this ultraviolet
$A_{5}$ scenario is the explicit non-abelian $A_{4}$ symmetry of the low-
energy theory, realized by a triplet of dGBs, as long as the scalar vev
corresponds to the MaNa minimum with the largest little group:

  * •

Three degenerate dGBs are expected. This is in stark contrast with the
abelian- symmetric ($Z_{N}$) spectra found in previous sections, which
predicted in all cases a doublet of degenerate dGBs.

  * •

No events with a single ALP-like particle are expected (as long as the SM
fields are singlets of the UV discrete symmetry). The experimental tell-tale
signals would be then the simultaneous emission of two, three or four
degenerate dGBs in SM initiated processes. In number of dGBs emitted, the
event topologies are therefore alike to those for the $Z_{3}$ symmetric low-
energy spectrum of an $A_{4}$ UV invariant theory, but the relative weights of
the amplitudes for multi-dGB emission could allow one to differentiate the two
cases, see Eq. (3.41) versus Eq. (5.9).

In order to illustrate this last point, consider the interaction between SM
fields (singlets of the discrete symmetry) and
$\mathcal{I}_{3}^{(\boldsymbol{4},A_{5})}$ ,

$\mathcal{L}^{\text{int}}\propto\frac{1}{M^{m}}\,\mathcal{O}^{SM}\mathcal{I}_{3}^{(\boldsymbol{4},A_{5})}+\cdots\,$
(5.12)

It follows from Eq. (3.51) that the $A_{4}$-symmetric spectrum (which has
resulted from the quadruplet of a UV $A_{5}$ invariant theory) lead to the
experimental ratios:

$\displaystyle\frac{\sigma(\text{SM}\rightarrow
2\pi)}{\sigma(\text{SM}\rightarrow 3\pi)}$
$\displaystyle=\frac{15f^{2}}{4}\frac{\Pi_{2}}{\Pi_{3}}\,,\qquad$
$\displaystyle\frac{\sigma(\text{SM}\rightarrow
3\pi)}{\sigma(\text{SM}\rightarrow 4\pi)}$
$\displaystyle=\frac{6f^{2}}{(41)^{2}}\frac{\Pi_{3}}{\Pi_{4}}\,,$ (5.13)
$\displaystyle\frac{\sigma\text{SM}\rightarrow
2\pi)}{\sigma(\text{SM}\rightarrow 3\pi)}$
$\displaystyle=120\pi^{2}\frac{f^{2}}{E_{\text{CM}}^{2}}\,,\qquad$
$\displaystyle\frac{\sigma(\text{SM}\rightarrow
3\pi)}{\sigma(\text{SM}\rightarrow 4\pi)}$
$\displaystyle=\left(\frac{24\pi}{41}\right)^{2}\frac{f^{2}}{E_{\text{CM}}^{2}}\,,$
(5.14)

to be compared with the rations expected for a $Z_{3}$-symmetric spectrum
(from a triplet of a UV $A_{4}$ invariant theory), see Eq. (3.54), and with
those expected for a $Z_{5}$-symmetric spectrum (from a triplet of a UV
$A_{5}$ invariant theory) in Eq. (4.14).

Analogous remarks to those in Sect. 3.3.3 for the signals expected from
interaction terms containing $\partial_{\mu}U$ components apply in this case.

##  6 Conclusions

The symmetry-protected masses of discrete Nambu-Goldstone bosons offer
promising theoretical avenues to soften the ultraviolet sensitivity of BSM
theories. We have discussed how exact –albeit non-linearly realized– discrete
symmetries allow one to give masses to those scalars without explicitly
breaking the discrete symmetry.

Examples of non-linearly realized UV abelian discrete symmetries ($Z_{N}$)
have been previously developed in depth, showing that a much-lighter than
usual axion can result [24, 25, 26]. Here we focus instead on the much richer
case of UV non-abelian symmetries. We illustrated the theoretical analysis for
various simple non-abelian discrete groups with a real scalar field in an
irreducible representation.

The non-linearity condition itself, quadratic in the scalar fields, implies
the absence of quadratic sensitivity to high scales for the dGB masses,
because no other quadratic form is allowed by the symmetry. By the same token,
the UV discrete invariance has the potential to strongly separate the dGB
masses from the scale of the effective theory. A key ingredient to assess the
smallness of dGB masses is the dimensionality of the first scalar operator in
the dGB potential that is invariant under the UV discrete symmetry. The higher
its dimension, the stronger the mass suppression expected. We showed how this
depends on the group and the representation, illustrating this effect with
some simple cases. The first discrete scalar invariant of the dGB potential
may already appear at cubic order, or quartic, or even appear only at the non-
renormalizable level, thus guaranteeing extremely suppressed dGB masses.

The model-independent approach of EFT and invariant analysis has been used. In
all cases, we have focused on the MaNa minima of the potential, that is, those
points that are guaranteed to be extrema and are independent of the values of
the parameters in the potential. All such extrema have been identified,
together with the manifolds spanned by the invariants, for various irreducible
real representations of $A_{4}$, $A_{5}$ and $S_{4}$, as representative
examples.

For the sake of theoretical illustration, we have discussed as well the
connection with the customary analysis of pion masses in spontaneously broken
global continuous theories, which always require additional arbitrary
potentials that break explicitly the continuous symmetry. The findings in this
paper could be then simply viewed as the result of requiring the potential for
explicit breaking to remain invariant under a discrete subgroup of the
continuous symmetry. That is, no discrete group is added ad hoc, while exact
invariance under a subgroup of the global symmetry itself is required (e.g.
$A_{4}$ or $A_{5}$ for $SO(3)$). The point is that exact discrete symmetries
often generate approximate continuous symmetries and, when non-linearly
realized, the corresponding pGB masses are protected: not only do the
quadratically divergent contributions to their masses from putative higher
scales cancel exactly but also the generated potential may acquire an
exponential suppression, which is controlled by the dimensionality of the
first invariant operator that breaks the continuous symmetry but preserves the
discrete one. Nevertheless, our results did not rely on any choice of
continuous group embedding. In order to better illustrate the mechanism, we
have also explicitly computed the cancellation of the quadratic divergent
diagrams in two UV complete models of dGBs whose realization involved either
fermions or scalars.

We have also shown that the low-energy theory around the MaNa extrema remains
explicitly invariant under a discrete subgroup of the UV discrete symmetry,
which is identified. That is, the spectra of dGBs are shown to exhibit an
explicit –i.e. à la Wigner– discrete symmetry. The latter may be abelian (e.g.
$Z_{3}$, $Z_{5}$ in some examples studied), or non-abelian (e.g. $A_{4}$,
$S_{3}$). This results in tantalizing tell-tale experimental signals:

  * •

Doublets of degenerate (dGB) scalars are to be expected for abelian symmetric
low-energy theories. In contrast, three or more degenerate dGBs are possible
for low-energy non-abelian symmetric spectra.242424Two degenerate pions can
also stem from very small non-abelian symmetries at low energy, such as
$S_{3}$. The key is the dimensionality of the irreducible representations of
the UV symmetry.

  * •

No event with single dGB emission is expected, as long as the SM fields are
singlets under the discrete symmetries. The simultaneous emission of two,
three or more dGBs are to be expected in this case from SM initiated
processes.

  * •

The relative weights of multi-dGB emission provide hints of the exact discrete
UV symmetry: the analysis of the low-energy dGB spectrum and interactions
cannot identify the UV symmetry unequivocally, but it can delineate the
possibilities by consistency.

While the first two points above are not exclusive signals of a discrete
symmetry (as they could be expected for exotic particles charged under an
exact and explicit new symmetry), it is the ensemble of the three points that
would allow one to see the presence in Nature of a protective discrete
symmetry.

Note as well that the second point above is at variance with usual ALP
searches, for which single ALP emission is the first signal hunted (e.g. a
single missing energy track if the ALP is stable or too long-lived to decay
within the detector). Would the SM fields be charged under the UV discrete
symmetry, the dGB degeneracy would remain untouched, while different
experimental detection patterns can follow (to be developed elsewhere). In the
flourishing realm of ALP searches, it is a pertinent quest to hunt for the
sets of observables indicated, as a possible trademark of an unbroken but
hidden discrete symmetry, and thus of a BSM theory with enhanced UV
protection.

Theoretical applications of the results in this paper could range from the
search for UV stable theories of multicomponent dark matter, to the solution
or alleviation of other fine-tuning and hierarchy issues in the known physics
laws.

## Acknowledgments

We thank Enrique Alvarez, Daniel Alvarez-Gavela, Quentin Bonnefoy, Ferruccio
Feruglio, Anson Hook, Aneesh Manohar, Luca Merlo and Mathias Pierre for
illuminating discussions. R.H. is supported by the STFC under Grant No.
ST/T001011/1. P.Q. acknowledges support by the Deutsche Forschungsgemeinschaft
under Germany’s Excellence Strategy - EXC 2121 Quantum Universe - 390833306
and by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) -
491245950. The work of P.Q. is supported in part by the U.S. Department of
Energy Grant No. DE-SC0009919. The work of V.E. was supported by the Spanish
MICIU through the National Program FPI-Severo Ochoa (grant number
PRE2020-094281). V.E. and B.G. acknowledge partial financial support by the
Spanish Research Agency (Agencia Estatal de Investigación) through the grant
IFT Centro de Excelencia Severo Ochoa No CEX2020-001007-S and by the grant
PID2019-108892RB-I00 funded by MCIN/AEI/ 10.13039/501100011033. All authors
acknowledge support by the European Union’s Horizon 2020 research and
innovation programme under the Marie Sklodowska-Curie grant agreement No
860881-HIDDeN. B.G. and P.Q. thank respectively DESY and Madrid IFT for their
warm hospitality where part of this work was carried out.

##  Appendix A Brief summary of $S_{4}$

As advanced at the end of Sec. 3, the scenario with an ultraviolet $S_{4}$
symmetry is closely related to that of $A_{4}$, but it presents some
suggestive differences, discussed next for its two triplet representations,
$\boldsymbol{3}$ and $\boldsymbol{3^{\prime}}$.

###  A.1 $S_{4}$ in its $\boldsymbol{3}$ representation

The Molien function for the $\boldsymbol{3}$ of $S_{4}$, reads

$\begin{split}\mathcal{F}_{S_{4}}(\boldsymbol{1};\boldsymbol{3};\lambda)&=\frac{1}{(1-\lambda^{2})(1-\lambda^{3})(1-\lambda^{4})}\,,\\\
\end{split}$ (A.1)

which indicates the same primary invariants as for the $\boldsymbol{3}$ of
$A_{4}$, of dimension two, three and four:
$\mathcal{I}_{2},\mathcal{I}_{3},\mathcal{I}_{4}$ in Eqs.(3.1)-(3.3). The
Molien function for the $\boldsymbol{3^{\prime}}$ representation is the same
except that it has a factor of $(1+\lambda_{6})$ in the numerator; in other
words, the difference is that the secondary invariant $\mathcal{I}_{6}$
defined in Eq. (3.4) by a syzygy of the $A_{4}$ triplet, is forbidden for
$S_{4}$. As a consequence, differences between both theories arise only at the
non-renormalizable level through this six-dimensional operator. It follows in
turn that the MaNa extrema of the potential are the same as those of $A_{4}$,
although their little groups are extended because of the larger size of the
group. That is, the manifold of invariants is that in Fig. 5 albeit with the
symmetries at the MaNa extrema changed as follows:

$\displaystyle\begin{array}[]{|c||c|}\hline\cr\text{Point}&\text{Little
group}\\\ \hline\cr\hline\cr A&V_{4}=Z_{2}\times Z_{2}\\\ B&V_{4}=Z_{3}\times
Z_{2}\\\ C&S_{3}=Z_{3}\times Z_{2}\\\ \hline\cr\end{array}$ Table 5:
Symmetries of the MaNa extrema for an invariant potential for the
$\boldsymbol{3}$ of $S_{4}$, where $A$, $B$ and $C$ are the points defined for
$A_{4}$ in Table 2 with the corresponding manifold as depicted in Fig 5.

In terms of the pions as defined by Eq. (5.8), the absence of
$\mathcal{I}_{6}$ translates into the exclusion of one of the
$Z_{3}$-invariant combinations of order 3 introduced by Eqs. (3.43) and
(3.44). Even though both expressions can be reshuffled as in Eq. (3.49) due to
pion degeneracy, this results in the presence of a single combination of both
operators in the Lagrangian regardless of the basis chosen. From another point
of view, this is a consequence of the extended $S_{3}$ invariance of the
formerly $Z_{3}$-symmetric MaNa extrema $B$ and $C$, which is more stringent
in forbidding combinations of the pions. Interestingly then, $S_{3}$ is a non-
abelian group —although the smallest existing one— and it will remain
explicitly realized in the spectrum in this case. Subtle differences at sub-
leading orders in the production rates or scattering cross sections predicted
by both theories might follow, but at leading order the physical predictions
stemming from the $\boldsymbol{3}$ of $A_{4}$ and $S_{4}$ are identical.

Figure 12: Geometrical distribution in field space of the $S_{3}$-symmetrical
MaNa extrema (B) with a $\boldsymbol{3^{\prime}}$ of $S_{4}$ .

###  A.2 The $\boldsymbol{3^{\prime}}$ of $S_{4}$

The $\boldsymbol{3^{\prime}}$ is of considerable more interest, with more
unique features that stem from its structural differences in relation to the
$\boldsymbol{3}$ of $A_{4}$ [52]. Looking at the Molien function,

$\begin{split}\mathcal{F}_{S_{4}}(\boldsymbol{1};\boldsymbol{3^{\prime}};\lambda)&=\frac{1+\lambda^{9}}{(1-\lambda^{2})(1-\lambda^{4})(1-\lambda^{6})},\\\
\end{split}$ (A.2)

one immediately spots the similarities and differences with the former cases.
First of all, the invariant of order 3 is forbidden, so non-trivial
contributions to the potential will appear only at the marginal level.
Explicitly, the invariants are

$\begin{split}\mathcal{I}_{2}^{(\boldsymbol{3^{\prime}},\,S_{4})}(\phi)&=\phi_{1}^{2}+\phi_{2}^{2}+\phi_{3}^{2},\\\
\mathcal{I}_{4}^{(\boldsymbol{3^{\prime}},\,S_{4})}(\phi)&=\phi_{1}^{4}+\phi_{2}^{4}+\phi_{3}^{4},\\\
\mathcal{I}_{6}^{(\boldsymbol{3^{\prime}},\,S_{4})}(\phi)&=(\phi_{1}\phi_{2}\phi_{3})^{2},\end{split}$
(A.3)

meaning that the third order invariant has been substituted by its square;
instead of $\mathcal{I}_{3}$ we have
$\mathcal{I}_{6}^{(\boldsymbol{3^{\prime}},\,S_{4})}=\mathcal{I}_{3}^{2}$. The
non-trivial secondary invariant is of order $9$, and thus extremely
suppressed. As it happened in $A_{4}$, it can be found to be proportional to
the determinant of the Jacobian of the primary invariants. We define

$\displaystyle\operatorname{det}\mathcal{J}_{(\boldsymbol{3^{\prime}},\,S_{4})}=-16\,\mathcal{I}_{9}^{(\boldsymbol{3^{\prime}},\,S_{4})}=-16\,\phi_{1}\phi_{2}\phi_{3}\left(\phi_{1}^{2}-\phi_{2}^{2}\right)\left(\phi_{2}^{2}-\phi_{3}^{2}\right)\left(\phi_{1}^{2}-\phi_{3}^{2}\right).$
(A.4)

The set of MaNa extrema is enlarged in this case; one can see from Fig. 13
that the structure of the invariant manifold has indeed changed. Now there are

  * •

6 MaNa extrema $A$ with $V_{4}=Z_{2}\times Z_{2}$ as their little group, which
are stable.

  * •

8 MaNa extrema $B$ with $S_{3}=Z_{2}\times Z_{3}$ as their little group, which
are stable.

  * •

12 MaNa extrema $C$ with $Z_{2}$ as their little group, which are saddles.

Table LABEL:eq:S4:3p:invariants:pions_S3 contains all the relevant information
about all these points, including some representatives for each case.

$\displaystyle\begin{array}[]{|c||cc|ccc|c|c|}\lx@intercol\hfil\text{{ MaNa
extrema for the} $\boldsymbol{3^{\prime}}$ {of}
$\boldsymbol{S_{4}}$}\hfil\lx@intercol\\\
\hline\cr\hbox{\multirowsetup\text{Point}}&\hbox{\multirowsetup$\mathcal{I}_{4}^{(\boldsymbol{3^{\prime}},\,S_{4})}$}&\hbox{\multirowsetup$\mathcal{I}_{6}^{(\boldsymbol{3^{\prime}},\,S_{4})}$}&\phi_{1}&\phi_{2}&\phi_{3}&\hbox{\multirowsetup\text{Little
group}}&\hbox{\multirowsetup\text{Nature}}\\\
&&&\lx@intercol\hfil\textit{(Representatives)}\hfil\lx<EMAIL_ADDRESS>\hline\cr\hline\cr\rule{0.0pt}{9.90276pt}\hbox{\multirowsetup
A}&\hbox{\multirowsetup$\frac{1}{3}$}&\hbox{\multirowsetup$\frac{1}{27}$}&\frac{1}{\sqrt{3}}&\frac{1}{\sqrt{3}}&\frac{1}{\sqrt{3}}&\hbox{\multirowsetup\text{$S_{3}$}}&\hbox{\multirowsetup\text{Minima}}\\\
&&&-\frac{1}{\sqrt{3}}&-\frac{1}{\sqrt{3}}&-\frac{1}{\sqrt{3}}&&\\\\[3.44444pt]
\hline\cr\rule{0.0pt}{9.90276pt}\hbox{\multirowsetup B}&\hbox{\multirowsetup
1}&\hbox{\multirowsetup 0}&0&0&\pm
1&\hbox{\multirowsetup\text{$V_{4}$}}&\hbox{\multirowsetup\text{Minima}}\\\
&&&0&\pm 1&0&&\\\ &&&\pm 1&0&0&&\\\
\hline\cr\rule{0.0pt}{9.90276pt}\hbox{\multirowsetup
C}&\hbox{\multirowsetup$\frac{1}{2}$}&\hbox{\multirowsetup$0$}&\pm\frac{1}{\sqrt{2}}&0&\pm\frac{1}{\sqrt{2}}&\hbox{\multirowsetup\text{$Z_{2}$}}&\hbox{\multirowsetup\text{Saddles}}\\\\[3.44444pt]
\hline\cr\end{array}$ Table 6:  MaNa critical points of the invariants of the
$\boldsymbol{3^{\prime}}$ of $S_{4}$. The manifold of primary invariants
relevant for the dGB potential is depicted in Fig. 13. Figure 13: Manifold
defined by $\mathcal{I}^{(\boldsymbol{3^{\prime}},\,S_{4})}_{4}(\Phi)$ and
$\mathcal{I}^{(\boldsymbol{3^{\prime}},\,S_{4})}_{6}(\Phi)$ invariants of
$S_{4}$ when $\Phi$ is in the $\boldsymbol{3^{\prime}}$ of $S_{4}$ and
fulfills $\phi_{1}^{2}+\phi_{2}^{2}+\phi_{3}^{2}=f^{2}$.

#### Expansion in terms of pions

There are two sets of stable MaNa extrema, denoted $A$ and $B$ in the manifold
in Fig. 13. Therefore, two different realizations in terms of the physical
dGBs are possible: either $A$ or $B$ is a minimum, depending on the sing of
the $\hat{c}_{i}$ coefficients. Assuming that the full potential can be
approximated by the first invariant $V_{\rm
dGB}\sim-\mathcal{I}_{4}^{(\boldsymbol{3},S_{3})}$, the $V_{4}$-symmetric
minima in $B$ are global minima, consistent with the Michel-Radicati result
[28].

##### Expanding around $A$.

On defining the dGBs around the $V_{4}$-invariant MaNa extremum
$\Phi_{B}=(0,0,1)^{\top}$, the contributions to the potential will read

$\displaystyle\mathcal{I}_{4}^{(\boldsymbol{3^{\prime}},\,S_{4})}$
$\displaystyle=f^{4}-2f^{2}(\pi_{1}^{2}+\pi_{2}^{2})+\frac{8}{3}(\pi_{1}^{2}+\pi_{2}^{2})^{2}-2\pi_{1}^{2}\pi_{2}^{2}+\mathcal{O}\left(\pi_{1}^{5}\right)\,,$
(A.5) $\displaystyle\mathcal{I}_{6}^{(\boldsymbol{3^{\prime}},\,S_{4})}$
$\displaystyle=\pi_{1}^{2}\pi_{2}^{2}+\mathcal{O}\left(\pi_{i}^{5}\right)\,,$
(A.6)

where the pions are arranged followed the pattern dictated by $V_{4}$
invariance. Using the Molien function formalism one can easily see that the
two primary invariants of the doublet of $V_{4}$ are indeed
$\mathcal{I}_{2}^{(\boldsymbol{2},V_{4})}=\pi_{1}^{2}+\pi_{2}^{2}$ and
$\mathcal{I}_{4}^{(\boldsymbol{2},V_{4})}=\pi_{1}^{2}\pi_{2}^{2}$.
Consequently, this case forbids the production of an odd number of pions to
all orders, distinguishing its predicted signal from all others that are
discussed in this paper.

##### Expanding around $B$ :

The expansion around a $S_{3}$-symmetric MaNa extremum yields a quite
different outcome:

$\displaystyle\mathcal{I}_{4}^{(\boldsymbol{3^{\prime}},\,S_{4})}$
$\displaystyle=\frac{f^{4}}{3}+\frac{4}{3}f^{2}\left(\pi 1^{2}+\pi
2^{2}\right)-\frac{2}{3}f\left(\pi_{1}^{3}-3\pi_{1}\pi_{2}^{2}\right)-\frac{29}{18}\left(\pi_{1}^{2}+\pi_{2}^{2}\right){}^{2}+\mathcal{O}\left(\pi_{i}^{5}\right)\,,$
(A.7) $\displaystyle\mathcal{I}_{6}^{(\boldsymbol{3^{\prime}},\,S_{4})}$
$\displaystyle=\frac{f^{6}}{27}-\frac{2}{9}f^{4}\left(\pi_{1}^{2}+\pi_{2}^{2}\right)-\frac{f^{3}}{27}\left(\pi_{1}^{3}-3\pi_{1}\pi_{2}^{2}\right)+\frac{53}{108}f^{2}\left(\pi_{1}^{2}+\pi_{2}^{2}\right){}^{2}+\mathcal{O}\left(\pi_{i}^{5}\right)\,,$
(A.8)

where the primary invariants of the doublet of $S_{3}$ are precisely
$\mathcal{I}_{2}^{(\boldsymbol{2},S_{3})}=\pi_{1}^{2}+\pi_{2}^{2}$ and
$\mathcal{I}_{3}^{(\boldsymbol{2},S_{3})}=\pi_{1}^{3}-3\pi_{1}\pi_{2}^{2}$.
Contrarily to what happens in the case of $Z_{3}$, there is no further allowed
combination of order 3, so subtle variances are expected again with respect to
the physical predictions coming from the $Z_{3}$ invariant points of $A_{4}$.
The pions are in fact arranged in the same structures as for the
$S_{3}$-symmetric minima from the $\boldsymbol{3}$ of $S_{3}$, but this case
is anyhow different in that the leading contribution comes from the marginal
order operator.

###  A.3 Phenomenological signals from a triplet of $S_{4}$

Since the lowest order invariant relevant for the dGB potential is now
$\mathcal{I}_{4}$, the leading interaction between SM fields and pure dGB
scalar invariants —under the same assumptions as used when considering the
$\boldsymbol{3}$ of $A_{4}$, see Eq.(3.50)— reads

$\mathcal{L}=\frac{1}{M^{m}}\mathcal{O}_{SM}\mathcal{I}_{4}\,,$ (A.10)

leading to a different prediction for the production ratios. The results
around the $S_{3}$-invariant minima have already been presented in Eq. (3.56).
If Nature settles instead around the $V_{4}$ minima, the expected production
rates are unique in that those for an odd number of pions are forbidden. The
first expected production ratio is then

$\displaystyle\frac{\sigma(\text{SM}\rightarrow
2\pi)}{\sigma(\text{SM}\rightarrow 4\pi)}$
$\displaystyle=\frac{27(8\pi^{2})}{19\cdot
2^{9}}\frac{f^{4}}{E_{\text{CM}}^{4}}.$ (A.11)

## References