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# General Discussions on the SU(2) Vector Boson Dark Matter Model with a
Single Higgs Multiplet — Lagrangian, Discrete Subgroups, and Scalar
Classifications

Chun-Xue Yuana Zhao Zhanga Chengfeng Caib Yi-Lei Tanga Hong-Hao Zhanga aSchool
of Physics, Sun Yat-Sen University, Guangzhou 510275, China  bSchool of
Science, Sun Yat-Sen University, Shenzhen 518107, China

###### Abstract

The vector boson dark matter particles which stem from some broken gauge
symmetries usually requires some unbroken symmetries to keep themselves
stable. In the previous literature, some simplest cases have been discussed,
in which the unbroken symmetry is provided by a remnant subgroup of the gauge
group. It would be interesting to ask whether all the possible remnant
subgroups as well as all the possible coupling forms can be enumerated.
Classifying all the Higgs components into different mass degenerate
representations to simplify the diagonalization processes is also necessary.
Rather than the ambitious target of providing a general solution to all kinds
of gauge groups configured with all forms of the Higgs multiplets, in this
paper, we concentrate on the case of $\text{SU(2)}_{\text{D}}$ gauge group
together with a single Higgs multiplet. We enumerate all possible discrete
subgroups that can survive up to $n=21$, where $n$ is the dimension of the
Higgs multiplet. We also provide the general algorithms to enumerate all
possible renormalizable operators, to write down the general forms of the
vacuum expectation value (VEV) configurations, and to give the detailed
results of all the mass degenerate irreducible representations embedded in the
Higgs multiplet.

###### Contents

  1. I Introduction
  2. II The Langrangian
  3. III Enumeration of the General Coupling Combinations
  4. IV Vacuum Patterns with a Remnant Discrete Symmetry
  5. V Scalar Classifications
  6. VI Several Examples
    1. VI.1 4-d scalar of SU(2)
    2. VI.2 9-d scalar of SU(2)
    3. VI.3 13-d scalar of SU(2)
  7. VII Subspaces for $(2n+1)$-d representation of SU(2)
  8. VIII Summary
  9. A Generators of the SU(2)
  10. B Discrete symmetry groups of SO(3)
  11. C Eigenvectors for 4, 9, 13-d representation
  12. D Character tables and generators of irreducible group
    1. D.1 $S_{4}$
    2. D.2 $A_{4}$
    3. D.3 $A_{5}$
    4. D.4 $K_{4}(Z_{2}\oplus Z_{2})$
    5. D.5 The characters of $n$-d representation of SU(2) related to discrete symmetry group

##  I Introduction

The dark matter (DM), which occupies about 26% of the energy density budget of
our current universe, still lies mysterious. In the literature, spin-0 and
spin-1/2 DM candidates have been sufficiently discussed, with the vector
bosons sometimes regarded as pure mediators to carry the interactions.
Although the vector boson itself does take the possibility to be the DM
candidate, most of the corresponding discussions in the literature seem to
focus on some specific models while losing some generalities.

The main procedure to discuss a particular vector boson DM model usually
commences with a Lagrangian accommodating some vector field multiplets as well
as the corresponding Higgs configurations. To keep the DM stable, some
discrete or continuous symmetry might be imposed by hand at the beginning
Lebedev:2011iq ; Abe:2012hb ; Farzan:2012hh ; Baek:2012se ; Domingo:2013tna ;
Yu:2014pra ; Chen:2014cbt ; Gross:2015cwa ; Duch:2015jta ; DiFranzo:2015nli ;
Barman:2018esi ; Azevedo:2018oxv ; Poulin:2018kap ; Saez:2018off ;
Barman:2019lvm ; Mohamadnejad:2019vzg ; Glaus:2019itb ; Arcadi:2020jqf ;
Abe:2020mph ; Delaunay:2020vdb ; Salehian:2020asa ; YaserAyazi:2022tbn ;
Frandsen:2022klh ; Das:2024tfe , or be satisfied accidentally Hambye:2008bq ;
Davoudiasl:2013jma ; Boehm:2014bia ; Gross:2015cwa ; DiChiara:2015bua ;
Karam:2015jta ; Khoze:2016zfi ; Karam:2016rsz ; Ko:2016fcd ; Arcadi:2016kmk ;
Poulin:2018kap ; Belyaev:2018xpf ; Ramos:2021omo ; Hu:2021pln ;
Baouche:2021wwa ; Chowdhury:2021tnm ; Cai:2021wmu ; Borah:2021ftr ;
Belyaev:2022shr ; Elahi:2022hgj ; Zhang:2022wek ; Frandsen:2022klh ; Benitez-
Irarrazabal:2024ich ; Acevedo:2024ava , or can be induced naturally after the
gauge symmetry spontaneously breaks down Chiang:2013kqa ; Baek:2013dwa ;
Khoze:2014woa ; Gross:2015cwa ; Karam:2016rsz ; Arcadi:2016kmk ;
Poulin:2018kap ; Choi:2019zeb ; Ghosh:2020ipy ; Nomura:2020zlm ; Zhang:2022wek
. After some Higgs components acquire the vacuum expectation value (VEV), one
then investigate in the parameter space so that vector bosons acquire masses
and the stable one(s) become the DM candidate. Phenomenologies are then
evaluated for the comparison with various experimental data, demonstrating the
viability of such a model.

In the literature, it has been so long realized that the gravity breaks the
global symmetry while keeps the gauge symmetry intact Giddings:1987cg ;
Abbott:1989jw ; Gilbert:1989nq ; Coleman:1989zu ; Kallosh:1995hi ;
Witten:2017hdv . It then becomes more natural that the dark matter stability
is kept by some subgroups remained from some broken gauge groups, which will
not be further demolished by the gravity effects. For a gauge symmetry beyond
the simplest U(1) case, a general discussion of the remained group is
difficult, and the diagonalizing processes of the mass matrix of all scalar
components is another trouble, since the analytical calculations become
extremely formidable.

Aiming at solving these problems at least in some simple cases, we study the
vector boson DM model stemming from a dark $\text{SU(2)}_{\text{D}}$ gauge
group accompanying with a single Higgs multiplet as an example. In this paper,
alternative to computing the phenomenologies based upon a particular
Lagrangian configuration, we try to achieve the following goals:

  * •

Give a general algorithm to enumerate the general renormalizable Lagrangian.

  * •

Enumerate all the possible subgroups as well as the corresponding VEV
configurations which is capable to keep the DM stable.

  * •

Classify all the scalar components into the irreducible representations of the
subgroup, and show how they are formulated by the combinations of the original
Higgs multiplet components.

Actually, Ref. Etesi:1996urw  has partly resolved the first and the second
goals in the odd-dimensional Higgs multiplet case, while Ref. Hamada:2015bra
has partly listed the potential for an SM $\text{SU}(2)_{\text{L}}$ scalar
multiplet with dimension from 4 to 7 with some particular settings of the
hypercharge. Ref. King:2018fke  also achieved parts of our results, but with a
quite different method for several specific cases without enumerating the
couplings. Alternatively we work on a more familiar format of the SU(2)
representations and follow a more intuitive process. In our paper, we adopt
the similar algorithms with Ref. Etesi:1996urw  to find the subgroups and the
corresponding VEV configurations. However, we develop a different method to
accomplish the complete set of the renormalizable couplings, regardless of the
intricate algorithms based upon the graph theory in Ref. Etesi:1996urw . In
our paper, our algorithms can be generalized so that all three goals can be
achieved for an arbitrary SU(2)${}_{\text{D}}$ Higgs multiplet in principle,
however we only show detailed processes for several typical examples, and
enumerate the results from $j=2$ up to $j=10$ multiplets.

##  II The Langrangian

As we have mentioned, we introduce a new $\text{SU(2)}_{\text{D}}$ gauge group
associated with the gauge field $A_{\text{D}}^{a\mu}$, where $a=1,2,3$ is the
$\text{SU(2)}_{\text{D}}$ index. One $n$-dimensional Higgs multiplet
$\Phi_{(n)}$ charged under this gauge group is also provided. Here $n=2j+1$ is
the dimension of the multiplet, where $j$ is the “angular quantum number” of
the irreducible representation that the $\Phi_{(n)}$ corresponds to.

For each $n$ or $j$, the most familiar basis for the components of
$\Phi_{(n)}$ is the “$J_{\text{D}}^{3}$ representation”, in which the
$\Phi_{(n)}$ components are arranged in accordance with the
“$|j,m\rangle_{\text{D}}$”, where $m$ indicates the $J_{\text{D}}^{3}$
eigenvalues. Here we just borrow the symbols from quantum mechanics, and the
“$|j,m\rangle$” means the eigenstates under the $\text{SU(2)}_{\text{D}}$
group, rather than the rotational group of the real space, therefore
additional “D” subscripts are imposed. $\hat{J}_{\text{D}}^{1,2,3}$ are the
corresponding generators of the $\text{SU(2)}_{\text{D}}$ group. For a
particular $j$, we write down the components of $\Phi_{(n)}$ as

$\displaystyle\Phi_{(n)}=\xi_{(n)}\left(\Delta_{j},\Delta_{j-1},\dots,\Delta_{-j+1},\Delta_{-j}\right)^{\mathrm{T}},$
(1)

where the “$\Delta_{m}$” here indicates the scalar component corresponding to
the “$|j,m\rangle_{\text{D}}$” eigenstate. The normalization factor
$\xi_{(n)}=1~{}(1/\sqrt{2})$ when $n$ is even (odd).

For an even $n$, $j=\frac{n-1}{2}$ is a half integer, and all $\Delta_{m}$’s
are complex. On the other hand, when $n$ is odd, and the $j=\frac{n-1}{2}$
therefore becomes an integer, the $n$-dimensional irreducible representation
can be real. In this paper, we would like a minimal configuration, so
$\Phi_{(n)}$ is appointed be the real $\text{SU(2)}_{\text{D}}$ multiplet when
$n$ is odd. It requires $\Phi_{(n)}$ to satisfy following constraint

$\displaystyle\Phi_{(n)}=T_{(n)}\Phi_{(n)}^{*},$ (2)

where

$\displaystyle T_{(n)}=[T_{(n)}]_{m_{1},m_{2}}$ (3)

is an $n\times n$ matrix, with all $(T_{(n)})_{m,-m}=1$ for any $m$, while all
other elements vanish.

Then we are ready to write down the general form of the renormalizable
Lagrangian. The dark sector is given by

$\displaystyle\mathcal{L}\supset-\frac{1}{4}F_{\text{D}\mu\nu}^{a}F_{\text{D}}^{a\mu\nu}+(D^{\mu}\Phi_{(n)})^{\dagger}D_{\mu}\Phi_{(n)}-V(\Phi_{(n)}^{\dagger},\Phi_{(n)},H^{\dagger},H),$
(4)

where
$F_{\text{D}\mu\nu}^{a}=\partial_{\mu}A_{\text{D}\nu}^{a}-\partial_{\nu}A_{\text{D}\mu}^{a}+g_{\text{D}}\epsilon^{abc}A_{\text{D}\mu}^{b}A_{\text{D}\nu}^{c}$,
and
$D_{\mu}\Phi_{(n)}=\partial_{\mu}\Phi_{(n)}-ig_{\text{D}}\tau_{(n)}^{a}A_{\text{D}\mu}^{a}\Phi_{(n)}$,
with $g_{D}$ the $\text{SU(2)}_{\text{D}}$ gauge coupling constant,
$\epsilon^{abc}$ the Levi-Civita symbol, and $\tau_{(n)}^{a}$ the generator
matrix of the $\text{SU(2)}_{\text{D}}$ group under the $n$-dimensional
irreducible representation.

$V(\Phi_{(n)}^{\dagger},\Phi_{(n)},H^{\dagger},H)$ is the effective potential,
and its general form can be formulated as

$\displaystyle V(\Phi_{(n)}^{\dagger},\Phi_{(n)},H^{\dagger},H)$ (5)
$\displaystyle=$
$\displaystyle[(\mu_{pq}^{2(2,0)}+\lambda_{Hpq}^{(2,0)}H^{\dagger}H)\Delta_{p}\Delta_{q}+(\mu_{pq}^{2(1,1)}+\lambda_{Hpq}^{(1,1)}H^{\dagger}H)\Delta_{p}\Delta_{q}^{*}+\text{h.c.}]$
$\displaystyle+$
$\displaystyle(\kappa^{(3,0)}_{pqr}\Delta_{p}\Delta_{q}\Delta_{r}+\kappa^{(2,1)}_{qpr}\Delta_{p}\Delta_{q}\Delta_{r}^{*}+\text{h.c.})$
$\displaystyle+$
$\displaystyle(\lambda^{(4,0)}_{pqrs}\Delta_{p}\Delta_{q}\Delta_{r}\Delta_{s}+\lambda^{(3,1)}_{pqrs}\Delta_{p}\Delta_{q}\Delta_{r}\Delta_{s}^{*}+\lambda^{(2,2)}_{pqrs}\Delta_{p}\Delta_{q}\Delta_{r}^{*}\Delta_{s}^{*}+\text{h.c.}),$

where $H$ is the standard model (SM) Higgs doublet, which is defined as

$\displaystyle H=\left(\begin{array}[]{c}G^{+}\\\
\frac{v_{\text{EW}}+\tilde{h}+iG^{0}}{\sqrt{2}}\end{array}\right),$ (8)

and $v_{\text{EW}}\approx 246$ GeV is the electroweak VEV, $\tilde{h}$ is the
SM-like Higgs field, and $G^{\pm,0}$ are the corresponding Goldstone bosons.
$\mu^{2(2,0)}_{pq}$, $\mu^{2(1,1)}_{pq}$, $\lambda^{(2,0)}_{Hpq}$,
$\lambda^{(1,1)}_{Hpq}$, $\kappa^{(3,0)}_{pqr}$, $\kappa^{(2,1)}_{pqr}$,
$\lambda^{(4,0)}_{pqrs}$, $\lambda^{(3,1)}_{pqrs}$, $\lambda^{(2,2)}_{pqrs}$
are the coupling constants with $p$, $q$, $r$, $s$ being integers running from
$-j$ to $j$. The two superscripts within the brackets indicate the number of
the original $\Delta$’s and the conjugated $\Delta^{*}$’s respectively
combined to follow the couplings. For example, $\lambda^{(3,1)}_{pqrs}$ is
joined with the $\Delta_{p}\Delta_{q}\Delta_{r}\Delta_{s}^{*}$ term, which is
combined by three original $\Delta$’s and one conjugated $\Delta^{*}$ to
formulate the $(3,1)$ in the superscript.

Gauge invariance requires that each term in Eq. (5) should be a
$\text{SU(2)}_{\text{D}}$ singlet. Then our purpose is to find out the general
possible $\mu^{2}$, $\lambda_{H}$, $\kappa$, $\lambda$ assignments, which can
be regarded as the combinations of the Clebsch–Gordan coefficients. Before
closing this section, we should point out that, for even or odd $n$, some
terms in above general potential obviously vanish or have simpler formations.
Firstly, when $n$ is an even number, $\mu^{2(2,0)}_{pq}$ and
$\lambda^{(2,0)}_{Hpq}$ finally disappears with our field configurations,
since there is only one way to contract a two-order tensor
$\Delta_{p}\Delta_{q}$ into a $\text{SU(2)}_{\text{D}}$ singlet —
$\mu^{2(2,0)}_{pq}\propto\lambda^{(2,0)}_{pq}\propto T_{(n)}$. The definition
of the $T_{(n)}$ for odd $n$ has been given around Eq. (3), however here we
have to define $T_{(n)}$ in the case of the even $n$,

$\displaystyle T_{(n)}=[T_{(n)}]_{m_{1},m_{2}},$ (9)

where $[T_{(n)}]_{m,-m}=-[T_{(n)}]_{-m,m}=1$ for all $m>0$, and all other
elements are set zero. Notice that $T[T_{(n)}]_{m_{1},m_{2}}$ is
antisymmetric, therefore contracting with the symmetric $\Delta_{p}\Delta_{q}$
terms kills both the $\mu^{2(2,0)}_{pq}$ and $\lambda^{(2,0)}_{pq}$ terms. For
the $\Delta_{p}\Delta_{q}\Delta_{r}$, $\Delta_{p}\Delta_{q}\Delta_{r}^{*}$
combinations, the direct product of three half integer spins cannot constitute
a singlet, implying that all the trilinear terms should also disappear.

On the other hand, when $n$ is an odd number, all the conjugated
$\Delta_{p}^{*}$ terms can be transformed into the original $\Delta_{p}$ by
utilizing the real field condition Eq. (2). Therefore, we can choose to
preserve the $\mu^{2(1,1)}_{pq}$, $\lambda^{(1,1)}_{Hpq}$,
$\kappa_{pqr}^{(3,0)}$, $\lambda^{(4,0)}_{pqrs}$ since all other contributions
can be attributed to them. In this case, it is convenient to neglect the
bracketed superscripts, and to write them down in the form $\mu^{2}_{pq}$,
$\lambda_{Hpq}$, $\kappa_{pqr}$, $\lambda_{pqrs}$.

It is easy to learn from the Littlewood Richardson Rule Littlewood:1934 , that
the only way to contract the bilinear term $\Delta_{p}\Delta_{q}^{*}$ into a
$\text{SU(2)}_{\text{D}}$ singlet is given by

$\displaystyle\Phi_{(n)}^{\dagger}\Phi_{(n)}=\xi_{(n)}^{2}\Delta_{m}\Delta_{m}^{*}.$
(10)

Therefore, Eq. (5) can be simplified to be

$\displaystyle V(\Phi_{(n)}^{\dagger},\Phi_{(n)},H^{\dagger},H)$ (11)
$\displaystyle=$
$\displaystyle[(\mu^{2}+\lambda_{H}H^{\dagger}H)\Delta_{p}\Delta_{p}^{*}+\text{h.c.}]+(\kappa^{(3,0)}_{pqr}\Delta_{p}\Delta_{q}\Delta_{r}+\text{h.c.})$
$\displaystyle+$
$\displaystyle(\lambda^{(4,0)}_{pqrs}\Delta_{p}\Delta_{q}\Delta_{r}\Delta_{s}+\lambda^{(3,1)}_{pqrs}\Delta_{p}\Delta_{q}\Delta_{r}\Delta_{s}^{*}+\lambda^{(2,2)}_{pqrs}\Delta_{p}\Delta_{q}\Delta_{r}^{*}\Delta_{s}^{*}+\text{h.c.}),$

where $\mu^{2}$ and $\lambda_{H}$ are the common factors extracted from the
$\mu^{2(1,1)}_{pq}\Delta_{p}\Delta_{q}^{*}$ and the
$\lambda^{(1,1)}_{Hpq}\Delta_{p}\Delta_{q}^{*}$ contractions.

##  III Enumeration of the General Coupling Combinations

In this section, we aim at enumerating all the possible combinations of the
$\kappa_{pqr}^{(3,0)}$, $\kappa_{pqr}^{(2,1)}$, $\lambda^{(4,0)}_{pqrs}$,
$\lambda^{(3,1)}_{pqrs}$ and $\lambda^{(2,2)}_{pqrs}$ from $j=0$ until $j=10$.
The Littlewood Richardson Rule can in principle help us to find all possible
contractions upon all the $\Delta_{p}\Delta_{q}\Delta_{r}$,
$\Delta_{p}\Delta_{q}\Delta_{r}\Delta_{s}$,
$\Delta_{p}\Delta_{q}\Delta_{r}\Delta_{s}^{*}$, and
$\Delta_{p}\Delta_{q}\Delta_{r}^{*}\Delta_{s}^{*}$ combinations to formulate
the $\text{SU(2)}_{\text{D}}$ singlets, since such a process is fundamentally
to find the trivial representation subspace for the direct product space of
the $\Delta_{s}^{*}$’s, just similar to the bilinear case. However,
manipulating the multiple miscellaneous Young diagrams becomes a cumbersome
task, and thus we take an alternative approach. In fact, the character table
method can at least tell us the multiplicity, or the dimension of the trivial
representation subspace. With the guide of this multiplicity as the bound of
the number of independent combination formulas, we then try to enumerate all
possible contractions by inserting various operators between
$\Phi_{(n)}^{\dagger}$ and $\Phi_{(n)}$ until the number of independent
contractions touches the bound.

Let us take the
$\lambda_{pqrs}^{(2,2)}\Delta_{p}\Delta_{q}\Delta_{r}^{*}\Delta_{s}^{*}$ term
as an example for illustration. Here
$\Delta_{p}\Delta_{q}\Delta_{r}^{*}\Delta_{s}^{*}$ is a 4-order tensor. When
an element $g\in\text{SU(2)}_{\text{D}}$ acts upon it, the tensor transforms
as

$\displaystyle\Delta_{p}\Delta_{q}\Delta_{r}^{*}\Delta_{s}^{*}\rightarrow\Delta^{\prime}_{p}\Delta^{\prime}_{q}\Delta^{\prime*}_{r}\Delta^{\prime*}_{s}=V_{(n)}(g)_{p\tilde{p}}V_{(n)}(g)_{q\tilde{q}}V_{(n)}(g)^{\dagger}_{\tilde{r}r}V_{(n)}(g)^{\dagger}_{\tilde{s}s}\Delta_{\tilde{p}}\Delta_{\tilde{q}}\Delta_{\tilde{r}}^{*}\Delta_{\tilde{s}}^{*},$
(12)

where $V_{(n)}(g)$ is the $n\times n$ representation matrix of the $g$ under
the $n$-dimensional irreducible representation of the
$\text{SU(2)}_{\text{D}}$ group. It is easy to realize that
$\Delta_{p}\Delta_{q}\Delta_{r}^{*}\Delta_{s}^{*}$ is symmetric by swapping
the $\\{p,q\\}$ and $\\{r,s\\}$ indices separately, so we can equivalently
study the decomposition of a tensor $T_{pqrs}$ which transforms under the
action of a $g\in\text{SU(2)}_{\text{D}}$ in the following way

$\displaystyle T_{pqrs}\rightarrow
T_{pqrs}=V_{(n)}(g)_{p\tilde{p}}V_{(n)}(g)_{q\tilde{q}}V_{(n)}(g)^{\dagger}_{\tilde{r}r}V_{(n)}(g)^{\dagger}_{\tilde{s}s}T_{\tilde{p}\tilde{q}\tilde{r}\tilde{s}},$
(13)

while satisfying the symmetric condition

$\displaystyle T_{pqrs}=T_{qprs}=T_{pqsr}=T_{qpsr}.$ (14)

In order to find out the dimension of the trivial representation subspace, we
need to compute the character table of the tensor space $T_{pqrs}$ with a
group of the normalized orthogonal basis
$B_{(n)}^{(2,2)}(p^{\prime},q^{\prime},r^{\prime},s^{\prime})_{pqrs}\in\\{T_{pqrs}\\}$
defined as

$\displaystyle
B_{(n)}^{(2,2)}(p^{\prime},q^{\prime},r^{\prime},s^{\prime})_{pqrs}=\frac{\delta_{p^{\prime}p}\delta_{q^{\prime}q}\delta_{r^{\prime}r}\delta_{s^{\prime}s}+\delta_{p^{\prime}q}\delta_{q^{\prime}p}\delta_{r^{\prime}r}\delta_{s^{\prime}s}+\delta_{p^{\prime}p}\delta_{q^{\prime}q}\delta_{r^{\prime}s}\delta_{s^{\prime}r}+\delta_{p^{\prime}q}\delta_{q^{\prime}p}\delta_{r^{\prime}s}\delta_{s^{\prime}r}}{\sqrt{2\times
2}},$ (15)

where the $\frac{1}{\sqrt{2\times 2}}$ factor is a normalized factor. The
“trace” of the transformation under the action of
$g\in\text{SU(2)}_{\text{D}}$ can then be computed by

$\displaystyle\sum_{p^{\prime}\leq q^{\prime},r^{\prime}\leq
s^{\prime}}B_{(n)}^{(2,2)}(p^{\prime},q^{\prime},r^{\prime},s^{\prime})_{pqrs}^{*}V_{(n)}(g)_{p\tilde{p}}V_{(n)}(g)_{q\tilde{q}}V_{(n)}(g)^{\dagger}_{\tilde{r}r}V_{(n)}(g)^{\dagger}_{\tilde{s}s}B_{(n)}^{(2,2)}(p^{\prime},q^{\prime},r^{\prime},s^{\prime})_{\tilde{p}\tilde{q}\tilde{r}\tilde{s}}.$
(16)

Since all rotation transformations with the same angle $\omega$ form the same
equivalence class, the simplest way to compute the character of this
equivalence class is to adopt the

$\displaystyle V_{(n)}(e^{i\omega
J_{z}})_{m\tilde{m}}=\delta_{m\tilde{m}}e^{im\omega}.$ (17)

Take this into Eq. (16) we acquire

$\displaystyle\chi_{(n)}^{(2,2)}(\omega)=\sum_{p\leq q,r\leq
s}e^{i(p+q-r-s)\omega},$ (18)

where $\chi_{(n)}^{(2,2)}(\omega)$ is the character of the equivalence class
$\omega$ under the tensor representation $T_{pqrs}$.

Then, the multiplicity of the trivial representation is given by the
continuous inner product of the character tables

$\displaystyle N_{(n)}^{(2,2)}$ $\displaystyle=$
$\displaystyle\frac{1}{4\pi^{2}}\int_{-\pi}^{\pi}d\phi\int_{0}^{\pi}d\theta\sin\theta\int_{0}^{2\pi}d\omega\sin^{2}\left(\frac{\omega}{2}\right)\chi_{1}^{*}(\omega)\chi_{(n)}^{(2,2)}(\omega)$
(19) $\displaystyle=$
$\displaystyle\frac{1}{\pi}\int_{0}^{2\pi}d\omega\left(\frac{1-\cos\omega}{2}\right)\chi_{(n)}^{(2,2)}(\omega),$

where $\chi_{1}^{*}(\omega)\equiv 1$ is the character table of the trivial
representation. Substitute Eq. (18) into Eq. (19), we have

$\displaystyle N_{(n)}^{(2,2)}=N_{(n)0}^{(2,2)}-N_{(n)1}^{(2,2)},$ (20)

where $N_{(n)i}^{(2,2)}$ is defined by the number of all possible $-j\leq
p\leq q\leq j$ and $-j\leq r\leq s\leq j$ combinations satisfying $p+q-r-s=i$.

As an example, when $n=4$ so that $j=\frac{n-1}{2}=\frac{3}{2}$, enumerating
all possible $p+q-r-s=1$ and $p+q-r-s=0$ when $-j\leq p\leq q\leq j$ and
$-j\leq r\leq s\leq j$ gives

$\displaystyle 1$ $\displaystyle=$
$\displaystyle-\frac{3}{2}-\frac{1}{2}+\frac{3}{2}+\frac{3}{2}=-\frac{3}{2}+\frac{1}{2}+\frac{3}{2}+\frac{1}{2}=-\frac{3}{2}+\frac{3}{2}+\frac{3}{2}-\frac{1}{2}=-\frac{3}{2}+\frac{3}{2}+\frac{1}{2}+\frac{1}{2}$
(21) $\displaystyle=$
$\displaystyle-\frac{1}{2}-\frac{1}{2}+\frac{3}{2}+\frac{1}{2}=-\frac{1}{2}+\frac{1}{2}+\frac{3}{2}-\frac{1}{2}=-\frac{1}{2}+\frac{1}{2}+\frac{1}{2}+\frac{1}{2}=-\frac{1}{2}+\frac{3}{2}+\frac{3}{2}-\frac{3}{2}$
$\displaystyle=$
$\displaystyle-\frac{1}{2}+\frac{3}{2}+\frac{1}{2}-\frac{1}{2}=\frac{1}{2}+\frac{1}{2}+\frac{3}{2}-\frac{3}{2}=\frac{1}{2}+\frac{1}{2}+\frac{1}{2}-\frac{1}{2}=\frac{1}{2}+\frac{3}{2}+\frac{1}{2}-\frac{3}{2}$
$\displaystyle=$
$\displaystyle\frac{1}{2}+\frac{3}{2}-\frac{1}{2}-\frac{1}{2}=\frac{3}{2}+\frac{3}{2}-\frac{1}{2}-\frac{3}{2},$
$\displaystyle 0$ $\displaystyle=$
$\displaystyle-\frac{3}{2}-\frac{3}{2}+\frac{3}{2}+\frac{3}{2}=-\frac{3}{2}-\frac{1}{2}+\frac{3}{2}+\frac{1}{2}=-\frac{3}{2}+\frac{1}{2}+\frac{3}{2}-\frac{1}{2}=-\frac{3}{2}+\frac{1}{2}+\frac{1}{2}+\frac{1}{2}$
(22) $\displaystyle=$
$\displaystyle-\frac{3}{2}+\frac{3}{2}+\frac{3}{2}-\frac{3}{2}=-\frac{3}{2}+\frac{3}{2}+\frac{1}{2}-\frac{1}{2}=-\frac{1}{2}-\frac{1}{2}+\frac{3}{2}-\frac{1}{2}=-\frac{1}{2}-\frac{1}{2}+\frac{1}{2}+\frac{1}{2}$
$\displaystyle=$
$\displaystyle-\frac{1}{2}+\frac{1}{2}+\frac{3}{2}-\frac{3}{2}=-\frac{1}{2}+\frac{1}{2}+\frac{1}{2}-\frac{1}{2}=-\frac{1}{2}+\frac{3}{2}+\frac{1}{2}-\frac{3}{2}=-\frac{1}{2}+\frac{3}{2}-\frac{1}{2}-\frac{1}{2}$
$\displaystyle=$
$\displaystyle\frac{1}{2}+\frac{1}{2}+\frac{1}{2}-\frac{3}{2}=\frac{1}{2}+\frac{1}{2}-\frac{1}{2}-\frac{1}{2}=\frac{1}{2}+\frac{3}{2}-\frac{1}{2}-\frac{3}{2}=\frac{3}{2}+\frac{3}{2}-\frac{3}{2}-\frac{3}{2}.$

Eq. (21) gives $N_{(7)1}^{(2,2)}=14$, and Eq. (22) gives
$N_{(4)1}^{(2,2)}=16$, so $N_{(4)}^{(2,2)}=2$. That is to say, there are only
two independent combinations of the
$\Delta_{p}\Delta_{q}\Delta_{r}^{*}\Delta_{s}^{*}$ to form the
$\text{SU(2)}_{\text{D}}$ singlets. Then the task is to look into the
following quartic polynomial chains

$\displaystyle\Phi_{(n)}^{\dagger}\Phi_{(n)}\Phi_{(n)}^{\dagger}\Phi_{(n)},~{}~{}\Phi_{(n)}^{\dagger}\tau_{(n)}^{a}\Phi_{(n)}\Phi_{(n)}^{\dagger}\tau_{(n)}^{a}\Phi_{(n)},~{}~{}\Phi_{(n)}^{\dagger}\tau_{(n)}^{a}\tau_{(n)}^{b}\Phi_{(n)}\Phi_{(n)}^{\dagger}\tau_{(n)}^{a}\tau_{(n)}^{b}\Phi_{(n)},$
(23)
$\displaystyle\Phi_{(n)}^{\dagger}\tau_{(n)}^{a}\tau_{(n)}^{b}\tau_{(n)}^{c}\Phi_{(n)}\Phi_{(n)}^{\dagger}\tau_{(n)}^{a}\tau_{(n)}^{b}\tau_{(n)}^{c}\Phi_{(n)},~{}~{}\Phi_{(n)}^{\dagger}\tau_{(n)}^{a}\tau_{(n)}^{b}\tau_{(n)}^{c}\tau_{(n)}^{d}\Phi_{(n)}\Phi_{(n)}^{\dagger}\tau_{(n)}^{a}\tau_{(n)}^{b}\tau_{(n)}^{c}\tau_{(n)}^{d}\Phi_{(n)},~{}\dots$

to select two linearly independent polynomials. The elements of the SU(2)
$n$-dimension irreducible representation generator matrices
$\tau^{a,b,c,\dots}_{(n)}$, with $a,b,c,\dots=1,2,3$, are illustrated in Eq.
(175) and Eq. (176). Selecting the linearly independent polynomials from Eq.
(23) can be easily accomplished by expanding each of them, and check the ranks
of the coefficient matrices composed by different polynomial combinations. The
simplest result of the independent quartic terms when $n=4$ is given by

$\displaystyle\Phi_{(4)}^{\dagger}\Phi_{(4)}\Phi_{(4)}^{\dagger}\Phi_{(4)},~{}~{}\Phi_{(4)}^{\dagger}\tau_{(4)}^{a}\Phi_{(4)}\Phi_{(4)}^{\dagger}\tau_{(4)}^{a}\Phi_{(4)}.$
(24)

For all the other $\Delta_{p}\Delta_{q}\Delta_{r}\Delta_{s}$,
$\Delta_{p}\Delta_{q}\Delta_{r}\Delta_{s}^{*}$ combinations under any
$n$-dimensional irreducible representation, one can modify and follow the
similar processes from Eq. (18) to (23) to write down the character table of a
tensor representation with a specific symmetry pattern of the indices, and
then compute its inner product with the trivial representation character
table. Finally, the multiplicity of the trivial representation
$N_{(n)}^{(a,b)}=N_{(n)0}^{(a,b)}-N_{(n)1}^{(a,b)}$ is acquired where $(a,b)$
corresponds to the superscripts appeared in the coefficients in Eq. (11).
Enumeration of the fields with the generator matrices contracted inside, like
the Eq. (23), should also be listed. Define the charge conjugated field

$\displaystyle\Phi_{(n)}^{C}=T_{(n)}\Phi_{(n)}^{*},$ (25)

then the following quartic polynomials

$\displaystyle\Phi_{(n)}^{C\dagger}\Phi_{(n)}\Phi_{(n)}^{C\dagger}\Phi_{(n)},~{}~{}\Phi_{(n)}^{C\dagger}\tau_{(n)}^{a}\Phi_{(n)}\Phi_{(n)}^{C\dagger}\tau_{(n)}^{a}\Phi_{(n)},~{}~{}\Phi_{(n)}^{C\dagger}\tau_{(n)}^{a}\tau_{(n)}^{b}\Phi_{(n)}\Phi_{(n)}^{C\dagger}\tau_{(n)}^{a}\tau_{(n)}^{b}\Phi_{(n)},$
(26)
$\displaystyle\Phi_{(n)}^{C\dagger}\tau_{(n)}^{a}\tau_{(n)}^{b}\tau_{(n)}^{c}\Phi_{(n)}\Phi_{(n)}^{C\dagger}\tau_{(n)}^{a}\tau_{(n)}^{b}\tau_{(n)}^{c}\Phi_{(n)},~{}~{}\Phi_{(n)}^{C\dagger}\tau_{(n)}^{a}\tau_{(n)}^{b}\tau_{(n)}^{c}\tau_{(n)}^{d}\Phi_{(n)}\Phi_{(n)}^{C\dagger}\tau_{(n)}^{a}\tau_{(n)}^{b}\tau_{(n)}^{c}\tau_{(n)}^{d}\Phi_{(n)},~{}\dots$

and

$\displaystyle\Phi_{(n)}^{\dagger}\Phi_{(n)}\Phi_{(n)}^{C\dagger}\Phi_{(n)},~{}~{}\Phi_{(n)}^{\dagger}\tau_{(n)}^{a}\Phi_{(n)}\Phi_{(n)}^{C\dagger}\tau_{(n)}^{a}\Phi_{(n)},~{}~{}\Phi_{(n)}^{\dagger}\tau_{(n)}^{a}\tau_{(n)}^{b}\Phi_{(n)}\Phi_{(n)}^{C\dagger}\tau_{(n)}^{a}\tau_{(n)}^{b}\Phi_{(n)},$
(27)
$\displaystyle\Phi_{(n)}^{\dagger}\tau_{(n)}^{a}\tau_{(n)}^{b}\tau_{(n)}^{c}\Phi_{(n)}\Phi_{(n)}^{C\dagger}\tau_{(n)}^{a}\tau_{(n)}^{b}\tau_{(n)}^{c}\Phi_{(n)},~{}~{}\Phi_{(n)}^{\dagger}\tau_{(n)}^{a}\tau_{(n)}^{b}\tau_{(n)}^{c}\tau_{(n)}^{d}\Phi_{(n)}\Phi_{(n)}^{C\dagger}\tau_{(n)}^{a}\tau_{(n)}^{b}\tau_{(n)}^{c}\tau_{(n)}^{d}\Phi_{(n)},~{}\dots$

are the possible singlet contractions of the
$\Delta_{p}\Delta_{q}\Delta_{r}\Delta_{s}$ and the
$\Delta_{p}\Delta_{q}\Delta_{r}\Delta_{s}^{*}$ respectively. Just utilize the
computer to expand each of the quartic contractions and select the anterior
independent polynomials up to the number of $N_{(n)}^{(4,0)}$ and
$N_{(n)}^{(3,1)}$, then the complete set of the independent
$\Delta_{p}\Delta_{q}\Delta_{r}\Delta_{s}$ and
$\Delta_{p}\Delta_{q}\Delta_{r}\Delta_{s}^{*}$ contractions are eventually
found.

For an odd-$n$ and even-$j$, the multiplicity of the trivial representation of
the $\Delta_{p}\Delta_{q}\Delta_{r}$ is calculated to be one. Since there is
no simple contractions of three $\Phi_{(n)}^{(\dagger)}$’s with the generator
matrices inserted inside, we could only resort the Clebsch–Gordan
coefficients. Fortunately, it is easy, since the only way to combine three
$n$-dimensional representations into a singlet is to contract two of them into
a $n$-dimensional representation, then pair with the rest $n$-dimensional
representation to form a singlet. Therefore, we have for the
odd-$n$-dimensional representations $\Phi_{(n)}$ with $n=2j+1$,

$\displaystyle\kappa_{pqr}^{(3,0)}\Delta_{p}\Delta_{q}\Delta_{r}\propto\xi_{pq(-r)}^{jjj}\Delta_{p}\Delta_{q}\Delta_{r},$
(28)

where $\xi_{pqr}^{j_{1}j_{2}j_{3}}$ are the Clebsch–Gordan coefficients
defined by

$\displaystyle\xi_{pq(-r)}^{j_{1}j_{2}j_{3}}=\langle
j_{3},m_{3}=r|(|j_{1},m_{1}=p\rangle\otimes|j_{2},m_{2}=q\rangle),$ (29)

for any integer $-|j_{1}-j_{2}|\leq j_{3}\leq j_{1}+j_{2}$.

As a summary, we list all the possible renormalizable independent contractions
of the $\Phi_{(n)}^{(\dagger)}$ fields up to $n=21$ in Tables 1 and 2 to close
this section.

$j$ | $\Delta_{p}\Delta_{q}\Delta_{r}$ | $\Delta_{p}\Delta_{q}\Delta_{r}\Delta_{s}$
---|---|---
$1$ | $0$ | $1$
| $(\Phi^{{\dagger}}\Phi)^{2}$
$2$ | $1$ | $1$
$\xi_{(5)}^{pqr}\Delta_{p}\Delta_{q}\Delta_{r}$ | $(\Phi^{{\dagger}}\Phi)^{2}$
$3$ | $0$ | $2$
| $(\Phi^{{\dagger}}\Phi)^{2},(\Phi^{{\dagger}}\tau^{a}\tau^{b}\Phi)^{2}$
$4$ | $1$ | $2$
$\xi_{(9)}^{pqr}\Delta_{p}\Delta_{q}\Delta_{r}$ | $(\Phi^{{\dagger}}\Phi)^{2},(\Phi^{{\dagger}}\tau^{a}\tau^{b}\Phi)^{2}$
$5$ | $0$ | $2$
| $(\Phi^{{\dagger}}\Phi)^{2},(\Phi^{{\dagger}}\tau^{a}\tau^{b}\Phi)^{2}$
$6$ | $1$ | $3$
$\xi_{(13)}^{pqr}\Delta_{p}\Delta_{q}\Delta_{r}$ | $(\Phi^{{\dagger}}\Phi)^{2},(\Phi^{{\dagger}}\tau^{a}\tau^{b}\Phi)^{2},(\Phi^{{\dagger}}\tau^{a}\tau^{b}\tau^{c}\tau^{d}\Phi)^{2}$
$7$ | $0$ | $3$
|
$(\Phi^{{\dagger}}\Phi)^{2},(\Phi^{{\dagger}}\tau^{a}\tau^{b}\Phi)^{2},(\Phi^{{\dagger}}\tau^{a}\tau^{b}\tau^{c}\tau^{d}\Phi)^{2}$
$8$ | $1$ | $3$
$\xi_{(17)}^{pqr}\Delta_{p}\Delta_{q}\Delta_{r}$ | $(\Phi^{{\dagger}}\Phi)^{2},(\Phi^{{\dagger}}\tau^{a}\tau^{b}\Phi)^{2},(\Phi^{{\dagger}}\tau^{a}\tau^{b}\tau^{c}\tau^{d}\Phi)^{2}$
$9$ | $0$ | $4$
|
$(\Phi^{{\dagger}}\Phi)^{2},(\Phi^{{\dagger}}\tau^{a}\tau^{b}\Phi)^{2},(\Phi^{{\dagger}}\tau^{a}\tau^{b}\tau^{c}\tau^{d}\Phi)^{2},(\Phi^{{\dagger}}\tau^{a}\tau^{b}\tau^{c}\tau^{d}\tau^{e}\tau^{f}\Phi)^{2}$
$10$ | $1$ | $4$
$\xi_{(21)}^{pqr}\Delta_{p}\Delta_{q}\Delta_{r}$ | $(\Phi^{{\dagger}}\Phi)^{2},(\Phi^{{\dagger}}\tau^{a}\tau^{b}\Phi)^{2},(\Phi^{{\dagger}}\tau^{a}\tau^{b}\tau^{c}\tau^{d}\Phi)^{2},(\Phi^{{\dagger}}\tau^{a}\tau^{b}\tau^{c}\tau^{d}\tau^{e}\tau^{f}\Phi)^{2}$
Table 1: Independent and renormalizable operators for odd dimension $n=3,5,7,\cdots,21$. The representation dimension subscripts $n=2j+1$ are omitted. The bilinear $\Delta_{p}\Delta_{q}$ forms only one gauge invariant operator $\Phi^{\dagger}\Phi$ in any $n$. $j$ | $\Delta_{p}\Delta_{q}\Delta_{r}\Delta_{s}$ | $\Delta_{p}\Delta_{q}\Delta_{r}\Delta_{s}^{*}$ | $\Delta_{p}\Delta_{q}\Delta_{r}^{*}\Delta_{s}^{*}$
---|---|---|---
$\frac{1}{2}$ | $0$ | $0$ | $1$
|  | $(\Phi^{{\dagger}}\Phi)^{2}$
$\frac{3}{2}$ | $1$ | $1$ | $2$
$(\Phi^{C\dagger}\tau^{a}\Phi)^{2}$ | $\Phi^{C\dagger}\tau^{a}\Phi\Phi^{{\dagger}}\tau^{a}\Phi$ | $(\Phi^{{\dagger}}\Phi)^{2},(\Phi^{{\dagger}}\tau^{a}\Phi)^{2}$
$\frac{5}{2}$ | $1$ | $1$ | $3$
$(\Phi^{C\dagger}\tau^{a}\Phi)^{2}$ | $\Phi^{C\dagger}\tau^{a}\Phi\Phi^{{\dagger}}\tau^{a}\Phi$ | $(\Phi^{{\dagger}}\Phi)^{2},(\Phi^{{\dagger}}\tau^{a}\Phi)^{2},(\Phi^{{\dagger}}\tau^{a}\tau^{b}\Phi)^{2}$
$\frac{7}{2}$ | $1$ | $1$ | $4$
$(\Phi^{C\dagger}\tau^{a}\Phi)^{2}$ | $\Phi^{C\dagger}\tau^{a}\Phi\Phi^{{\dagger}}\tau^{a}\Phi$ |  $(\Phi^{{\dagger}}\Phi)^{2},(\Phi^{{\dagger}}\tau^{a}\Phi)^{2},(\Phi^{{\dagger}}\tau^{a}\tau^{b}\Phi)^{2},(\Phi^{{\dagger}}\tau^{a}\tau^{b}\tau^{c}\Phi)^{2}$
$\frac{9}{2}$ | $2$ | $2$ | $5$
$(\Phi^{C\dagger}\tau^{a}\Phi)^{2},$ $(\Phi^{C\dagger}\tau^{a}\tau^{b}\tau^{c}\Phi)^{2}$ |  $\Phi^{C\dagger}\tau^{a}\Phi\Phi^{{\dagger}}\tau^{a}\Phi,$ $\Phi^{C\dagger}\tau^{a}\tau^{b}\tau^{c}\Phi\Phi^{{\dagger}}\tau^{a}\tau^{b}\tau^{c}\Phi$ |  $(\Phi^{{\dagger}}\Phi)^{2},(\Phi^{{\dagger}}\tau^{a}\Phi)^{2},(\Phi^{{\dagger}}\tau^{a}\tau^{b}\Phi)^{2},(\Phi^{{\dagger}}\tau^{a}\tau^{b}\tau^{c}\Phi)^{2},$ $(\Phi^{{\dagger}}\tau^{a}\tau^{b}\tau^{c}\tau^{d}\Phi)^{2}$
$\frac{11}{2}$ | $2$ | $2$ | $6$
$(\Phi^{C\dagger}\tau^{a}\Phi)^{2},$ $(\Phi^{C\dagger}\tau^{a}\tau^{b}\tau^{c}\Phi)^{2}$ |  $\Phi^{C\dagger}\tau^{a}\Phi\Phi^{{\dagger}}\tau^{a}\Phi$, $\Phi^{C\dagger}\tau^{a}\tau^{b}\tau^{c}\Phi\Phi^{{\dagger}}\tau^{a}\tau^{b}\tau^{c}\Phi$ |  $(\Phi^{{\dagger}}\Phi)^{2},(\Phi^{{\dagger}}\tau^{a}\Phi)^{2},\Phi^{{\dagger}}\tau^{a}\tau^{b}\Phi)^{2},(\Phi^{{\dagger}}\tau^{a}\tau^{b}\tau^{c}\Phi)^{2},$ $(\Phi^{{\dagger}}\tau^{a}\tau^{b}\tau^{c}\tau^{d}\Phi)^{2},(\Phi^{{\dagger}}\tau^{a}\tau^{b}\tau^{c}\tau^{d}\tau^{e}\Phi)^{2}$
$\frac{13}{2}$ | $2$ | $2$ | $7$
$(\Phi^{C\dagger}\tau^{a}\Phi)^{2},$ $(\Phi^{C\dagger}\tau^{a}\tau^{b}\tau^{c}\Phi)^{2}$ |  $\Phi^{C\dagger}\tau^{a}\Phi\Phi^{{\dagger}}\tau^{a}\Phi$, $\Phi^{C\dagger}\tau^{a}\tau^{b}\tau^{c}\Phi\Phi^{{\dagger}}\tau^{a}\tau^{b}\tau^{c}\Phi$ |  $(\Phi^{{\dagger}}\Phi)^{2},(\Phi^{{\dagger}}\tau^{a}\Phi)^{2},(\Phi^{{\dagger}}\tau^{a}\tau^{b}\Phi)^{2},(\Phi^{{\dagger}}\tau^{a}\tau^{b}\tau^{c}\Phi)^{2},$ $(\Phi^{{\dagger}}\tau^{a}\tau^{b}\tau^{c}\tau^{d}\Phi)^{2},(\Phi^{{\dagger}}\tau^{a}\tau^{b}\tau^{c}\tau^{d}\tau^{e}\Phi)^{2},$ $(\Phi^{{\dagger}}\tau^{a}\tau^{b}\tau^{c}\tau^{d}\tau^{e}\tau^{f}\Phi)^{2}$
$\frac{15}{2}$ | $3$ | $3$ | $8$
$(\Phi^{C\dagger}\tau^{a}\Phi)^{2},$ $(\Phi^{C\dagger}\tau^{a}\tau^{b}\tau^{c}\Phi)^{2},$ $(\Phi^{C\dagger}\tau^{a}\tau^{b}\tau^{c}\tau^{d}\tau^{e}\Phi)^{2}$ |  $\Phi^{C\dagger}\tau^{a}\Phi\Phi^{{\dagger}}\tau^{a}\Phi$, $\Phi^{C\dagger}\tau^{a}\tau^{b}\tau^{c}\Phi\Phi^{{\dagger}}\tau^{a}\tau^{b}\tau^{c}\Phi$ $\Phi^{C\dagger}\tau^{a}\tau^{b}\tau^{c}\tau^{d}\tau^{e}\Phi\Phi^{{\dagger}}\tau^{a}\tau^{b}\tau^{c}\tau^{d}\tau^{e}\Phi$ |  $(\Phi^{{\dagger}}\Phi)^{2},(\Phi^{{\dagger}}\tau^{a}\Phi)^{2},(\Phi^{{\dagger}}\tau^{a}\tau^{b}\Phi)^{2},(\Phi^{{\dagger}}\tau^{a}\tau^{b}\tau^{c}\Phi)^{2},$ $(\Phi^{{\dagger}}\tau^{a}\tau^{b}\tau^{c}\tau^{d}\Phi)^{2},(\Phi^{{\dagger}}\tau^{a}\tau^{b}\tau^{c}\tau^{d}\tau^{e}\Phi)^{2},$ $(\Phi^{{\dagger}}\tau^{a}\tau^{b}\tau^{c}\tau^{d}\tau^{e}\tau^{f}\Phi)^{2},(\Phi^{{\dagger}}\tau^{a}\tau^{b}\tau^{c}\tau^{d}\tau^{e}\tau^{f}\tau^{g}\Phi)^{2}$
$\frac{17}{2}$ | $3$ | $3$ | $9$
$(\Phi^{C\dagger}\tau^{a}\Phi)^{2},$ $(\Phi^{C\dagger}\tau^{a}\tau^{b}\tau^{c}\Phi)^{2},$ $(\Phi^{C\dagger}\tau^{a}\tau^{b}\tau^{c}\tau^{d}\tau^{e}\Phi)^{2}$ |  $\Phi^{C\dagger}\tau^{a}\Phi\Phi^{{\dagger}}\tau^{a}\Phi$, $\Phi^{C\dagger}\tau^{a}\tau^{b}\tau^{c}\Phi\Phi^{{\dagger}}\tau^{a}\tau^{b}\tau^{c}\Phi$ $\Phi^{C\dagger}\tau^{a}\tau^{b}\tau^{c}\tau^{d}\tau^{e}\Phi\Phi^{{\dagger}}\tau^{a}\tau^{b}\tau^{c}\tau^{d}\tau^{e}\Phi$ |  $(\Phi^{{\dagger}}\Phi)^{2},(\Phi^{{\dagger}}\tau^{a}\Phi)^{2},(\Phi^{{\dagger}}\tau^{a}\tau^{b}\Phi)^{2},(\Phi^{{\dagger}}\tau^{a}\tau^{b}\tau^{c}\Phi)^{2},$ $(\Phi^{{\dagger}}\tau^{a}\tau^{b}\tau^{c}\tau^{d}\Phi)^{2},(\Phi^{{\dagger}}\tau^{a}\tau^{b}\tau^{c}\tau^{d}\tau^{e}\Phi)^{2},$ $(\Phi^{{\dagger}}\tau^{a}\tau^{b}\tau^{c}\tau^{d}\tau^{e}\tau^{f}\Phi)^{2},(\Phi^{{\dagger}}\tau^{a}\tau^{b}\tau^{c}\tau^{d}\tau^{e}\tau^{f}\tau^{g}\Phi)^{2},$ $(\Phi^{{\dagger}}\tau^{a}\tau^{b}\tau^{c}\tau^{d}\tau^{e}\tau^{f}\tau^{g}\tau^{h}\Phi)^{2}$
$\frac{19}{2}$ | $3$ | $3$ | $10$
$(\Phi^{C\dagger}\tau^{a}\Phi)^{2},$ $(\Phi^{C\dagger}\tau^{a}\tau^{b}\tau^{c}\Phi)^{2},$ $(\Phi^{C\dagger}\tau^{a}\tau^{b}\tau^{c}\tau^{d}\tau^{e}\Phi)^{2}$ |  $\Phi^{C\dagger}\tau^{a}\Phi\Phi^{{\dagger}}\tau^{a}\Phi$, $\Phi^{C\dagger}\tau^{a}\tau^{b}\tau^{c}\Phi\Phi^{{\dagger}}\tau^{a}\tau^{b}\tau^{c}\Phi$ $\Phi^{C\dagger}\tau^{a}\tau^{b}\tau^{c}\tau^{d}\tau^{e}\Phi\Phi^{{\dagger}}\tau^{a}\tau^{b}\tau^{c}\tau^{d}\tau^{e}\Phi$ |  $(\Phi^{{\dagger}}\Phi)^{2},(\Phi^{{\dagger}}\tau^{a}\Phi)^{2},(\Phi^{{\dagger}}\tau^{a}\tau^{b}\Phi)^{2},(\Phi^{{\dagger}}\tau^{a}\tau^{b}\tau^{c}\Phi)^{2},$ $(\Phi^{{\dagger}}\tau^{a}\tau^{b}\tau^{c}\tau^{d}\Phi)^{2},(\Phi^{{\dagger}}\tau^{a}\tau^{b}\tau^{c}\tau^{d}\tau^{e}\Phi)^{2},$ $(\Phi^{{\dagger}}\tau^{a}\tau^{b}\tau^{c}\tau^{d}\tau^{e}\tau^{f}\Phi)^{2},(\Phi^{{\dagger}}\tau^{a}\tau^{b}\tau^{c}\tau^{d}\tau^{e}\tau^{f}\tau^{g}\Phi)^{2},$ $(\Phi^{{\dagger}}\tau^{a}\tau^{b}\tau^{c}\tau^{d}\tau^{e}\tau^{f}\tau^{g}\tau^{h}\Phi)^{2},(\Phi^{{\dagger}}\tau^{a}\tau^{b}\tau^{c}\tau^{d}\tau^{e}\tau^{f}\tau^{g}\tau^{h}\tau^{i}\Phi)^{2}$
Table 2: Independent and renormalizable operators for even dimension
$n=2,4,6,\cdots,20$. The representation dimension subscripts $n=2j+1$ are
omitted. The bilinear $\Delta_{p}\Delta_{q}^{*}$ forms only one gauge
invariant operator $\Phi^{\dagger}\Phi$ in any $n$. The trilinear
$\Delta_{p}\Delta_{q}\Delta_{r}$ and $\Delta_{p}\Delta_{q}\Delta_{r}^{*}$
cannot form a singlet in even dimension.

##  IV Vacuum Patterns with a Remnant Discrete Symmetry

For an $n$-dimension representation of SU(2) group, some discrete symmetry
group might remain so that the vacuum is invariant under it. The authors of
Ref. Etesi:1996urw  had enumerated all possible such discrete groups for
odd-$n$-dimensional multiplets up to $n=59$ by computing the product of the
character tables. What they have done is to try upon all known point groups in
a 3-dimensional space to see whether some trivial representations arise within
the $n$-dimensional scalar multiplets. However, the problem still remains for
an even-$n$-dimensional representation when $j$ is a half-integer, since the
multiplet does not restore to its original form after a $2\pi$-rotation along
any axis. Another thing that we want to do beyond the Ref. Etesi:1996urw  is
to show a systematic algorithm for finding the detailed formats of the VEV-
patterns invariant under the discrete groups. In this section, we try up to
$n=20$ to accomplish two goals:

  * •

To write down all possible VEV configurations which are invariant under the
transformations of the remained discrete groups.

  * •

To show that only the $Z_{N}$ group can remain when an even-$n$-dimensional
multiplet is adopted.

For an $n$-dimensional multiplet $\Phi_{(n)}$, let us assume it breaks the
$\text{SU(2)}_{\text{D}}$ symmetry into a nontrivial discrete symmetry $G$ by
a vacuum configuration $\vec{v}_{(n)}$. Recall that any nontrivial discrete
symmetry should include at least one $Z_{N}$ subgroup, without loss of the
generality, we can set the axis for $Z_{N}\subset G$ group to align with the
$z$-axis. The elements of the $\vec{v}_{(n)}$ can be denoted as

$\vec{v}=(a_{j},a_{j-1},...,a_{-j+1},a_{-j})^{T}\quad\text{or}\quad\vec{v}=\sum_{m=-j}^{m=j}a_{m}\vec{v}_{m},$
(30)

with $j=(n-1)/2$. $\vec{v}_{m}$ are the eigenvectors of $z$-axis rotation
generator $J_{3}$ and their corresponding expansion coefficients are denoted
as $a_{m}$. Since the $z$-axis and the rotation axis direction overlap, the
representation matrix of the generator of the $Z_{N}$ subgroup can be written
as

$U(\vec{z},\alpha)=\text{Diag}\\{e^{ij\alpha},e^{i(j-1)\alpha},...,e^{i(-j+1)\alpha},e^{i(-j)\alpha}\\},$
(31)

where $\alpha=\frac{4\pi}{N}$ when $n$ is even, and $\alpha=\frac{2\pi}{N}$
when $n$ is odd. Then the corresponding rotation operation is given by

$U(\vec{z},\alpha)\vec{v}=\sum_{m=-j}^{m=j}e^{im\alpha}a_{m}\vec{v}_{m}=\vec{v}.$
(32)

Then, the invariance of the $\vec{v}$ requires $a_{m}$ to vanish for all the
$m$ satisfying $e^{im\alpha}\neq 1$. Thereby, the vacuum configuration can be
written as

$\left\\{\begin{aligned} &\vec{v}=\sum_{m/N\in\mathbb{Z}~{}\&~{}|m|\leq
j}a_{m}\vec{v}_{m},\quad\text{when $n$ is odd}\\\
&\vec{v}=\sum_{2m/N\in\mathbb{Z}~{}\&~{}|m|\leq
j}a_{m}\vec{v}_{m},\quad\text{when $n$ is even}\end{aligned}\right.$ (33)

That is to say,

$\displaystyle\left\\{\begin{aligned} &{a_{m}=0,\quad\text{for
}\frac{m}{N}\notin\mathbb{Z}~{}\&~{}|m|\leq j}\text{ when $n$ is odd }\\\
&{a_{m}=0,\quad\text{for }\frac{2m}{N}\notin\mathbb{Z}~{}\&~{}|m|\leq j}\text{
when $n$ is even }\end{aligned}\right.$ (34)

In this paper, we would like to discuss about the non-vanishing VEV
configurations, and we also avoid the situation of a remnant U(1) subgroup, so
there should be at least one nonzero $|m|\leq j$ that satisfies
$m/N\in\mathbb{Z}$ or $2m/N\in\mathbb{Z}$ for the odd- or even-$n$
respectively. This forces the $N\leq j$ or $N\leq 2j$ for the odd- or even-$n$
respectively.

Suppose there are other elements in the $G$ except the ones in the $Z_{N}$,
i.e., $G$-$Z_{N}\neq\varnothing$, we can pick up another $Z_{N^{\prime}}\in G$
with the rotation axis aligned with another direction denoted by $\vec{n}$.
Without loss of generality, we can rotate this axis into the $x$-$z$ plane.
This allows us to set a normalized axis with a form as
$\vec{n}=(\sin\theta,0,\cos\theta)$. At the $J_{3}$ representation,
diagonalizing the $\vec{n}\cdot\vec{J}$ gives a group of the normalized
eigenvectors

$\displaystyle
V_{(n)}(\vec{n}\cdot\vec{J})\vec{v^{\prime}}_{m}(\theta)=m\vec{v^{\prime}}_{m}(\theta),$
(35)

where again, $m=-j,-j+1,\dots,j-1,j$, and $\vec{v^{\prime}}_{m}$’s with all
$\vec{v^{\prime}}^{\dagger}_{m}\vec{v^{\prime}}_{m^{\prime}}=\delta_{mm^{\prime}}$
are the corresponding eigenstates under the $\vec{n}\cdot\vec{J}$
representation.

We then try to expand the $\vec{v}$ with the basis $\vec{v^{\prime}}_{m}$,

$\vec{v}=\sum_{m=-j}^{m=j}b_{m}\vec{v^{\prime}}_{m}(\theta)$ (36)

where $b_{m}$’s are the coefficients. It is then easy to acquire the
relationship between $a_{m}$’s and $b_{m}$’s by

$\vec{v^{\prime}}^{\dagger}_{m^{\prime}}\vec{v}=\sum_{m=-j}^{m=j}a_{m}\vec{v^{\prime}}^{\dagger}_{m^{\prime}}\vec{v}_{m}=\sum_{m=-j}^{m=j}b_{m}\vec{v^{\prime}}^{\dagger}_{m^{\prime}}\vec{v^{\prime}}_{m}=\sum_{m=-j}^{m=j}b_{m}\delta_{mm^{\prime}}=b_{m^{\prime}}.$
(37)

Define

$v_{m^{\prime}m}(\theta)=\vec{v^{\prime}}^{\dagger}_{m^{\prime}}(\theta)\vec{v}_{m}$
(38)

we then have

$b_{m^{\prime}}=\sum_{m=-j}^{m=j}a_{m}v_{m^{\prime}m}(\theta).$ (39)

Again, similar to Eq. (33), the invariance of the VEV under the
$Z_{N^{\prime}}$ symmetry requires

$\left\\{\begin{aligned}
&\vec{v}=\sum_{m/{N^{\prime}}\in\mathbb{Z}~{}\&~{}|m|\leq
j}b_{m}\vec{v^{\prime}}_{m},\quad\text{when $n$ is odd}\\\
&\vec{v}=\sum_{2m/{N^{\prime}}\in\mathbb{Z}~{}\&~{}|m|\leq
j}b_{m}\vec{v^{\prime}}_{m},\quad\text{when $n$ is even}\end{aligned}\right..$
(40)

Therefore, we can obtain the following linear equations

$\displaystyle\left\\{\begin{aligned}
&{\sum_{m^{\prime}=-j}^{m^{\prime}=j}a_{m^{\prime}}v_{mm^{\prime}}(\theta)=b_{m}=0,\quad\text{for
}\frac{m}{N^{\prime}}\notin\mathbb{Z}~{}\&~{}|m|\leq j}\text{ when $n$ is
odd}\\\
&{\sum_{m^{\prime}=-j}^{m^{\prime}=j}a_{m^{\prime}}v_{mm^{\prime}}(\theta)=b_{m}=0,\quad\text{for
}\frac{2m}{N^{\prime}}\notin\mathbb{Z}~{}\&~{}|m|\leq j}\text{ when $n$ is
even}\end{aligned}\right.$ (41)

where the undetermined parameters are $a_{m^{\prime}}$’s and $\theta$. If the
solution space of the Eq. (41) is nontrivial, then there exist a nontrivial
VEV configurations invariant under both $Z_{N}$ and $Z_{N^{\prime}}$. In most
of the cases, this is the nontrivial VEV configurations which are invariant
under the entire $G$ as well.

We have tried to enumerate all the possible VEV configurations with some
remained discrete symmetry groups for $n$ from 4 to 21. We summarize the
results as bellows

  * •

When $n$ is odd, we adopt the discrete groups listed in Ref. Etesi:1996urw
and show them in Appendix B up to $n=21$. In this case, for the nontrivial
discrete groups $G=A_{4}/S_{4}/A_{5}$, two generators rotating along two
unparalleled axes corresponding to the two $Z_{N/N^{\prime}}$ subgroups
respectively are sufficient to generate all elements in $G$, so the solution
space of Eq. (41) is exactly the general VEV configurations. We just list the
results in the Section VII.

  * •

When $n$ is even, we were not able to find any information in the literature
about whether there can be some discrete group $G$ other than the simplest
$Z_{N}$ remained. In order to answer this question, we can study the
equivalent problem whether two nontrivial $Z_{N}$ and $Z_{N^{\prime}}$
symmetry along two different axes co-exist. This requires the nonzero solution
to the Eq. (41) for at least one specific $0<\theta<\pi$ value. The existence
of such a $\theta$ is equivalent to the vanishing determinants of all largest
sub-matrices of the coefficient matrix corresponding to the equation group Eq.
(41) for the given $\theta$. We have calculated these determinants
analytically for various $n$, and find that no such a $0<\theta<\pi$ exists up
to $n=20$. Therefore, when $n$ is even, the only discrete symmetries that can
remain after the spontaneously broken of $\text{SU(2)}_{\text{D}}$ are the
$Z_{N}$ groups for all possible $N\leq 2j$. Correspondingly, the general forms
of the VEV configurations are
$\vec{v}=\sum\limits_{2m/N\in\mathbb{Z}}a_{m}\vec{v}_{m}$, where $m$ runs all
through the half integers with the absolute values $|m|\leq j$.

##  V Scalar Classifications

After the Higgs multiplet acquires the VEV, the vector bosons become massive
by “eating” the Goldstone degrees of freedom. The next task is to diagonalize
all the mass matrices. In this paper, usually the mass matrix of the scalar
fields becomes too large for us to straightforwardly observe a simple
solution, although one might expect some systematic degeneracy among different
scalar combinations, catalogued by various irreducible representations of the
remnant discrete symmetry. The question then becomes: is it possible to
simplify the calculation by working out the detailed formats of these
irreducible representations under the discrete symmetry prior to the
diagonalizing processes? In order to solve it, in this section we suggest a
general algorithm to decompose the scalar multiplet into the irreducible
subspaces under the discrete symmetry.

For the half-integer $j$ situations when $n$ is even, as we have mentioned
before, up to $n=20$ the only discrete group which can remain from the broken
$\text{SU(2)}_{\text{D}}$ is the $Z_{N}$ group, and all $Z_{N}$ irreducible
representations are one-dimensional. Therefore, after Rotating the axis of the
$Z_{N}$ generator aligning with the $z$-axis, it is fairly easy for us to
observe that each element of $\Phi_{(n)}$ becomes one irreducible subspace of
$G$. When $j$ is an integer, so that $n$ is odd, discrete groups beyond the
$Z_{N}$ might appear, and at least two generators with the rotational axis
unparalleled to each other arise. In this section, we concentrate on this
situation.

The character table algorithm is again utilized. Suppose the
$\text{SU(2)}_{\text{D}}$ had been broken with a discrete $G$ remained, the
multiplicity of a irreducible representation $t$ in $\Phi_{(n)}$ is calculated
by

$n_{t,\Phi_{(n)}}=\frac{1}{|G|}\sum_{g\in
G}\bar{\chi}_{t}(g)\chi_{\Phi_{(n)}}(g),$ (42)

where $\chi_{t}(g)$ is the trace of the matrix of element $g\in G$ in the
irreducible representation $t$, and $\chi_{\Phi_{(n)}}(g)$ is the trace of the
matrix in the $\Phi_{(n)}$ space. The character tables of the irreducible
representations of the discrete groups appeared in this paper will be listed
in Appendix D. To compute $\chi_{\Phi_{(n)}}(g)$, if we regard the rotation
direction of the $g$ as the $z$-axis, we have

$\displaystyle
V_{(n)}(g)=\text{Diag}\\{e^{ij\varphi_{g}},e^{i(j-1)\varphi_{g}},\dots,e^{i(-j+1)\varphi_{g}},e^{i(-j)\varphi_{g}}\\},$
(43)

where $\varphi_{g}$ is the rotation angle of the $g$. Then

$\displaystyle\chi_{\varphi_{(n)}}(g)=e^{ij\varphi_{g}}+e^{i(j-1)\varphi_{g}}+\dots+e^{i(-j+1)\varphi_{g}}+e^{i(-j)\varphi_{g}}=\frac{\sin[(j+1/2)\varphi_{g}]}{\sin(\varphi_{g}/2)}.$
(44)

For the irreducible representation $t$, if $n_{t,\Phi_{(n)}}\neq 0$, then we
suggest the following processes to give the detailed format of the subspace(s)
under the $J_{3}$ representation.

Consider a projection operator $P_{t}$ which projects $\Phi_{(n)}$ into a
subspace which corresponds to the irreducible representation $t$ of the
discrete group $G$,

$P_{t}\Phi_{(n)}=\Phi_{t},$ (45)

where $\Phi_{t}$ is a $\mathrm{Dim}(t)$-dimensional vector, and
$\mathrm{Dim}(t)$ is the dimension of the irreducible representation $t$.
Therefore, $P_{t}$ is a $\mathrm{Dim}(t)\times n$ matrix. We would like that
the inner product of the $\Phi_{(n)}$ space be inherited by $\Phi_{t}$, so we
have

$\displaystyle P_{t}P_{t}^{\dagger}=I_{\mathrm{Dim}(t)\times\mathrm{Dim}(t)}.$
(46)

In this case, the columns of the $P_{t}^{\dagger}$ can be regarded as the
normalized orthogonal basis within the $\Phi_{(n)}$ space, which are projected
by the projector operator $P_{t}$ into the simplest
$(\vec{t}_{1},\vec{t}_{2},...,\vec{t}_{\text{Dim}(t)})$ basis in the
$\Phi_{t}$ space where $(\vec{t})_{j}=\delta_{ij}$.

Pick out two (or more) generators of $S$ and $T$ from $G$. The rotational axis
of them should not be parallel. Their corresponding matrices under the
$\Phi_{(n)}$ representation are formally written as

$\displaystyle
V_{(n)}(S)=e^{iV_{(n)}(\vec{J}\cdot\vec{n}_{S})\phi_{S}},~{}~{}~{}V_{(n)}(T)=e^{iV_{(n)}(\vec{J}\cdot\vec{n}_{T})\phi_{T}},$
(47)

where $\vec{n}_{S,T}$ are the normalized directions of the rotational axes of
the $S$ and $T$ respectively, while $\phi_{S,T}$ are the corresponding
rotational angles. The detailed forms of Eq. (47) can be simply calculated by
plugging in $V_{(n)}(\vec{J})$ according to Eq. (175) and (176).

Since the projection operation keeps the representation structures, for both
$S$ and $T$, we have

$P_{t}V_{(n)}(S)\Phi_{(n)}=S_{t}\Phi_{t}=S_{t}P_{t}\Phi_{(n)},~{}~{}P_{t}V_{(n)}(T)\Phi_{(n)}=T_{t}P_{t}\Phi_{(n)}.$
(48)

Note that Eq. (48) holds for any form of the $\Phi_{(n)}$, so that

$P_{t}V_{(n)}(S)=S_{t}P_{t},~{}~{}P_{t}V_{(n)}(T)=T_{t}P_{t}.$ (49)

In principle, solving Eq. (46) and (47) regarding the elements of $P_{t}$ as
unknown parameters can give the detailed results of the subspace of the
irreducible representation $t$. However, Eq. (46) are the quadratic equations,
making the analytic solutions too complicated to be acquired. In order to
resolve this problem, we can again select $\vec{n}_{S}$ along the
$z$-direction, so that $V_{(n)}(S)$ in Eq. (47) becomes diagonal. If we again
select a group of basis in the $t$-representation to let the $S_{t}$ become
diagonal, we therefore immediately realize that $P_{t}$ can only connect the
corresponding basis to the same eigenvalues within both $V_{(n)}(S)$ and
$S_{t}$, rendering most of the $P_{t}$ elements zero. Usually, after this
step, one can find out that now all the row vectors of $P_{t}$ automatically
become orthogonal with each other, and so that all the homogeneous functions
induced by Eq. (46) can be neglected.

Then, in principle $P_{t}V_{(n)}(T)=T_{t}P_{t}$ provides
$\mathrm{Dim}(t)\times n$ homogeneous equations. One can then find the
solution space of the remained nonzero $P_{t}$ elements, and finally normalize
each of the $P_{t}$ row vectors to fit the non-homogeneous equations in Eq.
(46), and eventually the detailed format of the irreducible representation $t$
subspace denoted in Eq. (45) is accomplished.

In this paper, we provide an algorithm which is a little bit different to
further diminish the number of equations. If we diagonalize $G_{t}$ and
$V_{(n)}(T)$ at the same time,

$\displaystyle
T^{\prime}_{t}=Q^{\dagger}T_{t}Q,~{}~{}T_{(n)}^{\prime}=R^{\dagger}V_{(n)}(T)R$
(50)

satisfying
$Q^{\dagger}Q=QQ^{\dagger}=I_{\mathrm{dim}(t)\times\mathrm{dim}(t)}$ and
$R^{\dagger}R=RR^{\dagger}=I_{n\times n}$, and $T^{\prime}_{t}$,
$T_{(n)}^{\prime}$ are the diagonalized matrices. $P_{t}V_{(n)}(T)=T_{t}P_{t}$
in Eq. (49) then can be written with

$\displaystyle
P_{t}RT_{(n)}^{\prime}R^{\dagger}=QT_{t}^{\prime}Q^{\dagger}P_{t}.$ (51)

Define

$\displaystyle P_{t}^{\prime}=Q^{\dagger}P_{t}R,$ (52)

therefore

$\displaystyle P_{t}^{\prime}T_{(n)}^{\prime}=T_{t}^{\prime}P_{t}^{\prime}.$
(53)

Similarly to the $P_{t}$ in the $P_{t}V_{(n)}(S)=S_{t}P_{t}$ in Eq. (49),
$P_{t}^{\prime}$ can only connect the corresponding vectors with the same
eigenvalues in $T_{(n)}^{\prime}$ and $T_{t}^{\prime}$ respectively, so some
elements in $P_{t}^{\prime}$ are again constrained to be zero. These vanishing
elements of the $P_{t}^{\prime}$ in Eq. (52) induce a group of homogeneous
equations with the $P_{t}$ nonzero elements being unknown parameters. Finding
the solution space of these equations again accomplishes the detailed format
of the $P_{t}$, as well as the detailed format of the irreducible
representation $t$ subspace denoted in Eq. (45).

##  VI Several Examples

In the previous sections we illustrate the general algorithm to find the VEVs
and to separate the degenerate scalars in a $\Phi_{(n)}$ for any $n$
assignment. These descriptions seem too “abstract” for one to understand and
follow. Therefore, in this section, we show some examples for clarify the
algorithm.

In subsection VI.1, we will take $n=4$ as a typical example for illustrating
the simplest case about the remnant $Z_{N}$ symmetry from the
$\text{SU(2)}_{\text{D}}$ group breaking. In subsection VI.2, we will take
$n=9$ as an example, to show the readers how our general algorithm described
in Section IV and V works when the $\text{SU(2)}_{\text{D}}$ breaks into an
$A_{4}$ group. These general algorithms can by principle generalized to any
value of $n$, however, we found that for some larger $n$, sometimes the simple
analytical results are not easy to acquire, since our general algorithm
usually gives the results written in the cumbersome tangled recursive radical
signs, which are extremely difficult to untangle. Numerical results with
decimal evaluations verify that our general algorithm does work well, and the
precision is sufficient if one only wants to input the results into some
numerical packages like micrOMEGAs Belanger:2001fz  or MadDM Backovic:2013dpa
. However in this paper, we dedicate to the more simplified analytical
results. We thereby develop some special however simpler algorithms other than
Section IV and V to find out the analytical results in some cases. One example
is shown in Subsection VI.3.

###  VI.1 4-d scalar of SU(2)

Consider the dark sector $\text{SU(2)}_{\text{D}}$ symmetry to be completely
broken by a scalar quartet. The eigenvectors of $\vec{n}\cdot\vec{J}$ with
eigenvalues 3/2, 1/2, -1/2, -3/2 are respectively given in Appendix C.
Assuming a vacuum invariant under a $Z_{3}$ symmetry and setting the rotation
axis to be $z$-axis, the vacuum configuration in basis of $J_{3}$ eigenvector
should be

$\vec{v}=a_{3/2}\vec{v}_{3/2}+a_{-3/2}\vec{v}_{-3/2}.$ (54)

where $m$ in $a_{m},v_{m}$ is eigenvalues of the basis.

If there exists another nontrivial $Z_{3}$ rotation, Eq.(40) yields

$\vec{v}=b_{3/2}\vec{v}^{\prime}_{3/2}+b_{-3/2}\vec{v}^{\prime}_{-3/2},$ (55)

where $\vec{v}^{\prime}_{m}$ are the eigenvectors of
$(J_{1}\sin\theta+J_{3}\cos\theta)$. Then Eq. (41) gives

$\left\\{\begin{aligned}
&a_{3/2}\cot\left(\frac{\theta}{2}\right)+a_{-3/2}=0\\\
&-a_{3/2}\tan\left(\frac{\theta}{2}\right)+a_{-3/2}=0\end{aligned}\right.,$
(56)

which have no nonzero solution so that no other $Z_{3}$ symmetry exists.

###  VI.2 9-d scalar of SU(2)

Known from Appendix B, the remnant symmetry of the vacuum in SU(2)
9-dimensional representation can be $S_{4}$, $D_{4}$, etc.. Since the $D_{4}$
results are simple, for brevity we only show the processes for the $S_{4}$,
which has six two-fold symmetry axes, four three-fold symmetry axes and three
four-fold symmetry axes. One can pick up the following generators

$\displaystyle S$ $\displaystyle=$ $\displaystyle e^{iJ_{3}\frac{\pi}{2}}$
$\displaystyle T$ $\displaystyle=$ $\displaystyle
e^{i(J_{1}\sin\theta+J_{3}\cos\theta)\frac{2\pi}{3}}$ $\displaystyle K$
$\displaystyle=$ $\displaystyle e^{iJ_{1}\pi}$ (57)

with one of the four-fold symmetry axes (or the axis of a $Z_{4}$-subgroup)
along the $z$-direction, and with one three-fold symmetry axis lying at the
$x$-$z$ plane. Here $\theta=\arctan\sqrt{2}$. Again, under the 9-dimensional
representation case, $V_{(9)}(S)$, $V_{(9)}(T)$, $V_{(9)}(K)$ are adopted
alternatively by replacing the $J_{1,2,3}$ in Eq. (57) with the
$V_{(9)}(J_{1,2,3})$. The concrete form of them can be found in Appendix D.

If the vacuum configuration remain invariant under the $V_{(9)}(S)$
transformations, according to Eq. (34) again, under the $J_{3}$ representation
basis the VEV is restricted in the form of

$\vec{v}=a_{4}\vec{v}_{4}+a_{0}\vec{v}_{0}+a_{-4}\vec{v}_{-4}.$ (58)

where $m$ in $a_{m},\vec{v}_{m}$ are the eigenvalues of the basis, $a_{0}$ is
real and $a_{-4}=a_{4}^{*}$. On rotation axis corresponding to $T$, following
the processes before Eq. (40), we therefore obtain the conditions

$\displaystyle b_{\pm 1}=b_{\pm 2}=b_{\pm 4}=0,$ (59)

inducing the equations

$\vec{v^{\prime}}^{\dagger}_{k}(\theta)\vec{v}=0,\quad(k=\pm 1,\pm 2,\pm 4),$
(60)

where $\vec{v^{\prime}}_{k}(\theta)$ are the eigenvectors of the
$V_{(9)}(J_{1}\sin\theta+J_{3}\cos\theta)$ satisfying
$V_{(9)}(T)\vec{v^{\prime}}_{k}(\theta)=e^{ik\frac{2\pi}{3}}\vec{v^{\prime}}_{k}(\theta)$.
Then, by solving Eq. (60) regarding $a_{0,\pm 4}$ as the unknown parameters,
we finally acquire

$\qquad a_{-4}=a_{4},\qquad a_{0}=-\sqrt{\frac{14}{5}}a_{4}.$ (61)

corresponding to the vacuum configuration

$\vec{v}=\left(a_{4},0,0,0,-\sqrt{\frac{14}{5}}a_{4},0,0,0,a_{4}\right)^{\mathrm{T}}$
(62)

From Eq. (57), (175) and (176), we can write down the matrix forms of the
$V_{(9)}(S,T,K)$. The character table algorithm tells us that the SU(2)
9-dimensional representation under the $S_{4}$ subgroup can be reduced
9=1+3+3’+2. The character of each irreducible representation of $S_{4}$ group
is listed in Appendix D. The 1 subspace is equivalent to the VEV configuration
Eq. (62). The other irreducible subspaces are then computed by following the
processes described in Section V. In this paper, we only show the detailed
process for finding the 3 subspace as an example for brevity.

Following Eq. (45), we consider a projection operator $P_{\textbf{3}}$
projecting the 9-d scalar $\Phi_{(9)}$ into the triplet representation 3. We
learn from Eq. (205) that $S_{\textbf{3}}$ takes three eigenvalues, $i$, $1$,
and $-i$. According to Eq. (57) we acquire

$\displaystyle V_{(9)}(S)$ $\displaystyle=$ $\displaystyle
e^{iV_{(9)}(J_{3})\frac{\pi}{2}}=\text{Diag}\\{e^{i\frac{4\pi}{2}},e^{i\frac{3\pi}{2}},e^{i\frac{2\pi}{2}},e^{i\frac{\pi}{2}},1,e^{-i\frac{\pi}{2}},e^{-i\frac{2\pi}{2}},e^{-i\frac{3\pi}{2}},e^{-i\frac{4\pi}{2}}\\}$
(63) $\displaystyle=$ $\displaystyle\text{Diag}\\{1,-i,-1,i,1,-i,-1,i,1\\}.$

Therefore, the projection operator $P_{\textbf{3}}$ should collect all the
$V_{(9)}(S)$’s eigenvectors of the eigenvalues $i$, $1$, $-i$ into the
corresponding $S_{\textbf{3}}$’s eigenvectors by taking the form of

$P_{\textbf{3}}=\begin{pmatrix}0&0&0&p_{1}&0&0&0&p_{2}&0\\\
p_{3}&0&0&0&p_{4}&0&0&0&p_{5}\\\ 0&p_{6}&0&0&0&p_{7}&0&0&0\end{pmatrix},$ (64)

where $p_{k}$ ($k=1,\dots,7$) are unknown parameters. Then Eq. (48) becomes

$P_{\textbf{3}}V_{(9)}(S)\Phi_{(9)}=S_{\textbf{3}}\Phi_{\textbf{3}},$ (65)

where $\Phi_{\textbf{3}}$ is the irreducible space of the 3 representation.

Following Eq. (50), we diagonalize $T_{\textbf{3}}$ in Eq. (206) by

$\displaystyle
T^{\prime}_{\textbf{3}}=Q^{\dagger}T_{\textbf{3}}Q=\text{Diag}\\{e^{-i\frac{2\pi}{3}},e^{i\frac{2\pi}{3}},1\\}$
(66)

where

$\displaystyle
Q=\left(\begin{array}[]{ccc}\frac{\sqrt{3}-3}{6}&-\frac{\sqrt{3}+3}{6}&\frac{1}{\sqrt{3}}\\\
-\frac{1}{\sqrt{3}}&\frac{1}{\sqrt{3}}&\frac{1}{\sqrt{3}}\\\
\frac{\sqrt{3}+3}{6}&\frac{3-\sqrt{3}}{6}&\frac{1}{\sqrt{3}}\\\
\end{array}\right).$ (70)

Then, $V_{(9)}(T)$ in Eq. (194) is also diagonalized by

$\displaystyle
T^{\prime}_{(9)}=R^{\dagger}V_{(9)}(T)R=\text{Diag}\\{e^{i\frac{2\pi}{3}},1,e^{-i\frac{2\pi}{3}},e^{i\frac{2\pi}{3}},1,e^{-i\frac{2\pi}{3}},e^{i\frac{2\pi}{3}},1,e^{-i\frac{2\pi}{3}}\\},$
(71)

where $R$ can be computed by listing all eigenvectors of the
$V_{(9)}(T)=e^{i\frac{2\pi}{3}V_{(9)}(J_{1}\sin\theta+J_{3}\cos\theta)}$. The
results are given by Eq. (183)-(186). Arranging them within $R$ gives

$R=\\{\vec{v}^{\prime}_{4},\vec{v}^{\prime}_{3},\vec{v}^{\prime}_{2},\vec{v}^{\prime}_{1},\vec{v}^{\prime}_{0},\vec{v}^{\prime}_{-1},\vec{v}^{\prime}_{-2},\vec{v}^{\prime}_{-3},\vec{v}^{\prime}_{-4}\\}.$
(72)

Therefore, according to Eq. (53), the $P_{\textbf{3}}^{\prime}$, projecting
the $\Phi_{(9)}$ into the 3 irreducible space from the
$V_{(9)}(T)=e^{i\frac{2\pi}{3}V_{(9)}(J_{1}\sin\theta+J_{3}\cos\theta)}$
representation, satisfying

$P^{\prime}_{\textbf{3}}T^{\prime}_{(9)}=T^{\prime}_{\textbf{3}}P^{\prime}_{\textbf{3}}.$
(73)

According to Eq. (52), the detailed formalism of the $P_{\textbf{3}}^{\prime}$
depending on $p_{1,2,3,4,5,6,7}$ is computed by

$\displaystyle P_{\textbf{3}}^{\prime}=Q^{\dagger}P_{\textbf{3}}R.$ (74)

However, Eq. (52) requires that the $P_{\textbf{3}}$ only connects the
corresponding eigenvectors with the same eigenvalues between
$T_{\textbf{3}}^{\prime}$ and $T_{(9)}^{\prime}$, restricting its expression
to be

$P^{\prime}_{\textbf{3}}=\left(\begin{array}[]{ccccccccc}0&0&p^{\prime}_{1}&0&0&p^{\prime}_{2}&0&0&p^{\prime}_{3}\\\
p^{\prime}_{4}&0&0&p^{\prime}_{5}&0&0&p^{\prime}_{6}&0&0\\\
0&p^{\prime}_{7}&0&0&p^{\prime}_{8}&0&0&p^{\prime}_{9}&0\\\
\end{array}\right),$ (75)

where $p^{\prime}_{k}$($k=1,\dots,9$) are unknown parameters. The zero
elements of the $P_{\textbf{3}}^{\prime}$ in Eqn. 75) prompt the eighteen
homogeneous equations

$\displaystyle(P^{\prime}_{\textbf{3}})_{(1,1)}=(P^{\prime}_{\textbf{3}})_{(1,2)}=(P^{\prime}_{\textbf{3}})_{(1,4)}=(P^{\prime}_{\textbf{3}})_{(1,5)}=(P^{\prime}_{\textbf{3}})_{(1,7)}=(P^{\prime}_{\textbf{3}})_{(1,8)}$
(76) $\displaystyle=$
$\displaystyle(P^{\prime}_{\textbf{3}})_{(2,2)}=(P^{\prime}_{\textbf{3}})_{(2,3)}=(P^{\prime}_{\textbf{3}})_{(2,5)}=(P^{\prime}_{\textbf{3}})_{(2,6)}=(P^{\prime}_{\textbf{3}})_{(2,8)}=(P^{\prime}_{\textbf{3}})_{(2,9)}$
$\displaystyle=$
$\displaystyle(P^{\prime}_{\textbf{3}})_{(3,1)}=(P^{\prime}_{\textbf{3}})_{(3,3)}=(P^{\prime}_{\textbf{3}})_{(3,4)}=(P^{\prime}_{\textbf{3}})_{(3,6)}=(P^{\prime}_{\textbf{3}})_{(3,7)}=(P^{\prime}_{\textbf{3}})_{(3,9)}=0$

depending on $p_{k},(k=1,\dots,7)$, where $(P^{\prime}_{\textbf{3}})_{(i,j)}$
indicates the element of the row $i$, column $j$ in $P^{\prime}_{\textbf{3}}$.
Solving Eq. (76) gives

$p_{2}=\frac{p_{1}}{\sqrt{7}},\quad p_{3}=\frac{2p_{1}}{\sqrt{7}},\quad
p_{4}=0,\quad p_{5}=-\frac{2p_{1}}{\sqrt{7}},\quad
p_{6}=-\frac{p_{1}}{\sqrt{7}},\quad p_{7}=-p_{1},$ (77)

with the only undetermined parameter $p_{1}$.

Finally, the requirement $P^{\dagger}_{\textbf{3}}P_{\textbf{3}}=I_{3\times
3}$ normalizes the $p_{1}$ by

$\frac{8}{7}p_{1}p_{1}^{*}=1.$ (78)

Selecting $p_{1}=i\sqrt{\frac{7}{8}}$, the triplet subspace
$\Phi_{\textbf{3}}$ is then given as

$\Phi_{\textbf{3}}=\frac{1}{\sqrt{2}}\begin{pmatrix}iG^{+}\\\ G^{0}\\\
-iG^{-}\end{pmatrix}=\left(\begin{array}[]{c}\frac{i\sqrt{7}}{4}\Delta_{1}+\frac{i}{4}\Delta_{-3}\\\
-\text{Im$\Delta$}_{4}\\\
-\frac{i\sqrt{7}}{4}\Delta_{-1}-\frac{i}{4}\Delta_{3}\\\ \end{array}\right)$
(79)

Here we use the symbols $G^{0,\pm}$ to denote the elements in
$\Phi_{\textbf{3}}$ due to the fact that these are actually three Goldstone
bosons to be eaten by the longitudinal polarizations of the vector fields.
This can also be verified by expanding the kinetic term
$D^{\mu}\Phi_{(9)}^{\dagger}D_{\mu}\Phi_{(9)}$ in Eq. (4) relative to the
vacuum configuration given by Eq. (62),

$\displaystyle\sqrt{2}\Phi_{(9)}=\left(a_{4}+\Delta_{4}^{\prime},\Delta_{3},\Delta_{2},\Delta_{1},-\sqrt{\frac{14}{5}}a_{4}+\Delta_{0}^{\prime},\Delta_{-1},\Delta_{-2},\Delta_{-3},a_{4}+\Delta^{\prime}_{-4}\right)^{\mathrm{T}},$
(80)

where

$\displaystyle\Delta_{4}$ $\displaystyle=$ $\displaystyle
a_{4}+\Delta_{4}^{\prime},$ $\displaystyle\Delta_{0}$ $\displaystyle=$
$\displaystyle-\sqrt{\frac{14}{5}}a_{4}+\Delta_{0}^{\prime}.$ (81)

Then, the bilinear terms connecting the partial derivatives of the three gauge
fields with $\Delta_{1,2,3}$ give the formalism of the Goldstone degrees of
freedom, which are exactly the same with Eq. (79). The real Goldstone triplet
formalism can also be written by defining

$G_{1}=\frac{G^{-}-G^{+}}{\sqrt{2}i},\qquad
G_{2}=-\frac{G^{-}+G^{+}}{\sqrt{2}},\qquad G_{3}=G_{0}.$ (82)

and therefore

$\begin{pmatrix}G_{1}\\\ G_{2}\\\
G_{3}\end{pmatrix}=\begin{pmatrix}-\frac{\sqrt{7}}{2}\text{Im}\Delta_{1}+\frac{1}{2}\text{Im}\Delta_{3}\\\
-\frac{\sqrt{7}}{2}\text{Re}\Delta_{1}-\frac{1}{2}\text{Re}\Delta_{3}\\\
-\sqrt{2}\text{Im}\Delta_{4}\end{pmatrix}.$ (83)

For the rest of the subspace classifications, we neglect the detailed
processes however only show the final results in Section VII.

###  VI.3 13-d scalar of SU(2)

As shown in Appendix B, the vacuum of the 13-dimensional representation
$\Phi_{(13)}$ of the $\text{SU(2)}_{\text{D}}$ group can be invariant under an
$A_{5}$ subgroup. In this case, we premise the vacuum has an $A_{5}$ symmetry
and then work out the concrete vacuum configuration using the method mentioned
in Section IV. The $A_{5}$ group contains five-fold axes, three-fold axes, and
two-fold axes. We can rotate to the direction that one five-fold axis aligns
to the $z$-axis, one three-fold axis and one two-fold axis lies within the
$x$-$z$ plane. With these configurations, the three generators of the $A_{5}$
group can be given by

$\displaystyle S$ $\displaystyle=$ $\displaystyle e^{iJ_{3}\frac{2\pi}{5}}$
$\displaystyle T$ $\displaystyle=$ $\displaystyle
e^{i(J_{1}\sin\theta_{53}+J_{3}\cos\theta_{53})\frac{2\pi}{3}}$ (84)
$\displaystyle K$ $\displaystyle=$ $\displaystyle
e^{i(J_{1}\sin\theta_{52}+J_{3}\cos\theta_{52})\pi},$ (85)

where $\theta_{53}$ is the angle between the $Z_{5}$ and $Z_{3}$ rotation
axes, while $\theta_{52}$ is the angle between the $Z_{5}$ and $Z_{2}$
rotation axes. There values are given by
$\theta_{53}=-\arctan\left(3-\sqrt{5}\right)$,
$\theta_{52}=-\arctan\left(\frac{\sqrt{5}+1}{2}\right)$. Therefore, the
invariance of the VEV under the $V_{(13)}(S)$ transformation constrains its
expression to be

$\vec{v}=a_{5}\vec{v}_{5}+a_{0}\vec{v}_{0}+a_{-5}\vec{v}_{-5},$ (86)

where $m$ in $a_{m},\vec{v}_{m}$ is the eigenvalue of the basis.

Again, similar to the processes described in Subsection VI.2, solving Eq. (41)
by replacing the $\vec{n}\cdot\vec{J}$ in Eq. (35) with $\theta=\theta_{53}$,
one finally arrives at

$\vec{v}=\left(0,a_{5},0,0,0,0,\sqrt{\frac{11}{7}}a_{5},0,0,0,0,a_{5},0\right)^{\mathrm{T}},$
(87)

where $a_{5}$ is an arbitrary real parameter. With the aid of the character
table displayed in Appendix D, the 13-dimensional target space of
$\text{SU(2)}_{\text{D}}$ can be decomposed into the direct sum of the
$\textbf{13}=\textbf{1}+\textbf{3}+\textbf{4}+\textbf{5}$ subspaces, which are
invariant under the $A_{5}$. The 1 subspace has been found in the form of Eq.
(87). One can then in principle follow the algorithm described in Section V to
solve all the detailed expressions of the 3, 4, 5 representations. However,
practical calculations show that the coefficients are extremely complicated,
and is very difficult for us to simplify the analytical expressions of the
recursive radical signs. Therefore, in this subsection, we adopt some other
flexible strategies in light of such kind of situations.

The only one triplet of $A_{5}$ group in this case is exactly the Goldstone
degrees of freedom eaten by gauge bosons. By shifting the $\Phi_{(13)}$
relative to the vacuum configuration in Eq. (87)

$\displaystyle\sqrt{2}\Phi_{(13)}=$
$\displaystyle\left(\Delta_{6},a_{5}+\Delta_{5}^{\prime},\Delta_{4},\Delta_{3},\Delta_{2},\Delta_{1},\sqrt{\frac{11}{7}}a_{5}+\Delta_{0}^{\prime},\Delta_{-1},\Delta_{-2},\Delta_{-3},\Delta_{-4},a_{5}+\Delta^{\prime}_{-5},\Delta_{-6}\right)^{\mathrm{T}},$

and substitute this into the $(D^{\mu}\Phi_{(9)})^{\dagger}D_{\mu}\Phi_{(9)}$
in Eq. (4), one can acquire the expressions of Goldstone fields by collecting
the bilinear mixing terms between the partial derivatives of the three gauge
fields with $\Delta_{m}$. The results are given by

$G^{+}=-\frac{\sqrt{3}}{5}\Delta_{6}-\frac{\sqrt{66}}{10}\Delta_{1}+\frac{\sqrt{22}}{10}\Delta_{-4},\qquad
G^{-}=(G^{+})^{*},\qquad G^{0}=-\sqrt{2}\text{Im}\Delta_{5},$ (89)

where $G^{\pm},G^{0}$ are eaten by $X^{\pm},Z^{\prime}$ respectively as shown
in Appendix A. This is equivalent to the projection operator

$P_{\textbf{3}}=\left(\begin{array}[]{ccccccccccccc}\frac{-i\sqrt{3}}{5}&0&0&0&0&\frac{-i\sqrt{66}}{10}&0&0&0&0&\frac{i\sqrt{22}}{10}&0&0\\\
0&\frac{i}{\sqrt{2}}&0&0&0&0&0&0&0&0&0&\frac{-i}{\sqrt{2}}&0\\\
0&0&\frac{-i\sqrt{22}}{10}&0&0&0&0&\frac{i\sqrt{66}}{10}&0&0&0&0&\frac{i\sqrt{3}}{5}\\\
\end{array}\right).$ (90)

Similarly with Eq. (82), the real triplet formalism can also be written as

$\begin{pmatrix}G_{1}\\\ G_{2}\\\
G_{3}\end{pmatrix}=\begin{pmatrix}\frac{\sqrt{6}}{5}\text{Im}\Delta_{6}+\frac{\sqrt{33}}{5}\text{Im}\Delta_{1}+\frac{\sqrt{11}}{5}\text{Im}\Delta_{4}\\\
\frac{\sqrt{6}}{5}\text{Re}\Delta_{6}+\frac{\sqrt{33}}{5}\text{Re}\Delta_{1}-\frac{\sqrt{11}}{5}\text{Re}\Delta_{4}\\\
-\sqrt{2}\text{Im}\Delta_{5}\end{pmatrix}.$ (91)

Then let us turn to the 5-subspace. Connecting the eigenvectors with the same
eigenvalues of both the matrix form of the $V_{13}(S)$ and the diagonalized
$S_{\textbf{5}}$ defined in Eq. (221) restricts the projection operator
$P_{\textbf{5}}$ satisfying

$P_{\textbf{5}}V_{(13)}(S)\Phi_{(13)}=S_{\textbf{5}}\Phi_{\textbf{5}},$ (92)

within the formalism of

$P_{\textbf{5}}=\left(\begin{array}[]{ccccccccccccc}0&0&0&0&p_{1}&0&0&0&0&p_{2}&0&0&0\\\
p_{3}&0&0&0&0&p_{4}&0&0&0&0&p_{5}&0&0\\\
0&p_{6}&0&0&0&0&p_{7}&0&0&0&0&p_{8}&0\\\
0&0&p_{9}&0&0&0&0&p_{10}&0&0&0&0&p_{11}\\\
0&0&0&p_{12}&0&0&0&0&p_{13}&0&0&0&0\\\ \end{array}\right),$ (93)

where $p_{1\text{-}13}$ are the undetermined parameters. Since all the
subspaces corresponding to different irreducible representations should be
orthogonal to each other, and each row of the projector operators can be
regarded as the basis within the $\Phi_{(15)}$ space that can be expanded into
the corresponding irreducible subspaces, each row of the Eq. (93) should be
perpendicular to the rows in Eq. (90) and (87). Then it is easy to obtain that
for the third row in Eq. (93), $p_{6,7,8}$ should satisfy

$p_{7}=-2\sqrt{\frac{7}{11}}p_{6},\quad p_{8}=p_{6}.$ (94)

As a result,

$P_{\textbf{5},3}\propto\left(0,1,0,0,0,0,-2\sqrt{\frac{7}{11}},0,0,0,0,1,0\\\
\right),$ (95)

where $P_{\textbf{5},k}$ means the $k$-th row of the projector operator
$P_{\textbf{5}}$.

As we have illustrated after Eq. (46), each column of the
$P_{\textbf{5}}^{\dagger}$ is a vector within the basis of the 5-subspace and
the condition that this subspace is closed under the transformation of any
element of the $A_{5}$ guarantees that

$\displaystyle
V_{(13)}(U)P_{\textbf{5},i}^{\dagger}=(U_{\textbf{5}})_{ij}P_{\textbf{5},j}^{\dagger}$
(96)

for any element $U\in A_{5}$, where $i,j$ run within $1$-$5$. Therefore,
specifically, if we adopt $U=e^{iJ_{3}\pi}e^{iJ_{2}\theta_{55}}$, where
$\theta_{55}=2\theta_{52}$ is the angle between two next-to-nearest neighbor
five-fold axes, one can verify by observing a regular dodecahedron that such a
$U$ rotates a five-fold axis aligning to the $z$-direction towards its next-
to-nearest neighbor by the $e^{iJ_{2}\theta_{55}}$ operation, then match all
the other three-fold axes and two-fold axes through the $e^{iJ_{3}\pi}$
operation to flip the direction of the pentagon at the top. Therefore,
$U=e^{iJ_{3}\pi}e^{iJ_{2}\theta_{55}}\in A_{5}$. With Eq. (95), one can
compute that

$\displaystyle
V_{(13)}(U)P_{\textbf{5},3}^{\dagger}=e^{iV_{(13)}(J_{3})\pi}e^{iV_{(13)}(J_{2})\theta_{55}}P_{\textbf{5},3}^{\dagger}$
(97) $\displaystyle\propto$
$\displaystyle\left(-\frac{\sqrt{66}}{25},-\frac{\sqrt{22}}{50},-\frac{9}{25},\frac{\sqrt{30}}{25},\frac{2\sqrt{30}}{25},-\frac{\sqrt{3}}{25},\frac{\sqrt{14}}{25},-\frac{\sqrt{3}}{25},\frac{2\sqrt{30}}{25},\frac{\sqrt{30}}{25},-\frac{9}{25},\right.$
$\displaystyle\left.-\frac{\sqrt{22}}{50},-\frac{\sqrt{66}}{25}\right)^{\text{T}}\in
P_{\textbf{5}}\Phi_{(13)}.$

We can express that Eq. (97) is a member within the 5-subspace, so it can be
expanded in terms of the combination of the basis $P_{\textbf{5},1}$,
$P_{\textbf{5},2}$, $P_{\textbf{5},3}$, $P_{\textbf{5},4}$, and
$P_{\textbf{5},5}$ as denoted in Eq. (96). Notice that the formalism of each
$P_{\textbf{5},i}$ is highly restricted by Eq. (93), therefore it is easy to
compare the coefficients and find out the relations among the $p_{1,2}$,
$p_{3,4,5}$, $p_{9,10,11}$ and $p_{12,13}$. After doing the normalization, one
acquire

$P_{\textbf{5}}=\left(\begin{array}[]{ccccccccccccc}0&0&0&0&\frac{2}{\sqrt{5}}&0&0&0&0&\frac{1}{\sqrt{5}}&0&0&0\\\
\frac{\sqrt{11}}{5}&0&0&0&0&\frac{\sqrt{2}}{10}&0&0&0&0&\frac{3\sqrt{6}}{10}&0&0\\\
0&\frac{\sqrt{22}}{10}&0&0&0&0&-\frac{\sqrt{14}}{5}&0&0&0&0&\frac{\sqrt{22}}{10}&0\\\
0&0&\frac{3\sqrt{6}}{10}&0&0&0&0&\frac{\sqrt{2}}{10}&0&0&0&0&\frac{\sqrt{11}}{5}\\\
0&0&0&\frac{1}{\sqrt{5}}&0&0&0&0&\frac{2}{\sqrt{5}}&0&0&0&0\\\
\end{array}\right),$ (98)

corresponding to a series of degenerate fields

$\Phi_{\textbf{5}}=\frac{1}{\sqrt{2}}\begin{pmatrix}D^{++}\\\ D^{+}\\\
D^{0}\\\ D^{-}\\\
D^{--}\end{pmatrix}=\left(\begin{array}[]{c}\sqrt{\frac{2}{5}}\Delta_{2}+\frac{1}{\sqrt{10}}\Delta_{-3}\\\
\frac{1}{10}\Delta_{1}+\frac{3\sqrt{3}}{10}\Delta_{-4}+\frac{1}{5}\sqrt{\frac{11}{2}}\Delta_{6}\\\
\frac{\sqrt{11}}{10}\Delta_{-5}+\frac{\sqrt{11}}{10}\Delta_{5}-\frac{\sqrt{7}}{5}\Delta_{0}\\\
\frac{1}{10}\Delta_{-1}+\frac{3}{10}\sqrt{3}\Delta_{4}+\frac{1}{5}\sqrt{\frac{11}{2}}\Delta_{-6}\\\
\sqrt{\frac{2}{5}}\Delta_{-2}+\frac{1}{\sqrt{10}}\Delta_{3}\\\
\end{array}\right).$ (99)

The corresponding real scalars can be defined by

$\displaystyle D_{1}=\frac{D^{++}+D^{--}}{\sqrt{2}},\qquad
D_{2}=\frac{D^{++}-D^{--}}{i\sqrt{2}},$ $\displaystyle
D_{3}=\frac{D^{+}+D^{-}}{\sqrt{2}},\qquad
D_{4}=\frac{D^{+}-D^{-}}{i\sqrt{2}},\qquad D_{5}=D^{0},$ (100)

so therefore

$\begin{pmatrix}D_{1}\\\ D_{2}\\\ D_{3}\\\ D_{4}\\\
D_{5}\end{pmatrix}=\left(\begin{array}[]{c}2\sqrt{\frac{2}{5}}\text{Re$\Delta$}_{2}+\sqrt{\frac{2}{5}}\text{Re$\Delta$}_{3}\\\
2\sqrt{\frac{2}{5}}\text{Im$\Delta$}_{2}-\sqrt{\frac{2}{5}}\text{Im$\Delta$}_{3}\\\
\frac{1}{5}\text{Re$\Delta$}_{1}+\frac{3\sqrt{3}}{5}\text{Re$\Delta$}_{4}+\frac{\sqrt{22}}{5}\text{Re$\Delta$}_{6}\\\
\frac{1}{5}\text{Im$\Delta$}_{1}-\frac{3\sqrt{3}}{5}\text{Im$\Delta$}_{4}+\frac{\sqrt{22}}{5}\text{Im$\Delta$}_{6}\\\
\frac{\sqrt{22}}{5}\text{Re$\Delta$}_{5}-\frac{\sqrt{14}}{5}\Delta_{0}\\\
\end{array}\right).$ (101)

The remained 4 irreducible subspace is hence acquired following the similar
algorithm, while in this subsection we neglect the detailed processes. The
final results are displayed in Section VII.

##  VII Subspaces for $(2n+1)$-d representation of SU(2)

In the previous sections, we have shown the algorithms adopted to calculate
the vacuum and subspace decomposition with the discrete symmetry broken from
the $\text{SU(2)}_{\text{D}}$ symmetry by a single $\Phi_{(n)}$ Higgs
multiplet. In this section, we list the results of breaking the
$\text{SU(2)}_{\text{D}}$ symmetry to the discrete groups for each odd
$n=5$-$21$, while neglect the detailed processes accomplishing them. As we
have illustrated, for all the even $n$ selections up to $n=20$, only the
simplest discrete $Z_{N}$ symmetries can be remained, so we do not list the
results here. For all the odd $n\leq 21$ values, all the possible non-trivial
discrete subgroups that the $\text{SU(2)}_{\text{D}}$ group can break into
have been calculated by Ref. Etesi:1996urw  and are repeated in Table 4 of
Appendix B as a bookkeeping. For all these cases we are going to show the
results of the detailed VEV configurations as well as the irreducible
subspaces of the Higgs multiplet, whereafter we will illustrate the
orientations of the axes of the $A_{4}$, $A_{5}$ and $S_{4}$ groups in the
Subsection VI.2 and VI.3, respectively. From Table 4 we learn that when
$n=13$, $17$, $19$ and $21$, the $\text{SU(2)}_{D}$ group can be broken into
the $S_{4}$. In this case we rotate toward a direction so that a four-fold
axis is along the $z$-axis, while one of the three-fold axis is within the
$x$-$z$ plain, just similar to the instructions illustrated in Subsection D.1.
Furthermore, when $n=13$, $19$ and $21$, the $\text{SU(2)}_{\text{D}}$ group
can be broken into the $A_{4}\subset S_{4}$ as well, making it convenient to
cast the $S_{4}$ results straightforwardly to the $A_{4}$ cases. Therefore, in
the cases of the $n=13$, $19$ and $21$, we adopt another orientation of the
$A_{4}$ axes in accordance with the $S_{4}$ assignments, so that one of the
two-fold axes of the $A_{4}$ group is set aligning to the $z$-direction, while
one of the three-fold axes is within the $x$-$z$ plain.

In the following, the multiplets are listed corresponding to the irreducible
subspace sequentially, which are presented in Table 3 for $n<13$. For all
these $n$-d representations of the SU(2), we give their characters related to
the discrete group in Appendix D.

5-d scalar | vacuum with $Z_{2}\oplus Z_{2}$ symmetry
---|---
$\Phi_{(5)}=\frac{1}{\sqrt{2}}\left(\Delta_{2},\Delta_{1},\Delta_{0},\Delta_{-1},\Delta_{-2}\right)^{\text{T}}$ | $\left(a,0,0,0,a\right)^{\text{T}}$
subspace 5= 1+1+$1_{a}$+$1_{b}$+$1_{c}$
$\sqrt{2}\text{Re}\Delta_{2},\quad\Delta_{0},\quad\sqrt{2}\text{Im}\Delta_{1}^{\text{G}},\quad-\sqrt{2}\text{Re}\Delta_{1}^{\text{G}},\quad-\sqrt{2}\text{Im}\Delta_{2}^{\text{G}}$
with the last three singlets to be Gosdstone bosons corresponding to
$G_{1},G_{2},G_{3}$ respectively.
7-d scalar | vacuum with $A_{4}$ symmetry
$\Phi_{(7)}=\frac{1}{\sqrt{2}}\left(\Delta_{3},\Delta_{2},\Delta_{1},\Delta_{0},\Delta_{-1},\Delta_{-2},\Delta_{-3}\right)^{\text{T}}$ | $\left(0,a,0,0,0,a,0\right)^{\text{T}}$
subspace 7= 1+3+3
$\sqrt{2}\text{Re}\Delta_{2},\quad\left(\begin{array}[]{c}\frac{\sqrt{5}}{2}\text{Im$\Delta$}_{1}+\frac{\sqrt{3}}{2}\text{Im$\Delta$}_{3}\\\
-\frac{\sqrt{5}}{2}\text{Re$\Delta$}_{1}+\frac{\sqrt{3}}{2}\text{Re$\Delta$}_{3}\\\
-\sqrt{2}\text{Im$\Delta$}_{2}\\\
\end{array}\right)^{\text{G}},\quad\begin{pmatrix}\frac{\sqrt{3}}{2}\text{Re}\Delta_{1}+\frac{\sqrt{5}}{2}\text{Re}\Delta_{3}\\\
\frac{\sqrt{3}}{2}\text{Im}\Delta_{1}-\frac{\sqrt{5}}{2}\text{Im}\Delta_{3}\\\
\Delta_{0}\end{pmatrix}.$
7-d scalar | vacuum with $D_{3}$ symmetry
$\Phi_{(7)}=\frac{1}{\sqrt{2}}\left(\Delta_{3},\Delta_{2},\Delta_{1},\Delta_{0},\Delta_{-1},\Delta_{-2},\Delta_{-3}\right)^{\text{T}}$ | $\left(a,0,0,0,0,0,a\right)^{\text{T}}$
subspace 7= 1+1’+1’+2+2
$\sqrt{2}\text{Re}\Delta_{3},\quad-\sqrt{2}\text{Im}\Delta_{3}^{\text{G}},\quad\Delta_{0},\quad\begin{pmatrix}\sqrt{2}\text{Im}\Delta_{2}\\\
-\sqrt{2}\text{Re}\Delta_{2}\end{pmatrix}^{\text{G}},\quad\begin{pmatrix}\sqrt{2}\text{Re}\Delta_{1}\\\
\sqrt{2}\text{Im}\Delta_{1}\end{pmatrix}$
with the second singlet and the first doublet to be Goldstone bosons
corresponding to $G_{3}$, $(G_{1},G_{2})^{T}$.
9-d scalar | vacuum with $S_{4}$ symmetry
$\Phi_{(9)}=\frac{1}{\sqrt{2}}\left(\Delta_{4},\Delta_{3},\Delta_{2},\Delta_{1},\Delta_{0},\Delta_{-1},\Delta_{-2},\Delta_{-3},\Delta_{-4}\right)^{\text{T}}$ | $\left(a,0,0,0,-\sqrt{\frac{14}{5}}a,0,0,0,a\right)^{\text{T}}$
subspace $\textbf{9}=\textbf{1}+\textbf{3}+\textbf{3'}+\textbf{2}$
$\left(\sqrt{\frac{5}{6}}\text{Re$\Delta$}_{4}-\frac{1}{2}\sqrt{\frac{7}{3}}\Delta_{0}\right),\quad\begin{pmatrix}-\frac{\sqrt{7}}{2}\text{Im}\Delta_{1}+\frac{1}{2}\text{Im}\Delta_{3}\\\
-\frac{\sqrt{7}}{2}\text{Re}\Delta_{1}-\frac{1}{2}\text{Re}\Delta_{3}\\\
-\sqrt{2}\text{Im}\Delta_{4}\end{pmatrix}^{\text{G}},\quad\begin{pmatrix}\frac{\sqrt{7}}{2}\text{Re$\Delta$}_{3}-\frac{1}{2}\text{Re$\Delta$}_{1}\\\
\frac{\sqrt{7}}{2}\text{Im$\Delta$}_{3}+\frac{1}{2}\text{Im$\Delta$}_{1}\\\
\sqrt{2}\text{Re$\Delta$}_{2}\end{pmatrix},\quad\left(\begin{array}[]{c}\frac{1}{2}\sqrt{\frac{5}{3}}\Delta_{0}+\sqrt{\frac{7}{6}}\text{Re$\Delta$}_{4}\\\
\sqrt{2}\text{Im$\Delta$}_{2}\\\ \end{array}\right)$

Table 3: The subspaces for $(2n+1)$-d representation of SU(2) with specific
remnant symmetry and vacuum configuration.

For $n\geq 13$, sometimes there are more than one non-Abelian discrete
symmetry groups that can remain. Since usually the results are too lenghthy to
be placed inside a table, we list them in the text below. We also have to note
that for all the multiplets both in Table 3 and below, the first triplet
representation of the discrete group corresponds to the three Goldstone
bosons.

  * •

For the 13-d representation, the general scalar field is given by

$\Phi_{(13)}=\frac{1}{\sqrt{2}}\left(\Delta_{6},\Delta_{5},\Delta_{4},\dots,\Delta_{-4},\Delta_{-5},\Delta_{-6}\right)^{\text{T}}.$
(102)

Specific vacuum configurations might break the $\text{SU(2)}_{\text{D}}$
symmetry into the $A_{4}$, $S_{4}$, and $A_{5}$ discrete groups. We are then
going to discuss these cases separately.

    * *

Let us begin with the $A_{5}$ case. The vacuum configuration is calculated to
be:

$\vec{v}=\left(0,a,0,0,0,0,\sqrt{\frac{11}{7}}a,0,0,0,0,a,0\right)^{\text{T}},$
(103)

where $a$ is an arbitrary real number. Such setting of $a$ applies all through
this section, including the VEV configurations in Table 3. The subspaces of
$A_{5}$ corresponding to $\bf{13}=\bf{1}+\bf{3}+\bf{4}+\bf{5}$ are given by:

$\left(\frac{\sqrt{11}}{5}\Delta_{0}+\frac{2\sqrt{7}}{5}\text{Re$\Delta$}_{5}\right)$
(104)
$\left(\begin{array}[]{c}\frac{\sqrt{33}}{5}\text{Im$\Delta$}_{1}+\frac{\sqrt{11}}{5}\text{Im$\Delta$}_{4}+\frac{\sqrt{6}}{5}\text{Im$\Delta$}_{6}\\\
\frac{\sqrt{33}}{5}\text{Re$\Delta$}_{1}+\frac{\sqrt{6}}{5}\text{Re$\Delta$}_{6}-\frac{\sqrt{11}}{5}\text{Re$\Delta$}_{4}\\\
-\sqrt{2}\text{Im$\Delta$}_{5}\\\ \end{array}\right)^{\text{G}},$ (105)
$\left(\begin{array}[]{c}\frac{2\sqrt{2}}{\sqrt{5}}\text{Re$\Delta$}_{3}-\sqrt{\frac{2}{5}}\text{Re$\Delta$}_{2}\\\
\sqrt{\frac{2}{5}}\text{Im$\Delta$}_{2}+\frac{2\sqrt{2}}{\sqrt{5}}\text{Im$\Delta$}_{3}\\\
-\frac{4}{5}\text{Re$\Delta$}_{1}+\frac{\sqrt{22}}{5}\text{Re$\Delta$}_{6}-\frac{2\sqrt{3}}{5}\text{Re$\Delta$}_{4}\\\
-\frac{4}{5}\text{Im$\Delta$}_{1}+\frac{2\sqrt{3}}{5}\text{Im$\Delta$}_{4}+\frac{\sqrt{22}}{5}\text{Im$\Delta$}_{6}\\\
\end{array}\right),$ (106)
$\left(\begin{array}[]{c}2\sqrt{\frac{2}{5}}\text{Re$\Delta$}_{2}+\sqrt{\frac{2}{5}}\text{Re$\Delta$}_{3}\\\
2\sqrt{\frac{2}{5}}\text{Im$\Delta$}_{2}-\sqrt{\frac{2}{5}}\text{Im$\Delta$}_{3}\\\
\frac{1}{5}\text{Re$\Delta$}_{1}+\frac{3\sqrt{3}}{5}\text{Re$\Delta$}_{4}+\frac{\sqrt{22}}{5}\text{Re$\Delta$}_{6}\\\
\frac{1}{5}\text{Im$\Delta$}_{1}-\frac{3\sqrt{3}}{5}\text{Im$\Delta$}_{4}+\frac{\sqrt{22}}{5}\text{Im$\Delta$}_{6}\\\
\frac{\sqrt{22}}{5}\text{Re$\Delta$}_{5}-\frac{\sqrt{14}}{5}\Delta_{0}\\\
\end{array}\right).$ (107)

    * *

Now let us turn to the $S_{4}$ symmetry. The vacuum configuration is
parametrized as:

$\vec{v}=\left(0,0,a,0,0,0,\sqrt{\frac{2}{7}}a,0,0,0,a,0,0\right)^{\text{T}}.$
(108)

The subspaces of the $S_{4}$ corresponding to
$\bf{13=1+1^{\prime}+2+3+3^{\prime}+3^{\prime}}$ are given by:

$\left(\frac{1}{2\sqrt{2}}\Delta_{0}+\frac{\sqrt{7}}{2}\text{Re$\Delta$}_{4}\right),$
(109)
$\left(\frac{1}{2}\sqrt{\frac{11}{2}}\text{Im$\Delta$}_{2}+\frac{1}{2}\sqrt{\frac{5}{2}}\text{Im$\Delta$}_{6}\right),$
(110)
$\left(\begin{array}[]{c}\frac{1}{2}\text{Re$\Delta$}_{4}-\frac{1}{2}\sqrt{\frac{7}{2}}\Delta_{0}\\\
\frac{1}{2}\sqrt{\frac{5}{2}}\text{Im$\Delta$}_{2}-\frac{1}{2}\sqrt{\frac{11}{2}}\text{Im$\Delta$}_{6}\\\
\end{array}\right),$ (111)
$\left(\begin{array}[]{c}\frac{1}{2}\sqrt{\frac{3}{2}}\text{Im$\Delta$}_{1}+\frac{\sqrt{15}}{4}\text{Im$\Delta$}_{3}+\frac{\sqrt{11}}{4}\text{Im$\Delta$}_{5}\\\
\frac{1}{2}\sqrt{\frac{3}{2}}\text{Re$\Delta$}_{1}-\frac{\sqrt{15}}{4}\text{Re$\Delta$}_{3}+\frac{\sqrt{11}}{4}\text{Re$\Delta$}_{5}\\\
-\sqrt{2}\text{Im$\Delta$}_{4}\\\ \end{array}\right)^{\text{G}},$ (112)
$\left(\begin{array}[]{c}\frac{1}{2}\sqrt{\frac{3}{2}}\text{Im$\Delta$}_{1}+\frac{\sqrt{15}}{4}\text{Im$\Delta$}_{3}+\frac{\sqrt{11}}{4}\text{Im$\Delta$}_{5}\\\
\frac{1}{2}\sqrt{\frac{3}{2}}\text{Re$\Delta$}_{1}-\frac{\sqrt{15}}{4}\text{Re$\Delta$}_{3}+\frac{\sqrt{11}}{4}\text{Re$\Delta$}_{5}\\\
-\sqrt{2}\text{Im$\Delta$}_{4}\\\ \end{array}\right),$ (113)
$\left(\begin{array}[]{c}-\frac{3}{\sqrt{14}}\text{Re$\Delta$}_{1}+\frac{1}{2}\sqrt{\frac{5}{7}}\text{Re$\Delta$}_{3}+\frac{1}{2}\sqrt{\frac{33}{7}}\text{Re$\Delta$}_{5}\\\
\frac{3}{\sqrt{14}}\text{Im$\Delta$}_{1}+\frac{1}{2}\sqrt{\frac{5}{7}}\text{Im$\Delta$}_{3}-\frac{1}{2}\sqrt{\frac{33}{7}}\text{Im$\Delta$}_{5}\\\
\frac{1}{2}\sqrt{\frac{11}{7}}\text{Re$\Delta$}_{6}-\frac{3}{2}\sqrt{\frac{5}{7}}\text{Re$\Delta$}_{2}\\\
\end{array}\right).$ (114)

    * *

Finally, we consider the case of $A_{4}$ symmetry. In principle, it can be
computed with the algorithm described in Section IV and V independently.
However, once the fact that $A_{4}\subset S_{4}$ is noticed, we can
straightforwardly utilize the Eq. (108)-(114) to write down the $A_{4}$
results. By observing the character in Table 5 and 6, one finds out the
relations between the representation of the two groups

$\displaystyle\bf{1},\bf{1^{\prime}}\ \text{of}\ S_{4}$
$\displaystyle\leftrightarrow\bf{1}\ \text{of}\ A_{4},$
$\displaystyle\bf{3},\bf{3^{\prime}}\ \text{of}\ S_{4}$
$\displaystyle\leftrightarrow\bf{3}\ \text{of}\ A_{4},$ $\displaystyle\bf{2}\
\text{of}\ S_{4}$
$\displaystyle\leftrightarrow\bf{1^{\prime}},\bf{1^{\prime\prime}}\ \text{of}\
A_{4}.$ (115)

One can therefore realize that Eq. (108), corresponding to the Eq. (109) can
also be a part of the VEV configuration in the $A_{4}$ case. However, pure Eq.
(108) respect the $S_{4}$ symmetry, so we consider another possible part of
the vacuum configuration as follows

$\vec{v}=i\left(b,0,0,0,\sqrt{\frac{11}{5}}b,0,0,0,-\sqrt{\frac{11}{5}}b,0,0,0,-b\right)^{\text{T}}.$
(116)

Here $b$ is another arbitrary real number. Since the subspace of the $A_{4}$
is decomposed into $\bf{13=1+1+1^{\prime}+1^{\prime\prime}+3+3+3}$, from Eq.
205 one can immediately realize that the two $\bf{1}$, and the three $\bf{3}$
representations of the $A_{4}$ group can be expressed by Eq. (109), (110),
(112), (113), and (114) straightforwardly. For the Eq.(111), it should be
decomposed into the $\bf{1^{\prime}}$ and $\bf{1^{\prime\prime}}$
representations of the $A_{4}$ group. The results are given by

$\left(-\frac{\sqrt{7}}{4}\Delta_{0}+\frac{1}{2\sqrt{2}}\text{Re$\Delta$}_{4}-i\frac{\sqrt{5}}{4}\text{Im$\Delta$}_{2}+i\frac{\sqrt{11}}{4}\text{Im$\Delta$}_{6}\right),$
(117)
$\left(\frac{\sqrt{7}}{4}\Delta_{0}-\frac{1}{2\sqrt{2}}\text{Re$\Delta$}_{4}-i\frac{\sqrt{5}}{4}\text{Im$\Delta$}_{2}+i\frac{\sqrt{11}}{4}\text{Im$\Delta$}_{6}\right),$
(118)

Again, the Eq. (112) is the Goldstone triplet corresponding to the VEV
configuration Eq. (108), and the Eq. (113) is the Goldstone triplet
corresponding to the VEV configuration Eq. (116).

  * •

For the 15-d representation, the complete scalar field is given by

$\Phi_{(15)}=\frac{1}{\sqrt{2}}\left(\Delta_{7},\Delta_{6},\Delta_{5},\Delta_{4},\Delta_{3},\Delta_{2},\Delta_{1},\Delta_{0},\Delta_{-1},\Delta_{-2},\Delta_{-3},\Delta_{-4},\Delta_{-5},\Delta_{-6},\Delta_{-7}\right)^{\text{T}}.$
(119)

When the $\text{SU(2)}_{\text{D}}$ group is broken into the $A_{4}$ symmetry,
the VEV configuration is given by

$\vec{v}=\left(0,a,0,0,\frac{1}{2}\sqrt{\frac{13}{11}}a,0,0,-\frac{1}{4}\sqrt{\frac{273}{11}}a,0,0,\frac{1}{2}\sqrt{\frac{13}{11}}a,0,0,a,0\right)^{\text{T}}.$
(120)

The subspaces of $A_{4}$ corresponding to the
$\bf{15}=\bf{1}+\bf{1^{\prime}}+\bf{1^{\prime\prime}}+\bf{3}+\bf{3}+\bf{3}+\bf{3}$
are given by

$\left(-\frac{1}{9}\sqrt{\frac{91}{3}}\Delta_{0}+\frac{4\sqrt{13}}{27}\text{Re$\Delta$}_{3}+\frac{8\sqrt{11}}{27}\text{Re$\Delta$}_{6}\right),$
(121)
$\left(-\frac{1}{9}\sqrt{\frac{22}{3}}\Delta_{-1}+\frac{4\sqrt{22}}{27}\Delta_{2}+\frac{1}{27}\Delta_{-4}+\frac{8\sqrt{2}}{27}\Delta_{5}+\frac{\sqrt{182}}{27}\Delta_{-7}\right),$
(122)
$\left(-\frac{1}{9}\sqrt{\frac{22}{3}}\Delta_{1}+\frac{4\sqrt{22}}{27}\Delta_{-2}+\frac{1}{27}\Delta_{4}+\frac{8\sqrt{2}}{27}\Delta_{-5}+\frac{\sqrt{182}}{27}\Delta_{7}\right),$
(123)
$\left(\begin{array}[]{c}-\frac{\sqrt{91}}{9}\text{Im$\Delta$}_{1}+\frac{5}{9}\sqrt{\frac{13}{21}}\text{Im$\Delta$}_{2}+\frac{1}{9}\sqrt{\frac{286}{21}}\text{Im$\Delta$}_{4}+\frac{2}{9}\sqrt{\frac{143}{21}}\text{Im$\Delta$}_{5}+\frac{2}{9}\sqrt{\frac{11}{3}}\text{Im$\Delta$}_{7}\\\
-\frac{\sqrt{91}}{9}\text{Re$\Delta$}_{1}-\frac{5}{9}\sqrt{\frac{13}{21}}\text{Re$\Delta$}_{2}+\frac{1}{9}\sqrt{\frac{286}{21}}\text{Re$\Delta$}_{4}-\frac{2}{9}\sqrt{\frac{143}{21}}\text{Re$\Delta$}_{5}+\frac{2}{9}\sqrt{\frac{11}{3}}\text{Re$\Delta$}_{7}\\\
-\frac{1}{3}\sqrt{\frac{26}{21}}\text{Im$\Delta$}_{3}-\frac{4}{3}\sqrt{\frac{22}{21}}\text{Im$\Delta$}_{6}\\\
\end{array}\right)^{\text{G}},$ (124)
$\left(\begin{array}[]{c}\frac{2}{3}\sqrt{\frac{11}{21}}\text{Im$\Delta$}_{2}-\frac{11}{3}\sqrt{\frac{2}{21}}\text{Im$\Delta$}_{4}-\frac{1}{3\sqrt{21}}\text{Im$\Delta$}_{5}+\frac{1}{3}\sqrt{\frac{13}{3}}\text{Im$\Delta$}_{7}\\\
\frac{2}{3}\sqrt{\frac{11}{21}}\text{Re$\Delta$}_{2}+\frac{11}{3}\sqrt{\frac{2}{21}}\text{Re$\Delta$}_{4}-\frac{1}{3\sqrt{21}}\text{Re$\Delta$}_{5}-\frac{1}{3}\sqrt{\frac{13}{3}}\text{Re$\Delta$}_{7}\\\
-\frac{4}{3}\sqrt{\frac{22}{21}}\text{Im$\Delta$}_{3}+\frac{1}{3}\sqrt{\frac{26}{21}}\text{Im$\Delta$}_{6}\\\
\end{array}\right),$ (125)
$\left(\begin{array}[]{c}-\frac{1}{\sqrt{47}}\text{Re$\Delta$}_{1}-3\sqrt{\frac{3}{47}}\text{Re$\Delta$}_{2}+\sqrt{\frac{22}{141}}\text{Re$\Delta$}_{4}+4\sqrt{\frac{11}{141}}\text{Re$\Delta$}_{5}\\\
-\frac{1}{\sqrt{47}}\text{Im$\Delta$}_{1}+3\sqrt{\frac{3}{47}}\text{Im$\Delta$}_{2}+\sqrt{\frac{22}{141}}\text{Im$\Delta$}_{4}-4\sqrt{\frac{11}{141}}\text{Im$\Delta$}_{5}\\\
\frac{2}{3}\sqrt{\frac{14}{47}}\Delta_{0}-\frac{23}{3}\sqrt{\frac{2}{141}}\text{Re$\Delta$}_{3}+\frac{2}{3}\sqrt{\frac{286}{141}}\text{Re$\Delta$}_{6}\\\
\end{array}\right),$ (126)
$\left(\begin{array}[]{c}\frac{10}{9}\sqrt{\frac{77}{141}}\text{Re$\Delta$}_{1}-\frac{4}{27}\sqrt{\frac{77}{47}}\text{Re$\Delta$}_{2}+\frac{31}{27}\sqrt{\frac{14}{47}}\text{Re$\Delta$}_{4}-\frac{17}{27}\sqrt{\frac{7}{47}}\text{Re$\Delta$}_{5}+\frac{\sqrt{611}}{27}\text{Re$\Delta$}_{7}\\\
\frac{10}{9}\sqrt{\frac{77}{141}}\text{Im$\Delta$}_{1}+\frac{4}{27}\sqrt{\frac{77}{47}}\text{Im$\Delta$}_{2}+\frac{31}{27}\sqrt{\frac{14}{47}}\text{Im$\Delta$}_{4}+\frac{17}{27}\sqrt{\frac{7}{47}}\text{Im$\Delta$}_{5}+\frac{\sqrt{611}}{27}\text{Im$\Delta$}_{7}\\\
\frac{16}{9}\sqrt{\frac{22}{141}}\Delta_{0}+\frac{14}{27}\sqrt{\frac{154}{47}}\text{Re$\Delta$}_{3}+\frac{5}{27}\sqrt{\frac{182}{47}}\text{Re$\Delta$}_{6}\\\
\end{array}\right).$ (127)

  * •

For the 17-d representation, the general scalar field is given by

$\Phi_{(17)}=\frac{1}{\sqrt{2}}\left(\Delta_{8},\Delta_{7},\Delta_{6},\Delta_{5},\dots,\Delta_{-5},\Delta_{-6},\Delta_{-7},\Delta_{-8}\right)^{\text{T}}.$
(128)

The vacuum configuration preserving the $S_{4}$ symmetry is given by:

$\vec{v}=\left(a,0,0,0,-2\sqrt{\frac{7}{65}}a,0,0,0,3\sqrt{\frac{22}{65}}a,0,0,0,-2\sqrt{\frac{7}{65}}a,0,0,0,a\right)^{\text{T}}$
(129)

The subspaces of the $S_{4}$ corresponding to
$\bf{17=1+2+2+3+3+3^{\prime}+3^{\prime}}$ are given by

$\left(\frac{\sqrt{33}}{8}\Delta_{0}-\frac{1}{2}\sqrt{\frac{7}{6}}\text{Re$\Delta$}_{4}+\frac{1}{4}\sqrt{\frac{65}{6}}\text{Re$\Delta$}_{8}\right),$
(130)
$\left(\begin{array}[]{c}3\sqrt{\frac{11}{82}}\text{Re$\Delta$}_{8}-\frac{1}{2}\sqrt{\frac{65}{41}}\Delta_{0}\\\
\sqrt{2}\text{Im$\Delta$}_{2}\\\ \end{array}\right),$ (131)
$\left(\begin{array}[]{c}2\sqrt{\frac{7}{127}}\Delta_{0}+3\sqrt{\frac{22}{127}}\text{Re$\Delta$}_{4}\\\
\sqrt{2}\text{Im$\Delta$}_{6}\\\ \end{array}\right),$ (132)
$\left(\begin{array}[]{c}\frac{3\sqrt{11}}{8}\text{Im$\Delta$}_{1}-\frac{1}{8}\sqrt{\frac{35}{3}}\text{Im$\Delta$}_{3}-\frac{\sqrt{91}}{24}\text{Im$\Delta$}_{5}+\frac{\sqrt{65}}{24}\text{Im$\Delta$}_{7}\\\
\frac{3\sqrt{11}}{8}\text{Re$\Delta$}_{1}+\frac{1}{8}\sqrt{\frac{35}{3}}\text{Re$\Delta$}_{3}-\frac{\sqrt{91}}{24}\text{Re$\Delta$}_{5}-\frac{\sqrt{65}}{24}\text{Re$\Delta$}_{7}\\\
\frac{\sqrt{7}}{6}\text{Im$\Delta$}_{4}-\frac{\sqrt{65}}{6}\text{Im$\Delta$}_{8}\\\
\end{array}\right)^{\text{G}},$ (133)
$\left(\begin{array}[]{c}\frac{1}{2}\sqrt{\frac{13}{3}}\text{Im$\Delta$}_{3}-\frac{\sqrt{5}}{6}\text{Im$\Delta$}_{5}+\frac{\sqrt{7}}{3}\text{Im$\Delta$}_{7}\\\
\frac{1}{2}\sqrt{\frac{13}{3}}\text{Re$\Delta$}_{3}+\frac{\sqrt{5}}{6}\text{Re$\Delta$}_{5}+\frac{\sqrt{7}}{3}\text{Re$\Delta$}_{7}\\\
\frac{\sqrt{65}}{6}\text{Im$\Delta$}_{4}+\frac{\sqrt{7}}{6}\text{Im$\Delta$}_{8}\\\
\end{array}\right),$ (134)
$\left(\begin{array}[]{c}-\frac{1}{2}\sqrt{\frac{13}{47}}\text{Re$\Delta$}_{1}-\sqrt{\frac{77}{47}}\text{Re$\Delta$}_{5}+\frac{1}{2}\sqrt{\frac{55}{47}}\text{Re$\Delta$}_{7}\\\
\frac{1}{2}\sqrt{\frac{13}{47}}\text{Im$\Delta$}_{1}+\sqrt{\frac{77}{47}}\text{Im$\Delta$}_{5}+\frac{1}{2}\sqrt{\frac{55}{47}}\text{Im$\Delta$}_{7}\\\
-\frac{1}{2}\sqrt{\frac{455}{94}}\text{Re$\Delta$}_{2}-\frac{3}{2}\sqrt{\frac{33}{94}}\text{Re$\Delta$}_{6}\\\
\end{array}\right),$ (135)
$\left(\begin{array}[]{c}-\sqrt{\frac{21}{71}}\text{Re$\Delta$}_{1}+\frac{1}{2}\sqrt{\frac{55}{71}}\text{Re$\Delta$}_{3}-\frac{1}{2}\sqrt{\frac{429}{71}}\text{Re$\Delta$}_{5}\\\
\sqrt{\frac{21}{71}}\text{Im$\Delta$}_{1}+\frac{1}{2}\sqrt{\frac{55}{71}}\text{Im$\Delta$}_{3}+\frac{1}{2}\sqrt{\frac{429}{71}}\text{Im$\Delta$}_{5}\\\
-\frac{3}{2}\sqrt{\frac{15}{142}}\text{Re$\Delta$}_{2}-\frac{1}{2}\sqrt{\frac{1001}{142}}\text{Re$\Delta$}_{6}\\\
\end{array}\right).$ (136)

  * •

For the 19-d representation, the general scalar field is given by

$\Phi_{(19)}=\frac{1}{\sqrt{2}}\left(\Delta_{9},\Delta_{8},\Delta_{7},\Delta_{6},\Delta_{5},\dots,\Delta_{-5},\Delta_{-6},\Delta_{-7},\Delta_{-8},\Delta_{-9}\right)^{\text{T}}.$
(137)

In this case, the $\text{SU(2)}_{\text{D}}$ group might be broken into the
$S_{4}$ or $A_{4}$ discrete group. We discuss them separately again.

    * *

The vacuum configuration preserving $S_{4}$ symmetry is given by

$\vec{v}=i\left(0,a,0,0,0,\sqrt{\frac{17}{7}}a,0,0,0,0,0,0,0,-\sqrt{\frac{17}{7}}a,0,0,0,-a,0\right)^{\text{T}}$
(138)

The subspaces of the $S_{4}$ corresponding to
$\bf{19=1+1^{\prime}+2+3+3+3+3^{\prime}+3^{\prime}}$ are given by

$\left(\frac{1}{2}\sqrt{\frac{17}{3}}\text{Im$\Delta$}_{4}+\frac{1}{2}\sqrt{\frac{7}{3}}\text{Im$\Delta$}_{8}\right),$
(139)
$\left(\frac{1}{2}\sqrt{\frac{3}{2}}\text{Re$\Delta$}_{2}+\frac{1}{2}\sqrt{\frac{13}{2}}\text{Re$\Delta$}_{6}\right),$
(140)
$\left(\begin{array}[]{c}\frac{1}{2}\sqrt{\frac{7}{3}}\text{Im$\Delta$}_{4}-\frac{1}{2}\sqrt{\frac{17}{3}}\text{Im$\Delta$}_{8}\\\
\frac{1}{2}\sqrt{\frac{3}{2}}\text{Re$\Delta$}_{6}-\frac{1}{2}\sqrt{\frac{13}{2}}\text{Re$\Delta$}_{2}\\\
\end{array}\right),$ (141)
$\left(\begin{array}[]{c}\frac{1}{4}\sqrt{\frac{221}{15}}\text{Re$\Delta$}_{3}+\frac{\sqrt{119}}{12}\text{Re$\Delta$}_{5}+\frac{1}{12}\sqrt{\frac{119}{5}}\text{Re$\Delta$}_{7}+\frac{1}{4}\sqrt{\frac{7}{5}}\text{Re$\Delta$}_{9}\\\
\frac{1}{4}\sqrt{\frac{221}{15}}\text{Im$\Delta$}_{3}-\frac{\sqrt{119}}{12}\text{Im$\Delta$}_{5}+\frac{1}{12}\sqrt{\frac{119}{5}}\text{Im$\Delta$}_{7}-\frac{1}{4}\sqrt{\frac{7}{5}}\text{Im$\Delta$}_{9}\\\
-\frac{1}{3}\sqrt{\frac{34}{5}}\text{Re$\Delta$}_{4}-\frac{2}{3}\sqrt{\frac{14}{5}}\text{Re$\Delta$}_{8}\\\
\end{array}\right)^{\text{G}},$ (142)
$\left(\begin{array}[]{c}\frac{3}{64}\sqrt{\frac{715}{2}}\text{Re$\Delta$}_{1}-\frac{13}{64}\sqrt{\frac{91}{15}}\text{Re$\Delta$}_{3}+\frac{763\text{Re$\Delta$}_{7}}{384\sqrt{5}}+\frac{29}{128}\sqrt{\frac{17}{5}}\text{Re$\Delta$}_{9}-\frac{\text{Re$\Delta$}_{5}}{192}\\\
\frac{3}{64}\sqrt{\frac{715}{2}}\text{Im$\Delta$}_{1}+\frac{13}{64}\sqrt{\frac{91}{15}}\text{Im$\Delta$}_{3}+\frac{29}{128}\sqrt{\frac{17}{5}}\text{Im$\Delta$}_{9}-\frac{\text{Im$\Delta$}_{5}}{192}-\frac{763\text{Im$\Delta$}_{7}}{384\sqrt{5}}\\\
\frac{1}{3}\sqrt{\frac{34}{5}}\text{Re$\Delta$}_{8}-\frac{2}{3}\sqrt{\frac{14}{5}}\text{Re$\Delta$}_{4}\\\
\end{array}\right),$ (143)
$\left(\begin{array}[]{c}\frac{21}{64}\sqrt{\frac{5}{2}}\text{Re$\Delta$}_{1}+\frac{\sqrt{1155}}{64}\text{Re$\Delta$}_{3}-\frac{3\sqrt{143}}{64}\text{Re$\Delta$}_{5}-\frac{3\sqrt{715}}{128}\text{Re$\Delta$}_{7}+\frac{\sqrt{12155}}{128}\text{Re$\Delta$}_{9}\\\
\frac{21}{64}\sqrt{\frac{5}{2}}\text{Im$\Delta$}_{1}-\frac{\sqrt{1155}}{64}\text{Im$\Delta$}_{3}-\frac{3\sqrt{143}}{64}\text{Im$\Delta$}_{5}+\frac{3\sqrt{715}}{128}\text{Im$\Delta$}_{7}+\frac{\sqrt{12155}}{128}\text{Im$\Delta$}_{9}\\\
\Delta_{0}\\\ \end{array}\right),$ (144)
$\left(\begin{array}[]{c}\frac{1}{2}\sqrt{\frac{11}{10}}\text{Im$\Delta$}_{1}+\frac{1}{4}\sqrt{\frac{21}{5}}\text{Im$\Delta$}_{3}+\frac{\sqrt{13}}{4}\text{Im$\Delta$}_{5}+\frac{1}{2}\sqrt{\frac{13}{5}}\text{Im$\Delta$}_{7}\\\
-\frac{1}{2}\sqrt{\frac{11}{10}}\text{Re$\Delta$}_{1}+\frac{1}{4}\sqrt{\frac{21}{5}}\text{Re$\Delta$}_{3}-\frac{\sqrt{13}}{4}\text{Re$\Delta$}_{5}+\frac{1}{2}\sqrt{\frac{13}{5}}\text{Re$\Delta$}_{7}\\\
-\frac{1}{2\sqrt{5}}\text{Im$\Delta$}_{2}-\frac{1}{2}\sqrt{\frac{39}{5}}\text{Im$\Delta$}_{6}\\\
\end{array}\right)$ (145)
$\left(\begin{array}[]{c}\frac{1}{8}\sqrt{\frac{429}{10}}\text{Im$\Delta$}_{1}-\frac{1}{8}\sqrt{\frac{91}{5}}\text{Im$\Delta$}_{3}+\frac{1}{16}\sqrt{\frac{3}{5}}\text{Im$\Delta$}_{7}-\frac{\sqrt{3}}{8}\text{Im$\Delta$}_{5}-\frac{\sqrt{255}}{16}\text{Im$\Delta$}_{9}\\\
\frac{1}{8}\sqrt{\frac{429}{10}}\text{Re$\Delta$}_{1}+\frac{1}{8}\sqrt{\frac{91}{5}}\text{Re$\Delta$}_{3}-\frac{1}{16}\sqrt{\frac{3}{5}}\text{Re$\Delta$}_{7}-\frac{\sqrt{3}}{8}\text{Re$\Delta$}_{5}-\frac{\sqrt{255}}{16}\text{Re$\Delta$}_{9}\\\
\frac{1}{2}\sqrt{\frac{39}{5}}\text{Im$\Delta$}_{2}-\frac{1}{2\sqrt{5}}\text{Im$\Delta$}_{6}\\\
\end{array}\right).$ (146)

    * *

Similarly to the case of 13-d representation, from Eq. (115) we know that the
$\bf{1^{\prime}}$ representation of the $S_{4}$ group corresponding to the Eq.
(140) induce another part of the VEV configuration proportional to

$\vec{v}=\left(0,0,0,b,0,0,0,\sqrt{\frac{3}{13}}b,0,0,0,\sqrt{\frac{3}{13}}b,0,0,0,b,0,0,0\right)^{\text{T}},$
(147)

to break the $S_{4}$ symmetry into the $A_{4}$ symmetry, where $b$ is an
arbitrary real number. Since the subspaces of the $A_{4}$ group is calculated
to be

$\bf{19=1+1+1^{\prime}+1^{\prime\prime}+3+3+3+3+3}$, so Eq. (139), (140),
(142), (143), (144) and (145) can be inherited, while the doublet Eq. (141) is
decomposed into the $\bf{1^{\prime}+1^{\prime\prime}}$ subspaces

$\left(\frac{\sqrt{13}}{4}\text{Re$\Delta$}_{2}-\frac{\sqrt{3}}{4}+i\frac{1}{2}\text{Re$\Delta$}_{6}\sqrt{\frac{7}{6}}\text{Im$\Delta$}_{4}-i\frac{1}{2}\sqrt{\frac{17}{6}}\text{Im$\Delta$}_{8}\right),$
(148)
$\left(\frac{\sqrt{13}}{4}\text{Re$\Delta$}_{2}-\frac{\sqrt{3}}{4}\text{Re$\Delta$}_{6}-i\frac{1}{2}\sqrt{\frac{7}{6}}\text{Im$\Delta$}_{4}+i\frac{1}{2}\sqrt{\frac{17}{6}}\text{Im$\Delta$}_{8}\right).$
(149)

The Eq. (142) is the Goldstone triplet corresponding to the VEV configuration
Eq. (138), and the Eq. (145) is the Goldstone triplet corresponding to the VEV
configuration Eq. (147).

  * •

For the 21-d representation, the scalar field is given by

$\Phi_{(21)}=\frac{1}{\sqrt{2}}\left(\Delta_{10},\Delta_{9},\Delta_{8},\Delta_{7},\Delta_{6},\Delta_{5},\dots,\Delta_{-5},\Delta_{-6},\Delta_{-7},\Delta_{-8},\Delta_{-9},\Delta_{-10}\right)^{\text{T}}.$
(150)

In this case the $\text{SU(2)}_{\text{D}}$ symmetry might be broken into the
$A_{4}$, $S_{4}$, $A_{5}$ subgroups. We again discuss them separately.

    * *

The VEV configuration preserving the $A_{5}$ symmetry is given by

$\vec{v}=\left(a,0,0,0,0,\sqrt{\frac{57}{17}}a,0,0,0,0,\sqrt{\frac{247}{187}}a,0,0,0,0,\sqrt{\frac{57}{17}}a,0,0,0,0,a\right)^{\text{T}}.$
(151)

The subspaces of $A_{5}$ corresponding to $\bf{21=1+3+3^{\prime}+5+5+4}$ are
given by

$\left(\frac{1}{25}\sqrt{\frac{247}{3}}\Delta_{0}+\frac{2\sqrt{209}}{25}\text{Re$\Delta$}_{5}+\frac{2}{25}\sqrt{\frac{187}{3}}\text{Re$\Delta$}_{10}\right),$
(152)
$\left(\begin{array}[]{c}\frac{\sqrt{247}}{25}\text{Im$\Delta$}_{1}+\frac{3\sqrt{57}}{25}\text{Im$\Delta$}_{4}+\frac{2\sqrt{114}}{25}\text{Im$\Delta$}_{6}+\frac{\sqrt{34}}{25}\text{Im$\Delta$}_{9}\\\
\frac{\sqrt{247}}{25}\text{Re$\Delta$}_{1}-\frac{3\sqrt{57}}{25}\text{Re$\Delta$}_{4}+\frac{2\sqrt{114}}{25}\text{Re$\Delta$}_{6}-\frac{\sqrt{34}}{25}\text{Re$\Delta$}_{9}\\\
-\frac{1}{5}\sqrt{\frac{114}{5}}\text{Im$\Delta$}_{5}-\frac{2}{5}\sqrt{\frac{34}{5}}\text{Im$\Delta$}_{10}\\\
\end{array}\right)^{\text{G}},$ (153)
$\left(\begin{array}[]{c}-\frac{\sqrt{221}}{25}\text{Im$\Delta$}_{2}-\frac{4\sqrt{34}}{25}\text{Im$\Delta$}_{3}-\frac{13\sqrt{2}}{25}\text{Im$\Delta$}_{7}-\frac{7\sqrt{3}}{25}\text{Im$\Delta$}_{8}\\\
\frac{\sqrt{221}}{25}\text{Re$\Delta$}_{2}-\frac{4\sqrt{34}}{25}\text{Re$\Delta$}_{3}+\frac{13\sqrt{2}}{25}\text{Re$\Delta$}_{7}-\frac{7\sqrt{3}}{25}\text{Re$\Delta$}_{8}\\\
\frac{2}{5}\sqrt{\frac{34}{5}}\text{Im$\Delta$}_{5}-\frac{1}{5}\sqrt{\frac{114}{5}}\text{Im$\Delta$}_{10}\\\
\end{array}\right),$ (154)
$\left(\begin{array}[]{c}-\frac{3}{5}\sqrt{\frac{51}{115}}\text{Im$\Delta$}_{2}+\frac{14}{15}\sqrt{\frac{442}{345}}\text{Im$\Delta$}_{3}-\frac{37}{15}\sqrt{\frac{26}{345}}\text{Im$\Delta$}_{7}-\frac{1}{15}\sqrt{\frac{299}{5}}\text{Im$\Delta$}_{8}\\\
\frac{3}{5}\sqrt{\frac{51}{115}}\text{Re$\Delta$}_{2}+\frac{14}{15}\sqrt{\frac{442}{345}}\text{Re$\Delta$}_{3}+\frac{37}{15}\sqrt{\frac{26}{345}}\text{Re$\Delta$}_{7}-\frac{1}{15}\sqrt{\frac{299}{5}}\text{Re$\Delta$}_{8}\\\
-\frac{16}{15}\sqrt{\frac{17}{115}}\text{Re$\Delta$}_{1}+\frac{8}{15}\sqrt{\frac{221}{345}}\text{Re$\Delta$}_{4}+\frac{13}{15}\sqrt{\frac{442}{345}}\text{Re$\Delta$}_{6}+\frac{2}{5}\sqrt{\frac{494}{115}}\text{Re$\Delta$}_{9}\\\
-\frac{16}{15}\sqrt{\frac{17}{115}}\text{Im$\Delta$}_{1}-\frac{8}{15}\sqrt{\frac{221}{345}}\text{Im$\Delta$}_{4}+\frac{13}{15}\sqrt{\frac{442}{345}}\text{Im$\Delta$}_{6}-\frac{2}{5}\sqrt{\frac{494}{115}}\text{Im$\Delta$}_{9}\\\
\frac{1}{3}\sqrt{\frac{494}{69}}\text{Re$\Delta$}_{10}-\frac{1}{3}\sqrt{\frac{374}{69}}\Delta_{0}\\\
\end{array}\right),$ (155)
$\left(\begin{array}[]{c}\frac{6}{5}\sqrt{\frac{39}{115}}\text{Re$\Delta$}_{2}+\frac{19}{15}\sqrt{\frac{2}{345}}\text{Re$\Delta$}_{3}+\frac{16}{15}\sqrt{\frac{34}{345}}\text{Re$\Delta$}_{7}+\frac{2}{15}\sqrt{\frac{391}{5}}\text{Re$\Delta$}_{8}\\\
\frac{6}{5}\sqrt{\frac{39}{115}}\text{Im$\Delta$}_{2}-\frac{19}{15}\sqrt{\frac{2}{345}}\text{Im$\Delta$}_{3}+\frac{16}{15}\sqrt{\frac{34}{345}}\text{Im$\Delta$}_{7}-\frac{2}{15}\sqrt{\frac{391}{5}}\text{Im$\Delta$}_{8}\\\
-\frac{37}{15}\sqrt{\frac{13}{115}}\text{Im$\Delta$}_{1}+\frac{277}{15\sqrt{345}}\text{Im$\Delta$}_{4}-\frac{62}{15}\sqrt{\frac{2}{345}}\text{Im$\Delta$}_{6}-\frac{1}{5}\sqrt{\frac{646}{115}}\text{Im$\Delta$}_{9}\\\
\frac{37}{15}\sqrt{\frac{13}{115}}\text{Re$\Delta$}_{1}+\frac{277}{15\sqrt{345}}\text{Re$\Delta$}_{4}+\frac{62}{15}\sqrt{\frac{2}{345}}\text{Re$\Delta$}_{6}-\frac{1}{5}\sqrt{\frac{646}{115}}\text{Re$\Delta$}_{9}\\\
\frac{19}{75}\sqrt{\frac{286}{69}}\Delta_{0}-\frac{3\sqrt{46}}{25}\text{Re$\Delta$}_{5}+\frac{22}{75}\sqrt{\frac{646}{69}}\text{Re$\Delta$}_{10}\\\
\end{array}\right),$ (156)
$\left(\begin{array}[]{c}-\frac{4\sqrt{39}}{25}\text{Re$\Delta$}_{2}-\frac{2}{25}\sqrt{\frac{2}{3}}\text{Re$\Delta$}_{3}+\frac{7}{25}\sqrt{\frac{34}{3}}\text{Re$\Delta$}_{7}+\frac{2\sqrt{17}}{25}\text{Re$\Delta$}_{8}\\\
-\frac{4\sqrt{39}}{25}\text{Im$\Delta$}_{2}+\frac{2}{25}\sqrt{\frac{2}{3}}\text{Im$\Delta$}_{3}+\frac{7}{25}\sqrt{\frac{34}{3}}\text{Im$\Delta$}_{7}-\frac{2\sqrt{17}}{25}\text{Im$\Delta$}_{8}\\\
-\frac{6\sqrt{13}}{25}\text{Im$\Delta$}_{1}-\frac{4}{25\sqrt{3}}\text{Im$\Delta$}_{4}+\frac{14}{25}\sqrt{\frac{2}{3}}\text{Im$\Delta$}_{6}+\frac{\sqrt{646}}{25}\text{Im$\Delta$}_{9}\\\
\frac{6\sqrt{13}}{25}\text{Re$\Delta$}_{1}-\frac{4}{25\sqrt{3}}\text{Re$\Delta$}_{4}-\frac{14}{25}\sqrt{\frac{2}{3}}\text{Re$\Delta$}_{6}+\frac{\sqrt{646}}{25}\text{Re$\Delta$}_{9}\\\
\end{array}\right).$ (157)

    * *

The VEV configuration preserving $S_{4}$ symmetry is given by

$\vec{v}=\left(0,0,a,0,0,0,-2\sqrt{\frac{3}{17}}a,0,0,0,-\sqrt{\frac{130}{187}}a,0,0,0,-2\sqrt{\frac{3}{17}}a,0,0,0,a,0,0\right)^{\text{T}}.$
(158)

The subspaces of $S_{4}$ corresponding to
$\bf{21=1+1^{\prime}+2+2+3+3+3^{\prime}+3^{\prime}+3^{\prime}}$ are given by

$\left(-\frac{1}{8}\sqrt{\frac{65}{6}}\Delta_{0}-\frac{\sqrt{11}}{4}\text{Re$\Delta$}_{4}+\frac{1}{8}\sqrt{\frac{187}{3}}\text{Re$\Delta$}_{8}\right),$
(159)
$\left(\frac{1}{8}\sqrt{\frac{247}{3}}\text{Im$\Delta$}_{2}-\frac{1}{8}\sqrt{\frac{19}{6}}\text{Im$\Delta$}_{6}-\frac{1}{8}\sqrt{\frac{85}{2}}\text{Im$\Delta$}_{10}\right),$
(160)
$\left(\begin{array}[]{c}\frac{1}{6}\sqrt{\frac{187}{7}}\Delta_{0}+\frac{1}{3}\sqrt{\frac{65}{14}}\text{Re$\Delta$}_{8}\\\
\frac{1}{3}\sqrt{\frac{2}{3}}\text{Im$\Delta$}_{2}+\frac{2}{3}\sqrt{\frac{13}{3}}\text{Im$\Delta$}_{6}\\\
\end{array}\right),$ (161)
$\left(\begin{array}[]{c}-\frac{1}{24}\sqrt{\frac{715}{14}}\Delta_{0}+\frac{\sqrt{21}}{4}\text{Re$\Delta$}_{4}+\frac{11}{24}\sqrt{\frac{17}{7}}\text{Re$\Delta$}_{8}\\\
\frac{1}{24}\sqrt{\frac{1105}{3}}\text{Im$\Delta$}_{2}-\frac{1}{24}\sqrt{\frac{85}{6}}\text{Im$\Delta$}_{6}+\frac{3}{8}\sqrt{\frac{19}{2}}\text{Im$\Delta$}_{10}\\\
\end{array}\right),$ (162)
$\left(\begin{array}[]{c}-\frac{1}{8}\sqrt{\frac{65}{2}}\text{Im$\Delta$}_{1}-\frac{7}{8}\sqrt{\frac{3}{5}}\text{Im$\Delta$}_{3}+\frac{3}{16}\sqrt{\frac{51}{5}}\text{Im$\Delta$}_{7}+\frac{1}{16}\sqrt{\frac{323}{5}}\text{Im$\Delta$}_{9}-\frac{3\sqrt{3}}{8}\text{Im$\Delta$}_{5}\\\
-\frac{1}{8}\sqrt{\frac{65}{2}}\text{Re$\Delta$}_{1}+\frac{7}{8}\sqrt{\frac{3}{5}}\text{Re$\Delta$}_{3}-\frac{3}{16}\sqrt{\frac{51}{5}}\text{Re$\Delta$}_{7}+\frac{1}{16}\sqrt{\frac{323}{5}}\text{Re$\Delta$}_{9}-\frac{3\sqrt{3}}{8}\text{Re$\Delta$}_{5}\\\
\sqrt{\frac{3}{10}}\text{Im$\Delta$}_{4}-\sqrt{\frac{17}{10}}\text{Im$\Delta$}_{8}\\\
\end{array}\right)^{\text{G}},$ (163)
$\left(\begin{array}[]{c}-\frac{1}{4}\sqrt{\frac{17}{5}}\text{Im$\Delta$}_{3}+\frac{\sqrt{17}}{4}\text{Im$\Delta$}_{5}-\frac{1}{4\sqrt{5}}\text{Im$\Delta$}_{7}+\frac{1}{4}\sqrt{\frac{57}{5}}\text{Im$\Delta$}_{9}\\\
-\frac{1}{4}\sqrt{\frac{17}{5}}\text{Re$\Delta$}_{3}-\frac{\sqrt{17}}{4}\text{Re$\Delta$}_{5}-\frac{1}{4\sqrt{5}}\text{Re$\Delta$}_{7}-\frac{1}{4}\sqrt{\frac{57}{5}}\text{Re$\Delta$}_{9}\\\
\sqrt{\frac{17}{10}}\text{Im$\Delta$}_{4}+\sqrt{\frac{3}{10}}\text{Im$\Delta$}_{8}\\\
\end{array}\right),$ (164)
$\left(\begin{array}[]{c}-\frac{3}{8}\sqrt{\frac{741}{110}}\text{Re$\Delta$}_{1}-\frac{13}{8}\sqrt{\frac{19}{55}}\text{Re$\Delta$}_{3}+\frac{1}{8}\sqrt{\frac{19}{11}}\text{Re$\Delta$}_{5}+\frac{1}{16}\sqrt{\frac{323}{55}}\text{Re$\Delta$}_{7}+\frac{1}{16}\sqrt{\frac{255}{11}}\text{Re$\Delta$}_{9}\\\
-\frac{3}{8}\sqrt{\frac{741}{110}}\text{Im$\Delta$}_{1}+\frac{13}{8}\sqrt{\frac{19}{55}}\text{Im$\Delta$}_{3}+\frac{1}{8}\sqrt{\frac{19}{11}}\text{Im$\Delta$}_{5}-\frac{1}{16}\sqrt{\frac{323}{55}}\text{Im$\Delta$}_{7}+\frac{1}{16}\sqrt{\frac{255}{11}}\text{Im$\Delta$}_{9}\\\
\frac{1}{4}\sqrt{\frac{247}{110}}\text{Re$\Delta$}_{2}-\frac{3}{8}\sqrt{\frac{19}{55}}\text{Re$\Delta$}_{6}-\frac{5}{8}\sqrt{\frac{51}{11}}\text{Re$\Delta$}_{10}\\\
\end{array}\right),$ (165)
$\left(\begin{array}[]{c}\frac{1}{8}\sqrt{\frac{255}{61}}\text{Re$\Delta$}_{1}-\frac{1}{2}\sqrt{\frac{221}{122}}\text{Re$\Delta$}_{5}+\frac{11}{8}\sqrt{\frac{65}{122}}\text{Re$\Delta$}_{7}+\frac{1}{8}\sqrt{\frac{3705}{122}}\text{Re$\Delta$}_{9}\\\
-\frac{1}{8}\sqrt{\frac{255}{61}}\text{Im$\Delta$}_{1}+\frac{1}{2}\sqrt{\frac{221}{122}}\text{Im$\Delta$}_{5}+\frac{11}{8}\sqrt{\frac{65}{122}}\text{Im$\Delta$}_{7}-\frac{1}{8}\sqrt{\frac{3705}{122}}\text{Im$\Delta$}_{9}\\\
\frac{9}{8}\sqrt{\frac{85}{61}}\text{Re$\Delta$}_{2}+\frac{1}{8}\sqrt{\frac{1105}{122}}\text{Re$\Delta$}_{6}+\frac{1}{8}\sqrt{\frac{741}{122}}\text{Re$\Delta$}_{10}\\\
\end{array}\right),$ (166)
$\left(\begin{array}[]{c}-\frac{89}{8}\sqrt{\frac{13}{3355}}\text{Re$\Delta$}_{1}+\frac{1}{2}\sqrt{\frac{183}{110}}\text{Re$\Delta$}_{3}+2\sqrt{\frac{6}{671}}\text{Re$\Delta$}_{5}+\frac{71}{8}\sqrt{\frac{51}{6710}}\text{Re$\Delta$}_{7}-\frac{5}{8}\sqrt{\frac{1615}{1342}}\text{Re$\Delta$}_{9}\\\
\frac{89}{8}\sqrt{\frac{13}{3355}}\text{Im$\Delta$}_{1}+\frac{1}{2}\sqrt{\frac{183}{110}}\text{Im$\Delta$}_{3}-2\sqrt{\frac{6}{671}}\text{Im$\Delta$}_{5}+\frac{71}{8}\sqrt{\frac{51}{6710}}\text{Im$\Delta$}_{7}+\frac{5}{8}\sqrt{\frac{1615}{1342}}\text{Im$\Delta$}_{9}\\\
-\frac{23}{8}\sqrt{\frac{39}{3355}}\text{Re$\Delta$}_{2}+\frac{509}{8}\sqrt{\frac{3}{6710}}\text{Re$\Delta$}_{6}-\frac{5}{8}\sqrt{\frac{323}{1342}}\text{Re$\Delta$}_{10}\\\
\end{array}\right).$ (167)

    * *

Again, besides the Eq. (158), the $\bf{1^{\prime}}$ representation of the
$S_{4}$ corresponding to the (160) induces

$\vec{v}=i\left(b,0,0,0,\sqrt{\frac{19}{255}}b,0,0,b,-\sqrt{\frac{494}{255}}b,0,0,0,\sqrt{\frac{494}{255}}b,0,0,0,-\sqrt{\frac{19}{255}}b,0,0,0,-b\right)^{\text{T}},$
(168)

to break the $S_{4}$ symmetry into the $A_{4}$ symmetry, where $b$ is an
arbitrary real number other than $a$. Since
$\bf{21}$=$\bf{1}$+$\bf{1}$+$\bf{1^{\prime}}$+$\bf{1^{\prime}}$+$\bf{1^{\prime\prime}}$+$\bf{1^{\prime\prime}}$\+
$\bf{3}$+$\bf{3}$+$\bf{3}$+$\bf{3}$+$\bf{3}$, and considering Eq.(115), Eq.
(159), (160), (163), (164), (165), (166) and (167) are inherited, while

$\left(\sqrt{\frac{66}{197}}\Delta_{0}-\sqrt{\frac{65}{197}}\text{Re$\Delta$}_{4}+i\frac{25}{4}\sqrt{\frac{5}{591}}\text{Im$\Delta$}_{2}-i\frac{1}{4}\sqrt{\frac{65}{1182}}\text{Im$\Delta$}_{6}+i\frac{1}{4}\sqrt{\frac{4199}{394}}\text{Im$\Delta$}_{10}\right),$
(169)
$\displaystyle\left(\frac{1}{16}\sqrt{\frac{12155}{591}}\Delta_{0}+\frac{11}{4}\sqrt{\frac{17}{394}}\text{Re$\Delta$}_{4}+\frac{1}{8}\sqrt{\frac{197}{6}}\text{Re$\Delta$}_{8}-i\frac{1}{8}\sqrt{\frac{663}{394}}\text{Im$\Delta$}_{2}\right.$
(170)
$\displaystyle\left.-i\frac{31}{16}\sqrt{\frac{51}{197}}\text{Im$\Delta$}_{6}+i\frac{1}{16}\sqrt{\frac{95}{197}}\text{Im$\Delta$}_{10}\right),$
(171)
$\left(-\sqrt{\frac{66}{197}}\Delta_{0}+\sqrt{\frac{65}{197}}\text{Re$\Delta$}_{4}+i\frac{25}{4}\sqrt{\frac{5}{591}}\text{Im$\Delta$}_{2}-i\frac{1}{4}\sqrt{\frac{65}{1182}}\text{Im$\Delta$}_{6}+i\frac{1}{4}\sqrt{\frac{4199}{394}}\text{Im$\Delta$}_{10}\right),$
(172)
$\displaystyle\left(-\frac{1}{16}\sqrt{\frac{12155}{591}}\Delta_{0}-\frac{11}{4}\sqrt{\frac{17}{394}}\text{Re$\Delta$}_{4}-\frac{1}{8}\sqrt{\frac{197}{6}}\text{Re$\Delta$}_{8}-i\frac{1}{8}\sqrt{\frac{663}{394}}\text{Im$\Delta$}_{2}\right.$
(173)
$\displaystyle\left.-i\frac{31}{16}\sqrt{\frac{51}{197}}\text{Im$\Delta$}_{6}+i\frac{1}{16}\sqrt{\frac{95}{197}}\text{Im$\Delta$}_{10}\right),$
(174)

are the $\bf{1^{\prime}}$, $\bf{1^{\prime}}$, $\bf{1^{\prime\prime}}$,
$\bf{1^{\prime\prime}}$ subspaces respectively decomposed from the
$S_{4}$-$\bf{2}$ representations denoted by Eq. (161) and (162). Deserved to
be mentioned, Eq. (163) is the Goldstone triplet corresponding to the VEV
configuration Eq. (151), while the Eq. (165) is the Goldstone triplet
corresponding to the VEV configuration Eq. (168).

##  VIII Summary

In this paper, we dissect the models of vector dark matter stemming from a
hidden $\text{SU(2)}_{\text{D}}$ gauge group, which is broken into some
remnant discrete subgroups by the VEV of a single Higgs multiplet
$\Phi_{(n)}$. We provide the general algorithms for writing down all possible
renormalizable couplings, and work out all the remnant discrete subgroups as
well as the corresponding VEV configurations and the detailed forms of the
Higgs multiplet’s subspaces. Up to $j=10$ or $n=21$, we find that the
nontrivial discrete subgroups (beyond $Z_{n}$) can only exist in some cases
with odd $n$. We list the results for each $n$ and find that the vector bosons
of the $\text{SU(2)}_{\text{D}}$ form a 3 representation and therefore become
degenerate in most of the cases. The remnant discrete groups can stabilize the
vector bosons and thus allow them to play the role of the dark matter
candidate. Since most of the non-abelian Lie groups include at least one SU(2)
as their subgroup, our algorithms can potentially be generalized to models
with other Lie groups, especially to the models with higher-dimensional
compact Lie groups. In those cases, the SU(2) subgroups play important roles
in enumerating all their possible finite dimensional irreducible
representations. We leave these as the direction of our future study.

###### Acknowledgements.

We thank to Zhao-Huan Yu, Pyungwon Ko for helpful discussions. C. C. is
supported by the National Natural Science Foundation of China (NSFC) under
Grant No. 11905300 and the Guangzhou Science and Technology Planning Project
under Grant No. 2023A04J0008. Y.L.T. is supported by NSFC under Grant No.
12005312. H.H.Z. is supported by NSFC under Grant No. 12275367. This work is
also supported by the Sun Yat-Sen University Science Foundation.

##  Appendix A Generators of the SU(2)

In our convention, the ladder operators $J_{+}$ and $J_{-}$ in $(2j+1)$-d
representation are defined as

$J_{+}\left|j,m\right\rangle=\left\\{\begin{aligned}
&-\sqrt{(j-m)(j+m+1)}\left|j,m+1\right\rangle\quad m\geq 0\\\
&\sqrt{(j-m)(j+m+1)}\left|j,m+1\right\rangle\quad m<0\end{aligned}\right..$
(175)

We denote the matrix forms of ladder operators $J_{+}$ and $J_{-}$ as
$\tau^{+}$ and $\tau^{-}$. Then the generators matrix of SU(2) are given by

$\tau_{(n)}^{1}=\frac{\tau_{(n)}^{+}+\tau_{(n)}^{-}}{2},\qquad\tau_{(n)}^{2}=\frac{\tau_{(n)}^{+}-\tau_{(n)}^{-}}{2i},\qquad\tau_{(n)}^{3}=\text{Diag}\\{j,j-1,...,-j+1,-j\\}.$
(176)

The scalar multiplet $\Phi_{(n)}$ in $\text{SU(2)}_{\text{D}}$ $n$-d
representation had been shown in Eq. (1),

$\Phi_{(n)}=\xi_{(n)}\left(\Delta_{j},\Delta_{j-1},\dots,\Delta_{-j+1},\Delta_{-j}\right)^{\mathrm{T}}.$
(177)

Its gauge covariant derivative term is given by

$\displaystyle\mathcal{L}_{\text{kin}}=$
$\displaystyle(D^{\mu}\Phi_{(n)})^{\dagger}D_{\mu}\Phi_{(n)}$ $\displaystyle=$
$\displaystyle(\partial^{\mu}\Phi_{(n)}^{\dagger})\partial_{\mu}\Phi_{(n)}+ig_{\text{D}}X^{a\mu}[\Phi_{(n)}^{\dagger}\tau_{(n)}^{a}\partial_{\mu}\Phi_{(n)}-(\partial_{\mu}\Phi_{(n)}^{\dagger})\tau_{(n)}^{a}\Phi_{(n)}]+g_{\text{D}}^{2}X_{\mu}^{a}X^{b\mu}\Phi_{(n)}^{\dagger}\tau_{(n)}^{a}\tau_{(n)}^{b}\Phi_{(n)}$
(178)

with
$D_{\mu}\Phi=\partial_{\mu}\Phi_{(n)}-ig_{\text{D}}X_{\mu}^{a}\tau_{(n)}^{a}\Phi_{(n)}$.
The $X^{a}(a=1,2,3)$ are three gauge bosons which can also be written as

$X^{1}=\frac{X^{-}+X^{+}}{\sqrt{2}},\qquad
X^{2}=\frac{X^{-}-X^{+}}{i\sqrt{2}},\qquad X^{3}=Z^{\prime}.$ (179)

##  Appendix B Discrete symmetry groups of SO(3)

The following table shows the discrete subgroups of the group SO(3) in an
arbitrary $2j+1$ dimensional irreducible representation.

The representation of weight $j$ contains the following stabilizer subgroups
systematically:

  * •

if $j$ is even and $j\geq 4$, then:

${\bf 1},Z_{2}\oplus Z_{2},Z_{3},...,Z_{j},D_{3},...D_{j}.$

  * •

if $j$ is odd and $j\geq 5$, then:

${\bf 1},Z_{2}\oplus Z_{2},Z_{3},...,Z_{j},D_{3},...,D_{j}.$

Beyond these the not systematical groups are showed in Table 4.

$\dim n$ | 5 | 7 | 9 | 11 | 13 | 15 | 17 | 19 | 21
---|---|---|---|---|---|---|---|---|---
subgroup | ${\bf Z}_{2}\oplus{\bf Z}_{2}$ | ${\bf 1}$ | $S_{4}$ |  | $A_{4}$ | $A_{4}$ | $S_{4}$ | $A_{4}$ | $A_{4}$
|  | $A_{4}$ |  |  | $S_{4}$ |  |  | $S_{4}$ | $S_{4}$
|  | ${\bf Z}_{3}$ |  |  | $A_{5}$ |  |  |  | $A_{5}$
|  | $D_{3}$ |  |  |  |  |  |  |
Table 4: Discrete subgroups that the $\text{SU(2)}_{D}$ groups can break into
by a $\Phi_{(n)}$ Higgs multiplet.

##  Appendix C Eigenvectors for 4, 9, 13-d representation

We list the eigenvectors $\vec{v}^{\prime}_{m^{\prime}}$ of the generator
$J_{1}\sin(\theta)+J_{3}\cos(\theta)$, in which the rotation axis lies within
the $x$-$z$ plane with an angle $\theta$ from the $z$ axis. All of them can be
expressed in the functions of the $\tan(\theta/2)$, and we will denote
$\tan(\theta/2)$ as $t_{\theta/2}$ for simplicity. In this appendix we do not
normalize the $\vec{v}^{\prime}_{m^{\prime}}$ since in our analysis it is not
necessary, e.g. in Eq. (41). The following vectors are proportional to the
normalized eigenvectors $\vec{v}^{\prime}_{m^{\prime}}$ with the last
components equal one. Then they satisfy

$v_{-m^{\prime},m}(t_{\theta/2})\propto(-1)^{j+m}\times
v_{m^{\prime},m}(t_{(\pi-\theta)/2}).$ (180)

where
$v_{m^{\prime},m}\equiv\vec{v^{\prime}}^{\dagger}_{m^{\prime}}(t_{\theta/2})\vec{v}_{m}$
is defined in Eq. (38). Here, the $m$ component
$(\vec{v}^{\prime}_{m^{\prime}})_{m}$ of the eigenvector
$\vec{v}^{\prime}_{m^{\prime}}$ is just $v_{m^{\prime},m}$.

The eigenvectors for 4-d representation:

$\vec{v}^{\prime}_{\frac{3}{2}}\propto\left(\begin{array}[]{c}-t^{-3}_{\theta/2}\\\
\sqrt{3}t^{-2}_{\theta/2}\\\ \sqrt{3}t^{-1}_{\theta/2}\\\
1\end{array}\right),\qquad\vec{v}^{\prime}_{\frac{1}{2}}\propto\left(\begin{array}[]{c}t^{-1}_{\theta/2}\\\
-\frac{1}{\sqrt{3}}(2-t^{-2}_{\theta/2})\\\
-\frac{1}{\sqrt{3}}(t_{\theta/2}-2t^{-1}_{\theta/2})\\\ 1\end{array}\right),$
(181)
$\vec{v}^{\prime}_{-\frac{1}{2}}\propto\left(\begin{array}[]{c}-t_{\theta/2}\\\
-\frac{1}{\sqrt{3}}(2-t^{2}_{\theta/2})\\\
\frac{1}{\sqrt{3}}(t^{-1}_{\theta/2}-2t_{\theta/2})\\\
1\end{array}\right),\qquad\vec{v}^{\prime}_{-\frac{3}{2}}\propto\left(\begin{array}[]{c}t^{3}_{\theta/2}\\\
\sqrt{3}t^{2}_{\theta/2}\\\ -\sqrt{3}t_{\theta/2}\\\ 1\end{array}\right).$
(182)

The eigenvectors for 9-d representation:

$\vec{v}^{\prime}_{4}\propto\left(\begin{array}[]{c}t^{-8}_{\theta/2}\\\
-2\sqrt{2}t^{-7}_{\theta/2}\\\ 2\sqrt{7}t^{-6}_{\theta/2}\\\
-2\sqrt{14}t^{-5}_{\theta/2}\\\ \sqrt{70}t^{-4}_{\theta/2}\\\
2\sqrt{14}t^{-3}_{\theta/2}\\\ 2\sqrt{7}t^{-2}_{\theta/2}\\\
2\sqrt{2}t^{-1}_{\theta/2}\\\
1\end{array}\right),\quad\vec{v}^{\prime}_{3}\propto\left(\begin{array}[]{c}-t^{-6}_{\theta/2}\\\
\frac{1}{2\sqrt{2}}(7t^{-5}_{\theta/2}-t^{-7}_{\theta/2})\\\
\frac{-\sqrt{7}}{2}(3t^{-4}_{\theta/2}-t^{-6}_{\theta/2})\\\
\frac{1}{2}\sqrt{\frac{7}{2}}(5t^{-3}_{\theta/2}-3t^{-5}_{\theta/2})\\\
-\sqrt{\frac{35}{2}}(t^{-2}_{\theta/2}-t^{-4}_{\theta/2})\\\
\frac{1}{2}\sqrt{\frac{7}{2}}(5t^{-3}_{\theta/2}-3t^{-1}_{\theta/2})\\\
\frac{\sqrt{7}}{2}(3t^{-2}_{\theta/2}-1)\\\
\frac{1}{2\sqrt{2}}(7t^{-1}_{\theta/2}-t_{\theta/2})\\\
1\end{array}\right),\quad\vec{v}^{\prime}_{2}\propto\left(\begin{array}[]{c}t^{-4}_{\theta/2}\\\
\frac{-1}{\sqrt{2}}(3t^{-3}_{\theta/2}-t^{-5}_{\theta/2})\\\
\frac{1}{2\sqrt{7}}(15t^{-2}_{\theta/2}-12t^{-4}_{\theta/2}+t^{-6}_{\theta/2})\\\
\frac{-1}{\sqrt{14}}(10t^{-1}_{\theta/2}-15t^{-3}_{\theta/2}+3t^{-5}_{\theta/2})\\\
\sqrt{\frac{5}{14}}(3-8t^{-2}_{\theta/2}+3t^{-4}_{\theta/2})\\\
\frac{1}{\sqrt{14}}(10t^{-3}_{\theta/2}-15t^{-1}_{\theta/2}+3t_{\theta/2})\\\
\frac{1}{2\sqrt{7}}(15t^{-2}_{\theta/2}-12+t^{2}_{\theta/2})\\\
\frac{1}{\sqrt{2}}(3t^{-1}_{\theta/2}-t_{\theta/2})\\\ 1\end{array}\right),$
(183)
$\vec{v}^{\prime}_{1}\propto\left(\begin{array}[]{c}-t^{-2}_{\theta/2}\\\
\frac{1}{2\sqrt{2}}(5t^{-1}_{\theta/2}-3t^{-3}_{\theta/2})\\\
\frac{-1}{2\sqrt{7}}(10-15t^{-2}_{\theta/2}+3t^{-4}_{\theta/2})\\\
\frac{1}{2\sqrt{14}}(10t_{\theta/2}-30t^{-1}_{\theta/2}+15t^{-3}_{\theta/2}-t^{-5}_{\theta/2})\\\
-\sqrt{\frac{5}{14}}(t^{2}_{\theta/2}-6+6t^{-2}_{\theta/2}-t^{-4}_{\theta/2})\\\
\frac{1}{2\sqrt{14}}(10t^{-3}_{\theta/2}-30t^{-1}_{\theta/2}+15t_{\theta/2}-t^{3}_{\theta/2})\\\
\frac{1}{2\sqrt{7}}(10t^{-2}_{\theta/2}-15+3t^{2}_{\theta/2})\\\
\frac{1}{2\sqrt{2}}(5t^{-1}_{\theta/2}-3t_{\theta/2})\\\
1\end{array}\right),\qquad\vec{v}^{\prime}_{0}\propto\left(\begin{array}[]{c}1\\\
2\sqrt{2}t^{-1}_{\theta}\\\ \frac{-2}{\sqrt{7}}(1-6t^{-2}_{\theta})\\\
-2\sqrt{\frac{2}{7}}(3t^{-1}_{\theta}-4t^{-3}_{\theta})\\\
\sqrt{\frac{2}{35}}(35-40t^{-2}_{\theta}+8t^{-4}_{\theta})\\\
-2\sqrt{\frac{2}{7}}(3t^{-1}_{\theta}-4t^{-3}_{\theta})\\\
\frac{-2}{\sqrt{7}}(1-6t^{-2}_{\theta})\\\ 2\sqrt{2}t^{-1}_{\theta}\\\
1\end{array}\right),$ (184)
$\vec{v}^{\prime}_{-1}\propto\left(\begin{array}[]{c}-t^{2}_{\theta/2}\\\
\frac{-1}{2\sqrt{2}}(5t_{\theta/2}-3t^{3}_{\theta/2})\\\
\frac{-1}{2\sqrt{7}}(10-15t^{2}_{\theta/2}+3t^{4}_{\theta/2})\\\
\frac{-1}{2\sqrt{14}}(10t^{-1}_{\theta/2}-30t_{\theta/2}+15t^{3}_{\theta/2}-t^{5}_{\theta/2})\\\
-\sqrt{\frac{5}{14}}(t^{-2}_{\theta/2}-6+6t^{2}_{\theta/2}-t^{4}_{\theta/2})\\\
\frac{-1}{2\sqrt{14}}(10t^{3}_{\theta/2}-30t_{\theta/2}+15t^{-1}_{\theta/2}-t^{-3}_{\theta/2})\\\
\frac{1}{2\sqrt{7}}(10t^{2}_{\theta/2}-15+3t^{-2}_{\theta/2})\\\
\frac{-1}{2\sqrt{2}}(5t_{\theta/2}-3t_{\theta/2}^{-1})\\\
1\end{array}\right),\qquad\vec{v}^{\prime}_{-2}\propto\left(\begin{array}[]{c}t^{4}_{\theta/2}\\\
\frac{1}{\sqrt{2}}(3t^{3}_{\theta/2}-t^{5}_{\theta/2})\\\
\frac{1}{2\sqrt{7}}(15t^{2}_{\theta/2}-12t^{4}_{\theta/2}+t^{6}_{\theta/2})\\\
\frac{1}{\sqrt{14}}(10t_{\theta/2}-15t^{3}_{\theta/2}+3t^{5}_{\theta/2})\\\
\sqrt{\frac{5}{14}}(3-8t^{2}_{\theta/2}+3t^{4}_{\theta/2})\\\
\frac{-1}{\sqrt{14}}(10t^{3}_{\theta/2}-15t_{\theta/2}+3t^{-1}_{\theta/2})\\\
\frac{1}{2\sqrt{7}}(15t^{2}_{\theta/2}-12+t^{-2}_{\theta/2})\\\
\frac{-1}{\sqrt{2}}(3t_{\theta/2}-t^{-1}_{\theta/2})\\\ 1\end{array}\right),$
(185)
$\vec{v}^{\prime}_{-3}\propto\left(\begin{array}[]{c}-t^{6}_{\theta/2}\\\
\frac{-1}{2\sqrt{2}}(7t^{5}_{\theta/2}-t^{7}_{\theta/2})\\\
\frac{-\sqrt{7}}{2}(3t^{4}_{\theta/2}-t^{6}_{\theta/2})\\\
\frac{-1}{2}\sqrt{\frac{7}{2}}(5t^{3}_{\theta/2}-3t^{5}_{\theta/2})\\\
-\sqrt{\frac{35}{2}}(t^{2}_{\theta/2}-t^{4}_{\theta/2})\\\
\frac{-1}{2}\sqrt{\frac{7}{2}}(5t^{3}_{\theta/2}-3t_{\theta/2})\\\
\frac{\sqrt{7}}{2}(3t^{2}_{\theta/2}-1)\\\
\frac{-1}{2\sqrt{2}}(7t_{\theta/2}-t^{-1}_{\theta/2})\\\
1\end{array}\right),\qquad\vec{v}^{\prime}_{-4}\propto\left(\begin{array}[]{c}t^{8}_{\theta/2}\\\
2\sqrt{2}t^{7}_{\theta/2}\\\ 2\sqrt{7}t^{6}_{\theta/2}\\\
2\sqrt{14}t^{5}_{\theta/2}\\\ \sqrt{70}t^{4}_{\theta/2}\\\
-2\sqrt{14}t^{3}_{\theta/2}\\\ 2\sqrt{7}t^{2}_{\theta/2}\\\
-2\sqrt{2}t_{\theta/2}\\\ 1\end{array}\right).$ (186)

The eigenvectors of 13-d representation:

$\vec{v}^{\prime}_{6}\propto\left(\begin{array}[]{c}t^{-12}_{\theta/2}\\\
-2\sqrt{3}t^{-11}_{\theta/2}\\\ \sqrt{66}t^{-10}_{\theta/2}\\\
-2\sqrt{55}t^{-9}_{\theta/2}\\\ 3\sqrt{55}t^{-8}_{\theta/2}\\\
-6\sqrt{22}t^{-7}_{\theta/2}\\\ 2\sqrt{231}t^{-6}_{\theta/2}\\\
6\sqrt{22}t^{-5}_{\theta/2}\\\ 3\sqrt{55}t^{-4}_{\theta/2}\\\
2\sqrt{55}t^{-3}_{\theta/2}\\\ \sqrt{66}t^{-2}_{\theta/2}\\\
2\sqrt{3}t^{-1}_{\theta/2}\\\
1\end{array}\right),\quad\vec{v}^{\prime}_{5}\propto\left(\begin{array}[]{c}-t^{-10}_{\theta/2}\\\
\frac{1}{2\sqrt{3}}(11t^{-9}_{\theta/2}-t^{-11}_{\theta/2})\\\
-\sqrt{\frac{11}{6}}(5t^{-8}_{\theta/2}-t^{-10}_{\theta/2})\\\
\frac{1}{2}\sqrt{55}(3t^{-7}_{\theta/2}-t^{-9}_{\theta/2})\\\
-\sqrt{55}(2t^{-6}_{\theta/2}-t^{-8}_{\theta/2})\\\
\sqrt{\frac{11}{2}}(7t^{-5}_{\theta/2}-5t^{-7}_{\theta/2})\\\
-\sqrt{231}(t^{-4}_{\theta/2}-t^{-6}_{\theta/2})\\\
\sqrt{\frac{11}{2}}(7t^{-5}_{\theta/2}-5t^{-3}_{\theta/2})\\\
\sqrt{55}(2t^{-4}_{\theta/2}-t^{-2}_{\theta/2})\\\
\frac{1}{2}\sqrt{55}(3t^{-3}_{\theta/2}-t^{-1}_{\theta/2})\\\
\sqrt{\frac{11}{6}}(5t^{-2}_{\theta/2}-1)\\\
\frac{1}{2\sqrt{3}}(11t^{-1}_{\theta/2}-t_{\theta/2})\\\
1\end{array}\right),\quad\vec{v}^{\prime}_{4}\propto\left(\begin{array}[]{c}t^{-8}_{\theta/2}\\\
\frac{-1}{\sqrt{3}}(5t^{-7}_{\theta/2}-t^{-9}_{\theta/2})\\\
\frac{1}{\sqrt{66}}(45t^{-6}_{\theta/2}-20t^{-8}_{\theta/2}+t^{-10}_{\theta/2})\\\
-\sqrt{\frac{5}{11}}(12t^{-5}_{\theta/2}-9t^{-7}_{\theta/2}+t^{-9}_{\theta/2})\\\
\sqrt{\frac{5}{11}}(14t^{-4}_{\theta/2}-16t^{-6}_{\theta/2}+3t^{-8}_{\theta/2})\\\
-\sqrt{\frac{2}{11}}(21t^{-3}_{\theta/2}-35t^{-5}_{\theta/2}+10t^{-7}_{\theta/2})\\\
\sqrt{\frac{21}{11}}(5t^{-2}_{\theta/2}-12t^{-4}_{\theta/2}+5t^{-6}_{\theta/2})\\\
\sqrt{\frac{2}{11}}(21t^{-5}_{\theta/2}-35t^{-3}_{\theta/2}+10t^{-1}_{\theta/2})\\\
\sqrt{\frac{5}{11}}(14t^{-4}_{\theta/2}-16t^{-2}_{\theta/2}+3)\\\
\sqrt{\frac{5}{11}}(12t^{-3}_{\theta/2}-9t^{-1}_{\theta/2}+t_{\theta/2})\\\
\frac{1}{\sqrt{66}}(45t^{-2}_{\theta/2}-20+t^{2}_{\theta/2})\\\
\frac{1}{\sqrt{3}}(5t^{-1}_{\theta/2}-t_{\theta/2})\\\ 1\end{array}\right),$
(187)
$\vec{v}^{\prime}_{3}\propto\left(\begin{array}[]{c}-t^{-6}_{\theta/2}\\\
\frac{\sqrt{3}}{2}(3t^{-5}_{\theta/2}-t^{-7}_{\theta/2})\\\
-\sqrt{\frac{3}{22}}(12t^{-4}_{\theta/2}-9t^{-6}_{\theta/2}+t^{-8}_{\theta/2})\\\
\frac{1}{2\sqrt{55}}(84t^{-3}_{\theta/2}-108t^{-5}_{\theta/2}+27t^{-7}_{\theta/2}-t^{-9}_{\theta/2})\\\
\frac{-3}{\sqrt{55}}(14t^{-2}_{\theta/2}-28t^{-4}_{\theta/2}+12t^{-6}_{\theta/2}-t^{-8}_{\theta/2})\\\
\frac{3}{\sqrt{22}}(7t^{-1}_{\theta/2}-21t^{-3}_{\theta/2}+14t^{-5}_{\theta/2}-2t^{-7}_{\theta/2})\\\
-\sqrt{\frac{21}{11}}(2-9t^{-2}_{\theta/2}+9t^{-4}_{\theta/2}-2t^{-6}_{\theta/2})\\\
\frac{-3}{\sqrt{22}}(2t_{\theta/2}-14t^{-1}_{\theta/2}+21t^{-3}_{\theta/2}-7t^{-5}_{\theta/2})\\\
\frac{-3}{\sqrt{55}}(t^{2}_{\theta/2}-12+28t^{-2}_{\theta/2}-14t^{-4}_{\theta/2})\\\
\frac{-1}{2\sqrt{55}}(t^{3}_{\theta/2}-27t_{\theta/2}+108t^{-1}_{\theta/2}-84t^{-3}_{\theta/2})\\\
\sqrt{\frac{3}{22}}(t^{2}_{\theta/2}-9+12t^{-2}_{\theta/2})\\\
-\frac{\sqrt{3}}{2}(t_{\theta/2}-3t^{-1}_{\theta/2})\\\
1\end{array}\right),\vec{v}^{\prime}_{2}\propto\left(\begin{array}[]{c}t^{-4}_{\theta/2}\\\
\frac{-2}{\sqrt{3}}(2t^{-3}_{\theta/2}-t^{-5}_{\theta/2})\\\
\sqrt{\frac{2}{33}}(14t^{-2}_{\theta/2}-16t^{-4}_{\theta/2}+3t^{-6}_{\theta/2})\\\
\frac{-2}{\sqrt{55}}(14t^{-1}_{\theta/2}-28t^{-3}_{\theta/2}+12t^{-5}_{\theta/2}-t^{-7}_{\theta/2})\\\
\frac{1}{3\sqrt{55}}(70-224t^{-2}_{\theta/2}+168t^{-4}_{\theta/2}-32t^{-6}_{\theta/2}+t^{-8}_{\theta/2})\\\
\frac{-2}{3}\sqrt{\frac{2}{11}}(7t_{\theta/2}-35t^{-1}_{\theta/2}+42t^{-3}_{\theta/2}-14t^{-5}_{\theta/2}+t^{-7}_{\theta/2})\\\
2\sqrt{\frac{7}{33}}(t_{\theta/2}^{2}-8+15t^{-2}_{\theta/2}-8t^{-4}_{\theta/2}+t^{-6}_{\theta/2})\\\
\frac{2}{3}\sqrt{\frac{2}{11}}(t^{3}_{\theta/2}-14t_{\theta/2}+42t^{-1}_{\theta/2}-35t^{-3}_{\theta/2}+7t^{-5}_{\theta/2})\\\
\frac{1}{3\sqrt{55}}(t^{4}_{\theta/2}-32t^{2}_{\theta/2}+168-224t^{-2}_{\theta/2}+70t^{-4}_{\theta/2})\\\
\frac{-2}{\sqrt{55}}(t^{3}_{\theta/2}-12t_{\theta/2}+28t^{-1}_{\theta/2}-14t^{-3}_{\theta/2})\\\
\sqrt{\frac{2}{33}}(3t^{2}_{\theta/2}-16+14t^{-2}_{\theta/2})\\\
\frac{-2}{\sqrt{3}}(t_{\theta/2}-2t^{-1}_{\theta/2})\\\ 1\end{array}\right),$
(188)
$\vec{v}^{\prime}_{1}\propto\left(\begin{array}[]{c}-t^{-2}_{\theta/2}\\\
\frac{1}{2\sqrt{3}}(7t^{-1}_{\theta/2}-5t^{-3}_{\theta/2})\\\
\frac{-1}{\sqrt{66}}(21-35t^{-2}_{\theta/2}+10t^{-4}_{\theta/2})\\\
\frac{1}{2}\sqrt{\frac{5}{11}}(14t_{\theta/2}-21t^{-1}_{\theta/2}+14t^{-3}_{\theta/2}-2t^{-5}_{\theta/2})\\\
\frac{-1}{3}\sqrt{\frac{5}{11}}(7t^{2}_{\theta/2}-35+42t^{-2}_{\theta/2}-14t^{-4}_{\theta/2}+t^{-6}_{\theta/2})\\\
\frac{1}{6\sqrt{22}}(21t^{3}_{\theta/2}-175t_{\theta/2}+350t^{-1}_{\theta/2}-210t^{-3}_{\theta/2}+35t^{-5}_{\theta/2}-t^{-7}_{\theta/2})\\\
\frac{-1}{2}\sqrt{\frac{7}{33}}(t^{4}_{\theta/2}-15t^{2}_{\theta/2}+50-50t^{-2}_{\theta/2}+15t^{-4}_{\theta/2}-t^{-6}_{\theta/2})\\\
\frac{-1}{6\sqrt{22}}(t^{5}_{\theta/2}-35t^{3}_{\theta/2}+210t_{\theta/2}-350t^{-1}_{\theta/2}+175t^{-3}_{\theta/2}-21t^{-5}_{\theta/2})\\\
\frac{1}{3}\sqrt{\frac{5}{11}}(t^{4}_{\theta/2}-14t^{2}_{\theta/2}+42-35t^{-2}_{\theta/2}+7t^{-4}_{\theta/2})\\\
\frac{-1}{2}\sqrt{\frac{5}{11}}(2t^{3}_{\theta/2}-14t_{\theta/2}+21t^{-1}_{\theta/2}-14t^{-3}_{\theta/2})\\\
\frac{1}{\sqrt{66}}(10t^{2}_{\theta/2}-35+21t^{-2}_{\theta/2})\\\
\frac{-1}{2\sqrt{3}}(5t_{\theta/2}-7t^{-1}_{\theta/2})\\\
1\end{array}\right),\vec{v}^{\prime}_{0}\propto\left(\begin{array}[]{c}1\\\
2\sqrt{3}t^{-1}_{\theta}\\\ -\sqrt{\frac{6}{11}}(1-10t^{-2}_{\theta})\\\
-2\sqrt{\frac{5}{11}}(3t^{-1}_{\theta}-8t^{-3}_{\theta})\\\
\sqrt{\frac{5}{11}}(1-16t^{-2}_{\theta}+16t^{-4}_{\theta})\\\
2\sqrt{\frac{2}{11}}(5t^{-1}_{\theta}-20t^{-3}_{\theta}+8t^{-5}_{\theta})\\\
\frac{-2}{\sqrt{231}}(5-90t^{-2}_{\theta}+120t^{-4}_{\theta}-16t^{-6}_{\theta})\\\
2\sqrt{\frac{2}{11}}(5t_{\theta}-20t^{3}_{\theta}+8t^{5}_{\theta})\\\
\sqrt{\frac{5}{11}}(1-16t^{-2}_{\theta}+16t^{-4}_{\theta})\\\
-2\sqrt{\frac{5}{11}}(3t^{-1}_{\theta}-8t^{-3}_{\theta})\\\
-\sqrt{\frac{6}{11}}(1-10t^{-2}_{\theta})\\\ 2\sqrt{3}t^{-1}_{\theta}\\\
1\end{array}\right),$ (189)