KYUSHU-HET-246

UME-PP-022

Pseudo-Nambu-Goldstone Dark Matter from
Non-Abelian Gauge Symmetry

Hajime<EMAIL_ADDRESS>Takashi
<EMAIL_ADDRESS>Koji
<EMAIL_ADDRESS>
Yoshiki<EMAIL_ADDRESS>and Naoki
<EMAIL_ADDRESS>
1Department of Physics, Kyushu University,
744 Motooka, Nishi-ku, Fukuoka, 819-0395, Japan
2Faculty of Education, University of Miyazaki,
1-1 Gakuen-Kibanadai-Nishi, Miyazaki 889-2192, Japan
3Department of Physics, National Taiwan University,
Taipei, Taiwan 10617, R.O.C.

##  1 Introduction

The standard model (SM) in particle physics is able to explain the results of
accelerator experiments with the exception of a few anomalies. However, some
phenomena that cannot be explained within the scope of the SM have emerged.
One of the important issues in modern particle physics and cosmology is the
search for the identity of dark matter (DM). The existence of dark matter (DM)
has been confirmed by several astronomical observations such as spiral
galaxies [1, 2], gravitational lensing [3], cosmic microwave background [4],
and collision of bullet cluster [5].

There are a lot of DM candidates since the nature of DM is not yet understood.
One such candidate is called the weakly interacting massive particle (WIMP).
An attractive property of WIMP DM is that it can be generated thermally, which
can be experimentally verified by introducing a non-gravitational effect. In
order to achieve the DM’s relic abundance, the mass of WIMPs is expected to be
in the range of $\mathcal{O}(10)$ GeV to $\mathcal{O}(100)$ TeV. Because of
the non-gravitational interactions of WIMPs, direct and indirect detections
are expected. There is still no clear signal for WIMPs, and hence direct
detections yield strong constraints on WIMP masses and interactions.

Several mechanisms in WIMP DM models are proposed to avoid the severe
constraints of the direct detection by considering e.g., a fermion DM with
pseudo-scalar interactions [6, 7, 8, 9, 10, 11] and a pseudo-Nambu-Goldstone
boson (pNGB) DM with additional global $U(1)$ group symmetry [12, 13, 14, 15,
16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27]. As pointed out in the
original pNGB DM model [14], the DM has the property of Nambu-Goldstone (NG)
mode, so the coupling of the DM with the SM Higgs boson is proportional to its
momentum. As a result, the scattering cross sections of the DM with the SM
particles via the Higgs bosons are strongly suppressed, while the annihilation
cross sections of the DM to the SM particles are kept.

Recently, a pNGB DM model based on gauged $G_{\rm SM}\times U(1)_{B-L}$
symmetry, which extends the softly broken $U(1)$ global symmetry to the gauged
$U(1)_{B-L}$ symmetry, was proposed [22, 23], where $G_{\rm
SM}:=SU(3)_{C}\times SU(2)_{W}\times U(1)_{Y}$. The DM direct detection cross
section is naturally suppressed as the same as the original pNGB DM model. On
the other hand, the pNGB DM decays into SM particles mediated by the
$U(1)_{B-L}$ gauge boson. As a result, the $U(1)_{B-L}$ symmetry breaking
scale is greater than $10^{13}$ GeV for the DM mass $<1$ TeV to escape the
constraint from DM stability, where the bound from gamma-ray observations is
strong as roughly the DM lifetime $\gtrsim 10^{27}\,\mathrm{s}$ for two body
decays [28]. In addition, the $G_{\rm SM}\times U(1)_{B-L}$ pNGB DM model has
been extended to $SO(10)$ grand unified theory (GUT) [29, 30]. In this model,
the vacuum expectation value (VEV) of the intermediate symmetry breaking scale
is greater than $10^{10}$ GeV and the DM mass is only allowed to be slightly
below half the Higgs boson mass from the requirements of DM stability and
grand unification and also the constraints of the Higgs invisible decay and
the gamma-ray observations for DM annihilations.

The purpose of this paper is to propose a new pNGB DM model based on non-
Abelian gauge symmetry $SU(2)_{D}$.111 A pNGB DM model based on non-Abelian
global symmetry $SU(2)_{g}$ and Abelian gauge symmetry $U(1)_{X}$ has been
proposed in Ref. [31]. This model can be regarded as a low-energy effective
model that is realized in a special parameter region in our model. Unlike the
$G_{\rm SM}\times U(1)_{B-L}$ and $SO(10)$ pNGB DM model, we will confirm that
the DM is stabilized due to the residual $U(1)$ symmetry of the $SU(2)$
custodial symmetry [32].222 A DM model using custodial symmetry emerging from
non-Abelian gauge symmetry $SU(2)_{D}$ for DM stability has been proposed in
Ref. [33], although it is not a pNGB DM model.  We will show that in our pNGB
DM model the VEV of $SU(2)_{D}$ breaking scale can be allowed to be roughly
${\cal O}(1)$ TeV without introducing very high energy scale.

The paper is organized as follows. In Sec. 2, we introduce an $SU(2)$ pNGB DM
model. In Sec. 3, we analyze vacuum structures and symmetry breaking patterns
of the model. In Sec. 4, we analyze the scalar potential of the system. In
Sec. 5, we investigate the mass spectra of scalar fields in this model. In
Sec. 6, we examine the constraints from direct detection experiments and the
thermal relic abundance of DM for our DM candidate. Section 7 is devoted to
summary and discussions. In Appendix A, we show the detailed calculation of
DM-quark scattering amplitude.

##  2 The model

The model consists of the SM gauge fields, an $SU(2)_{D}$ gauge field
${W}_{\mu}^{\prime a}$ $(a=1,2,3)$, a complex scalar field in ${\bf 2}$ of
$SU(2)_{D}$ $\Phi$, and a real scalar field in ${\bf 3}$ of $SU(2)_{D}$
$\Delta$. The matter content in the non-Abelian pNGB DM model is summarized in
Table 1.

| $Q$ | $u^{c}$ | $d^{c}$ | $L$ | $e^{c}$ | $H$ | $\Phi$ | $\Delta$ | $G_{\mu\nu}$ | $W_{\mu\nu}$ | $B_{\mu\nu}$ | $W_{\mu\nu}^{\prime}$
---|---|---|---|---|---|---|---|---|---|---|---|---
$SU(3)_{C}$ | ${\bf 3}$ | ${\bf\overline{3}}$ | ${\bf\overline{3}}$ | ${\bf 1}$ | ${\bf 1}$ | ${\bf 1}$ | ${\bf 1}$ | ${\bf 1}$ | ${\bf 8}$ | ${\bf 1}$ | ${\bf 1}$ | ${\bf 1}$
$SU(2)_{W}$ | ${\bf 2}$ | ${\bf 1}$ | ${\bf 1}$ | ${\bf 2}$ | ${\bf 1}$ | ${\bf 2}$ | ${\bf 1}$ | ${\bf 1}$ | ${\bf 1}$ | ${\bf 3}$ | ${\bf 1}$ | ${\bf 1}$
$U(1)_{Y}$ | $+1/6$ | $-2/3$ | $+1/3$ | $-1/2$ | $+1$ | $+1/2$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$
$SU(2)_{D}$ | ${\bf 1}$ | ${\bf 1}$ | ${\bf 1}$ | ${\bf 1}$ | ${\bf 1}$ | ${\bf 1}$ | ${\bf 2}$ | ${\bf 3}$ | ${\bf 1}$ | ${\bf 1}$ | ${\bf 1}$ | ${\bf 3}$
Table 1: The field content in the pNGB DM model is shown in the $G_{\rm
SM}\times SU(2)_{D}$ basis, where the fermions belong to $(1/2,0)$ under
$SL(2,\mathbb{C})$.

The Lagrangian is given by

$\displaystyle{\cal L}$
$\displaystyle=-\frac{1}{2}\mbox{tr}\left[{G}_{\mu\nu}{G}^{\mu\nu}\right]-\frac{1}{2}\mbox{tr}\left[{W}_{\mu\nu}{W}^{\mu\nu}\right]-\frac{1}{4}{B}_{\mu\nu}{B}^{\mu\nu}-\frac{1}{2}\mbox{tr}\left[{W}_{\mu\nu}^{\prime}{W}^{\prime\mu\nu}\right]$
$\displaystyle\hskip
10.00002pt+\left({D}_{\mu}H\right)^{\dagger}\left({D}^{\mu}H\right)+\left({D}_{\mu}{\Phi}\right)^{\dagger}\left({D}^{\mu}{\Phi}\right)+\frac{1}{2}\mbox{tr}\left[\left({D}_{\mu}\Delta\right)\left({D}^{\mu}\Delta\right)\right]$
$\displaystyle\hskip 10.00002pt-{\cal V}\left(H,\Phi,\Delta\right)$
$\displaystyle\hskip
10.00002pt+\overline{Q}i\cancel{{D}}Q+\overline{u^{c}}i\cancel{{D}}u^{c}+\overline{d^{c}}i\cancel{{D}}d^{c}+\overline{L}i\cancel{{D}}L+\overline{e^{c}}i\cancel{{D}}e^{c}$
$\displaystyle\hskip
10.00002pt-\left(y_{u}Qu^{c}H+y_{d}Qd^{c}H^{\dagger}+y_{e}Le^{c}H^{\dagger}+\mbox{h.c.}\right),$
(2.1)

where
${D}_{\mu}=\partial_{\mu}+ig_{s}{G}_{\mu}+ig_{2}W_{\mu}+ig_{1}B_{\mu}+ig_{2}^{\prime}W_{\mu}^{\prime}$;
${F}_{\mu\nu}=\partial_{\mu}{F}_{\nu}-\partial_{\nu}{F}_{\mu}+i{g}[{F}_{\mu},{F}_{\nu}]$,
where $F=G,W,B,W^{\prime}$ and $g_{s}$, $g_{2}$, $g_{1}$, $g_{1}^{\prime}$ are
gauge fields and gauge coupling constants of $SU(3)_{C}$, $SU(2)_{W}$,
$U(1)_{Y}$, $SU(2)_{D}$, respectively. The scalar potential ${\cal
V}\left(H,\Phi,\Delta\right)$ contains quadratic, cubic, and quartic coupling
terms,

$\displaystyle{\cal V}(H,\Phi,\Delta)$
$\displaystyle=-\mu_{H}^{2}H^{\dagger}H-\mu_{\Phi}^{2}\Phi^{\dagger}\Phi-\frac{1}{2}\mu_{\Delta}^{2}\mbox{Tr}\left(\Delta^{2}\right)$
$\displaystyle\hskip
10.00002pt+\sqrt{2}\left((\kappa_{1}+i\kappa_{2})\tilde{\Phi}^{\dagger}\Delta\Phi+(\kappa_{1}-i\kappa_{2}){\Phi}^{\dagger}\Delta\tilde{\Phi}\right)+2\sqrt{2}\kappa_{3}\Phi^{\dagger}\Delta\Phi$
$\displaystyle\hskip
10.00002pt+\lambda_{H}\left(H^{\dagger}H\right)^{2}+\lambda_{\Phi}\left(\Phi^{\dagger}\Phi\right)^{2}+\frac{1}{4}\lambda_{\Delta}\mbox{Tr}\left(\Delta^{2}\right)^{2}$
$\displaystyle\hskip
10.00002pt+\lambda_{H\Phi}\left(H^{\dagger}H\right)\left(\Phi^{\dagger}\Phi\right)+\lambda_{H\Delta}\left(H^{\dagger}H\right)\mbox{Tr}\left(\Delta^{2}\right)+\lambda_{\Phi\Delta}\left(\Phi^{\dagger}\Phi\right)\mbox{Tr}\left(\Delta^{2}\right),$
(2.2)

where $\tilde{\Phi}(x)=i\sigma_{2}\Phi(x)^{*}$; $\mu_{H}^{2}$,
$\mu_{\Phi}^{2}$, and $\mu_{\Delta}^{2}$ are real parameters with dimension 2,
$\kappa_{a}(a=1,2,3)$ are real parameters with dimension 1, and $\lambda_{H}$,
$\lambda_{\Phi}$, $\lambda_{\Delta}$ $\lambda_{H\Phi}$, $\lambda_{H\Delta}$,
and $\lambda_{\Phi\Delta}$ are dimensionless real parameters. We use the
following notation:

$\displaystyle\Delta$
$\displaystyle=\frac{1}{\sqrt{2}}\left(\begin{array}[]{cc}\eta_{3}&\eta_{1}-i\eta_{2}\\\
\eta_{1}+i\eta_{2}&-\eta_{3}\\\ \end{array}\right),$ (2.5) $\displaystyle\Phi$
$\displaystyle=\frac{1}{\sqrt{2}}\left(\begin{array}[]{c}\phi_{1}+i\phi_{2}\\\
\phi_{3}+i\phi_{4}\\\ \end{array}\right).$ (2.8)

Under the $SU(2)_{D}$ transformation, $\Phi(x)$ and $\Delta(x)$ behave as

$\displaystyle\Phi(x)\to U(x)\Phi(x),\ \ \Delta(x)\to
U(x)\Delta(x)U(x)^{{\dagger}},\ $ (2.9)

where $U(x)$ is the $SU(2)_{D}$ unitary transformation
$U(x)=\mbox{exp}\left[i\theta_{a}(x)\frac{\sigma_{a}}{2}\right]$;
$\theta_{a}(x)(a=1,2,3)$ are the parameters of the $SU(2)_{D}$ gauge
transformation and $\sigma_{a}$ stand for the Pauli matrices. Note that it is
easy to check invariant terms under $G_{\rm SM}\times SU(2)_{D}$ by using
GroupMath [34] and Sym2Int[35, 36]. (For Lie groups, see e.g., Ref. [37].)

We will analyze the relations between vacuum structures and symmetry breaking
patterns in the next section. In the model, the invariant terms that contain
only the scalar field $\Phi(x)$ in ${\bf 2}$ of $SU(2)_{D}^{\rm local}$ are
invariant under a larger global symmetry $SU(2)_{\Phi L}^{\rm global}\times
SU(2)_{\Phi R}^{\rm global}$. To check this extended global symmetry, it is
convenient to introduce a bi-doublet or $2\times 2$ matrix notation for
$\Phi(x)$ as

$\displaystyle{\Sigma}(x):=\left(\begin{array}[]{cc}\tilde{\Phi}(x)&\Phi(x)\\\
\end{array}\right).$ (2.11)

This notation is convenient to understand so-called $SU(2)$ custodial symmetry
[32]. We will find that the stability of the DM is realized by a “
$U(1)_{V}^{\rm global}$ custodial symmetry,” which is a $U(1)$ subgroup of the
$SU(2)_{\Phi L}^{\rm global}\times SU(2)_{\Phi R}^{\rm global}$ diagonal
subgroup $SU(2)_{V}^{\rm global}$.

By using the complex scalar field in ${\bf 2}$ of $SU(2)_{D}$ $\Sigma(x)$
instead of $\Phi(x)$, the scalar potential in Eq. (2.2) can be written as

$\displaystyle{\cal V}(H,{\Sigma},\Delta)$
$\displaystyle=-\mu_{H}^{2}H^{\dagger}H-\frac{\mu_{\Phi}^{2}}{2}\mbox{Tr}\left({\Sigma}^{\dagger}{\Sigma}\right)-\frac{1}{2}\mu_{\Delta}^{2}\mbox{Tr}\left(\Delta^{2}\right)$
$\displaystyle\hskip
10.00002pt-\sqrt{2}\kappa_{1}\mbox{Tr}\left(\sigma_{1}{\Sigma}^{\dagger}\Delta{\Sigma}\right)-\sqrt{2}\kappa_{2}\mbox{Tr}\left(\sigma_{2}{\Sigma}^{\dagger}\Delta{\Sigma}\right)-\sqrt{2}\kappa_{3}\mbox{Tr}\left(\sigma_{3}{\Sigma}^{\dagger}\Delta{\Sigma}\right)$
$\displaystyle\hskip
10.00002pt+\lambda_{H}\left(H^{\dagger}H\right)^{2}+\frac{\lambda_{\Phi}}{4}\left(\mbox{Tr}\left({\Sigma}^{\dagger}{\Sigma}\right)\right)^{2}+\frac{1}{4}\lambda_{\Delta}\mbox{Tr}\left(\Delta^{2}\right)^{2}$
$\displaystyle\hskip
10.00002pt+\frac{1}{2}\lambda_{H\Phi}\left(H^{\dagger}H\right)\mbox{Tr}\left({\Sigma}^{\dagger}{\Sigma}\right)+\frac{1}{2}\lambda_{H\Delta}\left(H^{\dagger}H\right)\mbox{Tr}\left(\Delta^{2}\right)+\frac{1}{2}\lambda_{\Phi\Delta}\mbox{Tr}\left({\Sigma}^{\dagger}{\Sigma}\right)\mbox{Tr}\left(\Delta^{2}\right),$
(2.12)

where we used a relation $(-i\sigma_{2})\Sigma^{*}(i\sigma_{2})=\Sigma$.

We verify what kind of global symmetry exists in the potential given in Eq.
(2.12). First, the potential ${\cal V}(0,{\Sigma},0)$ is invariant under
$SU(2)_{\Phi L}^{\rm global}\times SU(2)_{\Phi R}^{\rm global}$. $\Sigma$ is
the bi-doublet representation under $SU(2)_{\Phi L}^{\rm global}\times
SU(2)_{\Phi R}^{\rm global}$:

$\displaystyle{\Sigma}\to U_{\Phi L}{\Sigma}U_{\Phi R}^{\dagger},\ \ \ U_{\Phi
L}=e^{i\theta_{\Phi L}^{a}\sigma_{a}},\ \ U_{\Phi R}=e^{i\theta_{\Phi
R}^{a}\sigma_{a}},$ (2.13)

where $\theta_{\Phi L}^{a}$ and $\theta_{\Phi R}^{a}$ are parameters of
$SU(2)_{\Phi L}^{\rm global}$ and $SU(2)_{\Phi R}^{\rm global}$
transformations, respectively. Second, the potential ${\cal V}(0,0,\Delta)$ is
invariant under $SU(2)_{\Delta}^{\rm global}$ transformation. $\Delta$ is the
adjoint representation under $SU(2)_{\Delta}^{\rm global}$:

$\displaystyle\Delta\to U_{\Delta}\Delta U_{\Delta}^{\dagger},\ \
U_{\Delta}=e^{i\theta_{\Delta}^{a}\sigma_{a}},$ (2.14)

where $\theta_{\Delta}^{a}$ is a parameter of $SU(2)_{\Delta}^{\rm global}$,
and the global transformation corresponds to a global subgroup transformation
of the gauge group $SU(2)_{D}^{\rm local}$ transformation. Third, under
$SU(2)_{\Delta}^{\rm global}\times SU(2)_{\Phi L}^{\rm global}\times
SU(2)_{\Phi R}^{\rm global}$, each $\kappa_{a}$ term transforms as

$\displaystyle\mbox{Tr}\left(\sigma_{a}{\Sigma}^{\dagger}\Delta{\Sigma}\right)\to\mbox{Tr}\left(U_{\Phi
R}^{\dagger}\sigma_{a}U_{\Phi R}{\Sigma}^{\dagger}U_{\Phi
L}^{\dagger}U_{\Delta}\Delta U_{\Delta}^{\dagger}U_{\Phi L}{\Sigma}\right).$
(2.15)

This term is invariant under $SU(2)_{L}^{\rm global}\times U(1)_{\Phi
R{a}}^{\rm global}$, where $SU(2)_{L}^{\rm global}$ represents
$SU(2)_{\Delta}=SU(2)_{\Phi L}$, and $U(1)_{\Phi R{a}}^{\rm global}$
corresponds to the $\sigma_{a}$ direction of $SU(2)_{\Phi R}^{\rm global}$.
The combination of the $\kappa_{1}$, $\kappa_{2}$, and $\kappa_{3}$ terms is
also invariant under $SU(2)_{L}^{\rm global}\times U(1)_{\Phi R}^{\rm
global}$:

$\displaystyle\mbox{Tr}\left((\kappa_{1}\sigma_{1}+\kappa_{2}\sigma_{2}+\kappa_{3}\sigma_{3}){\Sigma}^{\dagger}\Delta{\Sigma}\right)\to\mbox{Tr}\left(U_{\Phi
R}^{\dagger}(\kappa_{1}\sigma_{1}+\kappa_{2}\sigma_{2}+\kappa_{3}\sigma_{3})U_{\Phi
R}{\Sigma}^{\dagger}\Delta{\Sigma}\right),$ (2.16)

where the $U(1)_{\Phi R}^{\rm global}$ transformation is associated with the
$(\kappa_{1}\sigma_{1}+\kappa_{2}\sigma_{2}+\kappa_{3}\sigma_{3})$ direction
in $SU(2)_{\Phi R}^{\rm global}$. Therefore, the potential is invariant under
$SU(2)_{L}^{\rm global}\times U(1)_{\Phi R}^{\rm global}$. Without losing
generality, we can choose the $\kappa_{a}\sigma_{a}$ direction associated with
the remaining $U(1)_{\Phi R}^{\rm global}$ symmetry. Therefore, in the
following we will take the $\kappa_{3}\sigma_{3}$ direction and denote
$\kappa_{3}$ as $\kappa$ and $U(1)_{\Phi R3}^{\rm global}$ as $U(1)_{R}^{\rm
global}$. In other words, we remove $\kappa_{1}$ and $\kappa_{2}$ by using the
$SU(2)_{\Phi R}^{\rm global}$ transformation. Therefore, the potential in Eq.
(2.12) is invariant under $SU(2)_{L}^{\rm global}\times U(1)_{R}^{\rm
global}$. The global symmetry $SU(2)_{\Delta}^{\rm global}\times SU(2)_{\Phi
L}^{\rm global}\times SU(2)_{\Phi R}^{\rm global}$ breaking pattern associated
with the explicit breaking terms is shown in Figure 1.

Figure 1: The global symmetry $SU(2)_{\Delta}^{\rm global}\times SU(2)_{\Phi
L}^{\rm global}\times SU(2)_{\Phi R}^{\rm global}$ breaking pattern is shown.
The $\kappa$ term stands for the soft symmetry breaking term. In the figure,
the superscript, global, is omitted.

##  3 Vacuum structure

We consider vacuum structures of ${\Sigma}(x)$ and $\Delta(x)$. The system we
are currently considering has $SU(2)_{D}^{\rm local}$ (or $SU(2)_{L}^{\rm
global}$) and $U(1)_{R}^{\rm global}$ symmetry. By using a total of four
degrees of freedom of $SU(2)_{D}^{\rm local}$ gauge and $U(1)_{R}^{\rm
global}$ transformations, without loss of generality, we take the VEVs of
${\Sigma}$ and $\Delta$ as
$\langle\Delta\rangle=(v_{\eta_{1}}\sigma_{1}+v_{\eta_{3}}\sigma_{3})/\sqrt{2}$
and $\langle\Sigma\rangle=v_{\Phi}I/\sqrt{2}$, i.e.,

$\displaystyle\langle\Delta\rangle$
$\displaystyle=\frac{1}{\sqrt{2}}\left(\begin{array}[]{cc}v_{\eta_{3}}&v_{\eta_{1}}\\\
v_{\eta_{1}}&-v_{\eta_{3}}\\\ \end{array}\right),\ \ \
\langle{\Sigma}\rangle=\frac{1}{\sqrt{2}}\left(\begin{array}[]{cc}v_{\Phi}&0\\\
0&v_{\Phi}\\\ \end{array}\right),$ (3.5)

where we remove the VEVs of $\eta_{2}$, $\phi_{1}$, $\phi_{2}$, and
$\phi_{4}$. The gauge symmetry $SU(2)_{D}^{\rm local}$ breaking patterns are
shown in Figure 2. As is well-known, the nonvanishing VEV of a complex scalar
field in ${\bf 2}$ of $SU(2)$ breaks $SU(2)$ symmetry completely, so a total
of three Nambu-Goldstone (NG) or pseudo NG (pNG) modes appear. More
specifically, when $SU(2)$ global symmetry is exact, three NG modes appear;
when $SU(2)$ global symmetry is softly broken to $U(1)$ global symmetry by
explicit breaking terms, one NG and two pNG modes appear; when $SU(2)$ global
symmetry is completely softly broken by explicit breaking terms, three pNG
modes appear. The nonvanishing VEV of a real scalar field in ${\bf 3}$ of
$SU(2)$ breaks $SU(2)$ symmetry to $U(1)$ symmetry, so a total of two NG or
pNG modes appear.

Figure 2: The gauge symmetry $SU(2)_{D}^{\rm local}$ breaking patterns are
shown. $\langle{\Sigma}\rangle\not=0$ and $\langle\Delta\rangle\not=0$
represent the spontaneous symmetry breaking (SSB) by the VEV of ${\Sigma}$ and
$\Delta$ in ${\bf 2}$ and ${\bf 3}$ of $SU(2)_{D}^{\rm local}$. $U(1)_{D}$ is
a local subgroup of $SU(2)_{D}^{\rm local}$. In the figure, the superscript,
local, is omitted.

Furthermore, when $\kappa=0$, the system has $SU(2)_{\Delta}^{\rm
global}\times SU(2)_{\Phi L}^{\rm global}\times SU(2)_{\Phi R}^{\rm global}$
symmetry. By using the degrees of freedom of $SU(2)_{\Delta}^{\rm global}$,
$SU(2)_{\Phi L}^{\rm global}$, and $SU(2)_{\Phi R}^{\rm global}$
transformations, we can take the VEVs of $\Sigma$ and $\Delta$ as

$\displaystyle\langle\Delta\rangle$
$\displaystyle=\frac{1}{\sqrt{2}}\left(\begin{array}[]{cc}v_{\eta_{3}}&0\\\
0&-v_{\eta_{3}}\\\ \end{array}\right),\ \ \
\langle{\Sigma}\rangle=\frac{1}{\sqrt{2}}\left(\begin{array}[]{cc}v_{\Phi}&0\\\
0&v_{\Phi}\\\ \end{array}\right),$ (3.10)

Here we check what kind of symmetry is preserved by the VEVs of ${\Sigma}$ and
$\Delta$ in Eqs. (3.5) and (3.10). First, we consider the VEV of ${\Sigma}$ as
$\langle{\Sigma}\rangle=v_{\Phi}/\sqrt{2}I$. Under $SU(2)_{\Phi L}^{\rm
global}\times SU(2)_{\Phi R}^{\rm global}$, the VEV transforms as

$\displaystyle\langle{\Sigma}\rangle\to U_{\Phi L}\langle{\Sigma}\rangle
U_{\Phi R}^{\dagger}=e^{i\theta_{\Phi
La}\sigma_{a}}\frac{v_{\Phi}}{\sqrt{2}}Ie^{-i\theta_{\Phi
Ra}\sigma_{a}}=\frac{v_{\Phi}}{\sqrt{2}}Ie^{i(\theta_{\Phi La}-\theta_{\Phi
Ra})\sigma_{a}}=\langle{\Sigma}\rangle e^{i(\theta_{\Phi La}-\theta_{\Phi
Ra})\sigma_{a}}.$ (3.11)

Therefore, only for $\theta_{\Phi La}=\theta_{\Phi Ra}$, the VEV is invariant.
That is, only $SU(2)_{\Phi V}^{\rm global}$ remains. In the case, a total of
three NG or pNG modes appear.

Next, we consider the VEV of $\Delta$ in Eqs. (3.5) and (3.10). When
$\kappa=0$, under $SU(2)_{\Delta}^{\rm global}$, the VEV in Eq. (3.10)
transforms as

$\displaystyle\langle\Delta\rangle$ $\displaystyle\to
U_{\Delta}\langle\Delta\rangle U_{\Delta}^{\dagger}.$ (3.12)

For $\theta_{\Delta 1}=\theta_{\Delta 2}=0$, the VEV is invariant because
$[U_{\Delta},\langle\Delta\rangle]=0$. That is, only $U(1)_{\Delta}^{\rm
global}$ associated with $\sigma_{3}$ remains. Under $SU(2)_{\Delta}^{\rm
global}$, the VEV in Eq. (3.5) transforms as

$\displaystyle\langle\Delta\rangle$ $\displaystyle\to
U_{\Delta}\langle\Delta\rangle U_{\Delta}^{\dagger}.$ (3.13)

When $\kappa\not=0$, for $\theta_{\Delta
3}=\frac{v_{\eta_{3}}}{v_{\eta_{1}}}\theta_{\Delta 1}$, $\theta_{\Delta 2}=0$,
the VEV is invariant because $[U_{\Delta},\langle\Delta\rangle]=0$. That is,
$U(1)_{\Delta}^{\rm global}$ associated with a linear combination of
$\sigma_{1}$ and $\sigma_{3}$ remains. From the above discussion, regardless
of the configuration of the VEV of $\Delta$, $SU(2)_{\Delta}^{\rm global}$
symmetry is broken to $U(1)_{\Delta}^{\rm global}$. Therefore, a total of two
NG or pNG modes appear.

We summarize NG or pNG modes in this model. When ${\Sigma}(x)$ in ${\bf 2}$ of
$SU(2)$ acquires a nonvanishing VEV, a total of three NG or pNG mode appear.
When $\Delta(x)$ in ${\bf 3}$ of $SU(2)$ acquires a nonvanishing VEV, a total
of two NG or pNG mode appear. In the dark $SU(2)_{D}$ sector, a total of up to
five NG or pNG modes appear.

##  4 Analyzing the potential

We summarize how to find the vacuum that satisfies the global minimum of the
potential for each set of model parameters below.

  * (1)

Write down the most general potential of fields such as $\Sigma(x)$ and
$\Delta(x)$. The potential ${\cal V}$ depends on some degrees of freedoms
$v_{X}$ such as $v_{\Phi}$ and $v_{\eta_{3}}$:

$\displaystyle{\cal V}(\\{v_{X}\\})={\cal
V}(v_{\Phi},v_{\eta_{1}},v_{\eta_{3}}).$ (4.1)

  * (2)

Calculate the first derivatives of the potential ${\cal V}(\\{v_{X}\\})$ with
respect to all the variables $\\{v_{X}\\}$.

We find three stationary conditions as

$\displaystyle\frac{\partial}{\partial v_{X}}{\cal V}(\\{v_{X}\\})=0.$ (4.2)

  * (3)

Solve the simultaneous equations derived from the stationary conditions.

We find that the variables $v_{X}$ are expressed as model parameters such as
$\mu_{\Phi}^{2}$ and $\lambda_{\Phi}$. Note that in some cases a VEV is
related with another VEV, and some flat directions exist. This situation
occurs when symmetry is unbroken.

  * (4)

Compare the values of the potential at all extrema and saddle points.

We find true vacua of the potential ${\cal V}(\\{v_{X}\\})$ at each parameter
region, where all the VEVs at the true vacuum must be real in our convention.

  * (5)

Check what kind of symmetry is realized at each vacuum for each parameter
region.

(Note that the same procedure is commonly used, e.g., to analyze the vacuum
structures of $SU(N)$ symmetry breaking by elementary scalar fields [38, 39]
and $E_{6}$, $SU(N)$ and $SO(N)$ symmetry breaking by composite scalar fields
[40, 41, 42, 43, 44].)

To understand the vacuum structure of this system, we first consider the case
$\kappa=0$. After that, we will discuss the case $\kappa\not=0$.

###  4.1 Without soft symmetry breaking term $(\kappa=0)$

We take the VEVs of $\Sigma(x)$ and $\Delta(x)$ given in Eq. (3.10).
Substituting the VEVs into the potential of $\Sigma(x)$ and $\Delta(x)$ given
in Eq. (2.12) with $\kappa=0$

$\displaystyle{\cal V}({\Sigma},\Delta)$
$\displaystyle=-\frac{\mu_{\Phi}^{2}}{2}\mbox{Tr}\left({\Sigma}^{\dagger}{\Sigma}\right)-\frac{1}{2}\mu_{\Delta}^{2}\mbox{Tr}\left(\Delta^{2}\right)$
$\displaystyle\hskip
10.00002pt+\frac{\lambda_{\Phi}}{4}\left(\mbox{Tr}\left({\Sigma}^{\dagger}{\Sigma}\right)\right)^{2}+\frac{1}{4}\lambda_{\Delta}\mbox{Tr}\left(\Delta^{2}\right)^{2}+\frac{1}{2}\lambda_{\Phi\Delta}\mbox{Tr}\left({\Sigma}^{\dagger}{\Sigma}\right)\mbox{Tr}\left(\Delta^{2}\right),$
(4.3)

the potential is given by

$\displaystyle{\cal V}(v_{\Phi},v_{\Delta})$
$\displaystyle=-\frac{1}{2}\mu_{\Phi}^{2}v_{\Phi}^{2}-\frac{1}{2}\mu_{\Delta}^{2}v_{\Delta}^{2}+\frac{1}{4}\lambda_{\Phi}v_{\Phi}^{4}+\frac{1}{4}\lambda_{\Delta}v_{\Delta}^{4}+\frac{1}{2}\lambda_{\Phi\Delta}v_{\Phi}^{2}v_{\Delta}^{2},$
(4.4)

where we denote $v_{\eta_{3}}$ as $v_{\Delta}$. This potential is invariant
under $SU(2)_{\Delta}^{\rm global}\times SU(2)_{\Phi L}^{\rm global}\times
SU(2)_{\Phi R}^{\rm global}$ and $SU(2)_{D}^{\rm local}$ transformations shown
in Figures 1 and 2.

Next, we calculate the first derivatives of the potential ${\cal
V}(v_{\Phi},v_{\Delta})$ with respect to $v_{\Phi}$ and $v_{\Delta}$.

$\displaystyle\frac{\partial}{\partial v_{\Phi}}{\cal V}(v_{\Phi},v_{\Delta})$
$\displaystyle=v_{\Phi}\left(-\mu_{\Phi}^{2}+\lambda_{\Phi}v_{\Phi}^{2}+\lambda_{\Phi\Delta}v_{\Delta}^{2}\right),\allowdisplaybreaks[1]$
$\displaystyle\frac{\partial}{\partial v_{\Delta}}{\cal
V}(v_{\Phi},v_{\Delta})$
$\displaystyle=v_{\Delta}\left(-\mu_{\Delta}^{2}+\lambda_{\Delta}v_{\Delta}^{2}+\lambda_{\Phi\Delta}v_{\Phi}^{2}\right).$
(4.5)

From the first derivatives, we find the following stationary conditions:

$\displaystyle 0$
$\displaystyle=v_{\Phi}\left(-\mu_{\Phi}^{2}+\lambda_{\Phi}v_{\Phi}^{2}+\lambda_{\Phi\Delta}v_{\Delta}^{2}\right),\allowdisplaybreaks[1]$
(4.6) $\displaystyle 0$
$\displaystyle=v_{\Delta}\left(-\mu_{\Delta}^{2}+\lambda_{\Delta}v_{\Delta}+\lambda_{\Phi\Delta}v_{\Phi}^{2}\right).$
(4.7)

We analytically solve the simultaneous equations given in Eqs. (4.6) and (4.7)
below.

  * •

From Eq. (4.6), we find

$\displaystyle v_{\Phi}=0\ \ \mbox{or}\ \
-\mu_{\Phi}^{2}+\lambda_{\Phi}v_{\Phi}^{2}+\lambda_{\Phi\Delta}v_{\Delta}^{2}=0.$
(4.8)

  * •

First, for $v_{\Phi}=0$ case, from Eqs. (4.6) and (4.7), we find

$\displaystyle v_{\Delta}=0\ \ \mbox{or}\ \
v_{\Delta}=\pm\sqrt{\frac{\mu_{\Delta}^{2}}{\lambda_{\Delta}}}.$ (4.9)

For the first case, $v_{\Delta}=0$, the VEVs are located at the origin

$\displaystyle v_{\Phi}=v_{\Delta}=0.$ (4.10)

$SU(2)_{D}^{\rm local}$ is unbroken, and $SU(2)_{\Delta}^{\rm global}\times
SU(2)_{\Phi L}^{\rm global}\times SU(2)_{\Phi R}^{\rm global}$ is also
unbroken.

For the second case, $v_{\Delta}=\pm\sqrt{\mu_{\Delta}^{2}/\lambda_{\Delta}}$,
the VEVs are given by

$\displaystyle v_{\Phi}=0,\ \
v_{\Delta}=\pm\sqrt{\frac{\mu_{\Delta}^{2}}{\lambda_{\Delta}}}.$ (4.11)

$SU(2)_{D}^{\rm local}$ is broken to its subgroup $U(1)_{D}^{\rm local}$, and
$U(1)_{\Delta}^{\rm global}\times SU(2)_{\Phi L}^{\rm global}\times
SU(2)_{\Phi R}^{\rm global}$ remains.

  * •

Next we consider the second condition in Eq. (4.8).

From Eq. (4.7), we find

$\displaystyle v_{\Delta}=0\ \ \mbox{or}\ \
-\mu_{\Delta}^{2}+\lambda_{\Delta}v_{\Delta}+\lambda_{\Phi\Delta}v_{\Phi}^{2}=0.$
(4.12)

For the first case $v_{\Delta}=0$, we find

$\displaystyle v_{\Phi}=\pm\sqrt{\frac{\mu_{\Phi}^{2}}{\lambda_{\Phi}}},\ \
v_{\Delta}=0.$ (4.13)

$SU(2)_{D}^{\rm local}$ is completely broken. $SU(2)_{\Phi L}^{\rm
global}\times SU(2)_{\Phi R}^{\rm global}$ is broken to the $SU(2)_{\Phi
V}^{\rm global}$ custodial symmetry that is the diagonal subgroup of
$SU(2)_{\Phi L}^{\rm global}\times SU(2)_{\Phi R}^{\rm global}$, so
$SU(2)_{\rm\Delta}^{\rm global}\times SU(2)_{\Phi V}^{\rm global}$ remains.

  * •

Finally, we consider the following simultaneous equations:

$\displaystyle
0=-\mu_{\Phi}^{2}+\lambda_{\Phi}v_{\Phi}^{2}+\lambda_{\Phi\Delta}v_{\Delta}^{2},\
\
0=-\mu_{\Delta}^{2}+\lambda_{\Delta}v_{\Delta}^{2}+\lambda_{\Phi\Delta}v_{\Phi}^{2}.$
(4.14)

Since the simultaneous equations can be decomposed into two quadratic
equations of $v_{\Phi}$ and $v_{\Delta}$, it can be solved as

$\displaystyle
v_{\Phi}=\pm\sqrt{\frac{\lambda_{\Delta}\mu_{\Phi}^{2}-\lambda_{\Phi\Delta}\mu_{\Delta}^{2}}{\lambda_{\Delta}\lambda_{\Phi}-\lambda_{\Phi\Delta}^{2}}},\
\
v_{\Delta}=\pm\sqrt{\frac{\lambda_{\Phi}\mu_{\Delta}^{2}-\lambda_{\Phi\Delta}\mu_{\Phi}^{2}}{\lambda_{\Delta}\lambda_{\Phi}-\lambda_{\Phi\Delta}^{2}}},$
(4.15)

where all sign combinations exist. $SU(2)_{D}^{\rm local}$ is completely
broken. $SU(2)_{\Delta}^{\rm global}$ is broken to $U(1)_{\Delta}^{\rm
global}$ and $SU(2)_{\Phi L}^{\rm global}\times SU(2)_{\Phi R}^{\rm global}$
is broken to the $SU(2)_{\Phi V}^{\rm global}$ custodial symmetry. Therefore
$U(1)_{\Delta}^{\rm global}\times SU(2)_{\Phi V}^{\rm global}$ remains.

We summarize the extrema and saddle points in the potential given in Eq. (4.4)
in Table 2. In the table, the potential energy at each extremum or saddle
point, remaining gauge and global symmetry, and a total number of NG modes are
also listed, where $V_{1}$, $V_{2}$, $V_{3}$, and $V_{4}$ represent the names
of the stationary points and the potential energies at $SU(2)_{D}^{\rm
local}$, $U(1)_{D}^{\rm local}$, $SU(2)_{\Delta}^{\rm global}\times
SU(2)_{\Phi V}^{\rm global}$, and $U(1)_{\Delta}^{\rm global}\times
SU(2)_{\Phi V}^{\rm global}$ stationary points, respectively.

Name | $V_{1}$ | $V_{2}$ | $V_{3}$ | $V_{4}$
---|---|---|---|---
$(v_{\Phi},v_{\Delta})$ | $\left(0,0\right)$ | $\left(0,\pm\sqrt{\frac{\mu_{\Delta}^{2}}{\lambda_{\Delta}}}\right)$ | $\left(\pm\sqrt{\frac{\mu_{\Phi}^{2}}{\lambda_{\Phi}}},0\right)$ | $\left(\pm\sqrt{\frac{\lambda_{\Delta}\mu_{\Phi}^{2}-\lambda_{\Phi\Delta}\mu_{\Delta}^{2}}{\lambda_{\Delta}\lambda_{\Phi}-\lambda_{\Phi\Delta}^{2}}},\pm(\Phi\leftrightarrow\Delta)\right)$
${\cal V}(v_{\Phi},v_{\Delta})$ | $0$ | $-\frac{\mu_{\Delta}^{4}}{4\lambda_{\Delta}}$ | $-\frac{\mu_{\Phi}^{4}}{4\lambda_{\Phi}}$ | $-\frac{\lambda_{\Phi}\mu_{\Delta}^{4}-2\lambda_{\Phi\Delta}\mu_{\Phi}^{2}\mu_{\Delta}^{2}+\lambda_{\Delta}\mu_{\Phi}^{4}}{4(\lambda_{\Delta}\lambda_{\Phi}-\lambda_{\Phi\Delta}^{2})}$
$\begin{array}[]{c}\mbox{Gauge}\\\ \mbox{symmetry}\\\ \end{array}$ | $SU(2)_{D}$ | $U(1)_{D}$ | None | None
$\begin{array}[]{c}\mbox{Global}\\\ \mbox{symmetry}\\\ \end{array}$ | $\begin{array}[]{c}SU(2)_{\Delta}\\\ \times SU(2)_{\Phi L}\\\ \times SU(2)_{\Phi R}\\\ \end{array}$ | $\begin{array}[]{c}U(1)_{\Delta}\\\ \times SU(2)_{\Phi L}\\\ \times SU(2)_{\Phi R}\\\ \end{array}$ | $\begin{array}[]{c}SU(2)_{\Delta}\\\ \times SU(2)_{\Phi V}\\\ \end{array}$ | $\begin{array}[]{c}U(1)_{\Delta}\times SU(2)_{\Phi V}\\\ \end{array}$
$\\#$ of NG | $0$ | $2$ | $3$ | $5$
Table 2: The extrema and saddle points in the potential given in Eq. (4.4) for
$\kappa=0$ are shown. The potential energy at each extremum or saddle point
and remaining gauge and global symmetry are also listed. $\\#$ of NG
represents the total number of NG modes. In the table, the superscript,
local/global, is omitted.

Next, we consider the correspondence between the parameter domain and the
symmetry realized in the vacuum. First of all, the quartic coupling constants
$\lambda_{\Delta}$, $\lambda_{\Phi}$, and $\lambda_{\Phi\Delta}$ must satisfy
the following conditions to stabilize the potential ${\cal
V}(v_{\Phi},v_{\Delta})$ with finite values of the VEVs:

$\displaystyle\lambda_{\Delta}>0,\ \ \lambda_{\Phi}>0,\ \ \
\lambda_{\Delta}\lambda_{\Phi}-\lambda_{\Phi\Delta}^{2}>0.$ (4.16)

There are four stationary points $V_{1,2,3,4}$ given in Table 2. They are not
always solutions in all parameter regions because the VEVs $v_{\Phi}$ and
$v_{\Delta}$ are defined as real numbers. In fact, $V_{1}$ is a solution in
any $\mu_{\Phi}^{2}$ and $\mu_{\Delta}^{2}$ region; $V_{2}$ is a solution for
$\mu_{\Delta}^{2}>0$; $V_{3}$ is a solution for $\mu_{\Phi}^{2}>0$; and
$V_{4}$ is a solution for $\mu_{\Delta}^{2}>0$ and $\mu_{\Phi}^{2}>0$.

We will find the true vacuum by comparing the potential energies of stationary
points. When $\mu_{\Delta}^{2}>0$ and $\mu_{\Phi}^{2}>0$, the potential energy
preserving $U(1)_{\Delta}^{\rm global}\times SU(2)_{\Phi V}$ is lower than the
other potential energies preserving $SU(2)_{D}^{\rm local}$, $U(1)_{D}^{\rm
local}$, and $SU(2)_{\Delta}^{\rm global}\times SU(2)_{\Phi V}^{\rm global}$
because

$\displaystyle V_{4}-V_{1}$
$\displaystyle=-\left\\{\frac{\lambda_{\Phi}}{4(\lambda_{\Delta}\lambda_{\Phi}-\lambda_{\Phi\Delta}^{2})}\left(\mu_{\Delta}^{2}-\frac{\lambda_{\Phi\Delta}}{\lambda_{\Phi}}\mu_{\Phi}^{2}\right)^{2}+\frac{\mu_{\Phi}^{4}}{4\lambda_{\Phi}}\right\\}<0,$
(4.17) $\displaystyle V_{4}-V_{2}$
$\displaystyle=-\frac{(\lambda_{\Delta}\mu_{\Phi}^{2}-\lambda_{\Phi\Delta}\mu_{\Delta}^{2})^{2}}{4\lambda_{\Delta}(\lambda_{\Delta}\lambda_{\Phi}-\lambda_{\Phi\Delta}^{2})}<0,$
(4.18) $\displaystyle V_{4}-V_{3}$
$\displaystyle=-\frac{(\lambda_{\Phi}\mu_{\Delta}^{2}-\lambda_{\Phi\Delta}\mu_{\Phi}^{2})^{2}}{4\lambda_{\Phi}(\lambda_{\Delta}\lambda_{\Phi}-\lambda_{\Phi\Delta}^{2})}<0$
(4.19)

from Eq. (4.16). For the other parameter spaces of
$(\mu_{\Phi}^{2},\mu_{\Delta}^{2})$, it is easy to find that for
$\mu_{\Phi}^{2}<0$ and $\mu_{\Delta}^{2}<0$, $SU(2)_{D}^{\rm local}$ is
realized at the vacuum; for $\mu_{\Phi}^{2}<0$ and $\mu_{\Delta}^{2}>0$,
$U(1)_{D}^{\rm local}$ is realized at the vacuum; for $\mu_{\Phi}^{2}>0$ and
$\mu_{\Delta}^{2}<0$, $SU(2)_{\Delta}^{\rm global}\times SU(2)_{\Phi V}^{\rm
global}$ is realized at the vacuum. The global symmetry $SU(2)_{\Delta}^{\rm
global}\times SU(2)_{\Phi L}^{\rm global}\times SU(2)_{\Phi R}^{\rm global}$
breaking patterns are shown in Figure 3.

Figure 3: The global symmetry $SU(2)_{\Delta}^{\rm global}\times SU(2)_{\Phi
L}^{\rm global}\times SU(2)_{\Phi R}^{\rm global}$ breaking patterns for
$\kappa=0$ are shown. $\langle\Sigma\rangle\not=0$ and
$\langle\Delta\rangle\not=0$ represent the SSB by the VEV of $\Sigma$ and
$\Delta$ in $({\bf 1,2,2})$ and $({\bf 3,1,1})$ of $SU(2)_{\Delta}^{\rm
global}\times SU(2)_{\Phi L}^{\rm global}\times SU(2)_{\Phi R}^{\rm global}$,
respectively. In the figure, the superscript, global, is omitted.

###  4.2 With soft symmetry breaking term $(\kappa\not=0)$

We will now begin analyzing the potential for the case $\kappa\not=0$. We take
the VEVs of $\Sigma(x)$ and $\Delta(x)$ given in Eq. (3.5). Substituting the
VEVs into the potential of $\Sigma(x)$ and $\Delta(x)$ given in Eq. (2.12)

$\displaystyle{\cal V}({\Sigma},\Delta)$
$\displaystyle=-\frac{\mu_{\Phi}^{2}}{2}\mbox{Tr}\left({\Sigma}^{\dagger}{\Sigma}\right)-\frac{1}{2}\mu_{\Delta}^{2}\mbox{Tr}\left(\Delta^{2}\right)-\sqrt{2}\kappa\mbox{Tr}\left(\sigma_{3}{\Sigma}^{\dagger}\Delta{\Sigma}\right)$
$\displaystyle\hskip
10.00002pt+\frac{\lambda_{\Phi}}{4}\left(\mbox{Tr}\left({\Sigma}^{\dagger}{\Sigma}\right)\right)^{2}+\frac{1}{4}\lambda_{\Delta}\mbox{Tr}\left(\Delta^{2}\right)^{2}+\frac{1}{2}\lambda_{\Phi\Delta}\mbox{Tr}\left({\Sigma}^{\dagger}{\Sigma}\right)\mbox{Tr}\left(\Delta^{2}\right),$
(4.20)

the potential is given by

$\displaystyle{\cal V}(v_{\Phi},v_{\eta_{1}},v_{\eta_{3}})$
$\displaystyle=-\frac{1}{2}\mu_{\Phi}^{2}v_{\Phi}^{2}-\frac{1}{2}\mu_{\Delta}^{2}\left(v_{\eta_{1}}^{2}+v_{\eta_{3}}^{2}\right)-\kappa
v_{\Phi}^{2}v_{\eta_{3}}$ $\displaystyle\hskip
10.00002pt+\frac{1}{4}\lambda_{\Phi}v_{\Phi}^{4}+\frac{1}{4}\lambda_{\Delta}\left(v_{\eta_{1}}^{2}+v_{\eta_{3}}^{2}\right)^{2}+\frac{1}{2}\lambda_{\Phi\Delta}v_{\Phi}^{2}\left(v_{\eta_{1}}^{2}+v_{\eta_{3}}^{2}\right).$
(4.21)

We recall that the $\kappa$ term is invariant under $SU(2)_{D}^{\rm local}$,
but breaks $SU(2)_{\Delta}^{\rm global}\times SU(2)_{\Phi L}^{\rm
global}\times SU(2)_{\Phi R}^{\rm global}$ to $SU(2)_{L}^{\rm global}\times
U(1)_{R}^{\rm global}$. Therefore, this system is invariant under
$SU(2)_{D}^{\rm local}$ gauge and $SU(2)_{L}^{\rm global}\times U(1)_{R}^{\rm
global}$ global transformations. Note that $\Sigma(x)$ and $\Delta(x)$ belong
to ${\bf 2}(\pm 1)$ and ${\bf 3}(0)$ of $SU(2)_{L}^{\rm global}\times
U(1)_{R}^{\rm global}$, respectively, where the numbers in boldface denote
$SU(2)_{L}^{\rm global}$ representation and numbers in parentheses denote
$U(1)_{R}^{\rm global}$ charges.

Next, we calculate the first derivatives of the potential ${\cal
V}(v_{\Phi},v_{\eta_{1}},v_{\eta_{3}})$ with respect to
$v_{\Phi},v_{\eta_{1}},v_{\eta_{3}}$.

$\displaystyle\frac{\partial}{\partial v_{\Phi}}{\cal
V}(v_{\Phi},v_{\eta_{1}},v_{\eta_{3}})$
$\displaystyle=v_{\Phi}\left(-\mu_{\Phi}^{2}-2\kappa
v_{\eta_{3}}+\lambda_{\Phi}v_{\Phi}^{2}+\lambda_{\Phi\Delta}\left(v_{\eta_{1}}^{2}+v_{\eta_{3}}^{2}\right)\right),\allowdisplaybreaks[1]$
$\displaystyle\frac{\partial}{\partial v_{\eta_{1}}}{\cal
V}(v_{\Phi},v_{\eta_{1}},v_{\eta_{3}})$
$\displaystyle=v_{\eta_{1}}\left(-\mu_{\Delta}^{2}+\lambda_{\Delta}\left(v_{\eta_{1}}^{2}+v_{\eta_{3}}^{2}\right)+\lambda_{\Phi\Delta}v_{\Phi}^{2}\right),\allowdisplaybreaks[1]$
$\displaystyle\frac{\partial}{\partial v_{\eta_{3}}}{\cal
V}(v_{\Phi},v_{\eta_{1}},v_{\eta_{3}})$
$\displaystyle=v_{\eta_{3}}\left(-\mu_{\Delta}^{2}+\lambda_{\Delta}\left(v_{\eta_{1}}^{2}+v_{\eta_{3}}^{2}\right)+\lambda_{\Phi\Delta}v_{\Phi}^{2}\right)-\kappa
v_{\Phi}^{2}.$ (4.22)

From the first derivatives, we find the following stationary conditions:

$\displaystyle 0$ $\displaystyle=v_{\Phi}\left(-\mu_{\Phi}^{2}-2\kappa
v_{\eta_{3}}+\lambda_{\Phi}v_{\Phi}^{2}+\lambda_{\Phi\Delta}\left(v_{\eta_{1}}^{2}+v_{\eta_{3}}^{2}\right)\right),\allowdisplaybreaks[1]$
(4.23) $\displaystyle 0$
$\displaystyle=v_{\eta_{1}}\left(-\mu_{\Delta}^{2}+\lambda_{\Delta}\left(v_{\eta_{1}}^{2}+v_{\eta_{3}}^{2}\right)+\lambda_{\Phi\Delta}v_{\Phi}^{2}\right),\allowdisplaybreaks[1]$
(4.24) $\displaystyle 0$
$\displaystyle=v_{\eta_{3}}\left(-\mu_{\Delta}^{2}+\lambda_{\Delta}\left(v_{\eta_{1}}^{2}+v_{\eta_{3}}^{2}\right)+\lambda_{\Phi\Delta}v_{\Phi}^{2}\right)-\kappa
v_{\Phi}^{2}.$ (4.25)

We analytically solve the simultaneous equations given by Eqs. (4.23), (4.24),
and (4.25) below.

  * •

From Eq. (4.23), we find

$\displaystyle v_{\Phi}=0\ \ \mbox{or}\ \ -\mu_{\Phi}^{2}-2\kappa
v_{\eta_{3}}+\lambda_{\Phi}v_{\Phi}^{2}+\lambda_{\Phi\Delta}\left(v_{\eta_{1}}^{2}+v_{\eta_{3}}^{2}\right)=0.$
(4.26)

  * •

First, for $v_{\Phi}=0$ case, from Eqs. (4.23), (4.24), and (4.25), we find

$\displaystyle v_{\eta_{1}}=v_{\eta_{3}}=0\ \ \mbox{or}\ \
-\mu_{\Delta}^{2}+\lambda_{\Delta}\left(v_{\eta_{1}}^{2}+v_{\eta_{3}}^{2}\right)^{2}=0.$
(4.27)

For the first case, the VEVs are located at the origin

$\displaystyle v_{\Phi}=v_{\eta_{1}}=v_{\eta_{3}}=0.$ (4.28)

$SU(2)_{D}^{\rm local}$ is unbroken, and $SU(2)_{L}^{\rm global}\times
U(1)_{R}^{\rm global}$ is also unbroken.

For the second case, we can take the following VEVs by using the
$SU(2)_{L}^{\rm global}$ transformation:

$\displaystyle v_{\Phi}=0,\ \ v_{\eta_{1}}=0,\ \
v_{\eta_{3}}=\pm\sqrt{\frac{\mu_{\Delta}^{2}}{\lambda_{\Delta}}}.$ (4.29)

$SU(2)_{D}^{\rm local}$ is broken to its subgroup $U(1)_{D}^{\rm local}$, and
$U(1)_{L}^{\rm global}\times U(1)_{R}^{\rm global}$ remains.

  * •

Next we consider the second condition in Eq. (4.26). From Eq. (4.24), we find

$\displaystyle v_{\eta_{1}}=0\ \ \mbox{or}\ \
-\mu_{\Delta}^{2}+\lambda_{\Delta}\left(v_{\eta_{1}}^{2}+v_{\eta_{3}}^{2}\right)+\lambda_{\Phi\Delta}v_{\Phi}^{2}=0.$
(4.30)

From Eq. (4.25), the above second condition leads to $\kappa=0$, but due to
$\kappa\not=0$, $v_{\eta_{1}}=0$. For $v_{\Phi}\not=0$ and $v_{\eta_{1}}=0$,
we need to solve the following simultaneous equations:

$\displaystyle 0$ $\displaystyle=-\mu_{\Phi}^{2}-2\kappa
v_{\eta_{3}}+\lambda_{\Phi}v_{\Phi}^{2}+\lambda_{\Phi\Delta}v_{\eta_{3}}^{2},$
(4.31) $\displaystyle 0$
$\displaystyle=v_{\eta_{3}}\left(-\mu_{\Delta}^{2}+\lambda_{\Delta}v_{\eta_{3}}^{2}+\lambda_{\Phi\Delta}v_{\Phi}^{2}\right)-\kappa
v_{\Phi}^{2}.$ (4.32)

The solutions of the simultaneous equations lead to $v_{\Phi}\not=0$ and
$v_{\eta_{3}}\not=0$, so the vacuum of these solutions breaks $SU(2)_{L}^{\rm
global}\times U(1)_{R}^{\rm global}$ to $U(1)_{V}^{\rm global}$.

The simultaneous equations in Eqs. (4.31) and (4.32) can be decomposed into a
cubic equations for $v_{\eta_{3}}$ and a quadratic equation for $v_{\Phi}$.
From the vacuum solutions listed in Table 2 and the soft symmetry breaking
$\kappa$ term, the three solutions of the cubic equation correspond to one
$SU(2)_{\Delta}^{\rm global}\times SU(2)_{\Phi V}^{\rm global}$ and two
$U(1)_{\Delta}^{\rm global}\times SU(2)_{\Phi V}^{\rm global}$ global symmetry
vacuum solutions in the $\kappa=0$ case. We can solve the exact solutions of
the simultaneous equations because of just cubic and quadratic equations, but
they are too complicated to show here. Instead, we can find approximate
solutions to the simultaneous equations by using the solutions around the
$SU(2)_{\Delta}^{\rm global}\times SU(2)_{\Phi V}^{\rm global}$ and
$U(1)_{\Delta}^{\rm global}\times SU(2)_{\Phi V}^{\rm global}$ vacuum
solutions. The detailed values are not important for the discussion here, so
we omit the particular form, but there are solutions around the $\kappa=0$
solution in Table 2 as follows. From the solution for the $SU(2)_{\Delta}^{\rm
global}\times SU(2)_{\Phi V}^{\rm global}$ vacuum in the $\kappa=0$ case,

$\displaystyle
v_{\Phi}=\pm\sqrt{\frac{\mu_{\Phi}^{2}}{\lambda_{\Phi}}}+O(\kappa),\ \
v_{\eta_{3}}=O(\kappa).$ (4.33)

From the solution for the $U(1)_{\Delta}^{\rm global}\times SU(2)_{\Phi
V}^{\rm global}$ vacuum in the $\kappa=0$ case,

$\displaystyle
v_{\Phi}=\pm\sqrt{\frac{\lambda_{\Delta}\mu_{\Phi}^{2}-\lambda_{\Phi\Delta}\mu_{\Delta}^{2}}{\lambda_{\Delta}\lambda_{\Phi}-\lambda_{\Phi\Delta}^{2}}}+O(\kappa),\
\
v_{\eta_{3}}=\pm\sqrt{\frac{\lambda_{\Phi}\mu_{\Delta}^{2}-\lambda_{\Phi\Delta}\mu_{\Phi}^{2}}{\lambda_{\Delta}\lambda_{\Phi}-\lambda_{\Phi\Delta}^{2}}}+O(\kappa).$
(4.34)

Name | $V_{\rm I}$ | $V_{\rm II}$ | $V_{\rm III}$ | $V_{\rm IV}$
---|---|---|---|---
$(v_{\Phi},v_{\eta_{3}})$ | $\left(0,0\right)$ | $\left(0,\pm\sqrt{\frac{\mu_{\Delta}^{2}}{\lambda_{\Delta}}}\right)$ | $\left(\pm\sqrt{\frac{\mu_{\Phi}^{2}}{\lambda_{\Phi}}},O(\kappa)\right)$ | $\left(\pm\sqrt{\frac{\lambda_{\Delta}\mu_{\Phi}^{2}-\lambda_{\Phi\Delta}\mu_{\Delta}^{2}}{\lambda_{\Delta}\lambda_{\Phi}-\lambda_{\Phi\Delta}^{2}}},\pm(\Phi\leftrightarrow\Delta)\right)$
${\cal V}(v_{\Phi},v_{\eta_{3}})$ | $0$ | $-\frac{\mu_{\Delta}^{4}}{4\lambda_{\Delta}}$ | $-\frac{\mu_{\Phi}^{4}}{4\lambda_{\Phi}}$ | $-\frac{\lambda_{\Phi}\mu_{\Delta}^{4}-2\lambda_{\Phi\Delta}\mu_{\Phi}^{2}\mu_{\Delta}^{2}+\lambda_{\Delta}\mu_{\Phi}^{4}}{4(\lambda_{\Delta}\lambda_{\Phi}-\lambda_{\Phi\Delta}^{2})}$
$\begin{array}[]{c}\mbox{Gauge}\\\ \mbox{symmetry}\\\ \end{array}$ | $SU(2)_{D}$ | $U(1)_{D}$ | None
$\begin{array}[]{c}\mbox{Global}\\\ \mbox{symmetry}\\\ \end{array}$ | $SU(2)_{L}\times U(1)_{R}$ | $U(1)_{L}\times U(1)_{R}$ | $U(1)_{V}$
$\\#$ of NG | $0$ | $2$ | $3$ | $5$
Table 3: The extrema and saddle points in the potential given in Eq. (4.21)
for $\kappa\not=0$ are shown. The potential energy at each extremum or saddle
point and remaining gauge and global symmetry are also listed, where
$v_{\eta_{1}}=0$. $\\#$ of NG represents the total number of NG and pNGB
modes. In the table, the superscript, local/global, is omitted. For
$U(1)_{V}^{\rm global}$ case, we omit $O(\kappa)$ if there is already a value
greater than $\kappa$.

Next, we consider the correspondence between the parameter domain and the
symmetry realized in the vacuum. Since the $\kappa$ term does not affect the
shape of the potential at infinity, the constraint of the parameter region
from the stability condition to potential is the same for $\kappa\not=0$ as
for $\kappa=0$, which is given in Eq. (4.16). In the region where $\kappa$ can
be treated perturbatively, the true vacuum does not change, so the results for
the case $\kappa=0$ are applicable. Therefore, for
${}^{\forall}\mu_{\Phi}^{2}$ and $\mu_{\Delta}^{2}>0$, $U(1)_{V}^{\rm global}$
is realized at the vacuum; for $\mu_{\Phi}^{2}<0$ and $\mu_{\Delta}^{2}<0$,
$SU(2)_{D}^{\rm local}$ is realized at the vacuum; for $\mu_{\Phi}^{2}<0$ and
$\mu_{\Delta}^{2}>0$, $U(1)_{D}^{\rm local}$ is realized at the vacuum. The
extrema and saddle points in the potential are summarized in Table 3. The
global symmetry $SU(2)_{\Delta}^{\rm global}\times SU(2)_{\Phi L}^{\rm
global}\times SU(2)_{\Phi R}^{\rm global}$ breaking patterns are shown in
Figure 4.

Figure 4: The global symmetry $SU(2)_{\Delta}^{\rm global}\times SU(2)_{\Phi
L}^{\rm global}\times SU(2)_{\Phi R}^{\rm global}$ breaking patterns for
$\kappa\not=0$ are shown. The $\kappa$ term stands for a soft symmetry
breaking term; $\langle\Sigma\rangle\not=0$ and $\langle\Delta\rangle\not=0$
represent the SSB by the VEVs of $\Sigma$ and $\Delta$ in $({\bf 1,2,2})$ and
$({\bf 3,1,1})$ of $SU(2)_{\Delta}^{\rm global}\times SU(2)_{\Phi L}^{\rm
global}\times SU(2)_{\Phi R}^{\rm global}$. $({\bf 1,2,2})$ and $({\bf
3,1,1})$ of $SU(2)_{\Delta}^{\rm global}\times SU(2)_{\Phi L}^{\rm
global}\times SU(2)_{\Phi R}^{\rm global}$ are decomposed into two ${\bf 2}$
and one ${\bf 3}$ of $SU(2)_{L}$, respectively. In the figure, the
superscript, global, is omitted.

Before investigating the mass spectra of the scalar sectors, we comment on the
would-be NG and pNG modes. For $V_{\rm I}$ of Table 3, there are no NG modes.
For $V_{\rm II}$, there are two NG modes, and they are absorbed by the
$SU(2)_{D}^{\rm local}/U(1)_{D}^{\rm local}$ gauge bosons. For $V_{\rm III}$,
there are three NG modes, and they are absorbed by the $SU(2)_{D}^{\rm local}$
gauge bosons. For $V_{\rm IV}$, there are five NG modes. Three of the five NG
modes are absorbed in the $SU(2)_{D}^{\rm local}$ gauge boson. The remaining
two NG modes are real scalar modes with $U(1)_{V}^{\rm global}$ charges and
are identified as one complex scalar.

As can be seen from the above discussion, a charged pNGB that can be regarded
as a DM appears only when $\mu_{\Phi}^{2}>0$ and $\mu_{\Delta}^{2}>0$. In the
following, we consider such a case.

##  5 Mass spectrum

Here we investigate the mass spectra of the scalar fields $\Sigma(x)$ (or
$\Phi(x)$), $\Delta(x)$, and $H(x)$ for the parameter region
$\mu_{\Phi}^{2}>0$ and $\mu_{\Delta}^{2}>0$, where $U(1)_{V}^{\rm global}$
symmetry is realized at a vacuum. In particular, we confirm that there is a
$U(1)_{V}^{\rm global}$ charged complex scalar with a mass proportional to the
$\kappa$ parameter. It corresponds to the pNG mode, which will be regarded as
a DM candidate.

First, we check the potential terms associated with $\Sigma$ and $\Delta$.
Here we consider the following field expression at the vacuum:

$\displaystyle\Delta(x)=\frac{1}{\sqrt{2}}\left(\begin{array}[]{cc}v_{\Delta}+\eta_{3}&\eta_{1}-i\eta_{2}\\\
\eta_{1}+i\eta_{2}&-v_{\Delta}-\eta_{3}\\\ \end{array}\right),\ \ \
\Sigma(x)=\frac{1}{\sqrt{2}}\left(\begin{array}[]{cc}v_{\Phi}+\phi_{3}-i\phi_{4}&\phi_{1}+i\phi_{2}\\\
\phi_{1}-i\phi_{2}&v_{\Phi}+\phi_{3}+i\phi_{4}\\\ \end{array}\right),$ (5.5)

where we denote the VEV of $\eta_{3}$ as $v_{\Delta}$, and the values of
$v_{\Delta}$ and $v_{\Phi}$ for $\kappa\to 0$ are given in Eq. (4.34). In our
convention, the generator of the remaining $U(1)_{V}^{\rm global}$ corresponds
to the $\sigma_{3}$ direction. Therefore, $\eta_{1}$, $\eta_{2}$, $\phi_{1}$,
and $\phi_{2}$ have the same $U(1)_{V}^{\rm global}$ charge, while $\eta_{3}$,
$\phi_{3}$, and $\phi_{4}$ have no $U(1)_{V}^{\rm global}$ charge. The
$U(1)_{V}^{\rm global}$ charge of fields can be checked by using, e.g., the
generator of the $U(1)_{V}^{\rm global}$ charge $\sigma_{3}$.

The stationary conditions in Eqs. (4.31) and (4.32) for
$v_{\Phi},v_{\Delta}\not=0$ can be written as

$\displaystyle\mu_{\Phi}^{2}$ $\displaystyle=-2\kappa
v_{\Delta}+\lambda_{\Phi}v_{\Phi}^{2}+\lambda_{\Phi\Delta}v_{\eta_{3}}^{2},$
(5.6) $\displaystyle\mu_{\Delta}^{2}$
$\displaystyle=\lambda_{\Delta}v_{\Delta}^{2}+\lambda_{\Phi\Delta}v_{\Phi}^{2}-\kappa\frac{v_{\Phi}^{2}}{v_{\Delta}},$
(5.7)

where we replaced $v_{\eta_{3}}$ as $v_{\Delta}$.

Substituting Eqs. (5.5), (5.6), and (5.7) into the potential ${\cal
V}(\Sigma,\Delta)$ given in Eq. (4.20), the potential is given as

$\displaystyle{\cal
V}(\eta_{1},\eta_{2},\eta_{3},\phi_{1},\phi_{2},\phi_{3},\phi_{4})={\cal
V}_{0}+{\cal V}_{2}+{\cal V}_{3}+{\cal V}_{4},$ (5.8)

where ${\cal V}$ stands for $V_{\rm IV}$ in Table 3, and the subscript of
${\cal V}_{j}(j=0,2,3,4)$ denotes the mass dimension of the operator. Note
that the tadpole term of the potential ${\cal V}_{1}$ disappears from the
stationary conditions. The constant terms of the potential of the real scalar
fields $(\eta_{1},\eta_{2},\eta_{3},\phi_{1},\phi_{2},\phi_{3},\phi_{4})$ are
given by

$\displaystyle{\cal
V}_{0}=-\frac{1}{4}\left(\lambda_{\Delta}v_{\Delta}^{4}+\lambda_{\Phi}v_{\Phi}^{4}+2\lambda_{\Phi\Delta}v_{\Delta}^{2}v_{\Phi}^{2}-2\kappa
v_{\Delta}v_{\Phi}^{2}\right).$ (5.9)

The quadratic terms are given by

$\displaystyle{\cal V}_{2}$
$\displaystyle=\frac{1}{2}\left(\begin{array}[]{cc}\phi_{1}&\eta_{1}\\\
\end{array}\right)\left(\begin{array}[]{cc}4\kappa v_{\Delta}&2\kappa
v_{\Phi}\\\ 2\kappa v_{\Phi}&\frac{v_{\Phi}^{2}}{v_{\Delta}}\kappa\\\
\end{array}\right)\left(\begin{array}[]{c}\phi_{1}\\\ \eta_{1}\\\
\end{array}\right)+\frac{1}{2}\left(\begin{array}[]{cc}\phi_{2}&\eta_{2}\\\
\end{array}\right)\left(\begin{array}[]{cc}4\kappa v_{\Delta}&-2\kappa
v_{\Phi}\\\ -2\kappa v_{\Phi}&\frac{v_{\Phi}^{2}}{v_{\Delta}}\kappa\\\
\end{array}\right)\left(\begin{array}[]{c}\phi_{2}\\\ \eta_{2}\\\
\end{array}\right)$ (5.20) $\displaystyle\hskip
10.00002pt+\frac{1}{2}\left(\begin{array}[]{cc}\phi_{3}&\eta_{3}\\\
\end{array}\right)\left(\begin{array}[]{cc}2\lambda_{\Phi}v_{\Phi}^{2}&2\lambda_{\Phi\Delta}v_{\Phi}v_{\Delta}-2v_{\Phi}\kappa\\\
2\lambda_{\Phi\Delta}v_{\Phi}v_{\Delta}-2v_{\Phi}\kappa&2\lambda_{\Delta}v_{\Delta}^{2}+\frac{v_{\Phi}^{2}}{v_{\Delta}}\kappa\\\
\end{array}\right)\left(\begin{array}[]{c}\phi_{3}\\\ \eta_{3}\\\
\end{array}\right)+0\times\phi_{4}^{2}.$ (5.26)

Since the determinant of $(\eta_{1},\phi_{1})$ is zero, one of the eigenvalues
is zero. Furthermore, if $\kappa$ is set to zero, both eigenvalues are zero.
The same is true for $(\eta_{2},\phi_{2})$. Since the determinant of
$(\eta_{3},\phi_{3})$ is non-zero, even for $\kappa\to 0$ zero eigenvalues do
not appear. Instead of it, a $U(1)_{V}^{\rm global}$ neutral scalar field
$\phi_{4}$ is always massless. Therefore, we find that one of the two linear
combinations of $\eta_{1}$ and $\phi_{1}$ is an NG mode that is absorbed by an
$SU(2)_{D}^{\rm local}$ gauge boson, and the other is a pNG mode. The same is
true for $\eta_{2}$ and $\phi_{2}$. $\phi_{4}$ is an NG mode, and $\eta_{3}$
and $\phi_{3}$ are Higgs modes. Similarly, the cubic and quartic terms are
given by

$\displaystyle{\cal V}_{3}$
$\displaystyle=2\kappa\left(\eta_{1}\phi_{1}\phi_{3}-\eta_{2}\phi_{2}\phi_{3}+\eta_{2}\phi_{1}\phi_{4}+\eta_{1}\phi_{2}\phi_{4}\right)+\kappa\eta_{3}\left(\phi_{1}^{2}+\phi_{2}^{2}-\phi_{3}^{2}-\phi_{4}^{2}\right)$
$\displaystyle+\lambda_{\Delta}v_{\Delta}\eta_{3}\left(\eta_{1}^{2}+\eta_{2}^{2}+\eta_{3}^{2}\right)+\lambda_{\Phi}v_{\Phi}\phi_{3}\left(\phi_{1}^{2}+\phi_{2}^{2}+\phi_{3}^{2}+\phi_{4}^{2}\right)$
$\displaystyle+\lambda_{\Phi\Delta}v_{\Delta}\eta_{3}\left(\phi_{1}^{2}+\phi_{2}^{2}+\phi_{3}^{2}+\phi_{4}^{2}\right)+\lambda_{\Phi\Delta}v_{\Phi}\phi_{3}\left(\eta_{1}^{2}+\eta_{2}^{2}+\eta_{3}^{2}\right),$
(5.27) $\displaystyle{\cal V}_{4}$
$\displaystyle=\frac{1}{4}\lambda_{\Delta}\left(\eta_{1}^{2}+\eta_{2}^{2}+\eta_{3}^{2}\right)^{2}+\frac{1}{4}\lambda_{\Phi}\left(\phi_{1}^{2}+\phi_{2}^{2}+\phi_{3}^{2}+\phi_{4}^{2}\right)^{2}$
$\displaystyle\ \
+\frac{1}{2}\lambda_{\Phi\Delta}\left(\eta_{1}^{2}+\eta_{2}^{2}+\eta_{3}^{2}\right)\left(\phi_{1}^{2}+\phi_{2}^{2}+\phi_{3}^{2}+\phi_{4}^{2}\right).$
(5.28)

For the calculation of the scattering amplitudes of SM particles and a charged
pNGB in the next section, here we switch on the scalar field $H$ in ${\bf 2}$
of $SU(2)_{W}^{\rm local}$ listed in Table 1 and rewrite charged scalar fields
as follows:

$\displaystyle H=\frac{1}{\sqrt{2}}\left(\begin{array}[]{c}0\\\ v+h\\\
\end{array}\right),\ \ \ \phi_{(\pm)}:=\frac{1}{\sqrt{2}}\left(\phi_{1}\pm
i\phi_{2}\right),\ \ \ \eta_{(\pm)}:=\frac{1}{\sqrt{2}}\left(\eta_{1}\mp
i\eta_{2}\right),$ (5.31)

where the subscripts (positive and negative signs) enclosed in parentheses
indicate the sign of the $U(1)_{V}$ charge, and $v$ stands for the VEV of the
SM Higgs boson, which breaks $SU(2)_{W}^{\rm local}\times U(1)_{Y}^{\rm
local}$ into $U(1)_{\rm EM}^{\rm local}$. In the rest of this section, we will
examine the mass matrix when a $U(1)_{V}^{\rm global}$ neutral scalar field
$h$ is added to the mass matrix given in Eq. (5.26), where the field $h$ is a
main component of the SM Higgs boson.

First, we consider the mass eigenstates for the $U(1)_{V}^{\rm global}$
charged scalar fields $\phi_{1,2}$ and $\eta_{1,2}$. When we rewrite
$\phi_{1,2}$ and $\eta_{1,2}$ in terms of $\phi_{(\pm)}$ and $\eta_{(\pm)}$ as
in Eq. (5.31), the mass term of $\phi_{1,2}$ and $\eta_{1,2}$ given in Eq.
(5.26) can be rewritten as

$\displaystyle{\cal
V}_{2}\ni\frac{1}{2}\left(\begin{array}[]{cc}\phi_{(\mp)}&\eta_{(\mp)}\\\
\end{array}\right)M_{C}^{2}\left(\begin{array}[]{c}\phi_{(\pm)}\\\
\eta_{(\pm)}\\\ \end{array}\right),\ \
M_{C}^{2}:=\left(\begin{array}[]{cc}4\kappa v_{\Delta}&2\kappa v_{\Phi}\\\
2\kappa v_{\Phi}&\frac{v_{\Phi}^{2}}{v_{\Delta}}\kappa\\\ \end{array}\right).$
(5.37)

The above mass matrix can be easily diagonalized, and the mass and mixing
matrix are given by

$\displaystyle m_{G_{(\pm)}}=0,\ \
m_{\varphi}^{2}=\kappa\left(4v_{\Delta}+\frac{v_{\Phi}^{2}}{v_{\Delta}}\right),\
\ \left(\begin{array}[]{c}G_{(\pm)}\\\ \varphi_{(\pm)}\\\
\end{array}\right):=\left(\begin{array}[]{cc}\cos\beta&\sin\beta\\\
-\sin\beta&\cos\beta\\\
\end{array}\right)\left(\begin{array}[]{c}\phi_{(\pm)}\\\ \eta_{(\pm)}\\\
\end{array}\right),$ (5.44)

where

$\displaystyle\sin\beta=\frac{v_{\Phi}}{\sqrt{v_{\Phi}^{2}+4v_{\Delta}^{2}}},\
\ \cos\beta=\frac{2v_{\Delta}}{\sqrt{v_{\Phi}^{2}+4v_{\Delta}^{2}}},$ (5.45)

$G_{(\pm)}$ is the would-be NG modes of $SU(2)_{D}^{\rm local}$, and
$\varphi_{(\pm)}$ is a $U(1)_{V}^{\rm global}$ charged scalar mode that is a
pNG mode of $SU(2)_{V}^{\rm global}/U(1)_{V}^{\rm global}$, which will be
identified as a DM.

Second, even when $h$ exists, there is no mass mixing of $\phi_{4}$ and $h$,
so $\phi_{4}$ remains massless. That is, $\phi_{4}$ is a would-be NG mode of
$SU(2)_{D}^{\rm local}$. Further, $\phi_{4}$ has no charge of $U(1)_{D}^{\rm
local}$ and $U(1)_{V}^{\rm global}$. Therefore, $\phi_{4}$ can be identified
with the neutral NG mode $G_{0}$.

Finally, the mass matrix of a $U(1)_{V}^{\rm global}$ neutral sector
$(\eta_{3},\phi_{3},h)$ is given by

$\displaystyle
M_{H}^{2}:=\left(\begin{array}[]{ccc}2\lambda_{H}v^{2}&\lambda_{H\Phi}vv_{\Phi}&2\lambda_{H\Delta}vv_{\Delta}\\\
\lambda_{H\Phi}vv_{\Phi}&2\lambda_{\Phi}v_{\Phi}^{2}&2\lambda_{\Phi\Delta}v_{\Phi}v_{\Delta}-2v_{\Phi}\kappa\\\
2\lambda_{H\Delta}vv_{\Delta}&2\lambda_{\Phi\Delta}v_{\Phi}v_{\Delta}-2v_{\Phi}\kappa&2\lambda_{\Delta}v_{\Delta}^{2}+\frac{v_{\Phi}^{2}}{v_{\Delta}}\kappa-2v_{\Phi}\kappa\\\
\end{array}\right).$ (5.49)

Since this mass matrix is a real symmetric matrix, it can be diagonalized by a
unitary matrix (orthogonal matrix) $U_{H}$, where $U_{H}^{\dagger}U_{H}=I$ and
$U_{H}^{\dagger}=U_{H}^{T}$. That is,
$U_{H}M_{H}^{2}U_{H}^{\dagger}=(M_{H}^{2})^{\rm diag}$, where
$(M_{H}^{2})^{\rm diag}$ is a $3\times 3$ diagonal matrix. The mass
eigenstates can be expressed from the original basis as follows:

$\displaystyle\left(\begin{array}[]{c}h_{1}\\\ h_{2}\\\ h_{3}\\\
\end{array}\right):=U_{H}\left(\begin{array}[]{c}h\\\ \phi_{3}\\\ \eta_{3}\\\
\end{array}\right),$ (5.56)

where $h_{j}(j=1,2,3)$ are mass eigenstates with no $U(1)_{V}^{\rm global}$
charge, and $h_{1}$ is identified as the observed SM Higgs mode with a mass of
about 125 GeV. The exact eigenvalues and eigenvectors are too complicated to
show here. Instead of it, we show the approximate mass eigenvalues and mass
mixing matrix when $v_{\Delta}$ is sufficiently larger than $v$ and
$v_{\Phi}$. For $v_{\Delta}\gg v_{\Phi},v$, the mixing matrix $U_{H}$ is given
by

$\displaystyle U_{H}$
$\displaystyle=\left(\begin{array}[]{ccc}1&0&-\frac{\lambda_{H\Delta}v}{\lambda_{\Delta}v_{\Delta}}\\\
0&1&-\frac{\lambda_{\Phi\Delta}v_{\Phi}}{\lambda_{\Delta}v_{\Delta}}\\\
\frac{\lambda_{H\Delta}v}{\lambda_{\Delta}v_{\Delta}}&\frac{\lambda_{\Phi\Delta}v_{\Phi}}{\lambda_{\Delta}v_{\Delta}}&1\\\
\end{array}\right)\left(\begin{array}[]{ccc}\cos\alpha&\sin\alpha&0\\\
-\sin\alpha&\cos\alpha&0\\\ 0&0&1\\\
\end{array}\right)+O\left(\frac{v^{2}}{v_{\Delta}^{2}},\frac{v_{\Phi}^{2}}{v_{\Delta}^{2}}\right),$
(5.63) $\displaystyle\tan
2\alpha\simeq\frac{2vv_{\Phi}\left(\lambda_{H\Phi}\lambda_{\Phi}-\lambda_{H\Delta}\lambda_{\Phi\Delta}\right)}{v^{2}\left(\lambda_{H\Delta}^{2}-\lambda_{H}\lambda_{\Delta}\right)-v_{\Phi}^{2}(\lambda_{S\Phi}^{2}-\lambda_{\Phi}\lambda_{\Delta})}.$
(5.64)

The mass eigenvalues for $h_{i}$ are given by

$\displaystyle m_{h_{1}}^{2}$
$\displaystyle\simeq\lambda_{H}v^{2}-\frac{\lambda_{H\Delta}^{2}\lambda_{\Phi}-2\lambda_{H\Phi}\lambda_{H\Delta}\lambda_{\Phi\Delta}+\lambda_{\Delta}\lambda_{H\Phi}^{2}}{\lambda_{\Phi}\lambda_{\Delta}-\lambda_{\Phi\Delta}^{2}}v_{\Phi}^{2},$
$\displaystyle m_{h_{2}}^{2}$
$\displaystyle\simeq\frac{\lambda_{\Phi}\lambda_{\Delta}-\lambda_{\Phi\Delta}^{2}}{\lambda_{\Delta}}v_{\Phi}^{2}+\frac{\left(\lambda_{\Delta}\lambda_{H\Phi}-\lambda_{H\Delta}\lambda_{\Phi\Delta}\right)^{2}}{\lambda_{\Delta}(\lambda_{\Phi}\lambda_{\Delta})-\lambda_{\Delta\Phi}^{2}}v^{2},$
$\displaystyle m_{h_{3}}^{2}$
$\displaystyle\simeq\lambda_{\Delta}v_{\Delta}^{2}.$ (5.65)

Here we comment on the mass matrix in Eq. (5.49). The similar mass matrix has
been analyzed in $G_{\rm SM}\times U(1)_{B-L}$ and $SO(10)$ pNGB DM models
Refs. [22, 23, 29, 30], but only the case $v_{\Delta}\gg v_{\Phi},v$ such as
$v_{\Delta}>O(10^{10})$ GeV and $v_{\Phi},v=O(10^{2})$ GeV is allowed in those
models due to the DM stability problem. In this model, the stability of the DM
is guaranteed by $U(1)$, but when we identify $\varphi_{(\pm)}$ as the DM, the
direct detection leads to some constraints, which we will discuss in the next
section.

##  6 Direct detection and relic abundance

In this section, we will show how the model introduced in Sec. 2 is
constrained by various DM experiments. Firstly, we study the scattering
amplitudes of a DM candidate $\varphi_{(\pm)}$ and SM fermions via the SM
Higgs and additional scalar fields shown in Figure 5.

Figure 5: Tree-level scattering of $\varphi_{(\pm)}$ and the SM fermions $f$
via the scalar fields $h,\phi_{3},\eta_{3}$ is shown.

In the original pNGB DM model (Ref. [14] for Abelian and Ref. [31] for non-
Abelian case), the soft breaking term is a scalar bilinear term and gives only
an origin of pNGB DM mass term. Such soft breaking terms preserve a nature of
NGB for DM and gives derivative portal interactions, resulting in vanishing
DM-nucleon scattering amplitudes in $t\to 0$ limit. In the model we introduced
in Sec. 2, however, the soft breaking term found in Eq. (4.20) is a scalar
trilinear term, and gives not only an origin of pNGB DM mass term but also
additional $h_{i}\varphi_{(+)}\varphi_{(-)}$ portal interactions proportional
to soft breaking parameter $\kappa$, just like Refs. [22] and [29]. These
portal interactions give rise to new contribution in addition to canceling
diagrams, resulting in a nonzero DM-nucleon scattering process even in $t\to
0$ limit. Therefore, we must look into parameter regions that escape direct
detection constraints.

Apart from the vanishing part in $t\to 0$ limit, the DM-nucleon scattering
amplitude $\mathcal{A}_{dd}$ in our model is proportional to soft breaking
parameter $\kappa$, which is replaced with the DM mass $m_{\varphi}$ by using
Eq. (5.44) as

$\displaystyle{\cal A}_{dd}$
$\displaystyle\underset{\sim}{\propto}\frac{m_{\varphi}^{2}}{4v_{\Delta}^{2}}\frac{m_{f}}{v}\left[v_{\Phi}\sin
2\alpha_{z}\left(\frac{1}{m_{h_{1}}^{2}}-\frac{1}{m_{h_{2}}^{2}}\right)+4v_{\Delta}\left(-\frac{1}{m_{h_{1}}^{2}}\alpha_{y}\cos\alpha_{z}+\frac{1}{m_{h_{2}}^{2}}\alpha_{x}\sin\alpha_{z}\right)\right],$
(6.1)

where $m_{f}$ denotes the mass of SM fermions $f$; $\alpha_{x}$, $\alpha_{y}$,
and $\alpha_{z}$ stand for the mixing angles of $\phi_{3}\mathchar
45\relax\eta_{3}$, $h\mathchar 45\relax\eta_{3}$, and $h\mathchar
45\relax\phi_{3}$, respectively;

$\displaystyle\left(\begin{array}[]{c}h_{1}\\\ h_{2}\\\
h_{3}\end{array}\right)=\left(\begin{array}[]{ccc}1&0&0\\\
0&\cos\alpha_{x}&\sin\alpha_{x}\\\
0&-\sin\alpha_{x}&\cos\alpha_{x}\end{array}\right)\left(\begin{array}[]{ccc}\cos\alpha_{y}&0&\sin\alpha_{y}\\\
0&1&0\\\
-\sin\alpha_{y}&0&\cos\alpha_{y}\end{array}\right)\left(\begin{array}[]{ccc}\cos\alpha_{z}&\sin\alpha_{z}&0\\\
-\sin\alpha_{z}&\cos\alpha_{z}&0\\\
0&0&1\end{array}\right)\left(\begin{array}[]{c}h\\\ \phi_{3}\\\
\eta_{3}\end{array}\right).$ (6.17)

Note that the mixing angles $\alpha_{x}$ and $\alpha_{y}$ are expressed in
terms of VEVs of the scalar fields and four point interaction coefficients as

$\displaystyle\alpha_{x}\simeq-\frac{\lambda_{H\Delta}v}{\lambda_{\Delta}v_{\Delta}},\quad\alpha_{y}\simeq-\frac{\lambda_{\Phi\Delta}v_{\Phi}}{\lambda_{\Delta}v_{\Delta}}.$
(6.18)

Note also that we retain only the first order term for $v/v_{\Delta}$ or
$v_{\Phi}/v_{\Delta}$ in Eq. (6.1). See Appendix A for the detailed
derivation.

As commented in the previous section, the previous models such as $G_{\rm
SM}\times U(1)_{B-L}$ and $SO(10)$ pNGB DM models required a very high
$v_{\Delta}$ due to DM longevity. The high $v_{\Delta}$ also brings about a
small DM-nucleon scattering amplitude, because it is suppressed by
$1/v_{\Delta}^{2}$. In this model, on the other hand, the stability of the DM
is guaranteed by $U(1)_{V}$, so $v_{\Delta}$ is expected to be allowed to be a
much smaller scale than $O(10^{10})$ GeV, as we will see later.

In the remainder of this section, we compare the spin-independent (SI) DM-
nucleon cross section $\sigma_{\rm SI}$ and show limitations on $v_{\Delta}$
from recent DM experiments [45, 46]. In the model, the SI cross section
$\sigma_{\rm SI}$ is approximately given by

$\displaystyle\sigma_{\rm SI}$
$\displaystyle\simeq\frac{1}{16\pi}\Bigg{(}\frac{m_{\varphi}^{2}}{2v_{\Delta}^{2}}\frac{m_{f}}{v}\Bigg{[}v_{\Phi}\sin
2\alpha_{z}\left(\frac{1}{m_{h_{1}}^{2}}-\frac{1}{m_{h_{2}}^{2}}\right)$
$\displaystyle\qquad\qquad\qquad\qquad+4v_{\Delta}\left(-\frac{1}{m_{h_{1}}^{2}}\alpha_{y}\cos\alpha_{z}+\frac{1}{m_{h_{2}}^{2}}\alpha_{x}\sin\alpha_{z}\right)\Bigg{]}\Bigg{)}^{2}\frac{m_{p}^{4}f_{N}^{2}}{\left(m_{\varphi}+m_{p}\right)^{2}},$
(6.19)

where the proton mass $m_{p}\simeq 0.938$ GeV, and $f_{N}\simeq 0.3$. Note
that we retain only the first order term for $v/v_{\Delta}$ or
$v_{\Phi}/v_{\Delta}$ in Eq. (6.19). For the conversion formula from DM-quark
scattering to DM-nucleon scattering, see e.g., Ref. [47].

The thermal relic abundance of DM in a model that can be regarded as a low-
energy effective description of this model has been calculated in Ref. [31],
and it has been shown that the observed value can be reproduced when the DM
mass satisfies the condition $m_{\rm DM}\gtrsim m_{h_{1}}/2$. Furthermore,
there is a constraint from Higgs invisible decay when the DM mass is less than
half of the Higgs boson mass. Therefore, in the following, we mainly focus on
regions where the DM mass is more than half the Higgs boson mass:
$m_{\varphi}\gtrsim m_{h_{1}}/2$.

Figure 6: Constraints from a direct detection experiment and a prediction for
the relic abundance in our $SU(2)$ pNGB DM model. Solid lines express a
parameter contour corresponding to $\Omega h^{2}=0.12$, while dashed line
represents a direct detection constraints from the LUX-ZEPLIN experiment[46].
The gray shaded region is the region where the $SU(2)_{D}$ neutral gauge boson
can also become DM candidate, which is not dealt with in this analysis for
simplicity. The purple shaded region is excluded by Higgs invisible decay
constraints[48].

As a benchmark parameter set, we fix mass parameters for the second and third
neutral Higgs fields as $m_{h_{2}}=300$ GeV and $m_{h_{3}}=500$ GeV,
respectively. We take a sample set
$(\sin\alpha_{x},\,\sin\alpha_{y},\,\sin\alpha_{z})=(0.06,0.05,0.1)$. We
assume that $SU(2)_{D}$ gauge bosons are heavy.

In Figure 6, we show allowed parameter regions consistent with various
experimental constraints, varying the ratio of the $SU(2)_{D}$ doublet and
triplet VEV as $v_{\Phi}/v_{\Delta}=1/4$ and $1/10$. Solid lines express
parameter contours reproducing an observed DM energy density, $\Omega
h^{2}=0.12$. Three dips in solid lines correspond to the resonance
contributions from $h_{1}$, $h_{2}$, and $h_{3}$. Therefore, they locate at
the half of their masses, $m_{h_{1}}/2=$62.5 GeV, $m_{h_{2}}/2=$150 GeV, and
$m_{h_{3}}/2=$250 GeV, respectively. The small depression at $m_{\varphi}=300$
GeV is due to the opening of a new annihilation channel,
$\varphi_{(+)}\varphi_{(-)}\to h_{2}h_{2}$. Dashed lines represent constraints
from a direct detection LUX-ZEPLIN experiment[46] recasted to the upper limit
for the VEV ratio $v/v_{\Phi}$. The direct detection constraints become tight
at large DM mass region. That is because the DM-nucleon scattering amplitude
is proportional to a soft-breaking parameter, namely, the DM mass square
$m_{\varphi}^{2}$, as we show in Eq. (6.1). The gray shaded region satisfies
$m_{Z^{\prime}}<2m_{\varphi}$ with $m_{Z^{\prime}}$ being mass of the
$SU(2)_{D}$ neutral gauge boson. In this region, the $SU(2)_{D}$ neutral gauge
boson can also become DM candidate, which is not dealt with in this analysis
for simplicity. The purple shaded region is excluded by Higgs invisible decay
constraints[48]. The Higgs invisible decay width in this model shows
$v_{\Phi}/v_{\Delta}$-dependence only through sub-leading terms. Therefore,
the excluded region colored in purple is common for VEV ratio
$v_{\Phi}/v_{\Delta}=1/4$ and $1/10$.

We apply the same method in Refs. [49, 47, 50] to calculate the DM-nucleon
scattering cross-sections and thermally averaged total annihilation cross
sections. We find that the relic abundance does not change so much when we
vary the VEV ratio $v_{\Phi}/v_{\Delta}$. We also find that there are plenty
of allowed parameter regions which escape the direct detection constraints and
reproduce a correct DM relic abundance at the same time.

##  7 Summary and discussions

We proposed a new pNGB DM model based on non-Abelian gauge symmetry
$SU(2)_{D}$, in which scalars in ${\bf 2}$ and ${\bf 3}$ of $SU(2)_{D}$ are
introduced. We analyzed the structure of the symmetry and its breaking
patterns in detail by analyzing the scalar potential. We found that when the
mass parameters of the scalars $\mu_{\Phi}^{2}$ and $\mu_{\Delta}^{2}$ are
positive in our convention, the $SU(2)_{D}$ gauge symmetry is spontaneously
broken to the exact $U(1)_{V}$ global symmetry by the VEVs of the scalars in
${\bf 2}$ and ${\bf 3}$ of $SU(2)_{D}$. The charged pNGB under the $U(1)_{V}$
custodial symmetry appears, and is identified as DM. The stability of the DM
is guaranteed by the exact $U(1)_{V}$ custodial symmetry. From Figure 6, we
showed that the relic abundance is correctly reproduced while escaping the
severe constraints from the direct detection experiments.

We comment on the additional $SU(2)_{D}$ gauge symmetry breaking scale. In the
$G_{\rm SM}\times U(1)_{B-L}$ and $SO(10)$ pNGB DM models [22, 23, 29, 30],
the VEV of the additional $U(1)_{B-L}$ gauge symmetry breaking scale must be
higher than ${\cal O}(10^{10})$ GeV to suppress DM decay rate, while in our
new pNGB DM model the VEV of $SU(2)_{D}$ breaking scale is allowed to be
roughly ${\cal O}(1)$ TeV or higher due to the stability of DM guaranteed by
the $U(1)_{V}$ custodial symmetry. Complementary verification by accelerator
experiments may be possible in some parameter regions in our model.

In the $SU(2)_{g}\times U(1)_{X}$ pNGB DM model [31], it has been pointed out
that an additional gauge coupling of $U(1)_{X}$ is not asymptotically free and
there is a Landau pole in the high-energy region, so this problem can be
tackled by extending to $SU(2)$ gauge theory. However, it is not enough to
extend the additional gauge sector part to non-Abelian gauge symmetry because
the SM gauge group includes an Abelian gauge symmetry $U(1)_{Y}$. To address
this issue, we have to discuss extensions to grand unified theory (GUT) [51,
52, 37]. The extension of this model to GUT models will be left as future
work.

## Acknowledgments

This work was supported in part by the MEXT Grant-in-Aid for Scientific
Research on Innovation Areas Grant No. JP18H05543 (T.S., K.T., and Y.U.), JSPS
Grant-in-Aid for Scientific Research KAKENHI Grant Nos. JP20K14477 (H.O.),
JP18K03651 (T.S.), JP18H01210 (T.S.), JP22K03622 (T.S.), JP22K03620 (K.T.),
the Ministry of Science and Technology of Taiwan under Grant No.
MOST-111-2811-M-002-047-MY2 (N.Y.), and the Education and Research Program for
Mathematical and Data Science from the Kyushu University (H.O.).

##  Appendix A DM-quark scattering amplitude

We will derive DM-quark scattering amplitude given in Eq. (6.1). The scalar
kinetic terms $\mathcal{K}$ and scalar potential $\mathcal{V}$ parts of the
Lagrangian in Eq. (2.1) are given as

$\displaystyle\mathcal{K}=$
$\displaystyle~{}{(D_{\mu}H)^{\dagger}D^{\mu}H}+\frac{1}{2}\mbox{Tr}\left({(D_{\mu}\Sigma)^{\dagger}D^{\mu}\Sigma}\right)+\frac{1}{2}\mbox{Tr}\left({D_{\mu}\Delta
D^{\mu}\Delta}\right),$ (A.1) $\displaystyle\mathcal{V}=$
$\displaystyle~{}{-\mu_{H}^{2}H^{\dagger}H}-\frac{1}{2}\mu_{\Phi}^{2}\mbox{Tr}\left({\Sigma^{\dagger}\Sigma}\right)-\frac{1}{2}\mu_{\Delta}^{2}\mbox{Tr}\left({\Delta^{2}}\right)-\sqrt{2}\kappa\mbox{Tr}\left({\sigma_{3}\Sigma^{\dagger}\Delta\Sigma}\right)$
$\displaystyle~{}{+\lambda_{H}\left(H^{\dagger}H\right)^{2}}+\frac{\lambda_{\Phi}}{4}\left(\mbox{Tr}\left({\Sigma^{\dagger}\Sigma}\right)\right)^{2}+\frac{\lambda_{\Delta}}{4}\left(\mbox{Tr}\left({\Delta}^{2}\right)\right)^{2}$
$\displaystyle~{}{+\frac{\lambda_{H\Phi}}{2}\left(H^{\dagger}H\right)\mbox{Tr}\left({\Sigma}^{\dagger}{\Sigma}\right)+\frac{\lambda_{H\Delta}}{2}\left(H^{\dagger}H\right)\mbox{Tr}\left(\Delta^{2}\right)}+\frac{\lambda_{\Phi\Delta}}{2}\mbox{Tr}\left({\Sigma^{\dagger}\Sigma}\right)\mbox{Tr}\left({\Delta^{2}}\right),$
(A.2)

where $\Sigma=(\tilde{\Phi},\Phi)$ and $\Delta$ are $SU(2)_{D}^{\rm local}$
bi-doublet and real triplet fields, respectively.

For deriving relevant interactions for DM-quark scattering amplitude, we adopt
non-linear basis, because in the non-linear basis, the Higgs-portal
interactions only come from kinetic terms in Eq. (A.1) and the soft breaking
term in the first line of Eq. (A), which makes derivation of DM-quark
scattering amplitude much easier than that in the linear basis. Note that the
result Eq. (6.1) is the same regardless of whether we choose a linear or no-
linear basis. The polar decomposition for bi-doublet and real triplet fields
are given as

$\displaystyle\Sigma=\frac{v_{\Phi}+\phi_{3}}{\sqrt{2}}{\xi}_{\Sigma}\qquad\qquad\quad\mbox{with}\qquad{\xi}_{\Sigma}=\exp\left(\frac{i}{v_{\Phi}}(\phi_{(+)}\sigma_{(+)}+\phi_{(-)}\sigma_{(-)}+\phi_{4}\sigma_{3})\right),$
(A.3)
$\displaystyle\Delta=\frac{v_{\Delta}+\eta_{3}}{\sqrt{2}}{\xi}_{\Delta}^{\dagger}\sigma_{3}{\xi}_{\Delta}\qquad~{}\,\mbox{with}\qquad{\xi}_{\Delta}=\exp\left(-\frac{i}{2v_{\Delta}}(\eta_{(+)}\sigma_{(+)}+\eta_{(-)}\sigma_{(-)})\right),$
(A.4)

where $v_{\Phi}$, and $v_{\Delta}$ are VEVs for each scalar field.
$\sigma_{(\pm)}$ is expressed in terms of the first and second Pauli matrices
as $\sigma_{(\pm)}=({\sigma_{1}\mp i\sigma_{2}})/{\sqrt{2}}$. Note that
$SU(2)$ generators $\sigma_{(\pm)}$, $\sigma_{3}$ satisfy the normalization
conditions
$\mbox{Tr}\left({\sigma_{(+)}\sigma_{(+)}}\right)=\mbox{Tr}\left({\sigma_{(-)}\sigma_{(-)}}\right)=0$,
$\mbox{Tr}\left({\sigma_{(+)}\sigma_{(-)}}\right)=\mbox{Tr}\left({\sigma_{3}\sigma_{3}}\right)=2$,
and bi-doublet and real triplet fields satisfy

$\displaystyle\mbox{Tr}\left({\Sigma^{\dagger}\Sigma}\right)=(v_{\Phi}+\phi_{3})^{2}\,,\qquad\mbox{Tr}\left({\Delta^{2}}\right)=(v_{\Delta}+\eta_{3})^{2}\,.$
(A.5)

We evaluate scattering amplitudes of the DM $\varphi_{{(\pm)}}$ and the SM
fermions $f$ shown in Figure 5. Substituting polar decompositions Eqs. (A.3)
and (A.4) into the potential $\mathcal{V}$ and kinetic terms $\mathcal{K}$
given in Eq. (A.1) and Eq. (A), and extracting cubic scalar interactions
relevant to DM-fermion scattering, we get

$\displaystyle-\mathcal{V}_{\rm
NL}\supset~{}-\frac{2}{v_{\Phi}}\phi_{3}~{}(\phi_{(-)},\eta_{(-)})M_{C}^{2}\left(\begin{array}[]{c}\phi_{(+)}\\\
\eta_{(+)}\end{array}\right)-\frac{1}{v_{\Delta}}\eta_{3}~{}(\phi_{(-)},\eta_{(-)})M_{C}^{2}\left(\begin{array}[]{c}\phi_{(+)}\\\
\eta_{(+)}\end{array}\right),$ (A.10) $\displaystyle\mathcal{K}_{\rm
NL}\supset~{}\left(1+\frac{\phi_{3}}{v_{\Phi}}\right)^{2}\partial_{\mu}\phi_{(+)}\partial^{\mu}\phi_{(-)}+\left(1+\frac{\eta_{3}}{v_{\Delta}}\right)^{2}\partial_{\mu}\eta_{(+)}\partial^{\mu}\eta_{(-)}\,.$
(A.11)

where $M_{C}^{2}$ is given in Eq. (5.37). The cubic interactions in Eq. (A.10)
can be rewritten in terms of mass eigenstate $\varphi_{(\pm)}$ as

$\displaystyle-\mathcal{V}_{\rm NL}\supset$
$\displaystyle~{}\left(-2\frac{m_{\varphi}^{2}}{v_{\Phi}}\phi_{3}-\frac{m_{\varphi}^{2}}{v_{\Delta}}\eta_{3}\right)\varphi_{(+)}\varphi_{(-)}\,.$
(A.12)

The cubic interactions in Eq. (A.11) can be rewritten in terms of
$\varphi_{(\pm)}$ as

$\displaystyle\mathcal{K}_{\rm NL}$ $\displaystyle\supset
2\left(\frac{\sin^{2}\beta}{v_{\Phi}}\phi_{3}+\frac{\cos^{2}\beta}{v_{\Delta}}\eta_{3}\right)\partial_{\mu}\varphi_{(+)}\partial^{\mu}\varphi_{(-)}$
$\displaystyle=\left(\frac{2}{v_{\Phi}}\phi_{3}+\frac{1}{v_{\Delta}}\eta_{3}\right)\partial_{\mu}\varphi_{(+)}\partial^{\mu}\varphi_{(-)}-\left(2\,\frac{v_{\Phi}}{v_{\Phi}^{2}+4v_{\Delta}^{2}}\phi_{3}+\frac{1}{v_{\Delta}}\frac{4v_{\Delta}^{2}-v_{\Phi}^{2}}{v_{\Phi}^{2}+4v_{\Delta}^{2}}\eta_{3}\right)\partial_{\mu}\varphi_{(+)}\partial^{\mu}\varphi_{(-)}\,.$
(A.13)

Combining the first term in the last line of Eq. (A) with Eq. (A.12), we get
333 To obtain Eq. (A), we used
$\displaystyle\rho\,\partial_{\mu}\varphi_{(+)}\partial^{\mu}\varphi_{(-)}=$
$\displaystyle~{}\frac{1}{2}\partial_{\mu}\Big{[}\rho\,(\partial^{\mu}\varphi_{(+)})\varphi_{(-)}+\rho\,\varphi_{(+)}(\partial^{\mu}\varphi_{(-)})-(\partial^{\mu}\rho)\varphi_{(+)}\varphi_{(-)}\Big{]}$
$\displaystyle+\frac{1}{2}(\partial^{2}\rho)\varphi_{(+)}\varphi_{(-)}-\frac{1}{2}\rho\,(\partial^{2}\varphi_{(+)})\varphi_{(-)}-\frac{1}{2}\rho\,\varphi_{(+)}(\partial^{2}\varphi_{(-)})$
(A.14) with $\rho=\phi_{3},\eta_{3}$. Total derivative terms in the first line
of Eq. (A.14) are irrelevant, so we dropped them in Eq. (A).

$\displaystyle\left(\frac{2}{v_{\Phi}}\phi_{3}+\frac{1}{v_{\Delta}}\eta_{3}\right)\partial_{\mu}\varphi_{(+)}\partial^{\mu}\varphi_{(-)}-\left(2\frac{m_{\varphi}^{2}}{v_{\Phi}}\phi_{3}+\frac{m_{\varphi}^{2}}{v_{\Delta}}\eta_{3}\right)\varphi_{(+)}\varphi_{(+)}$
$\displaystyle=\left(\frac{2}{v_{\Phi}}\phi_{3}+\frac{1}{v_{\Delta}}\eta_{3}\right)\bigg{[}-\frac{1}{2}\Big{(}[\partial^{2}+m_{\varphi}^{2}]\,\varphi_{(+)}\Big{)}\varphi_{(-)}-\frac{1}{2}\varphi_{(+)}\Big{(}[\partial^{2}+m_{\varphi}^{2}]\,\varphi_{(-)}\Big{)}\bigg{]}$
$\displaystyle\ \ \
+\left(\frac{2}{v_{\Phi}}(\partial^{2}\phi_{3})+\frac{1}{v_{\Delta}}(\partial^{2}\eta_{3})\right)\varphi_{(+)}\varphi_{(-)}.$
(A.15)

We find that all the terms appearing in Eq. (A) are irrelevant to the DM-
fermion scattering in direct detection; the first line vanishes due to on-
shell conditions for pNGB DM; the second line gives contributions proportional
to momentum-transfer $t=(p_{2}-p_{1})^{2}$ with $p_{1,2}$ being in-coming and
out-going DM momentum. This will also vanish when we take $t\to 0$ limit.

The remaining cubic interaction relevant to DM-fermion scattering shown in
Figure 5 is the second term of Eq. (A):

$\displaystyle\mathcal{K}_{\rm NL}\supset$
$\displaystyle~{}-\left(2\,\frac{v_{\Phi}}{v_{\Phi}^{2}+4v_{\Delta}^{2}}\phi_{3}+\frac{1}{v_{\Delta}}\frac{4v_{\Delta}^{2}-v_{\Phi}^{2}}{v_{\Phi}^{2}+4v_{\Delta}^{2}}\eta_{3}\right)\partial_{\mu}\varphi_{(+)}\partial^{\mu}\varphi_{(-)}.$
(A.16)

Note that this term decouples when we assume $v_{\Delta}\gg v_{\Phi}$.
Further, by using the relation in Eq. (A.14), we obtain

$\displaystyle\mathcal{L}_{\rm NL}\supset$
$\displaystyle~{}-{\frac{1}{2}}\left(2\,\frac{v_{\Phi}}{v_{\Phi}^{2}+4v_{\Delta}^{2}}\partial^{2}\phi_{3}+\frac{1}{v_{\Delta}}\frac{4v_{\Delta}^{2}-v_{\Phi}^{2}}{v_{\Phi}^{2}+4v_{\Delta}^{2}}\partial^{2}\eta_{3}\right)\varphi_{(+)}\varphi_{(-)}\,$
$\displaystyle~{}+\frac{1}{2}\left(2\,\frac{v_{\Phi}}{v_{\Phi}^{2}+4v_{\Delta}^{2}}\phi_{3}+\frac{1}{v_{\Delta}}\frac{4v_{\Delta}^{2}-v_{\Phi}^{2}}{v_{\Phi}^{2}+4v_{\Delta}^{2}}\eta_{3}\right)\left(\left(\partial^{2}\varphi_{(+)}\right)\varphi_{(-)}+\varphi_{(+)}\left(\partial^{2}\varphi_{(-)}\right)\right),$
(A.17)

where we ignored the total derivative. The first term vanishes for $t\to 0$
limit. By replacing $\partial^{2}\varphi_{(\pm)}$ to
$m_{\varphi}^{2}\varphi_{(\pm)}$, the effective DM-scalar interaction for the
DM-fermion scattering becomes for $t\to 0$

$\displaystyle\left(2\,\frac{v_{\Phi}}{v_{\Phi}^{2}+4v_{\Delta}^{2}}\phi_{3}+\frac{1}{v_{\Delta}}\frac{4v_{\Delta}^{2}-v_{\Phi}^{2}}{v_{\Phi}^{2}+4v_{\Delta}^{2}}\eta_{3}\right)m_{\varphi}^{2}\varphi_{(+)}\varphi_{(-)}$
$\displaystyle=:\left(\begin{array}[]{ccc}\kappa_{\varphi\varphi
h}&\kappa_{\varphi\varphi\phi_{3}}&\kappa_{\varphi\varphi\eta_{3}}\\\
\end{array}\right)\left(\begin{array}[]{c}h\\\ \phi_{3}\\\ \eta_{3}\\\
\end{array}\right)\varphi_{(+)}\varphi_{(-)},$ (A.22)

where

$\displaystyle\left(\begin{array}[]{c}\kappa_{\varphi\varphi h}\\\
\kappa_{\varphi\varphi\phi_{3}}\\\ \kappa_{\varphi\varphi\eta_{3}}\\\
\end{array}\right)=\frac{m_{\varphi}^{2}}{v_{\Phi}^{2}+4v_{\Delta}^{2}}\left(\begin{array}[]{c}0\\\
2{v_{\Phi}}\\\ \frac{4v_{\Delta}^{2}-v_{\Phi}^{2}}{v_{\Delta}}\\\
\end{array}\right).$ (A.29)

Next, Yukawa interaction terms of the scalar fields and SM fermions $f$ on the
vacuum are given by

$\displaystyle\frac{m_{f}}{v}h\bar{f}f=\frac{m_{f}}{v}\left(\begin{array}[]{ccc}1&0&0\\\
\end{array}\right)\left(\begin{array}[]{c}h\\\ \phi_{3}\\\ \eta_{3}\\\
\end{array}\right)\bar{f}f,$ (A.34)

where $m_{f}$ stands for a mass parameter of the SM fermion $f$.

By using the DM-scalar and scalar-fermion interactions in Eqs. (A.29) and
(A.34), we find that the scattering amplitude shown in Figure 5 for $t\to 0$
is given by

$\displaystyle{\cal A}_{dd}$
$\displaystyle\underset{\sim}{\propto}\left(\begin{array}[]{ccc}\kappa_{\varphi\varphi
h}&\kappa_{\varphi\varphi\phi_{3}}&\kappa_{\varphi\varphi\eta_{3}}\end{array}\right)\left(M_{H}^{2}\right)^{-1}\frac{m_{f}}{v}\left(\begin{array}[]{c}1\\\
0\\\ 0\\\ \end{array}\right),$ (A.39)

where $\left(M_{H}^{2}\right)^{-1}=U_{H}^{-1}\left[\left(M_{H}^{2}\right)^{\rm
diag}\right]^{-1}U_{H}$; the approximate form of $U_{H}$ for $v_{\Delta}\gg
v_{\Phi},v$ is given in Eq. (5.64). Substituting Eq. (A.29) into Eq. (A.39),
we get

$\displaystyle{\cal A}_{dd}$
$\displaystyle\underset{\sim}{\propto}\frac{m_{\varphi}^{2}}{4v_{\Delta}^{2}}\frac{m_{f}}{v}\Bigg{[}v_{\Phi}\sin
2\alpha_{z}\left(\frac{1}{m_{h_{1}}^{2}}-\frac{1}{m_{h_{2}}^{2}}\right)+4v_{\Delta}\left(-\frac{1}{m_{h_{1}}^{2}}\alpha_{y}\cos\alpha_{z}+\frac{1}{m_{h_{2}}^{2}}\alpha_{x}\sin\alpha_{z}\right)\Bigg{]},$
(A.40)

which is identical to Eq. (6.1).

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