#
CP-like Symmetry
with Discrete and Continuous Groups
and CP Violation/Restoration

Hiroshi Ohki1, Shohei Uemura2,

1 Department of Physics, Nara Women’s University, Nara 630-8506, Japan

2 Research organization of Integrative STEAM Education,

Nara Women’s University, Nara 630-8506, Japan

<EMAIL_ADDRESS>
( Abstract
We study physical implications of general CP symmetry including CP-like
symmetry. Various scattering amplitudes of CP asymmetry are calculated in CP-
like symmetric models. We explicitly show that the CP-like transformation
leads to a specific relation between different CP asymmetries. The resultant
relation is similar to the one obtained in GUT baryogenesis and sphaleron
processes, where we also obtain a required condition for generating particle
number asymmetry in CP-like symmetric models. In addition, we propose a
generalization of a CP-like transformation for continuous symmetry groups.
Since the CP transformation is an outer automorphism, which depends on the
internal symmetry group, it turns out that the physical CP and CP-like
symmetries can be mutually converted through the spontaneous symmetry breaking
(SSB) of the internal symmetry. We investigate properties of physical CP
asymmetry in both CP and CP-like symmetric phases, and find that the
spontaneous CP violation and restoration can be observed even in models with
continuous groups. We demonstrate that CP-like symmetric models with
continuous Lie groups can be naturally realized in physical CP symmetric
models through the SSB. )

##  1 Introduction

Violation of CP symmetry is a key property of the Standard Model (SM). CP
violation in the SM has been confirmed by several experiments, and the
observed CP asymmetries are consistently described by the complex phases in
the Yukawa couplings [1]. In addition, CP violation is a necessary condition
for the matter/antimatter asymmetry of the universe [2]. However the CP
violation of the SM is not sufficient to reproduce the baryon number density
of the nature, and hence CP violation should be related to new physics beyond
the SM. It is important to study the property of CP transformations in such an
underlying theory.

If an underlying theory has other (flavor) symmetry groups a generalized CP
transformation is considered as an extension of the CP transformation in the
SM [3, 4, 5]. There are vast phenomenological applications of generalized CP.
For instance, generalized CP is introduced as $\mu-\tau$ reflection symmetry
[6, 7] 111For review of $\mu-\tau$ symmetry see [8] and references therein..
It is also studied in flavor symmetric models with discrete groups such as
$S_{4}$ [9, 10, 11, 12, 13], $A_{4}$ [13, 14, 15, 16], $A_{5}$ [13, 17, 18,
19, 20, 21], $T^{\prime}$ [13, 22], $\Delta(27)$ [23], $\Delta(48)$ [24, 25],
$\Delta(96)$ [26], $\Delta(3n^{2})$ [27, 28, 29], $\Delta(6n^{2})$ [30, 28,
31, 32, 29], and other discrete groups [33, 34, 35, 36] 222For review of non-
Abelian discrete groups in particle physics, see [37, 38, 40, 39, 41] and
references therein.. Recently generalized CP symmetries (and its spontaneous
breaking) in modular symmetric models have been investigated [42, 43, 44, 45,
46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56], where the generalized CP
transformation also acts on the moduli space [58, 57]. String compactification
also gives rise to conventional flavor symmetries [59, 60, 61], and hence, it
provides a unified origin of generalized CP, modular, and discrete flavor
symmetries [57, 62, 63, 64, 65, 66].

From a theoretical viewpoint, a generalized CP transformation is characterized
by its action on the internal symmetry group $G$. Its action on $G$ is given
by $u_{CP}:\,G\to CP\circ G\circ CP^{-1}$, and $u_{CP}$ must be an outer
automorphism if the model is invariant under $G$ and CP [5, 67]. This is a
consistency condition for generalized CP transformation. As for a continuous
symmetry it is known that most of semi-simple Lie algebras 333The only
exception is $\mathfrak{so}(8)$, which exhibit the triality. have an outer
automorphism of $\mathbb{Z}_{2}$ or trivial group, which corresponds to the
charge conjugation CP transformation, and hence the physical CP is uniquely
defined [67]. By contrast, outer automorphisms of a discrete group have a more
variety, e.g. ${\rm Out}(\Delta(54))\cong S_{4}$ [68]. Thus in principle any
of $\mathbb{Z}_{2}$ outer automorphisms can be a generalized CP
transformation. Among them, proper CP transformation is defined to be a
transformation which maps a complex representation to its complex conjugate
representation. This type of automorphism is referred to as a class-inverting
automorphism (CIA) [69]. On the other hand, a general CP transformation which
does not induce CIA is called a CP-like transformation. In fact, there are
groups without CIA, which are referred to as the Type I group in [69]. Thus
any of outer automorphisms of Type I group should correspond to a CP-like
transformation. Since the CP-like transformation includes non-trivial
interchanges of particles and antiparticles which are not related by complex
conjugations, it has been shown that the physical CP symmetry is violated in
models with a Type I symmetry group [69, 71, 70]. Despite its interesting
property, phenomenological aspects of the CP-like symmetric models with a
general symmetry group other than type I have not been investigated
comprehensively so far. In addition, the dynamical origins of the CP-like
symmetric models and its relation to the proper CP symmetry for general and
continuous symmetry groups have not been carefully investigated in the
previous works.

In this paper, we study physical implications of general CP transformation
including CP-like symmetry with discrete and continuous global symmetries. For
our purposes, we first propose a generalization of a CP-like transformation
for general continuous symmetry groups. Following the previous work [69], we
compute various physical observables of CP asymmetry including 4-point
scattering amplitudes in CP-like symmetric models. We then explicitly show
that in CP-like symmetric models there exists a characteristic relation
between two different CP violating amplitudes. The resultant relation is
similar to the $B-L$ in GUT baryogenesis [72] and sphaleron process [73], so
that the CP-like model can provide us with an alternative mechanism for
generating matter asymmetry as well as asymmetric dark matter [74]. In
addition, we find that there exists a CP-like eigenstate which does not have
CP-like partner, and hence its number conservation is violated too. We also
study the effects of the spontaneous symmetry breaking (SSB) of the internal
symmetry on the CP(-like) symmetric models. We show that the proper CP and CP-
like symmetries are mutually changeable through the SSB of the internal
symmetry 444CP $\to$ CP-like symmetry has been already studied in [71].. We
will point out that the emergence of CP-like symmetry is not directly related
to the absence of the CIA of the discrete group. In fact, we demonstrate that
various CP-like models with discrete and continuous Lie groups can be
naturally realized in proper CP symmetric models through the SSB.

This paper is organized as follows. In Sec. 2, we briefly review CP
transformation and study general CP transformation from the view point of
internal symmetry. We clarify the difference between proper CP symmetry and
CP-like symmetry. We then propose a generalization of CP-like transformation
for continuous groups. In Sec. 3, we consider general CP symmetric model with
internal symmetry. We use a toy model with $\Delta(27)$ symmetry for
illustrative purposes. We calculate scattering amplitude in CP-like symmetric
models, and study physical implications of the CP-like invariance. Then we
consider general CP transformation with continuous group and introduce CP-like
symmetry. In Sec. 4, we study CP symmetric model with SSB of internal
symmetry. We consider models with various internal symmetry including discrete
and continuous groups. We show CP-like symmetric model can be originated from
physical CP symmetric model and vice versa. Section 5 is devoted to
conclusion. In App. A, we summarize group property of $\Delta(54)$. In App. B,
we show part of Clebsch-Gordan (CG) coefficients of $\Delta(27)$ and
$\Delta(54)$ relevant to our discussions. In App. C, we explicitly compute
one-loop amplitude of a CP violating process. In App. D, we study sufficient
condition for general CP violation.

##  2 CP Transformation

We first review CP transformation with the internal symmetry denoted by a
discrete group $G$. We concentrate on global symmetries. We introduce a
definition of general CP transformation for multiple fields. A general CP
transformation is characterized by its actions on the internal symmetry, i.e.
automorphisms of $G$. Among them, there exists an automorphism which is not
related to the matter/antimatter symmetry (CP-like). We then extend these CP
and CP-like symmetries to general (continuous) symmetry groups. Finally we
classify general CP transformation into three classes. Our notation is
basically based on [69, 75] and references therein.

###  2.1 General CP Transformation for Discrete Symmetry

Particle and antiparticle states of a complex scalar field with momentum
$\mathbf{p}$ are given by
$a_{\mathbf{p}}^{\dagger}\ket{0},b_{\mathbf{p}}^{\dagger}\ket{0}$, where
$a_{\mathbf{p}}^{\dagger}$ and $b_{\mathbf{p}}^{\dagger}$ denote the creation
operators of a particle and antiparticle respectively.
$a_{\mathbf{p}}^{\dagger}$ and $b_{\mathbf{p}}^{\dagger}$ are given as Fourier
coefficients of a complex field $\phi(x)$,

$\displaystyle\phi(\mathbf{x},t)=$
$\displaystyle\int\frac{d^{3}p}{(2\pi)^{3}}\frac{1}{\sqrt{2E_{\mathbf{p}}}}\left.\left(a_{\mathbf{p}}e^{-ip\cdot
x}+b_{\mathbf{p}}^{\dagger}e^{ip\cdot
x}\right)\right|_{p^{0}=E_{\mathbf{p}}}.$

CP transformation transforms particle with momentum $\mathbf{p}$ into
antiparticle with inverse momentum. Hence CP transformation acts on the
quantum state space as

$\displaystyle\mathcal{CP}a_{\mathbf{p}}^{\dagger}\ket{0}$
$\displaystyle=\eta_{a}b_{-\mathbf{p}}^{\dagger}\ket{0},$
$\displaystyle\mathcal{CP}b_{\mathbf{p}}^{\dagger}\ket{0}$
$\displaystyle=\eta_{b}a_{-\mathbf{p}}^{\dagger}\ket{0}.$

where $\eta_{a}$ and $\eta_{b}$ are arbitrary $U(1)$ phases. This
transformation can be reinterpreted as an action on the operator space. We
find

$\displaystyle\mathcal{CP}a_{\mathbf{p}}^{\dagger}\mathcal{CP}^{-1}$
$\displaystyle=\eta_{a}b_{-\mathbf{p}}^{\dagger},$
$\displaystyle\mathcal{CP}b_{\mathbf{p}}^{\dagger}\mathcal{CP}^{-1}$
$\displaystyle=\eta_{b}a_{-\mathbf{p}}^{\dagger},$

where we assume $\mathcal{CP}\ket{0}=\ket{0}$. $\mathcal{CP}$ on the
annihilation operators are similarly given by
$\mathcal{CP}a_{\mathbf{p}}\mathcal{CP}^{-1}=\eta_{a}^{\prime}b_{-\mathbf{p}}$
and
$\mathcal{CP}b_{\mathbf{p}}\mathcal{CP}^{-1}=\eta_{b}^{\prime}a_{-\mathbf{p}}$.
CP transformation of the canonical commutation relation is given by

$\displaystyle\mathcal{CP}[a_{\mathbf{p}},a_{\mathbf{p}^{\prime}}^{\dagger}]\mathcal{CP}^{-1}=[\eta_{a}^{\prime}b_{-\mathbf{p}},\eta_{a}b_{-\mathbf{p}^{\prime}}^{\dagger}]=\eta_{a}^{\prime}\eta_{a}(2\pi)^{3}\delta^{(3)}(-\mathbf{p}+\mathbf{p}^{\prime}).$

Not to violate the canonical commutation relations of the scalar fields, we
set $\eta_{a}^{\prime}=\eta_{a}^{*}$ and $\eta_{b}^{\prime}=\eta_{b}^{*}$. We
also require CP transformation of $\phi$ is properly defined. In other words,
we require $\eta_{a}=\eta_{b}^{*}=\eta$. Then, we obtain CP transformation for
the field operator $\phi(x)$

$\displaystyle\mathcal{CP}\phi(x)\mathcal{CP}^{-1}=\eta\phi^{\dagger}(\tilde{x}),$

where $\tilde{x}=(t,-\mathbf{x})^{T}$. It is equivalent to that
$\textrm{CP}:\phi(x)\to\eta\phi^{*}(\tilde{x})$ for the complex scalar field.
$\eta$ can be absorbed by field redefinition of $\phi$ and we usually set
$\eta=1$. CP transformations for fermions and gauge bosons should be modified
depending on its spin structures. In this section we concentrate on scalar
fields for simplicity.

If there are multiple fields in the theory, CP transformation can be
generalized. Especially with the internal symmetry such as gauge and discrete
flavor symmetries, the definition of CP transformation as well as that of CP
invariance are extended. Let us consider a quantum field theory with $N$
fields denoted by $\phi_{i=1,...,N}$. If the model is invariant under the
internal symmetry group $G$,
$\bm{\phi}=(\phi_{1},\phi_{2},\cdots,\phi_{N})^{T}$ forms $N$ dimensional (in
general) reducible representation of $G$;

$\displaystyle g\in G:$
$\displaystyle\bm{\phi}(x)\to\rho_{\phi}(g)\bm{\phi}(x),$ $\displaystyle g\in
G:$ $\displaystyle\bm{\phi}^{*}(x)\to\rho_{\phi}(g)^{*}\bm{\phi}^{*}(x),$

where $\rho_{\phi}(g)$ is a unitary representation of $g$. The most general CP
transformation for $\phi$ is given by [5, 4, 39]

$\displaystyle\textrm{CP}:$ $\displaystyle\bm{\phi}(x)\to
U\bm{\phi}^{*}(\tilde{x}),$ CP $\displaystyle:\bm{\phi}^{*}(x)\to
U^{*}\bm{\phi}(\tilde{x}),$ (2.1)

where $U$ is an $N\times N$ unitary matrix 555We distinguish $\mathcal{CP}$
from CP in this paper. We use $\mathcal{CP}$ to denote an operator which acts
on the quantum state space as well as operator space. CP represents more
general CP transformation..

We have a consistency condition of CP and $G$ for symmetry of the theory.
Since $CP^{-1}\circ g\circ CP$ does not change space-time coordinates, and it
should act on the internal space. If $CP$ and $G$ are symmetries of the
theory, $CP^{-1}\circ g\circ CP\in G$. We refer to a general CP transformation
which satisfies

$\displaystyle CP^{-1}\circ G\circ CP\subset G,$ (2.2)

as a consistent CP transformation. It is obvious that $CP^{-1}\circ G\circ
CP=G$ since this relation is equivalent to $G\subset CP\circ G\circ CP^{-1}$,
hence it is bijection. On the other hand, an inconsistent CP transformation is
referred to as a general CP transformation which does not satisfy the above
relation, that is

$\displaystyle CP^{-1}\circ G\circ CP\not\subset G.$

It is also obvious that the inconsistent CP symmetry is not compatible with
the symmetry group $G$. It follows that either an inconsistent CP is not a
symmetry or $G$ is enlarged to $\tilde{G}$ such that
$CP^{-1}\circ\tilde{G}\circ CP\subset\tilde{G}$.

We should note that the consistency condition in Eq. (2.2) is a necessary
condition for the CP invariance of a theory, but not a sufficient condition.
In fact, canonical CP transformation for the SM, which is a general CP
transformation associated with the identity matrix $I$ for each generation of
quarks and leptons, is a consistent CP transformation for the SM gauge group,
but the SM is not invariant under the canonical CP transformation.

Next, we clarify the physical CP transformation. We denote
$\Phi=(\phi_{\mathbf{r}_{1}},\cdots,\phi_{\mathbf{r}_{N^{\prime}}})$ as a
direct sum of irreducible representations of $N$-multiplet field $\bm{\phi}$,
where $\mathbf{r}_{i}\ (i=1,\cdots N^{\prime})$ is an irreducible
representation of $G$. The physical CP transformation transforms particles to
their own antiparticles with opposite momenta. Therefore the physical CP
transformation is defined to transform a field $\phi_{\mathbf{r}_{i}}$ in
irreducible representation $\mathbf{r}_{i}$ to its complex conjugate;

$\displaystyle\textrm{physical CP}:\phi_{\mathbf{r}_{i}}(x)\to
U_{\mathbf{r}_{i}}\phi_{\mathbf{r}_{i}}^{*}(\tilde{x}),\ \
(i=1,\cdots,N^{\prime}),$ (2.3)

where $U_{\mathbf{r}_{i}}$ is a unitary matrix associated with the physical CP
transformation 666If there is $F$-fold multiplicity in representation
$\mathbf{r}_{i}$, $U_{\mathbf{r}_{i}}$ is given as a
$(\dim\mathbf{r}_{i}\times F)^{2}$ unitary matrix.. Note that in our
terminology the set of representations
$\mathbf{r}_{1},\cdots,\mathbf{r}_{N^{\prime}}$ does not necessarily include
all the irreducible representations of $G$. The general CP transformation is
not restricted solely to transformations among the same irreducible
representations, so that the group associated with the general CP
transformation in Eq. (2.1) should include the one corresponding to the
physical CP transformation in Eq. (2.3).

Let us consider the consistent CP transformation in detail. If there is a
consistent CP transformation in a given theory, for any $g\in G$ we have $h\in
G$ satisfying the following relation,

$\displaystyle U_{ij}\rho(g)^{*}_{jk}U^{\dagger}_{kl}=\rho(h)_{il}.$ (2.4)

This means that the consistent CP transformation induces an automorphism
$u_{CP}:G\to G$ such that $u_{CP}(g)\equiv h$, since

$\displaystyle
U\rho(g_{1}g_{2})^{*}U^{\dagger}=U\rho(g_{1})^{*}U^{\dagger}U\rho(g_{2})^{*}U^{\dagger}=\rho(h_{1})\rho(h_{2})=\rho(h_{1}h_{2}).$

Note that in general $u_{CP}$ has to be a CIA if the CP transformation is a
physical CP transformation [69]. This is shown as follows. Since a physical CP
transformation transforms $\phi_{\mathbf{r}_{i}}$ to
$\phi_{\mathbf{r}_{i}}^{*}$, and $\Phi$ can be represented as a direct sum of
the irreducible representations, $U$ must be block diagonalized as

$\displaystyle\Phi\to U\Phi,\ \
U=\left(\begin{array}[]{ccc|c|ccc}\nwarrow&&\nearrow&&&&\\\
&U_{\mathbf{r}_{1}}&&&&0&\\\ \swarrow&&\searrow&&&&\\\
\hline\cr&&&\ddots&&&\\\ \hline\cr&&&&\nwarrow&&\nearrow\\\
&0&&&&U_{\mathbf{r}_{N^{\prime}}}&\\\ &&&&\swarrow&&\searrow\\\
\end{array}\right),$ (2.12)

for each subspace $\phi_{\mathbf{r}_{i}}$. From the consistency condition Eq.
(2.2) we see that the following relations,

$\displaystyle
U_{\mathbf{r}_{i}}\rho_{\mathbf{r}_{i}}(g)^{*}U^{\dagger}_{\mathbf{r}_{i}}=\rho_{\mathbf{r}_{i}}(h)$
(2.13)

should be satisfied for every irreducible representation $\mathbf{r}_{i}$.
This implies

$\displaystyle\mathrm{tr}\,\rho_{\mathbf{r}_{i}}(h)=\mathrm{tr}\,\rho_{\mathbf{r}_{i}}(g)^{*}=\mathrm{tr}\,\rho_{\mathbf{r}_{i}}(g)^{\dagger}=\mathrm{tr}\,\rho_{\mathbf{r}_{i}}(g^{-1}),$

where we use that $\rho(g)$ is unitary. Since $G$ is a discrete group, we have

$\displaystyle\chi_{\mathbf{r}_{i}}(h)=\chi_{\mathbf{r}_{i}}(g^{-1}),$ (2.14)

where $\chi_{\mathbf{r}}$ is the character of $\mathbf{r}$. Conjugacy class is
specified by its character. This relation implies that $g^{-1}$ and $h$ belong
to the same conjugacy class. The CIA is an automorphism on a discrete group
$G$ which maps any element of $G$ to an element which is conjugate to the
inverse of the original one. Therefore $u_{CP}$ is a CIA, and the
corresponding CP transformation is a class-inverting CP transformation.

We should note that the CIA of $G$ is a sufficient condition for the existence
of a physical and consistent CP transformation. If $u_{CI}$ is a CIA of $G$,

$\displaystyle u_{CI}:G\to G,~{}~{}g\mapsto h=h^{\prime}g^{-1}h^{\prime-1},$

$\rho_{\mathbf{r}}\circ u_{CI}(g)=\rho_{\mathbf{r}}(u_{CI}(g))$ is equivalent
to $\mathbf{r}^{*}$ due to the one-to-one correspondence between character and
irreducible representation of discrete groups. From Eq. (2.14) there exists a
unitary matrix $U_{\mathbf{r}}$ which satisfy

$\displaystyle\rho_{\mathbf{r}}(u_{CI}(g))=U_{\mathbf{r}}\rho_{\mathbf{r}}(g)^{*}U^{\dagger}_{\mathbf{r}}.$

Therefore the CIA is consistent with the physical CP transformation in Eq.
(2.3).

###  2.2 CP(-like) Transformation for General (Continuous) Group

A consistent CP transformation does not always induce the CIA. In fact, a type
of discrete groups do not have any CIA 777This type of group is referred to as
Type I in [69].. In this case, a consistent CP transformation becomes to CP-
like transformation [69], since it has been considered that a CP-like symmetry
has not been directly related to the matter/antimatter symmetry. We note,
however, that a CP-like symmetry can be obtained by SSB of $G$ in an
underlying theory with a physical CP symmetry, where $G$ could be either a
discrete group or a continuous group such as $SU(N)$ [71].

For our purposes, we consider extending the CP(-like) transformation to
encompass general (continuous) symmetry groups. In the remaining of this
section, we consider $G$ as either a continuous or discrete group. As in the
case with discrete groups, a CP-like transformation of $N$-multiplet field
$\Phi(x)$ is defined as,

$\displaystyle\textrm{CP-like}:\Phi(x)\to U_{\textrm{CP-
like}}\Phi^{*}(\tilde{x}),$ (2.15) $\displaystyle U_{\textrm{CP-
like}}\rho_{\Phi}(g)^{*}U_{\textrm{CP-
like}}^{\dagger}=\rho_{\Phi}(h),\quad\quad(g,h\in G)$ (2.16)

where $U_{\textrm{CP-like}}$ is a unitary matrix. This transformation induces
an automorphism denoted as $u_{\textrm{CP-like}}:g\mapsto h$. Since we are
interested in $\mathbb{Z}_{2}$ transformations which possibly originate from a
physical and consistent CP transformation, we restrict ourselves to the
following form of $U_{\textrm{CP-like}}$ which includes anti-diagonal block
elements as 888For an extension to non-CP ($Z_{N\geq 3}$) transformations, see
[77, 76].,

$\displaystyle U_{\textrm{CP-
like}}=\left(\begin{array}[]{cc|cc|cc}U_{\mathbf{r}_{1}}&\mathbf{0}&\cdots&&&\\\
\mathbf{0}&\ddots&&&&\\\ \hline\cr\vdots&&\mathbf{0}&U_{\mathbf{r}_{ij}}&&\\\
&&U_{\mathbf{r}_{ji}}&\mathbf{0}&&\vdots\\\ \hline\cr&&&&\ddots&\mathbf{0}\\\
&&&\cdots&\mathbf{0}&U_{\mathbf{r}_{N^{\prime}}}\end{array}\right).$ (2.23)

Here we note that the irreducible representations of
$\mathbf{r}_{1},...,\mathbf{r}_{N^{\prime}}$ are restricted to those existing
in $\Phi$. From the consistency condition in Eq. (2.16) the unitary matrices
should satisfy

$\displaystyle
U_{\mathbf{r}_{ij}}\rho_{\mathbf{r}_{j}}(g)^{*}U^{\dagger}_{\mathbf{r}_{ij}}=\rho_{\mathbf{r}_{i}}(h),\quad\quad
U_{\mathbf{r}_{ji}}\rho_{\mathbf{r}_{i}}(g)^{*}U^{\dagger}_{\mathbf{r}_{ji}}=\rho_{\mathbf{r}_{j}}(h).$

From the above conditions, we see that $\mathbf{r}_{i}$ and $\mathbf{r}_{j}$
have the same dimension. Therefore the CP-like transformation in Eq. (2.23)
exchanges a particle in representation $\mathbf{r}_{i}$ to an antiparticles in
different representation $\mathbf{r}_{j}^{*}$. In addition, there are
irreducible representations which are properly transformed to their complex
conjugates under CP-like transformation 999 If irreducible representations of
all the fields contents are restricted to such properly transformed
representations, CP-like transformation can be interpreted as a physical CP..
We are particularly interested in the form of Eq. (2.23), since we will find
that this type of CP-like symmetry can be obtained from models with physical
CP symmetry by SSB. More importantly, as will be shown later, a physical CP
symmetry and the CP-like symmetry can be mutually converted through the SSB.
The details will be discussed in Sec. 4.

It should be noted that the unitary matrix $U_{\textrm{CP-like}}$ in Eq.
(2.23) is applicable to the case of discrete groups. We show some examples.
One simple example is $G=\mathbb{Z}_{3}$ group ($a^{3}=e,a\in\mathbb{Z}_{3}$),
which has three irreducible representations of $\mathbf{1}_{0,1,2}$. Let us
consider a model with a triplet field
$\Phi=(\phi_{\mathbf{1}_{0}},\phi_{\mathbf{1}_{1}},\phi_{\mathbf{1}_{2}})^{T}$,
where $\mathbf{1}_{0,1,2}$ are the irreducible representations of
$\mathbb{Z}_{3}$. A group action of $\mathbb{Z}_{3}$ on $\Phi$ is given as

$\displaystyle\rho_{\Phi}(a)=\begin{pmatrix}1&0&0\\\ 0&\omega&0\\\
0&0&\omega^{2}\\\ \end{pmatrix},$

where $\omega=e^{\frac{2\pi i}{3}}$. We find that the following CP-like
transformation,

$\displaystyle\textrm{CP-like}:\Phi(x)\to U_{\textrm{CP-
like}}\Phi^{*}(\tilde{x}),\quad\quad U_{\textrm{CP-like}}=$
$\displaystyle\begin{pmatrix}1&0&0\\\ 0&0&1\\\ 0&1&0\\\ \end{pmatrix},\ \ $
(2.24)

satisfies the consistency condition $U_{\textrm{CP-
like}}\rho(a)^{*}U_{\textrm{CP-like}}^{\dagger}=\rho(a)$. The corresponding
automorphism is given as an identity map; $u_{\textrm{CP-
like}}^{\mathbb{Z}_{3}}(g)=g$ for any $g\in\mathbb{Z}_{3}$. In this case, it
should be noted that the two exchanged fields $\phi_{\mathbf{1}_{1}}$ and
$\phi_{\mathbf{1}_{2}}^{*}$ have the same charge under $\mathbb{Z}_{3}$. If
the theory is CP-like symmetric, it implies that these two fields can not be
distinguished from the quantum theory point of view, and the particle
eigenstates should be parity (CP-like) eigenstates that are obtained by linear
combinations of these two fields. We will carefully study their physical
amplitudes based on the the energy eigenstates for this particular situation
in Sec. 3.

The situation changes if we consider a direct product group of
$\mathbb{Z}_{3}\times U(1)$. Let us consider a model with a triplet field
$\Phi=(\phi_{\mathbf{1}_{0}}^{q},\phi_{\mathbf{1}_{1}}^{q},\phi_{\mathbf{1}_{2}}^{q})^{T}$,
where $q\neq 0$ is a $U(1)$ charge. A group action of $\mathbb{Z}_{3}\times
U(1)$ is given as

$\displaystyle\rho_{\Phi}(a,\theta)=\begin{pmatrix}e^{iq\theta}&0&0\\\
0&\omega e^{iq\theta}&0\\\ 0&0&\omega^{2}e^{iq\theta}\\\ \end{pmatrix},$

where $\theta$ is a parameter of $U(1)$. Obviously the same CP-like
transformation as in Eq. (2.24) satisfies the consistency condition. Thus the
induced automorphism $u_{\textrm{CP-like}}$ is given by a direct product of
two specific automorphisms as

$\displaystyle u_{\textrm{CP-like}}^{\mathbb{Z}_{3}\times
U(1)}=u^{\mathbb{Z}_{3}}_{\textrm{CP-like}}\otimes u^{U(1)}_{CP},$ (2.25)
$\displaystyle u^{\mathbb{Z}_{3}}_{\textrm{CP-like}}:a\mapsto a,\quad\quad
u^{U(1)}_{CP}:q\mapsto-q.$ (2.26)

On the contrary to the previous case, both the two exchanged fields of
$\phi_{\mathbf{1}_{1}}$ and $\phi_{\mathbf{1}_{2}}^{*}$ have the same
$\mathbb{Z}_{3}$ charge, but opposite $U(1)$ charges. Therefore these two
particles should be distinguished under the CP-like transformation.

As shown above, the CP-like transformation uniquely determines the group
automorphisms. However, whether or not $U_{\textrm{CP-like}}$ in Eq. (2.23)
exists depends on the structure of group (inner and outer) automorphisms of a
given symmetry group $G$. Ensuring the presence of such a matrix is
challenging, especially when dealing with general (non-simple) groups. We find
such non-trivial examples of the CP-like model for a continuous group which
will be discussed later in Sec. 3.

As for the charge conjugation CP transformation, the same unitary matrix in
Eq. (2.12) is also used for a general continuous group. This CP transformation
induces a complex conjugation automorphism, which is consistent with a CIA for
discrete groups. We call this CP transformation as a proper CP transformation.

Finally we emphasize that it is important to consider the inconsistent but
physical CP transformation. If a group $G$ does not have any complex
conjugation automorphism, the physical CP transformation should be an
inconsistent CP transformation in general. Even if a group has complex
conjugation automorphisms, as will discussed later, if a theory has a CP-like
symmetry, the theory is not invariant under the physical CP transformation.
The detailed discussion will be given in Sec. 4.

###  2.3 Classification of General CP transformations

Based on the previous discussions, we summarize classification of general CP
transformation in Eq. (2.1) with internal symmetry represented by $G$:

  * 1.

Consistent CP transformation: general CP transformation satisfying consistency
condition in Eq. (2.2). This class of CP transformation induces automorphism
of $G$ via $CP^{-1}\circ G\circ CP\subset G$. It is possible to impose
invariance under this type of CP transformation and $G$ simultaneously. We
consider the following two cases.

    * a.

Proper CP: physical and consistent CP, whose transformation matrix is given in
Eq. (2.12). In the case of a discrete group, it induces a CIA, which
transforms $g\in G\to h\sim g^{-1}$. If a model is proper CP invariant, all
the particles are properly transformed to their complex conjugates under the
CP transformation.

    * b.

CP-like: consistent but (in general) unphysical CP, whose transformation
matrix is given in Eq. (2.23). In the case of a discrete group, it does not
induce any CIA. CP-like transformation includes an exchange between a particle
and a different antiparticle. It can be physical CP if the model does not have
such fields.

  * 2.

Inconsistent CP: general CP not satisfying consistency condition Eq. (2.2). We
can not impose invariance under inconsistent CP transformation, or $G$ is
enlarged to $\tilde{G}$ which satisfies Eq. (2.2). If there is no charge
conjugation automorphism of $G$, physical CP transformation is inconsistent CP
transformation in general. CP is not a symmetry in such a model.

We also summarize the classification in Tab. 1. It should be emphasized that
this classification of generic CP transformations is obtained independently of
the group property to serve our specific purposes. Consequently, some of CP
transformations may not be well-defined for a given $G$. See [69] for a
classification of discrete groups in terms of possible CP transformations.

| consistency with $G$ | automorphism | physical/unphysical
---|---|---|---
Proper CP |  | consistent
---
| charge conjugation
---
(class-inverting)
| physical
---
CP-like | consistent |  | non charge conjugation
---
(non class-inverting)
| physical/unphysical
---
inconsistent CP | inconsistent | no | physical/unphysical
Table 1: Classification of general CP transformations. We also show their
properties inside parenthesis () in the case of discrete groups.

##  3 General CP Symmetries and CP Asymmetry

We study the general CP symmetric models with internal symmetry. In the
following, we first show constructions of various models with general CP
symmetries. We use a discrete symmetry group for illustration purposes, where
a special attention is paid to the CP-like symmetry. To explore the impact of
the CP-like symmetry in comparison with the proper CP symmetry, we study
physical scattering amplitudes of the CP asymmetry. Although the physical CP
symmetry is in general violated in CP-like symmetric models, it is important
to notice that different CP violating amplitudes can be related to each other
through the CP-like transformations. We will find that such correspondences
can be classified by representations of initial and final states in the
scattering amplitudes. We then discuss a possibility of particle number
generations through the CP violating processes along with these
correspondences. Subsequently we consider extending CP-like transformation to
general continuous symmetry groups. An outer automorphism of order two is used
to define a CP-like transformation. We then propose to construct a new class
of CP-like symmetric model with continuous symmetry groups which include
$U(1)$ and general non-Abelian Lie groups.

###  3.1 General CP Transformation in $\Delta(27)$ Model

We study an explicit model with $\Delta(27)$ symmetry for illustration
purposes. $\Delta(27)$ is isomorphic to
$(\mathbb{Z}_{3}\times\mathbb{Z}_{3})\rtimes\mathbb{Z}_{3}$, and all elements
of $\Delta(27)$ is represented by $a^{i}a^{\prime j}b^{k}$, where
$i,j,k=0,1,2$, where $a,a^{\prime},$ and $b$ satisfy
$ba^{\prime}=ab,ba=a^{2}a^{\prime 2}b,aa^{\prime}=a^{\prime}a$ and
$a^{3}=a^{\prime 3}=b^{3}=e$. There are 11 conjugacy classes in $\Delta(27)$:

$\displaystyle C_{1a}:$ $\displaystyle\\{e\\},$ $\displaystyle C_{3a}:$
$\displaystyle\\{a,a^{\prime},a^{2}a^{\prime 2}\\},$ $\displaystyle C_{3b}:$
$\displaystyle\\{a^{2},a^{\prime 2},aa^{\prime}\\},$ $\displaystyle C_{3c}:$
$\displaystyle\\{b,aa^{\prime 2}b,a^{2}a^{\prime}b\\},$ $\displaystyle
C_{3d}:$ $\displaystyle\\{b^{2},a^{2}a^{\prime}b^{2},aa^{\prime 2}b^{2}\\},$
$\displaystyle C_{3e}:$ $\displaystyle\\{aa^{\prime}b,a^{2}b,a^{\prime
2}b\\},$ $\displaystyle C_{3f}:$
$\displaystyle\\{a^{\prime}b^{2},a^{2}a^{\prime 2}b^{2},ab^{2}\\},$
$\displaystyle C_{3g}:$ $\displaystyle\\{ab,a^{\prime}b,a^{2}a^{\prime
2}b\\},$ $\displaystyle C_{3h}:$ $\displaystyle\\{a^{\prime
2}b^{2},a^{2}b^{2},aa^{\prime}b^{2}\\},$ $\displaystyle C_{3i}:$
$\displaystyle\\{aa^{\prime 2}\\},$ $\displaystyle C_{3j}:$
$\displaystyle\\{a^{2}a^{\prime}\\},$

and hence there are nine singlets $\mathbf{1}_{i}$ and two triplets,
$\mathbf{3}$ and $\mathbf{3}^{*}$. $\mathbf{3}^{*}$ is the complex conjugate
representation of $\mathbf{3}$. The complex conjugate representations of the
singlets are also given by

$\displaystyle\mathbf{1}^{*}_{1}$ $\displaystyle=\mathbf{1}_{2},$
$\displaystyle\mathbf{1}^{*}_{3}$ $\displaystyle=\mathbf{1}_{6},$
$\displaystyle\mathbf{1}^{*}_{4}$ $\displaystyle=\mathbf{1}_{8},$
$\displaystyle\mathbf{1}^{*}_{5}$ $\displaystyle=\mathbf{1}_{7},$

and vice versa. $\mathbf{1}_{0}$ is the trivial singlet. Character table of
$\Delta(27)$ is summarized in Tab. 2. As is known, the group $\Delta(27)$ has
no CIA [69]. Therefore there is no proper CP transformation in models with
$\Delta(27)$ symmetry.

| $C_{1a}$ | $C_{3a}$ | $C_{3b}$ | $C_{3c}$ | $C_{3d}$ | $C_{3e}$ | $C_{3f}$ | $C_{3g}$ | $C_{3h}$ | $C_{3i}$ | $C_{3j}$
---|---|---|---|---|---|---|---|---|---|---|---
| $1$ | $3$ | $3$ | $3$ | $3$ | $3$ | $3$ | $3$ | $3$ | $1$ | $1$
$\Delta(27)$ | $e$ | $a$ | $a^{2}$ | $b$ | $b^{2}$ | $a^{2}b$ | $ab^{2}$ | $ab$ | $a^{2}b^{2}$ | $aa^{\prime 2}$ | $a^{2}a^{\prime}$
$\mathbf{1}_{0}$ | $1$ | $1$ | $1$ | $1$ | $1$ | $1$ | $1$ | $1$ | $1$ | $1$ | $1$
$\mathbf{1}_{1}$ | $1$ | $1$ | $1$ | $\omega^{2}$ | $\omega$ | $\omega^{2}$ | $\omega$ | $\omega^{2}$ | $\omega$ | $1$ | $1$
$\mathbf{1}_{2}$ | $1$ | $1$ | $1$ | $\omega$ | $\omega^{2}$ | $\omega$ | $\omega^{2}$ | $\omega$ | $\omega^{2}$ | $1$ | $1$
$\mathbf{1}_{3}$ | $1$ | $\omega^{2}$ | $\omega$ | $1$ | $1$ | $\omega$ | $\omega^{2}$ | $\omega^{2}$ | $\omega$ | $1$ | $1$
$\mathbf{1}_{4}$ | $1$ | $\omega^{2}$ | $\omega$ | $\omega^{2}$ | $\omega$ | $1$ | $1$ | $\omega$ | $\omega^{2}$ | $1$ | $1$
$\mathbf{1}_{5}$ | $1$ | $\omega^{2}$ | $\omega$ | $\omega$ | $\omega^{2}$ | $\omega^{2}$ | $\omega$ | $1$ | $1$ | $1$ | $1$
$\mathbf{1}_{6}$ | $1$ | $\omega$ | $\omega^{2}$ | $1$ | $1$ | $\omega^{2}$ | $\omega$ | $\omega$ | $\omega^{2}$ | $1$ | $1$
$\mathbf{1}_{7}$ | $1$ | $\omega$ | $\omega^{2}$ | $\omega^{2}$ | $\omega$ | $\omega$ | $\omega^{2}$ | $1$ | $1$ | $1$ | $1$
$\mathbf{1}_{8}$ | $1$ | $\omega$ | $\omega^{2}$ | $\omega$ | $\omega^{2}$ | $1$ | $1$ | $\omega^{2}$ | $\omega$ | $1$ | $1$
$\mathbf{3}$ | $3$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $3\omega$ | $3\omega^{2}$
$\mathbf{3}^{*}$ | $3$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $3\omega^{2}$ | $3\omega$
Table 2: Character table of $\Delta(27)$.

####  3.1.1 CP-like transformation for $\Delta(27)$

There are several CP-like transformations for $\Delta(27)$ model. One example
of CP-like transformations we examine here is

$\displaystyle\mathbf{3}$ $\displaystyle\to U_{s}\mathbf{3}^{*},$ (3.1)

where $U_{s}$ is given by [68]

$\displaystyle U_{s}$ $\displaystyle=\begin{pmatrix}\omega^{2}&0&0\\\ 0&0&1\\\
0&1&0\end{pmatrix}.$

Since $\mathbf{3}$ is a faithful representation of $\Delta(27)$, the outer
automorphism induced by this CP-like transformation is specified without
ambiguity. It is given by

$\displaystyle u:(a,a^{\prime},b)\mapsto(a^{2}a^{\prime
2},a^{\prime},a^{2}b^{2}),$

and this automorphism transforms the conjugacy classes as

$\displaystyle C_{3c}$ $\displaystyle\leftrightarrow C_{3h},$ $\displaystyle
C_{3d}$ $\displaystyle\leftrightarrow C_{3g},$ $\displaystyle C_{3e}$
$\displaystyle\leftrightarrow C_{3f},$ $\displaystyle C_{3i}$
$\displaystyle\leftrightarrow C_{3j}.$ (3.2)

$C_{3a}$ and $C_{3b}$ are fixed under the map $u$. To satisfy this relation,
the CP-like transformation for the singlets are determined as follows,

$\displaystyle\mathbf{1}_{0}$ $\displaystyle\to\mathbf{1}_{0}^{*},$
$\displaystyle\mathbf{1}_{1}$ $\displaystyle\to\mathbf{1}_{1}^{*},$
$\displaystyle\mathbf{1}_{2}$ $\displaystyle\to\mathbf{1}_{2}^{*},$
$\displaystyle\mathbf{1}_{3}$ $\displaystyle\to\mathbf{1}_{7}^{*},$
$\displaystyle\mathbf{1}_{4}$ $\displaystyle\to\mathbf{1}_{8}^{*},$
$\displaystyle\mathbf{1}_{5}$ $\displaystyle\to\mathbf{1}_{6}^{*},$
$\displaystyle\mathbf{1}_{6}$ $\displaystyle\to\mathbf{1}_{5}^{*},$
$\displaystyle\mathbf{1}_{7}$ $\displaystyle\to\mathbf{1}_{3}^{*},$
$\displaystyle\mathbf{1}_{8}$ $\displaystyle\to\mathbf{1}_{4}^{*}.$ (3.3)

This CP-like transformation is represented by a unitary matrix $A$,

$\displaystyle\mathbf{1}_{i}$ $\displaystyle\to
A_{ij}\mathbf{1}_{j}^{*},~{}~{}~{}~{}~{}\mathrm{with}~{}~{}~{}~{}~{}A=\begin{pmatrix}1&0&0&0&0&0&0&0&0\\\
0&1&0&0&0&0&0&0&0\\\ 0&0&1&0&0&0&0&0&0\\\ 0&0&0&0&0&0&0&1&0\\\
0&0&0&0&0&0&0&0&1\\\ 0&0&0&0&0&0&1&0&0\\\ 0&0&0&0&0&1&0&0&0\\\
0&0&0&1&0&0&0&0&0\\\ 0&0&0&0&1&0&0&0&0\\\ \end{pmatrix}.$

As shown above, this transformation does not transform all singlets to their
own conjugate. While this is not a physical CP transformation,
$\mathbf{1}_{0,1,2}$ are properly transformed to their complex conjugates
under the CP-like transformation. Thus this CP-like transformation can be
identified as a physical CP transformation for a model without
$\mathbf{1}_{3,...,8}$. Generally speaking, if a model has only two
representations, one singlet representation and its complex conjugate
representation, there exists a certain CP-like transformation which can be
regarded as a physical CP transformation [28].

####  3.1.2 Inconsistent CP transformation for $\Delta(27)$

As explained above, if a model has a sufficient number of irreducible
representations, the CP-like transformation ceases to be a physical CP
transformation. Thus the following physical CP transformation,

$\displaystyle\textrm{CP}_{\textrm{inc}}:$
$\displaystyle\begin{cases}\mathbf{1}_{i}\to\mathbf{1}_{i}^{*},\ \ \
(i=0,\cdots,8)\\\ \mathbf{3}\to U_{s}\mathbf{3}^{*},\end{cases}$ (3.4)

should be an inconsistent CP transformation. This is because there is no
element $h\in G$ satisfying
$U_{s}\rho^{*}_{\mathbf{3}}(g)U_{s}^{\dagger}=\rho_{\mathbf{3}}(h)$, and
$\rho^{*}_{{\mathbf{1}}_{i}}(g)=\rho_{{\mathbf{1}}_{i}}(h)$ for any $g$ and
$i$. Thus we can not realize physical CP invariance in models with the
internal symmetry of $\Delta(27)$.

To see differences of the CP-like and the inconsistent physical CP
transformations, we consider a simple model. We introduce nine singlet scalars
$\phi_{i},\ (i=0,...,8)$ and one triplet Dirac fermion
$\Psi=(\psi_{1},\psi_{2},\psi_{3})^{T}$, where $\phi_{i}$ and $\Psi$ are in
representations of $\mathbf{1}_{i}$, and $\mathbf{3}$ under $\Delta(27)$,
respectively. The $\Delta(27)$-invariant Yukawa interactions are given as

$\displaystyle\mathcal{L}_{yukawa}=$
$\displaystyle\sum_{\begin{subarray}{c}i=0,...,8\end{subarray}}y_{i}[\phi_{i}\otimes(\bar{\Psi}\otimes\Psi)_{\mathbf{1}^{*}_{i}}]+(h.c.)$
$\displaystyle=$ $\displaystyle\sum_{\begin{subarray}{c}i=0,...,8\\\
j,k=1,2,3\end{subarray}}y_{i}(M_{i^{*}})_{jk}\phi_{i}\bar{\psi}_{j}\psi_{k}+\sum_{\begin{subarray}{c}i=0,...,8\\\
j,k=1,2,3\end{subarray}}y_{i}^{*}(M_{i^{*}}^{\dagger})_{jk}\phi_{i}^{\dagger}\bar{\psi}_{j}\psi_{k}$
(3.5)

where $y_{i}$ is a complex constant and $(M_{i^{*}})_{jk}$ is the CG
coefficients of $\Delta(27)$, which are summarized in Appendix B.1. Imposing
the CP-like invariance in the model, the coupling constants should satisfy the
following conditions,

$\displaystyle y_{0}$ $\displaystyle=y_{0}^{*},$ $\displaystyle y_{1}$
$\displaystyle=y_{1}^{*},$ $\displaystyle y_{2}$ $\displaystyle=y_{2}^{*},$
$\displaystyle\omega y_{7}$ $\displaystyle=y_{3}^{*},$ $\displaystyle\omega
y_{8}$ $\displaystyle=y_{4}^{*},$ $\displaystyle\omega y_{6}$
$\displaystyle=y_{5}^{*},$ (3.6) $\displaystyle\omega y_{5}$
$\displaystyle=y_{6}^{*},$ $\displaystyle\omega y_{3}$
$\displaystyle=y_{7}^{*},$ $\displaystyle\omega y_{4}$
$\displaystyle=y_{8}^{*}.$

On the other hand, imposing inconsistent physical CP invariance given in Eq.
(3.4), we obtain vanishing Yukawa couplings. This is because some Yukawa terms
are transformed to prohibited Yukawa terms under the inconsistent CP
transformation. For example,
$\phi_{3}\otimes(\bar{\Psi}\otimes\Psi)_{\mathbf{1}^{*}_{3}}=\phi_{3}\otimes(\bar{\Psi}\otimes\Psi)_{{\mathbf{1}}_{6}}$
is transformed to
$\phi_{3}^{\dagger}\otimes(\omega^{2}\bar{\Psi}_{j}\otimes\Psi_{k})_{{\mathbf{1}}_{7}}$
under the inconsistent CP transformation. This term is not $\Delta(27)$
invariant. Thus allowed Yukawa couplings are restricted to real values as
follows,

$\displaystyle y_{0}$ $\displaystyle=y_{0}^{*},$ $\displaystyle y_{1}$
$\displaystyle=y_{1}^{*},$ $\displaystyle y_{2}$ $\displaystyle=y_{2}^{*},$
(3.7)

and the other couplings of $y_{3,\cdots,8}$ are zero. Obviously the
inconsistent CP symmetry is different from the CP-like symmetry. We notice,
however, that the conditions for $y_{0,1,2}$ in Eq. (3.7) are the same as the
one obtained in the CP-like symmetry in Eq. (3.6). Thus CP-like transformation
is identified as the physical one if the model does not have $\phi_{3,...,8}$.

###  3.2 Physical Implications of CP-like Symmetry

We study physical implications of CP-like symmetry in detail. Let us first
review the CP transformation of the scattering amplitude of multi-particle
states
$\\{\phi_{\mathbf{r}_{1}},\phi_{\mathbf{r}_{2}},...\\}\to\\{\phi_{\mathbf{r}^{\prime}_{1}},\phi_{\mathbf{r}_{2}^{\prime}},...\\}$,
where ${\mathbf{r}_{i}}$ is an irreducible representation of the internal
symmetry group $G$. For concreteness, $G$ is assumed to be a discrete group.
In the following analysis we restrict the asymptotic states to a direct
product of single particle states made of elementary fields for simplicity
101010It may be interesting to consider extending to more general states
including non-product representations such as composite and entangled states.
For a recent study of the CP transformation of composite state, see [76].
Neglecting the internal (wave function) structure, the asymptotic states can
be simply decomposed into those in the irreducible representations. Thus we
only consider that the initial and final states of $i$ and $f$ are in the
irreducible representation of $\mathbf{r}_{i}$ and $\mathbf{r}_{f}$ of $G$,
where we note that $\mathbf{r}_{i}=\mathbf{r}_{f}$ thanks to the symmetry.

The cross section of this process is given by the Lorentz invariant $S$-matrix
elements $i\mathcal{M}$ given as

$\displaystyle
i(2\pi)^{4}\mathcal{M}_{\\{\phi_{\mathbf{r}_{1}}(p_{1}),\phi_{\mathbf{r}_{2}}(p_{2}),...\\}\to\\{\phi_{\mathbf{r}^{\prime}_{1}}(p^{\prime}_{1}),\phi_{\mathbf{r}_{2}^{\prime}}(p^{\prime}_{2}),...\\}}\delta^{(4)}(p_{\textrm{init}}-p_{\textrm{final}})$
$\displaystyle=\lim_{T\to\infty}\bra{p^{\prime}_{1},p^{\prime}_{2},...}e^{-2i\mathcal{H}T}\ket{p_{1},p_{2},...},$

where $p_{i}$ and $p^{\prime}_{i}$ represent the momenta of the initial and
final particles in representation of ${\mathbf{r}_{i}}$ and
${\mathbf{r}^{\prime}_{i}}$, respectively. $p_{\textrm{init}}$ and
$p_{\textrm{final}}$ denote the total momenta of the initial and final states.
The physical CP transformation relates a scattering amplitude and the
identical process but with antiparticles of momenta
$\mathbf{k}_{i}=-\mathbf{p}_{i},\mathbf{k}_{i}^{\prime}=-\mathbf{p}_{i}^{\prime}$
as follows,

$\displaystyle\lim_{T\to\infty}\bra{p^{\prime}_{1},p^{\prime}_{2},...}\mathcal{CP}^{-1}\mathcal{CP}e^{-2i\mathcal{H}T}\mathcal{CP}^{-1}\mathcal{CP}\ket{p_{1},p_{2},...}$
$\displaystyle=\lim_{T\to\infty}\bra{k^{\prime}_{1},k^{\prime}_{2},...}e^{-2i\mathcal{CP}\mathcal{H}\mathcal{CP}^{-1}T}\ket{k_{1},k_{2},...},$

where $\mathcal{CP}$ is the physical CP transformation operator. If the theory
is invariant under the physical CP transformation, the operator $\mathcal{CP}$
satisfies $[\mathcal{H},\mathcal{CP}]=0$, and hence we obtain

$\displaystyle\mathcal{M}_{\\{\phi^{\mathrm{CP}}_{\mathbf{r}_{1}}(k_{1}),\phi^{\mathrm{CP}}_{\mathbf{r}_{2}}(k_{2}),...\\}\to\\{\phi^{\mathrm{CP}}_{\mathbf{r}^{\prime}_{1}}(k^{\prime}_{1}),\phi^{\mathrm{CP}}_{\mathbf{r}_{2}^{\prime}}(k^{\prime}_{2}),...\\}}=\mathcal{M}_{\\{\phi_{\mathbf{r}_{1}}(p_{1}),\phi_{\mathbf{r}_{2}}(p_{2}),...\\}\to\\{\phi_{\mathbf{r}^{\prime}_{1}}(p^{\prime}_{1}),\phi_{\mathbf{r}_{2}^{\prime}}(p^{\prime}_{2}),...\\}},$
(3.8)

where $\phi_{\mathbf{r}_{i}}^{\textrm{CP}}(k_{i})$ is the antiparticle of
$\phi_{\mathbf{r}_{i}}(p_{i})$ with inverse momentum. Therefore the scattering
amplitude is invariant under exchanging particles and antiparticles with
inverse momenta. It is worth noting that the physical CP transformation flips
the sign of the particle numbers, so that the physical CP invariance implies
that the two processes of particle generations and particle annihilation can
occur with the same probability. Thus the CP violation is necessary for number
generation mechanisms such as baryogenesis.

Next, we consider the CP-like transformation of the scattering amplitudes. If
a model is invariant under a CP-like transformation, we obtain

$\displaystyle\mathcal{M}_{\\{\phi^{\textrm{CP-
like}}_{\mathbf{r}_{1}}(k_{1}),\phi^{\textrm{CP-
like}}_{\mathbf{r}_{2}}(k_{2}),...\\}\to\\{\phi^{\textrm{CP-
like}}_{\mathbf{r}^{\prime}_{1}}(k^{\prime}_{1}),\phi^{\textrm{CP-
like}}_{\mathbf{r}_{2}^{\prime}}(k^{\prime}_{2}),...\\}}=\mathcal{M}_{\\{\phi_{\mathbf{r}_{1}}(p_{1}),\phi_{\mathbf{r}_{2}}(p_{2}),...\\}\to\\{\phi_{\mathbf{r}^{\prime}_{1}}(p^{\prime}_{1}),\phi_{\mathbf{r}_{2}^{\prime}}(p^{\prime}_{2}),...\\}},$
(3.9)

where $\phi^{\textrm{CP-like}}_{\mathbf{r}_{i}}(k_{i})$ denotes the CP-like
transformed particle of $\phi_{\mathbf{r}_{i}}(p_{i})$ with inverse momentum.
By definition, we have at least one particle $\phi_{\mathbf{r}_{i}}$ which is
transformed to ${\phi}^{*}_{\mathbf{r}_{j\neq i}}$ under the CP-like
transformation. Therefore Eq. (3.9) does not guarantee the vanishing CP
asymmetry.

From the perspective of the CP-like transformation, it is convenient to
classify the representations for the asymptotic states of the physical
amplitudes into the following three classes:

  1. A.

Representation which is properly transformed to its complex conjugate
representation under the CP-like transformation. For instance
$\mathbf{3},\mathbf{3}^{*}$ and $\mathbf{1}_{0,1,2}$ in the previous
$\Delta(27)$ model are properly transformed to their complex conjugate
representations under Eq. (3.2).

  2. B.

Representation which is transformed to the same representation under the CP-
like transformation, i.e. the CP-like eigenstate. For instance,
$\mathbf{1}_{4}$ in the $\Delta(27)$ model is transformed
$\mathbf{1}_{4}\to\mathbf{1}^{*}_{8}=\mathbf{1}_{4}$.

  3. C.

Representation which is transformed to an irrelevant representation under the
CP-like transformation different from neither its complex conjugate
representation nor the original representation. $\mathbf{1}_{3,5,6,7}$ in the
$\Delta(27)$ model are classified into this class.

In the case of class A, the CP-like transformation of the initial state $i$ is
$i\to i^{\textrm{CP-like}}=\bar{i}$ (up to phase), where $\bar{i}$ is the
complex conjugate state of $i$ with inverse momentum. Thus the final state $f$
is also transformed to its complex conjugate state as $f\to f^{\textrm{CP-
like}}=\bar{f}$ (up to phase). The CP-like symmetry relates two different
matrix elements as follows,

$\displaystyle|\mathcal{M}_{\\{i\\}\to\\{f\\}}|=|\mathcal{M}_{\\{i^{\textrm{CP-
like}}\\}\to\\{f^{\textrm{CP-
like}}\\}}|=|\mathcal{M}_{\\{\bar{i}\\}\to\\{\bar{f}\\}}|.$ (3.10)

The physical CP violation can not be observed from this process.

As for the case of class B, the field $\phi_{\mathbf{r}_{i}}$ is transformed
as $\phi_{\mathbf{r}_{i}}\to\phi_{\mathbf{r}_{i}^{*}}^{*}$. Since
$\phi_{\mathbf{r}_{i}}$ and $\phi_{\mathbf{r}_{i}^{*}}^{*}$ have the same
representation under $G$, both two fields could be kinematically mixed. In
fact, from the symmetry point of view the following mass terms should be
generally allowed,

$\displaystyle
m^{2}\left(\phi_{\mathbf{r}_{i}}^{*}\phi_{\mathbf{r}_{i}}^{\phantom{*}}+\phi_{\mathbf{r}^{*}_{i}}^{*}\phi_{\mathbf{r}^{*}_{i}}^{\phantom{*}}\right)+\mu^{2}\phi_{\mathbf{r}_{i}^{*}}\phi_{\mathbf{r}_{i}}+(c.c.)=(\phi_{\mathbf{r}_{i}}^{*},\phi_{\mathbf{r}^{*}_{i}}^{\phantom{*}})\begin{pmatrix}m^{2}&\mu^{2*}\\\
\mu^{2}&m^{2}\end{pmatrix}\begin{pmatrix}\phi_{\mathbf{r}_{i}}^{\phantom{*}}\\\
\phi_{{\mathbf{r}}^{*}_{i}}^{*}\end{pmatrix}.$

This mass matrix is diagonalized by the CP-like eigenstates
$\phi_{\mathbf{r}_{i}}^{\pm}=\frac{1}{\sqrt{2}}(\phi_{\mathbf{r}_{i}}\pm\phi_{{\mathbf{r}}^{*}_{i}}^{*})$,
and the particle (energy) eigenstates should be also given by the CP-like
eigenstates $\phi_{\mathbf{r}_{i}}^{\pm}$ rather than $\phi_{\mathbf{r}_{i}}$.
If the states $i$ and $f$ are CP-like eigenstates, the CP-like symmetry
implies that

$\displaystyle|\mathcal{M}_{\\{i\\}\to\\{f\\}}|=|\mathcal{M}_{\\{i^{\textrm{CP-
like}}\\}\to\\{f^{\textrm{CP-
like}}\\}}|=|\mathcal{M}_{\\{\bar{i}\\}\to\\{\bar{f}\\}}|.$ (3.11)

There is no relationship between $\mathcal{M}_{\\{i\\}\to\\{f\\}}$ and
$\mathcal{M}_{\\{\bar{i}\\}\to\\{\bar{f}\\}}$. Therefore in principle we can
observe the physical CP violation from this process.

In the last case of class C, if the initial and final states transform as
$i\to\bar{i^{\prime}}$ and $f\to\bar{f}^{\prime}$ by the CP-like
transformation, where
$\mathbf{r}_{i^{\prime}}=\mathbf{r}_{f^{\prime}}\neq\mathbf{r}_{i}=\mathbf{r}_{f}$.
We obtain

$\displaystyle\mathcal{M}_{\\{i\\}\to\\{f\\}}=\mathcal{M}_{\\{\bar{i^{\prime}}\\}\to\\{\bar{f^{\prime}}\\}}.$
(3.12)

From this process the physical CP violation can also be observed. In addition,
this relation means that the two CP-violating amplitudes are equivalent.

The differences of the above three amplitudes are clearly obtained when we
consider a three-point decay process as depicted in Fig. 1, which will be
studied in models with $\Delta(27)$ in the next subsection.

$\phi_{\mathbf{r}_{1}}$${\psi}_{\mathbf{r}_{2}}$$\bar{\psi}_{\mathbf{r}_{3}}$
Figure 1: 3-point decay.

####  3.2.1 CP-like symmetry in $\Delta(27)$ model: class B

To illustrate the physical implications of the CP-like symmetry, we calculate
a decay amplitude whose initial state belongs to class B in a model with
$\Delta(27)$ symmetry. We introduce one complex scalar singlet $\phi_{4}^{+}$,
a triplet scalar fields $\Phi_{\mathbf{3}}$, two singlet fermions $\psi_{5},$
$\psi_{6}$, and a triplet fermion $\Psi_{\mathbf{3}}$. The field contents are
summarized in Tab. 3. The CP-like transformation of the fields are given as

$\displaystyle\textrm{CP-like}:~{}$
$\displaystyle\phi_{4}^{+}\to\phi_{4}^{+},$ $\displaystyle\Phi_{\mathbf{3}}$
$\displaystyle\to U_{s}\Phi_{\mathbf{3}}^{*},$ $\displaystyle\psi_{5}\to
C\psi_{6}^{*},$ $\displaystyle\psi_{6}$ $\displaystyle\to C\psi_{5}^{*},$
$\displaystyle\Psi_{\mathbf{3}}$ $\displaystyle\to
U_{s}C\Psi_{\mathbf{3}}^{*},$

where $C=-i\gamma_{2}\gamma_{0}$ is the standard charge conjugation matrix for
spinors. $\phi_{4}^{+}$ is the CP-like eigenstate. The Yukawa terms are given
by

$\displaystyle\mathcal{L}_{\textrm{Yukawa}}=$ $\displaystyle
y_{1}\phi_{4}^{+}\bar{\psi}_{6}\psi_{5}+y_{2}\phi_{4}^{+}\otimes(\bar{\Psi}_{\mathbf{3}}\otimes\Psi_{\mathbf{3}})_{\mathbf{1}_{8}}+y_{3}\bar{\psi}_{5}\otimes(\Phi_{\mathbf{3}}^{*}\otimes\Psi_{\mathbf{3}})_{\mathbf{1}_{5}}$
$\displaystyle+y_{4}\bar{\psi}_{6}\otimes(\Phi_{\mathbf{3}}^{*}\otimes\Psi_{\mathbf{3}})_{\mathbf{1}_{6}}+(h.c.).$

The CP-like invariance constrains $y_{4}=y_{3}^{*}$, while $y_{1}$ and $y_{2}$
are the free parameters. The scalar potential is irrelevant to discussion in
this section, and we omit it here. Assuming that $\phi_{4}^{+}$ is much
heavier than $\psi_{5}$ and $\bar{\psi}_{6}$, we calculate the decay amplitude
of $\phi_{4}^{+}\to\psi_{6}\bar{\psi}_{5}$. Since the product state of
${\mathbf{1}_{6}}\otimes\mathbf{1}^{*}_{5}={\mathbf{1}_{4}}$, the initial and
final states are in the same representation under $\Delta(27)$ as expected. As
explained in Sec. 3.1.2, the Yukawa term is not invariant under the physical
CP transformation in Eq. (3.4), we expect that the nonzero value of a CP
asymmetry
$\epsilon_{\phi_{4}^{+}\to\psi_{6}\bar{\psi}_{5}}\equiv|\mathcal{M}_{\phi_{4}^{+}\to\psi_{6}\bar{\psi}_{5}}|^{2}-|\bar{\mathcal{M}}_{\bar{\phi}_{4}^{+}\to\bar{\psi}_{6}{\psi}_{5}}|^{2}$
can be observed. At one-loop order the matrix element
$\mathcal{M}_{\phi_{4}^{+}\to\psi_{6}\bar{\psi}_{5}}$ is given by the
following diagrams as

$\displaystyle\leavevmode\hbox to0pt{\vbox
to0pt{\pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower 0.0pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ } \feynhand \vertex(a) at (-1,0)
{$\phi_{4}^{+}$}; \vertex(b) at (1,-1) {$\bar{\psi}_{5}$}; \vertex(c) at (1,1)
{$\psi_{6}$}; \vertex[dot] (o) at (0,0); \propag[sca] (a) to (o);
\propag[fermion] (o) to (c); \propag[anti fermion] (o) to (b);
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}+\leavevmode\hbox
to0pt{\vbox to0pt{\pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower 0.0pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ } \feynhand \vertex(a) at (-1,0)
{$\phi_{4}^{+}$}; \vertex(b) at (1,-1) ; \vertex(c) at (1,1) ; \vertex(d) at
(2, -1) {$\bar{\psi}_{5}$}; \vertex(e) at (2, 1) {$\psi_{6}$}; \vertex[dot]
(o) at (0,0); \propag[sca] (a) to (o); \propag[fermion] (b) to [edge label =
$\Psi_{\mathbf{3}}$](o); \propag[anti fermion] (c) to [edge label' =
$\Psi_{\mathbf{3}}$] (o); \propag[anti fermion] (b) to (d); \propag[fermion]
(c) to (e); \propag[sca] (b) to [edge label'= $\Phi_{\mathbf{3}}$] (c);
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}=$
$\displaystyle-i\left\\{y_{1}-y_{2}y_{3}^{*}y_{4}\textrm{tr}\left(M_{6}M_{8}M_{5}^{\dagger}\right)I\right\\}\bar{u}^{s}(p)v^{s^{\prime}}(p^{\prime})$
$\displaystyle=$
$\displaystyle-i\left(y_{1}-3\omega^{2}y_{2}y_{3}^{*}y_{4}I\right)\bar{u}^{s}(p)v^{s^{\prime}}(p^{\prime}),$

where $I$ is the one-loop integral, and $p,p^{\prime}$ are the momenta of the
final state fermions. The explicit form of $I$ is given in Appendix C. $M_{i}$
are the CG coefficients of $\Delta(27)$ given in Appendix B.1. The physical CP
asymmetry is observed as

$\displaystyle\epsilon_{\phi_{4}^{+}\to\psi_{6}\bar{\psi}_{5}}\propto\mathrm{Im}\left(3\omega^{2}y_{1}^{*}y_{2}y_{3}^{*}y_{4}\right)\mathrm{Im}\,I\neq
0.$ (3.13)

Thus decay process of class B state is CP asymmetric.

| $\phi_{4}^{+}$ | $\Phi_{\mathbf{3}}$ | $\psi_{5}$ | $\psi_{6}$ | $\Psi_{\mathbf{3}}$
---|---|---|---|---|---
$\Delta(27)$ | $\mathbf{1}_{4}$ | $\mathbf{3}$ | $\mathbf{1}_{5}$ | $\mathbf{1}_{6}$ | $\mathbf{3}$
mass | $m_{4}$ | $m_{s}$ | $m_{5}$ | $m_{6}$ | $m_{f}$
Table 3: Field contents of $\Delta(27)$ model.

We also find that this decay process can generate the particle numbers of
$\psi_{5}$ and $\psi_{6}$, since $\psi_{5}$ and $\psi_{6}$ are in different
representations and are different particles. We note, however, that the sum of
the particle numbers of $\psi_{5,6}$ is conserved, i.e.

$\displaystyle N_{\psi_{5}}+N_{\psi_{6}}=0,$ (3.14)

where $N_{\psi_{5,6}}\neq 0$ are the particle numbers of $\psi_{5,6}$. The
relation in Eq. (3.14) is analogous to other particle number generation
processes such as the GUT baryogenesis [72] and the sphaleron process [73], in
which the number of $B-L$ is conserved thanks to the internal (gauge)
symmetry. In this model the conservation of $N_{\psi_{5}}+N_{\psi_{6}}$ is
guaranteed by the CP-like symmetry. The decay of the CP-like eigenstate could
be the origin of matter generations.

####  3.2.2 CP-like Symmetry in $\Delta(27)$ Model: Class C

We consider a model with nine singlet scalars $\phi_{i}$ and $N$ triplet
fermions $\Psi^{I}_{\mathbf{3}}$,

$\displaystyle\Psi^{I}_{\mathbf{3}}$
$\displaystyle=(\psi_{1}^{I},\psi_{2}^{I},\psi_{3}^{I})^{T},$ $\displaystyle
I$ $\displaystyle=1,...,N.$

The field contents are summarized in Tab. 4. The mass terms of $\phi_{i}$ and
$\Psi_{\mathbf{3}}^{I}$ are given as

$\displaystyle\mathcal{L}_{\textrm{mass}}=m_{i}^{2}\phi_{i}^{*}\phi_{i}+m_{I}\bar{\Psi}_{\mathbf{3}}^{I}\Psi_{\mathbf{3}}^{I},$

where we omit the mixing mass terms, since they are irrelevant to the
following analysis. The Yukawa terms are almost the same as those in Eq.
(3.5), but Yukawa couplings are generalized to $y_{i}^{IJ}$. At the one-loop
order, the matrix element
$\mathcal{M}_{\phi_{i}\to\psi^{I}_{j}\bar{\psi}^{J}_{k}}$ $(i\neq 0,1,2,4,8)$
is given by the following diagrams as

$\displaystyle\leavevmode\hbox to0pt{\vbox
to0pt{\pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower 0.0pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ } \feynhand \vertex(a) at (-1,0)
{$\phi_{i}$}; \vertex(b) at (1,-1) {$\bar{\psi}_{k}^{J}$}; \vertex(c) at (1,1)
{$\psi_{j}^{I}$}; \vertex[dot] (o) at (0,0); \propag[sca] (a) to (o);
\propag[fermion] (o) to (c); \propag[anti fermion] (o) to (b);
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}+\leavevmode\hbox
to0pt{\vbox to0pt{\pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower 0.0pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ } \feynhand \vertex(a) at (-1,0)
{$\phi_{i}$}; \vertex(b) at (1,-1) ; \vertex(c) at (1,1) ; \vertex(d) at (2,
-1) {$\bar{\psi}_{k}^{J}$}; \vertex(e) at (2, 1) {$\psi_{j}^{I}$};
\vertex[dot] (o) at (0,0); \propag[sca] (a) to (o); \propag[fermion] (b) to
[edge label = $\bar{\psi}_{n}^{L}$](o); \propag[anti fermion] (c) to [edge
label' = $\psi_{m}^{K}$] (o); \propag[anti fermion] (b) to (d);
\propag[fermion] (c) to (e); \propag[sca] (b) to [edge label'= $\phi_{l}$]
(c); \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}$
$\displaystyle=-i\left((Y_{i}^{IJ})_{jk}-\sum_{\begin{subarray}{c}l,K,L\end{subarray}}\left[Y_{l}^{IK\dagger}Y^{KL}_{i}Y^{LJ}_{l}+Y_{l}^{IK}Y^{KL}_{i}Y^{LJ\dagger}_{l}\right]_{jk}I^{KL}_{l}\right)\bar{u}^{s}(p)v^{s^{\prime}}(p^{\prime}),$

where $I_{l}^{KL}$ is the loop integral with
$\phi_{l},\psi_{\mathbf{3}}^{K},\psi_{\mathbf{3}}^{L}$. $Y_{i}^{IJ}$ is the
Yukawa matrix given by $Y_{i}^{IJ}=y^{IJ}_{i}M_{i^{*}}$. As a result, we
calculate a CP asymmetry,

$\displaystyle\epsilon_{\phi_{i}\to\Psi_{\mathbf{3}}^{I}\bar{\Psi}_{\mathbf{3}}^{J}}\equiv$
$\displaystyle\sum_{\begin{subarray}{c}j,k=1\end{subarray}}^{3}\left(|\mathcal{M}_{\phi_{i}\to\psi^{I}_{j}\bar{\psi}^{J}_{k}}|^{2}-|\bar{\mathcal{M}}_{\bar{\phi}_{i}\to\bar{\psi}^{I}_{i}{\psi}^{J}_{k}}|^{2}\right)$
$\displaystyle=$
$\displaystyle-4\sum_{\begin{subarray}{c}l,K,L\end{subarray}}\mathrm{Im}\left(\textrm{tr}\,Y_{i}^{IJ\dagger}\left[Y_{l}^{IK\dagger}Y^{KL}_{i}Y^{LJ}_{l}+Y_{l}^{IK}Y^{KL}_{i}Y^{LJ\dagger}_{l}\right]\right)\mathrm{Im}\,I^{KL}_{l}$
$\displaystyle=$
$\displaystyle-8\sum_{\begin{subarray}{c}l,K,L\end{subarray}}\mathrm{Re}\left(y_{l}^{IK*}y_{l}^{LJ}\textrm{tr}\,M_{{l}^{*}}^{\dagger}M_{i^{*}}M_{{l}^{*}}M_{{i}^{*}}^{\dagger}\right)\mathrm{Im}\,y_{i}^{IJ*}y_{i}^{KL}\mathrm{Im}\,I^{KL}_{l},$
(3.15)

where we sum over the indices of $j,k$ from 1 to 3 for the final states
$\mathbf{3}$, since those three components are equivalent under the
$\Delta(27)$ symmetry. This summation is important to prove that there is no
general CP transformation that cancels the CP asymmetry for a specific channel
111111See Appendix D.. In terms of $\Delta(27)$ symmetry this summation should
correspond to extracting $\mathbf{1}_{i}$ from the tensor products of
$\mathbf{3}\otimes{\mathbf{3}}^{*}$. From the conditions for CP-like symmetry
in Eq. (3.6), $y_{i}^{IJ}$ is complex. Hence when $N\geq 2$, even though the
theory is CP-like invariant, $y_{i}^{IJ*}y_{i}^{KL}$ is complex, and we
observe the physical CP violation of
$\epsilon_{\phi_{i}\to\Psi^{I}\bar{\Psi}^{J}}\neq 0$.

| $\phi_{i}$ | $\Psi^{I}_{\mathbf{3}}$
---|---|---
$\Delta(27)$ | $\mathbf{1}_{i}$ | $\mathbf{3}$
Table 4: Matter contents of $\Delta(27)$ model. $i=0,...,8,$ and $I=1,...,N$.

In addition, it follows from Eq. (3.12) that the CP-like transformation
relates $\epsilon_{\phi_{i}\to\Psi^{I}\bar{\Psi}^{J}}$ to the other CP
asymmetry of $\epsilon_{\phi_{i^{\prime}}\to\Psi^{I}\bar{\Psi}^{J}}$, where
$i^{\prime}$ is the index of the CP-like conjugated state of $\phi_{i}$ given
in Eq. (3.3). For instance, $\phi_{3}$ is related to $\phi_{7}^{*}$. Using the
following relations

$\displaystyle\mathrm{tr}\,M_{l^{*}}^{\dagger}M_{3^{*}}M_{l^{*}}M_{3^{*}}^{\dagger}=(\mathrm{tr}\,M_{l^{\prime*}}^{\dagger}M_{7^{*}}M_{l^{\prime*}}M_{7^{*}}^{\dagger})^{*},\
\ \ y_{l}^{IK*}y_{l}^{LJ}=(y_{l^{\prime}}^{IK*}y_{l^{\prime}}^{LJ})^{*},$

we obtain

$\displaystyle\epsilon_{\phi_{3}\to\Psi^{I}_{\mathbf{3}}\bar{\Psi}_{\mathbf{3}}^{J}}=$
$\displaystyle-8\sum_{l^{\prime},K,L}\mathrm{Re}\,\left((y_{l^{\prime}}^{IK*}y_{l^{\prime}}^{LJ})^{*}(\mathrm{tr}\,M_{l^{\prime*}}^{\dagger}M_{7^{*}}M_{{l}^{\prime*}}M_{7^{*}}^{\dagger})^{*}\right)\mathrm{Im}\,(y_{7}^{IJ*}y_{7}^{KL})^{*}\mathrm{Im}\,I^{KL}_{l^{\prime}}$
$\displaystyle=$
$\displaystyle-\epsilon_{\phi_{7}\to\Psi^{I}_{\mathbf{3}}\bar{\Psi}^{J}_{\mathbf{3}}},$

where we use $I_{l}^{KL}=I_{l^{\prime}}^{KL}$, because the masses of
$\phi_{l}$ and $\phi_{l^{\prime}}$ are the same due to the CP-like symmetry,
and the minus sign in the last line comes from
$\mathrm{Im}\,(y_{7}^{IJ*}y_{7}^{KL})^{*}=-\mathrm{Im}\,y_{7}^{IJ*}y_{7}^{KL}$.
Thus we obtain

$\displaystyle\epsilon_{\phi_{3}\to\Psi^{I}_{\mathbf{3}}\bar{\Psi}_{\mathbf{3}}^{J}}+\epsilon_{\phi_{7}\to\Psi_{\mathbf{3}}^{I}\bar{\Psi}_{\mathbf{3}}^{J}}=0.$

The CP asymmetry of the two different amplitudes are related by the CP-like
transformation as expected.

For both cases of class B and C, the physical CP violation can be observed. In
addition, since two particles in different representations are exchanged by
the CP-like transformation, two different particle numbers can be generated
through the corresponding CP-violating scattering process, although the sum of
two exchanged particle numbers are conserved by the CP-like transformation. It
is noted that our result can also be applied to the pair annihilation process
of two fermions to one complex scalar field. Thus the CP asymmetry also
implies the particle number violation of the complex scalar fields such as
$\phi_{4}^{+}$ and $\phi_{3,7}$ in the models presented here. The only
difference between class B and C is that the CP-eigenstate has no CP-like
partner, and its particle number is not related to other antiparticles nor
conserved, i.e. $N_{\phi_{4}^{+}}\neq 0$. In any case, it is important for a
number generation to consider a state of either class B or C. This mechanism
might be useful for phenomenological purpose, e.g. for baryogenesis and
asymmetric DM [78].

###  3.3 CP-like Symmetry for Continuous Groups

We discuss a possible extension of CP-like transformations to general groups.
As is known, since most of popular simple Lie groups such as $SU(N)\ (N\geq
3)$ and Abelian group $U(1)$ have the unique outer automorphism group of
$\mathbb{Z}_{2}$ which identified with the proper CP transformation [67], the
CP-like transformation is not well defined for such simple groups. Let us
provide a specific explanation by using a $U(1)$ symmetry. For a multiplet
field $\Phi=(\phi_{p},\phi_{q})^{T}$, where $\phi_{p,q}$ have $U(1)$ charges
of $p,q\neq 0$, a $U(1)$ transformation is represented as

$\displaystyle\Phi\to\Phi^{\prime}=\rho_{\Phi}(\theta)\Phi,\quad\quad\rho_{\Phi}(\theta)=\begin{pmatrix}e^{ip\theta}&0\\\
0&e^{iq\theta}\end{pmatrix},\ \ \theta\in\mathbb{R}.$ (3.16)

A candidate CP-like transformation is given as

$\displaystyle\Phi(x)\to U\Phi^{*}(\tilde{x}),\ \ U=\begin{pmatrix}0&1\\\
1&0\end{pmatrix}.$ (3.17)

The consistency condition of
$U\rho_{\Phi}^{*}(\theta)U=\rho_{\Phi}(\theta^{\prime})$ leads to the
following relation as

$\displaystyle\begin{pmatrix}e^{-iq\theta}&0\\\
0&e^{-ip\theta}\end{pmatrix}=\begin{pmatrix}e^{ip\theta^{\prime}}&0\\\
0&e^{iq\theta^{\prime}}\end{pmatrix}\ \Rightarrow\ p=\pm q.$

As a result, two automorphisms are allowed. In the case of $p=q$, this is a
proper CP transformation, which induces the unique outer automorphism
$u^{U(1)}_{\textrm{CP}}$ as given in Eq. (2.26). On the other hand in the case
of $p=-q$ it induces an identity map

$\displaystyle u^{U(1)}_{\textrm{CP-like}}:p\mapsto p,$ (3.18)

where two exchanged fields of $\phi_{p}$ and $\phi_{-p}^{*}$ are in the same
irreducible representation (class B). These two are kinematically mixed to be
CP-like basis

$\displaystyle\phi_{\pm}=\phi_{p}+\phi_{-p}^{*}.$

Under this automorphism there is no representation belongs to class A or C
except for the trivial singlet, so that this is not a genuine CP-like
transformation.

For the reasons above, we shall extend slightly further to consider non-simple
groups of $G=G_{1}\times G_{2}$, where we assume that both $G_{1}$ and $G_{2}$
have a charge conjugation outer automorphism. We provide different ways to
construct models with CP-like transformations as follows.

One trivial way to construct a CP-like symmetric model is to use the following
direct product of two automorphisms as

$\displaystyle u_{\textrm{CP-like}}^{G_{1}\times G_{2}}=u^{G_{1}}_{\textrm{CP-
like}}\times u^{G_{2}}_{\textrm{CP}},$

where $u^{G_{1}}_{\textrm{CP-like}}$ is the identify map of the group $G_{1}$,
while $u^{G_{2}}_{\textrm{CP}}$ corresponds to the $\mathbb{Z}_{2}$ outer
automorphism of the group $G_{2}$. Thus the CP-like transformation for a
direct product representation of $(\mathbf{r}_{1},\mathbf{r}_{2})$ is given as

$\displaystyle\textrm{CP-
like}:(\mathbf{r}_{1},\mathbf{r}_{2})\to(\mathbf{r}_{1},\mathbf{r}^{*}_{2}),$

where $\mathbf{r}_{1}$ and $\mathbf{r}_{2}$ are some non-trivial irreducible
representations of $G_{1}$ and $G_{2}$, respectively, and $\mathbf{r}^{*}_{2}$
is the complex conjugate representation of $\mathbf{r}_{2}$. Accordingly the
direct product representations are categorized into the aforementioned three
classes in the following manner,

$\displaystyle\textrm{Class
A}:(\mathbf{1}_{1},\mathbf{r}_{2}),\quad\quad\textrm{Class
B}:(\mathbf{r}_{1},\mathbf{1}_{2}),\quad\quad\textrm{Class
C}:(\mathbf{r}_{1},\mathbf{r}_{2}),$

where $\mathbf{1}_{1}$ and $\mathbf{1}_{2}$ are the trivial singlet
representations of $G_{1}$ and $G_{2}$, respectively.

An example for this type of CP-like transformation is easily constructed in a
model with $U(1)_{1}\times U(1)_{2}$ symmetry. Let us introduce a multiplet
field $\Phi=(\phi_{p},\phi_{q})^{T}$, where $\phi_{p}$ and $\phi_{q}$ have two
$U(1)$ charges $p_{1,2}$, and $q_{1,2}$ under $U(1)_{1}\times U(1)_{2}$. A
group action on $\Phi$ is represented as

$\displaystyle\Phi\to\Phi^{\prime}=\rho_{\Phi}(\theta_{1},\theta_{2})\Phi,\quad\quad\rho_{\Phi}(\theta_{1},\theta_{2})=\begin{pmatrix}e^{i(p_{1}\theta_{1}+p_{2}\theta_{2})}&0\\\
0&e^{i(q_{1}\theta_{1}+q_{2}\theta_{2})}\end{pmatrix},\ \
\theta_{1,2}\in\mathbb{R}.$ (3.19)

For example, when $q_{1}=-p_{1}$ and $q_{2}=p_{2}$ the following CP-like
transformation

$\displaystyle\Phi(x)\to U_{\textrm{CP-like}}^{U(1)_{1}\times
U(1)_{2}}\Phi^{*}(\tilde{x}),\ \ U_{\textrm{CP-like}}^{U(1)_{1}\times
U(1)_{2}}=\begin{pmatrix}0&1\\\ 1&0\end{pmatrix},$ (3.20)

satisfies the consistency conditions. The corresponding automorphism
$u_{\textrm{CP-like}}^{U(1)_{1}\times U(1)_{2}}$ is

$\displaystyle u_{\textrm{CP-like}}^{U(1)_{1}\times U(1)_{2}}$
$\displaystyle:(p_{1},p_{2})\mapsto(p_{1},-p_{2}).$ (3.21)

Thus the CP-like partners of $\phi_{p}$ and $\phi_{q}$ have the opposite
$U(1)_{1}$ charges but the same $U(1)_{2}$ charge. If the model has this CP-
like symmetry, it follows from the CPT theorem that it requires the existence
of a CP-like partner in addition to their complex conjugate fields. In the
same way, it is possible to construct a CP-like symmetric model with internal
symmetry group $G_{1}\times G_{2}$.

In addition, in the case of $U(1)$ two independent $U(1)$ transformations can
mix, so that it is possible to find a non-trivial CP-like transformation for
generic values of $U(1)$ charges. When $p_{1}q_{2}-p_{2}q_{1}\neq 0$, a
general solution to the consistency condition is as follows,

$\displaystyle\begin{pmatrix}0&1\\\
1&0\end{pmatrix}\rho_{\Phi}(\theta_{1},\theta_{2})^{*}\begin{pmatrix}0&1\\\
1&0\end{pmatrix}^{\dagger}=\rho_{\Phi}(\theta_{1}^{\prime},\theta_{2}^{\prime})$
$\displaystyle\quad\quad\Rightarrow\quad\quad$
$\displaystyle\begin{pmatrix}\theta_{1}^{\prime}\\\
\theta_{2}^{\prime}\end{pmatrix}=A\begin{pmatrix}\theta_{1}\\\
\theta_{2}\end{pmatrix},\quad\quad
A=\frac{1}{p_{1}q_{2}-p_{2}q_{1}}\begin{pmatrix}p_{1}p_{2}-q_{1}q_{2}&p_{2}^{2}-q_{2}^{2}\\\
q_{1}^{2}-p_{1}^{2}&q_{1}q_{2}-p_{1}p_{2}\end{pmatrix}.$ (3.22)

This is a CP-like transformation since it exchanges the states of linearly
independent charges. To see it more clearly we diagonalize the automorphism
induced by the CP-like transformation in Eq. (3.22) as,

$\displaystyle\begin{pmatrix}\tilde{\theta}_{1}\\\
\tilde{\theta}_{2}\end{pmatrix}=M\begin{pmatrix}\theta_{1}\\\
\theta_{2}\end{pmatrix}$ (3.23)

where $M$ is a diagonalization matrix of $A$, and $\tilde{\theta}_{1}$ and
$\tilde{\theta}_{2}$ are transformation parameters of
$\tilde{U}(1)_{1}\times\tilde{U}(1)_{2}\simeq U(1)_{1}\times U(1)_{2}$ (linear
combinations of original two $U(1)$ groups). From the facts that $\det{A}=-1$,
in the new basis the automorphism is given as

$\displaystyle\begin{pmatrix}\tilde{\theta}_{1}^{\prime}\\\
\tilde{\theta}_{2}^{\prime}\end{pmatrix}=\begin{pmatrix}1&0\\\
0&-1\end{pmatrix}\begin{pmatrix}\tilde{\theta}_{1}\\\
\tilde{\theta}_{2}\end{pmatrix}.$

The $\tilde{U}(1)_{2}$ charge reverses, while the $\tilde{U}(1)_{1}$ charge is
unchanged. Thus we obtain a CP-like transformation for $U(1)_{1}\times
U(1)_{2}$ symmetry.

##  4 CP/CP-like Symmetry and Spontaneous Symmetry Breaking

In this section we first discuss the origin of the CP-like symmetry in the CP-
like symmetric model. In the previous sections, we have shown that models with
CP-like symmetry for type I group (i.e. absence of CIA) exhibit a physical CP
violation. It is important to notice that such the type I group can be
embedded into simpler groups such as $SU(N)\ (N\geq 3)$, in which a proper CP
transformation can be defined. This fact suggests a possibility that in some
dynamical models when a vacuum expectation value (VEV) gives rise to a
symmetry breaking of $G\to H$, a physical CP is simultaneously violated due to
a spontaneous change of the proper CP $\to$ CP-like, even if the VEV does not
break the CP symmetry, i.e. the vacuum is invariant under the original CP
transformation. The spontaneous CP violation based on the mechanism was first
proposed in a model with $SU(3)\to T_{7}$ [71], where $T_{7}$ is a type I
group. Thus, it might be expected that a limited class of groups, e.g. type I,
can exhibit such the spontaneous CP violation. We note, however, that as will
be shown later, a wider class of CP-like symmetric models can be derived from
a proper CP symmetric model, even if the subgroup $H$ has a CIA.

If $H$ does not have any CIA, the physical CP transformation in broken phase
should be an inconsistent CP transformation, and CP asymmetry should be
observed. There are two possibilities for CP violation in the broken phase. If
the inconsistent physical CP transformation in the broken phase is given by a
product of the original CP transformation and a broken element $\in
G\backslash H$, the CP violation is directly caused by the SSB of the internal
symmetry, and hence the CP asymmetry is proportional to the VEV giving rise to
the SSB. On the other hand, if not the case, the inconsistent physical CP in
the broken phase should be an inconsistent CP even in the symmetric phase,
even though the theory is invariant under the same CP(-like) transformation in
both phases. Therefore the CP asymmetry is not necessary to be proportional to
the VEV. We will show that the latter type of CP asymmetry really occurs
through SSB. In this case, the CP violation genuinely comes from the group
structure. We will carefully study the CP asymmetry of several physical
observables in the broken phases, by which we elucidate the relations between
the physical CP violation in the broken phase and the CP invariance in
symmetric phase in detail.

Next, we discuss the fate of the CP-like symmetry. Let us assume that $H$ is a
type I group. As mentioned above, if a model is invariant under $H$ and CP-
like transformations, the physical CP is violated in general. Here one
question arises immediately: is there a possibility that the CP-like
transformation would become a proper CP transformation after further symmetry
breaking of $H\to H^{\prime}$? If such the VEV does not break the CP-like
symmetry, it appears to be possible. A trivial example is the case where $H$
is completely broken down to $H^{\prime}=I$, where all fields would belong to
the only one representation of the trivial singlet (multiplet merging), and
the CP-like partners can not be distinguished from each other. Accordingly the
CP-like transformation will spontaneously change to a proper CP
transformation. This result is also evident from the definitions of both
proper CP and CP-like transformations in Eqs. (2.12) and (2.23). Considering
more general case of $H^{\prime}\neq I$, when two mutually exchanged
representations of $\mathbf{r}_{i}$ and $\mathbf{r}_{j}$ belong to the same
representation after SSB, the transformation matrix in Eq. (2.23) should
become to the one for the proper CP transformation in Eq. (2.12). On the
contrary, the spontaneous CP violation may occur when two mutually exchanged
fields under a proper CP transformation split into two distinct
representations (multiplet splitting). The details of such examples will be
shown in what follows.

###  4.1 $\Delta(54)\times U(1)$ Model

In this subsection, we study symmetry breaking pattern of model with
$\Delta(54)\times U(1)$. The discrete group $\Delta(54)$ has 10 irreducible
representations of two real singlets $\mathbf{1}_{0,1}$, four pseudoreal
doublets $\mathbf{2}_{1,2,3,4}$ and four complex triplets
$\mathbf{3}_{1,2},\mathbf{3}^{*}_{1,2}$. See Appendix A for the details of
$\Delta(54)$. Since there is no CIA of $\Delta(54)$, we consider the following
CP-like transformation,

$\displaystyle\mathbf{3}_{i}$ $\displaystyle\to U_{s}\mathbf{3}_{i}^{*},$
$\displaystyle\mathbf{2}_{1}$ $\displaystyle\to\mathbf{2}_{1}^{*},$
$\displaystyle\mathbf{2}_{2}$ $\displaystyle\to\mathbf{2}_{4}^{*},$
$\displaystyle\mathbf{2}_{3}$ $\displaystyle\to S_{2}\mathbf{2}_{3}^{*},$
$\displaystyle\mathbf{2}_{4}$ $\displaystyle\to\mathbf{2}_{2}^{*},$
$\displaystyle\mathbf{1}_{0,1}\to\mathbf{1}_{0,1}^{*},$ (4.1)

where $U_{s}$ and $S_{2}$ are unitary matrices given by

$\displaystyle U_{s}$ $\displaystyle=\begin{pmatrix}\omega^{2}&0&0\\\ 0&0&1\\\
0&1&0\end{pmatrix},\ $ $\displaystyle S_{2}$
$\displaystyle=\begin{pmatrix}0&1\\\ 1&0\end{pmatrix}.$ (4.2)

This CP-like transformation induces an automorphism

$\displaystyle u^{\Delta(54)}_{\textrm{CP-
like}}~{}:~{}(a,a^{\prime},b,c)\mapsto(a^{2}a^{\prime
2},a^{\prime},a^{2}b^{2},c),$ (4.3)

on $\Delta(54)$ 121212It is straightforward to check that
$U_{s}\rho_{\mathbf{3}_{i}}(g)^{*}U_{s}^{\dagger}=\rho_{\mathbf{3}_{i}}(u_{cp}(g))$,
and
$S_{2}\rho_{\mathbf{2}_{3}}(g)^{*}S_{2}^{\dagger}=\rho_{\mathbf{2}_{3}}(u_{cp}(g))$.,
and $e^{i\theta}\to e^{-i\theta}$ on $U(1)$.

| $\Psi^{1}$ | $\Psi^{2}$ | $\Psi^{3}$ | $\Psi^{4}$ | $S$ | $T$ | $R$
---|---|---|---|---|---|---|---
$\Delta(54)$ | $\mathbf{3}_{1}$ | $\mathbf{3}_{1}$ | $\mathbf{3}_{1}$ | $\mathbf{3}_{1}$ | $\mathbf{1}_{1}$ | $\mathbf{2}_{3}$ | $\mathbf{2}_{1}$
$U(1)$ | $1$ | $2$ | $3$ | $4$ | $0$ | $-1$ | $-2$
Table 5: Irreducible representations and $U(1)$ charges of the fields.
$\Psi^{I}$ are Dirac fermions, $T,R$ are complex scalars, $S$ is a real
scalar.

We consider a Lagrangian with

$\displaystyle\mathcal{L}=\bar{\Psi}^{I}(i\not{\partial}-m_{\Psi}^{I})\Psi^{I}+\partial_{\mu}S\partial^{\mu}S+\partial_{\mu}T^{\dagger}\partial^{\mu}T+\partial_{\mu}R^{\dagger}\partial^{\mu}R-\mathcal{L}_{Yukawa}-\mathcal{V}(S,T,R),$

where we introduce two complex doublet scalar fields $T,R$, one real singlet
scalar $S$, and have four triplet Dirac fermions $\Psi^{1,2,3,4}$. Their
charges and representations are summarized in Tab. 5. The interaction term
$\mathcal{L}_{Yukawa}$ is given by

$\displaystyle\mathcal{L}_{Yukawa}=$ $\displaystyle
t_{1}[T\otimes(\bar{\Psi}^{1}\otimes\Psi^{2})_{\mathbf{2}_{3}}]_{\mathbf{1}_{0}}+t_{2}[T\otimes(\bar{\Psi}^{3}\otimes\Psi^{4})_{\mathbf{2}_{3}}]_{\mathbf{1}_{0}}$
$\displaystyle+r_{1}[R\otimes(\bar{\Psi}^{1}\otimes\Psi^{3})_{\mathbf{2}_{1}}]_{\mathbf{1}_{0}}+r_{2}[R\otimes(\bar{\Psi}^{2}\otimes\Psi^{4})_{\mathbf{2}_{1}}]_{\mathbf{1}_{0}}+(h.c.),$

where $t_{i}$ and $r_{j}$ are Yukawa couplings. Since
$\mathbf{3}^{*}_{1}\otimes\mathbf{3}_{1}$ does not include $\mathbf{1}_{1}$,
there are no Yukawa couplings between $S$ and the triplet fermions. The tensor
products are written as

$\displaystyle[T\otimes(\bar{\Psi}^{1}\otimes\Psi^{2})_{\mathbf{2}_{3}}]_{\mathbf{1}_{0}}=$
$\displaystyle M^{\prime}_{ijk}T_{i}\bar{\Psi}_{j}^{1}\Psi_{k}^{2}$
$\displaystyle[R\otimes(\bar{\Psi}^{1}\otimes\Psi^{3})_{\mathbf{2}_{1}}]_{\mathbf{1}_{0}}=$
$\displaystyle N^{\prime}_{ijk}R_{i}\bar{\Psi}_{j}^{1}\Psi_{k}^{3},$

where

$\displaystyle M^{\prime}_{1jk}$ $\displaystyle=\begin{pmatrix}0&0&1\\\
\omega&0&0\\\ 0&\omega^{2}&0\end{pmatrix},~{}~{}$ $\displaystyle
M^{\prime}_{2jk}$ $\displaystyle=\begin{pmatrix}0&1&0\\\ 0&0&\omega^{2}\\\
\omega&0&0\end{pmatrix},$ $\displaystyle N^{\prime}_{1jk}$
$\displaystyle=\begin{pmatrix}1&0&0\\\ 0&\omega&0\\\
0&0&\omega^{2}\end{pmatrix},~{}~{}$ $\displaystyle N^{\prime}_{2jk}$
$\displaystyle=\begin{pmatrix}1&0&0\\\ 0&\omega^{2}&0\\\
0&0&\omega\end{pmatrix}.$

We note that the above CG coefficients are the same as those of $\Delta(27)$:
$M^{\prime}_{1}=M_{4},M^{\prime}_{2}=M_{8},N^{\prime}_{1}=M_{1}$ and
$N^{\prime}_{2}=M_{2}$ in App. B.1. The scalar potential is given by

$\displaystyle\mathcal{V}=\,$ $\displaystyle
m_{S}^{2}S^{2}+m_{T}^{2}(T^{\dagger}\otimes
T)_{\mathbf{1}_{0}}+m_{R}^{2}(R^{\dagger}\otimes
R)_{\mathbf{1}_{0}}+\kappa_{1}S(T^{\dagger}\otimes
T)_{\mathbf{1}_{1}}+\kappa_{2}S(R^{\dagger}\otimes R)_{\mathbf{1}_{1}}$
$\displaystyle+\frac{\lambda_{1}}{4}(T^{\dagger}\otimes
T)_{\mathbf{1}_{0}}^{2}+\frac{\lambda_{2}}{4}(T^{\dagger}\otimes
T)_{\mathbf{1}_{1}}^{2}+\frac{\lambda_{3}}{4}(R^{\dagger}\otimes
R)_{\mathbf{1}_{0}}^{2}+\frac{\lambda_{4}}{4}(R^{\dagger}\otimes
R)_{\mathbf{1}_{1}}^{2}$ $\displaystyle+\lambda_{5}(T^{\dagger}\otimes
T)_{\mathbf{1}_{0}}(R^{\dagger}\otimes
R)_{\mathbf{1}_{0}}+\lambda_{6}(T^{\dagger}\otimes
T)_{\mathbf{1}_{1}}(R^{\dagger}\otimes R)_{\mathbf{1}_{1}}$
$\displaystyle+\lambda_{7}S^{2}(T^{\dagger}\otimes
T)_{\mathbf{1}_{0}}+\lambda_{8}S^{2}(R^{\dagger}\otimes
R)_{\mathbf{1}_{0}}+\lambda_{9}S^{4}+(h.c.),$ (4.4)

where $\kappa_{i}$ and $\lambda_{j}$ are complex parameters. The tensor
products are given by

$\displaystyle(T^{\dagger}\otimes T)_{\mathbf{1}_{0}}$
$\displaystyle=|T_{1}|^{2}+|T_{2}|^{2},$ $\displaystyle(R^{\dagger}\otimes
R)_{\mathbf{1}_{0}}$ $\displaystyle=|R_{1}|^{2}+|R_{2}|^{2},$
$\displaystyle(T^{\dagger}\otimes T)_{\mathbf{1}_{1}}$
$\displaystyle=|T_{1}|^{2}-|T_{2}|^{2},$ $\displaystyle(R^{\dagger}\otimes
R)_{\mathbf{1}_{1}}$ $\displaystyle=|R_{1}|^{2}-|R_{2}|^{2}.$

All the fields in the model are properly transformed to their complex
conjugate fields under the CP-like transformation (See Eq. (4.1)), and hence
this is regarded as a physical CP transformation 131313Since $S$ is real, $S$
transforms to $S$ itself under the CP-like transformation.. By imposing the
CP-like invariance in this model we obtain the following constrains on the
couplings as

$\displaystyle\kappa_{1}=$ $\displaystyle 0,$ $\displaystyle\lambda_{6}=$
$\displaystyle 0,$ $\displaystyle t_{i}=$ $\displaystyle|t_{i}|\omega,$
$\displaystyle r_{1,2},\kappa_{2},\lambda_{i}(i\neq 6)\in\mathbb{R},$

where we notice that $(T^{\dagger}\otimes T)_{\mathbf{1}_{1}}$ is CP(-like)
odd and $(R^{\dagger}\otimes R)_{\mathbf{1}_{1}}$ is CP(-like) even. It should
be noted here that this model has complex couplings, while it is invariant
under the physical CP transformation by construction. In fact, for example,
the CP asymmetry of the partial decay width of $T\to\Psi^{1}\bar{\Psi}^{2}$ at
one-loop level is given by the interference of diagrams written in Fig. 2. It
is given as

$\displaystyle\epsilon_{T\to\Psi^{1}\bar{\Psi}^{2}}\propto$
$\displaystyle\sum_{\begin{subarray}{c}a,b=1,2\end{subarray}}\mathrm{Im}\left(t_{1}^{*}r_{1}t_{2}r_{2}^{*}\mathrm{tr}M^{\prime\dagger}_{a}N^{\prime}_{b}M^{\prime}_{a}N^{\prime\dagger}_{b}\right)$
$\displaystyle=$
$\displaystyle\mathrm{Im}\left(-6t_{1}^{*}r_{1}t_{2}r_{2}^{*}\right)=0.$

In the last line, we use the fact that $r_{1,2}$ are real and the phases of
$t_{1,2}$ are the same.

$T$$\Psi^{2}$$\Psi^{1}$

$T$$\Psi^{2}$$\Psi^{1}$$\Psi^{4}$$\Psi^{3}$$R_{1,2}$

Figure 2: Decay diagram of $T\to\Psi^{1}\bar{\Psi}^{2}$.

####  4.1.1 Spontaneous CP Violation: $\Delta(54)\times
U(1)\to\Delta(27)\times U(1)$

We consider a spontaneous symmetry breaking of $\Delta(54)\times
U(1)\to\Delta(27)\times U(1)$. Since $\Delta(27)$ is inconsistent with a
proper CP transformation, we expect that the physical CP transformation for
$\Delta(54)$ turns into CP-like transformation in the broken phase. The
irreducible representations of $\Delta(54)$ are decomposed to those of
$\Delta(27)$ as follows,

$\displaystyle\mathbf{2}_{1}$
$\displaystyle=\mathbf{1}_{2}\oplus\mathbf{1}_{1},$
$\displaystyle\mathbf{1}_{i}$ $\displaystyle=\mathbf{1}_{0},$
$\displaystyle\mathbf{2}_{2}$
$\displaystyle=\mathbf{1}_{5}\oplus\mathbf{1}_{7},$
$\displaystyle\mathbf{3}_{i}$ $\displaystyle=\mathbf{3},$
$\displaystyle\mathbf{2}_{3}$
$\displaystyle=\mathbf{1}_{8}\oplus\mathbf{1}_{4},$
$\displaystyle\mathbf{3}^{*}_{i}$ $\displaystyle=\mathbf{3}^{*},$
$\displaystyle\mathbf{2}_{4}$
$\displaystyle=\mathbf{1}_{6}\oplus\mathbf{1}_{3},$

where the left-hand side corresponds to the irreducible representations of
$\Delta(54)$, and the right-hand side $\Delta(27)$. Thus $\Delta(54)\times
U(1)$ is broken down to $\Delta(27)\times U(1)$ by a nonzero VEV for the non-
trivial singlet $S$,

$\displaystyle\braket{S}$ $\displaystyle=v\in\mathbb{R}.$ (4.5)

This vacuum does not violate the original CP symmetry, since
$\braket{S}\in\mathbb{R}$ is invariant under the CP transformation defined in
Eq. (4.1). This vacuum is stable if $m_{S}^{2}$ is negative and $\lambda_{9}$
is positive. The doublets of $T,R$ are decomposed into the singlets of
$\Delta(27)$. The degenerate masses of $R_{1}$ and $R_{2}$ split by
$\braket{S}$. On the other hand, the masses of $T_{1}$ and $T_{2}$ remain
degenerate even after SSB, because they are related by the CP-like
transformation. We summarize the representations and mass eigenstates in Tab.
6.

| $\Psi^{I}$ | $S$ | $T_{1}$ | $T_{2}$ | $R_{1}$ | $R_{2}$
---|---|---|---|---|---|---
$\Delta(27)$ | $\mathbf{3}$ | $\mathbf{1}_{0}$ | $\mathbf{1}_{8}$ | $\mathbf{1}_{4}$ | $\mathbf{1}_{2}$ | $\mathbf{1}_{1}$
$U(1)$ | $q^{I}$ | $0$ | $-1$ | $-1$ | $-2$ | $-2$
$m$ | $m_{\Psi}^{I}$ | $\sqrt{-\lambda_{9}m_{S}^{2}}$ | $m_{T}$ | $m_{T}$ | $\sqrt{m_{R}^{2}+\kappa_{2}v}$ | $\sqrt{m_{R}^{2}-\kappa_{2}v}$
Table 6: Irreducible representations and $U(1)$ charges of the fields, and
mass eigenvalues.

This model is inconsistent with the physical CP transformation because there
are more than three fields in different singlet and triplet representations.
Thus physical CP transformation turns into a CP-like transformation in the
broken phase while the vacuum is invariant under the original CP
transformation. The CP-like transformation for $\Delta(27)\times U(1)$ model
are exactly the same as those in Eqs. (3.1), (3.3), namely

$\displaystyle\mathbf{3}$ $\displaystyle\to U_{s}\mathbf{3}^{*},$
$\displaystyle\mathbf{1}_{0,1,2}\to$ $\displaystyle\mathbf{1}_{0,1,2}^{*},$
$\displaystyle\mathbf{1}_{4}\to$ $\displaystyle\mathbf{1}_{8}^{*},$

The field contents in the broken phase are similar to those of the model
studied in Sec. 3.2.2 (up to $U(1)$ charges). The CP asymmetry of the partial
decay width of the complex scalar fields in the broken phase is similarly
calculable. For instance, the CP asymmetry of the
$T_{1}\to\Psi^{1}\bar{\Psi}^{2}$ is given by

$\displaystyle\epsilon_{T_{1}\to\Psi^{1}\bar{\Psi}^{2}}=$
$\displaystyle\sum_{a=1,2}|t_{1}||t_{2}|r_{1}r_{2}\mathrm{Im}\left(\mathrm{tr}M_{1}^{\prime\dagger}N^{\prime}_{a}M^{\prime}_{1}N_{a}^{\prime\dagger}\right)\mathrm{Im}\,I_{a}^{34}$
$\displaystyle=$
$\displaystyle\frac{3}{2}|t_{1}||t_{2}|r_{1}r_{2}\left(-\mathrm{Im}\,I_{1}^{34}+\mathrm{Im}\,I_{2}^{34}\right)$

where $I_{a}^{34}\ (a=1,2)$ corresponds to the one-loop integral with $R_{a}$
and $\Psi^{3,4}$ (See Fig. 2). Since the masses of $R_{1}$ and $R_{2}$ are not
degenerate in the broken phase, and hence,
$-\mathrm{Im}\,I_{1}^{34}+\mathrm{Im}\,I_{2}^{34}\neq 0$, we observe the
physical CP violation $\epsilon_{T_{1}\to\Psi^{1}\bar{\Psi}^{2}}\neq 0$. On
the other hand, $\epsilon_{T_{2}\to\Psi^{1}\bar{\Psi}^{2}}$ is given by

$\displaystyle\epsilon_{T_{2}\to\Psi^{1}\bar{\Psi}^{2}}=$
$\displaystyle\sum_{a=1,2}|t_{1}||t_{2}|r_{1}r_{2}\mathrm{Im}\left(\mathrm{tr}M_{2}^{\prime\dagger}N^{\prime}_{a}M^{\prime}_{2}N_{a}^{\prime\dagger}\right)\mathrm{Im}\,I_{a}^{34}$
$\displaystyle=$
$\displaystyle\frac{3}{2}|t_{1}||t_{2}|r_{1}r_{2}\left(\mathrm{Im}\,I_{1}^{34}-\mathrm{Im}\,I_{2}^{34}\right)\neq
0.$

Thus we obtain

$\displaystyle\epsilon_{T_{1}\to\Psi^{1}\bar{\Psi}^{2}}+\epsilon_{T_{2}\to\Psi^{1}\bar{\Psi}^{2}}=0.$

The magnitudes of both CP asymmetries for $T_{1}$ and $T_{2}$ are the same,
while their signs are opposite due to the CP-like symmetry, as shown
previously in Sec. 3.2.2.

Here we should comment on the role of the VEV in the physical CP asymmetry. As
shown above, the CP asymmetry $\epsilon_{T_{i}\to\Psi^{1}\bar{\Psi}^{2}}$ is
proportional to the mass difference of $R_{1}$ and $R_{2}$ in the one-loop
integrals, thus $\epsilon_{T_{i}\to\Psi^{1}\bar{\Psi}^{2}}$ should be
proportional to $\braket{S}$. This result might imply that the physical CP is
violated by the VEV. In other words all the physical CP asymmetries would
vanish when $\braket{S}\to 0$. We note, however, that this is not always the
case, since as stated above the vacuum does not violate the original CP
symmetry 141414A VEV dependent CP violation was first studied in a CP-like
$T_{7}$ model in [71], where the decay asymmetry for a particle in CP-like
eigenstate (class B) was calculated.. In fact, if physical CP transformation
for $\Delta(27)\times U(1)$ is not given by a product of the CP-like
transformation and a broken element, physical CP is violated even in the
symmetric phase, and hence there is no reason to make the CP asymmetry
proportional to $\braket{S}$.

To show this point clearly, we also calculate the CP asymmetry of $2\to 2$
scattering amplitudes of four complex scalars as depicted in Fig. 3
151515There are other one-loop diagrams contributing to the 4-point
scattering, but the other diagrams does not contribute to the CP asymmetry
because there are no complex couplings without Yukawa couplings in this model.
Therefore the CP asymmetry should be obtained from the diagrams including
fermions at one-loop level..

$T_{b}$$\bar{R}_{c}$$\bar{R}_{a}$$T_{d}$

$T_{b}$$\bar{R}_{c}$$\bar{R}_{a}$$T_{d}$$\psi^{4}$$\psi^{3}$$\psi^{1}$$\psi^{2}$

$T_{b}$$\bar{R}_{c}$$\bar{R}_{a}$$T_{d}$$\psi^{1}$$\psi^{2}$$\psi^{4}$$\psi^{3}$

Figure 3: 4-point scattering of $T_{b}\bar{R}_{c}\to T_{d}\bar{R}_{a}$.

The tree level and one-loop diagrams are calculated as

$\displaystyle\mathcal{M}^{\textrm{tree}}_{b\bar{c}\to d\bar{a}}=$
$\displaystyle-i\lambda_{5}\delta_{ac}\delta_{bd}$
$\displaystyle\mathcal{M}^{\textrm{1-loop}}_{b\bar{c}\to d\bar{a}}=$
$\displaystyle(-i)^{4}(r_{2}t_{2}r_{1}^{*}t_{1}^{*}\textrm{tr}N_{a}^{\prime}M_{b}^{\prime}N_{c}^{\prime\dagger}M_{d}^{\prime\dagger}+r_{1}t_{1}r_{2}^{*}t_{2}^{*}\textrm{tr}N_{a}^{\prime}M_{b}^{\prime}N_{c}^{\prime\dagger}M_{d}^{\prime\dagger})I_{\textrm{1-loop}}$
$\displaystyle=$
$\displaystyle(-i)^{4}2r_{2}|t_{2}|r_{1}|t_{1}|\textrm{tr}N_{a}^{\prime}M_{b}^{\prime}N_{c}^{\prime\dagger}M_{d}^{\prime\dagger}I_{\textrm{1-loop}}$
$\displaystyle=$
$\displaystyle\begin{cases}6r_{1}r_{2}|t_{1}t_{2}|\omega^{2}\delta_{ac}\delta_{bd}I_{\textrm{1-loop}}&(a=b)\\\
6r_{1}r_{2}|t_{1}t_{2}|\omega\delta_{ac}\delta_{bd}I_{\textrm{1-loop}}&(a\neq
b)\end{cases},$

where we use that $r_{1},r_{2}$ are real, and the phases of $t_{1}$ and
$t_{2}$ are the same. As a result, the CP asymmetry at one-loop level is given
by

$\displaystyle\epsilon_{T_{b}\bar{R}_{c}\to
T_{d}\bar{R}_{a}}=\begin{cases}3\sqrt{3}\lambda_{5}r_{1}r_{2}|t_{1}t_{2}|\delta_{ac}\delta_{bd}\textrm{Im}\,I_{\textrm{1-loop}}&(a=b)\\\
-3\sqrt{3}\lambda_{5}r_{1}r_{2}|t_{1}t_{2}|\delta_{ac}\delta_{bd}\textrm{Im}\,I_{\textrm{1-loop}}&(a\neq
b)\end{cases}.$ (4.6)

Thus the CP violation is observed in the $2\to 2$ scattering processes. We
should emphasize that the CP asymmetry $\epsilon_{T_{b}\bar{R}_{c}\to
T_{d}\bar{R}_{a}}$ does not depend on the mass splitting of the doublets, nor
scalar interaction parameters that include $\braket{S}$. Thus
$\epsilon_{T_{b}\bar{R}_{c}\to T_{d}\bar{R}_{a}}$ is not proportional to
$\braket{S}$ at one-loop level. The spontaneous CP violation is caused by the
splitting of multiplicity rather than VEV of scalar field. Actually this
specific CP asymmetry is non-zero even in the symmetric phase. Nevertheless,
this process does not violate the CP symmetry in the symmetric phase. This is
because $T_{1,2}$ and $R_{1,2}$ are in the same multiplet in $\Delta(54)$, and
hence we can sum over the flavor indices $a,b$ of $T_{a}$ and $\bar{R}_{b}$ in
external lines in the scattering amplitudes. Thus the cross section should be
given by

$\displaystyle\left(\frac{d\sigma}{d\Omega}\right)_{T\bar{R}\to
T\bar{R}}\propto\sum_{a,b,c,d}|\mathcal{M}_{T_{b}\bar{R}_{c}\to
T_{d}\bar{R}_{a}}|^{2}.$

This value is CP symmetric because

$\displaystyle\left(\frac{d\sigma}{d\Omega}\right)_{T\bar{R}\to
T\bar{R}}-\left(\frac{d\bar{\sigma}}{d\Omega}\right)_{\bar{T}{R}\to\bar{T}{R}}$
$\displaystyle\propto\sum_{a,b,c,d}\left(|\mathcal{M}_{T_{b}\bar{R}_{c}\to
T_{d}\bar{R}_{a}}|^{2}-|\mathcal{M}_{\bar{T}_{b}{R}_{c}\to\bar{T}_{d}{R}_{a}}|^{2}\right)$
$\displaystyle=\epsilon_{T_{1}\bar{R}_{1}\to
T_{1}\bar{R}_{1}}+\epsilon_{T_{1}\bar{R}_{2}\to
T_{1}\bar{R}_{2}}+\epsilon_{T_{2}\bar{R}_{1}\to
T_{2}\bar{R}_{1}}+\epsilon_{T_{2}\bar{R}_{2}\to T_{2}\bar{R}_{2}}$
$\displaystyle=2\left(\epsilon_{T_{1}\bar{R}_{1}\to
T_{1}\bar{R}_{1}}+\epsilon_{T_{2}\bar{R}_{1}\to
T_{2}\bar{R}_{1}}\right)\quad\quad(\because\Delta(54))$ $\displaystyle=0,$

where $\epsilon_{T_{1}\bar{R}_{1}\to T_{1}\bar{R}_{1}}$ and
$\epsilon_{T_{2}\bar{R}_{1}\to T_{2}\bar{R}_{1}}$ cancel out each other (See
Eq. (4.6)). This cancellation is also consistent with the fact that there
exists a general CP transformation associated with a unitary matrix $S_{2}$ in
Eq. (4.2). As shown here, the summation over the internal indices is important
to prove the existence of the CP violation (see Appendix D for a general
proof).

This result is also understood from a calculation of the matrix elements in a
diagonal basis for the CP-like transformations. We take the following CP basis

$\displaystyle\tilde{T}_{\pm}$ $\displaystyle=\frac{1}{\sqrt{2}}(T_{1}\pm
T_{2}),\quad\tilde{R}_{1,2}=R_{1,2}$

on which the CP-like transformation in Eq. (4.1) diagonally acts as

$\displaystyle\textrm{CP-
like}:\begin{cases}\tilde{T}_{\pm}\to\tilde{T}_{\pm}^{\textrm{CP-
like}}=\pm\tilde{T}^{*}_{\pm},\\\ R_{1,2}\to R_{1,2}^{\textrm{CP-
like}}=R^{*}_{1,2}.\end{cases}$ (4.7)

From the properties in Eq. (4.7) we see that the CP-like symmetry make a
connection between two amplitudes as

$\displaystyle\mathcal{M}_{\tilde{T}_{x}\bar{R}_{c}\to\tilde{T}_{y}\bar{R}_{a}}=\mathcal{M}_{\tilde{T}_{x}^{\textrm{CP-
like}}\bar{R}_{c}^{\textrm{CP-like}}\to\tilde{T}_{y}^{\textrm{CP-
like}}\bar{R}_{a}^{\textrm{CP-
like}}}=xy\mathcal{M}_{\bar{\tilde{T}}_{x}R_{c}\to\bar{\tilde{T}}_{y}R_{a}}$
(4.8)

where $x,y=\pm$. Thus the matrix element in this CP-basis is invariant under
the CP-like transformation. In the broken phase, however, we have to
distinguish two fields $T_{1}$ and $T_{2}$ due to a multiplet splitting.
Therefore we specify either $T_{1}$ or $T_{2}$ as the initial and final
states. For example, a scattering amplitude of $T_{1}\bar{R}_{a}\to
T_{1}\bar{R}_{c}$ can be obtained as a sum of the CP invariant matrix elements
as

$\displaystyle\mathcal{M}_{T_{1}\bar{R}_{a}\to T_{1}\bar{R}_{c}}$
$\displaystyle=\frac{1}{2}\left(\mathcal{M}_{\tilde{T}_{+}\bar{R}_{a}\to\tilde{T}_{+}\bar{R}_{c}}+\mathcal{M}_{\tilde{T}_{+}\bar{R}_{a}\to\tilde{T}_{-}\bar{R}_{c}}+\mathcal{M}_{\tilde{T}_{-}\bar{R}_{a}\to\tilde{T}_{+}\bar{R}_{c}}+\mathcal{M}_{\tilde{T}_{-}\bar{R}_{a}\to\tilde{T}_{-}\bar{R}_{c}}\right).$

We note that while each term is CP invariant, there are interferences between
them, and hence it is not guaranteed that $\mathcal{M}_{T_{1}\bar{R}_{a}\to
T_{1}\bar{R}_{c}}$ is also CP invariant. In fact, from Eq. (4.8) we easily see
that there is a relationship between two different matrix elements for
different states as

$\displaystyle\mathcal{M}_{T_{1}\bar{R}_{a}\to T_{1}\bar{R}_{c}}=$
$\displaystyle\mathcal{M}_{\bar{T}_{2}R_{a}\to\bar{T}_{2}R_{c}},$

but there is no relation between $\mathcal{M}_{T_{1}\bar{R}_{a}\to
T_{1}\bar{R}_{c}}$ and $\mathcal{M}_{\bar{T}_{1}R_{a}\to\bar{T}_{1}R_{c}}$.
This is also consistent with the general result obtained in Eq. (3.12).

As shown above, we explicitly confirm that when the vacuum does not violate
the original CP(-like) symmetry, it remains after the SSB. Nevertheless the
spontaneous CP violation occurs due to multiplet splitting. To be specific,
$S_{2}$ can be regarded as a physical CP transformation matrix at the level of
$\Delta(54)$, however it becomes to a CP-like transformation after the SSB,
since $S_{2}$ exchanges $T_{1}$ and $T_{2}$ in different irreducible
representations at the level of $\Delta(27)$. Here we note that a physical CP
transformation in the broken phase is an inconsistent CP transformation, since
it should be defined as

$\displaystyle\mathbf{3}_{i}$ $\displaystyle\to U\mathbf{3}_{i}^{*},$
$\displaystyle\mathbf{2}_{1}$ $\displaystyle\to\mathbf{2}_{1}^{*},$
$\displaystyle\mathbf{2}_{2}$ $\displaystyle\to\mathbf{2}_{4}^{*},$
$\displaystyle\mathbf{2}_{3}$ $\displaystyle\to\mathbf{2}_{3}^{*},$
$\displaystyle\mathbf{2}_{4}$ $\displaystyle\to\mathbf{2}_{2}^{*},$
$\displaystyle\mathbf{1}_{0,1}\to\mathbf{1}_{0,1}^{*}.$ (4.9)

Obviously this transformation is inconsistent with any automorphisms of
$\Delta(54)$ and $\Delta(27)$, and it is not given by the product of original
CP transformation and a group element. We also note that this model is not
invariant under the physical CP transformation (Eq. (4.9)) from the beginning.
Therefore for some scattering amplitudes we could observe a “CP violation” in
both symmetric and broken phases, e.g. $\epsilon_{T_{b}\bar{R}_{c}\to
T_{d}\bar{R}_{a}}$, where we recall that the complex conjugate states are
defined by using the physical (but inconsistent) CP transformation defined in
Eq. (4.9). In fact we explicitly confirm that this CP asymmetry is not
proportional to the VEV at leading order. Remarkably enough, in the broken
phase this CP asymmetry proves evidence of the physical CP violation, while in
the symmetric phase this does not mean the physical CP violation due to the
existence of a general CP transformation. It should be noted that the results
presented here do not depend on basis for the transformation matrix $(S_{2})$
as explicitly shown in the calculations with a different (CP-like) basis.

In the next section we will further investigate the spontaneous breaking of
the internal symmetry group of $\Delta(27)\times U(1)$ in light of the
relationship between the CP-like and physical CP transformations. We will see
that when $\Delta(27)\times U(1)$ symmetry is broken down further by the
scalar fields VEV of $T,R$, the role of the original CP transformation can
change depending on the symmetry breaking pattern. We study the following
three CP-like invariant vacua: $\braket{R_{i}}\in\mathbb{R}$,
$\braket{T_{1}}=\braket{T_{2}}^{*}$, and their combination. The first one
breaks $U(1)$ and the internal symmetry is broken down to $\Delta(27)$. Since
there is no CIA in $\Delta(27)$, it is still CP-like symmetric, and the
physical CP symmetry remains violated as well. On the other hand, in other two
cases we will see some interesting phenomena will occur such as the
spontaneous CP restorations due to a multiplet merging.

####  4.1.2 Spontaneous CP restoration: $\Delta(27)\times
U(1)\to\mathbb{Z}_{3}^{a^{2}a^{\prime}}\times\mathbb{Z}^{a^{2}b}_{3}$

We consider the VEVs of the scalar fields

$\displaystyle\braket{T_{1}}=\braket{T_{2}}^{*}=\frac{v_{T}}{\sqrt{2}}e^{i\theta},$

and $\braket{R_{i}}=0$. This vacuum does not violate the CP-like symmetry.
$v_{T}$ is given by

$\displaystyle v_{T}=\pm\sqrt{\frac{-2m_{T}^{2}}{\lambda_{1}}},$

where $m_{T}$ and $\lambda_{1}$ is the real parameter of the scalar potential
Eq. (4.4). This vacuum is stable if $\lambda_{2}>0$ and
$m_{R}^{2}-\frac{4\lambda_{5}}{\lambda_{1}}m_{T}^{2}\pm
2\kappa_{2}\braket{S}>0$. The vacuum is invariant under $a^{2}a^{\prime}$ and
$a^{2}b$, which are $\mathbb{Z}_{3}$ generators of $\Delta(27)$, and the
$U(1)$ is completely broken down. Thus symmetry is broken to
$\mathbb{Z}_{3}^{a^{2}a^{\prime}}\times\mathbb{Z}_{3}^{a^{2}b}$. The
irreducible decompositions are given as

$\displaystyle\mathbf{3}_{1}=$
$\displaystyle\mathbf{1}_{20}\oplus\mathbf{1}_{21}\oplus\mathbf{1}_{22},$
$\displaystyle\mathbf{1}_{4,8}=$ $\displaystyle\mathbf{1}_{00},$
$\displaystyle\mathbf{1}_{1}=$ $\displaystyle\mathbf{1}_{01},$
$\displaystyle\mathbf{1}_{2}=$ $\displaystyle\mathbf{1}_{02},$

where the left-hand side corresponds to the irreducible representations of
$\Delta(27)$, and the right-hand side
$\mathbb{Z}_{3}^{a^{2}a^{\prime}}\times\mathbb{Z}_{3}^{a^{2}b}$. The lower
indices of $\mathbf{1}_{ij}$ denote the charges of
$\mathbb{Z}_{3}^{a^{2}a^{\prime}}\times\mathbb{Z}_{3}^{a^{2}b}$,

$\displaystyle a^{2}a^{\prime}:$
$\displaystyle\mathbf{1}_{ij}\to\omega^{i}\mathbf{1}_{ij},$ $\displaystyle
a^{2}b:$ $\displaystyle\mathbf{1}_{ij}\to\omega^{j}\mathbf{1}_{ij}.$

The representations are summarized in Tab. 7. Since the vacuum is invariant
under the CP-like transformation at $\Delta(27)$, it induces an automorphism
of $\mathbb{Z}_{3}\times\mathbb{Z}_{3}$,

$\displaystyle
u^{\mathbb{Z}_{3}^{a^{2}a^{\prime}}\times\mathbb{Z}_{3}^{a^{2}b}}:(a^{2}a^{\prime},a^{2}b)\mapsto(aa^{\prime
2},a^{\prime}b^{2})=((a^{2}a^{\prime})^{-1},(a^{2}b)^{-1}).$

Thus all group elements in
$\mathbb{Z}_{3}^{a^{2}a^{\prime}}\times\mathbb{Z}_{3}^{a^{2}b}$ are
transformed to their own inverses, this is the proper CP transformation (CIA)
at the level of
$\mathbb{Z}_{3}^{a^{2}a^{\prime}}\times\mathbb{Z}_{3}^{a^{2}b}$. This result
is also consistent with the fact that the two exchanged fields $T_{1}$ and
$T_{2}$ under the CP-like transformation belong to the same representation
$\mathbf{1}_{00}$ as shown in Tab. 7. Due to the multiplet merging, the CP-
like transformation given by $S_{2}$ spontaneously changes to the proper CP
transformation.

| $\Psi^{i}$ | $S,T_{1},T_{2}$ | $R_{1}$ | $R_{2}$
---|---|---|---|---
$\mathbb{Z}_{3}^{a^{2}a^{\prime}}\times\mathbb{Z}_{3}^{a^{2}b}$ | $\mathbf{1}_{20}\oplus\mathbf{1}_{21}\oplus\mathbf{1}_{22}$ | $\mathbf{1}_{00}$ | $\mathbf{1}_{01}$ | $\mathbf{1}_{02}$
Table 7: The representations of the matter fields.

####  4.1.3 CP-like symmetry for group with CIA: $\Delta(27)\times
U(1)\to\mathbb{Z}_{3}^{a^{2}a^{\prime}}\times\mathbb{Z}^{a^{\prime}}_{3}$

If $\braket{R_{1}}$ and $\braket{R_{2}}$ develop non-zero real VEVs, the
unbroken symmetry can change. We consider the following VEVs

$\displaystyle\braket{R_{1}}=v_{1},\ \ \braket{R_{2}}=v_{2},\ \ \braket{T}=0,\
\ v_{i}\in\mathbb{R}.$

This vacuum is realized if $m_{R}^{2}<0$, $\lambda_{3}>0$, and
$v_{1}^{2}-v_{2}^{2}=-\frac{2\kappa_{2}}{\lambda_{4}}\braket{S}$. This vacuum
has two flat directions, but is free from tachyon. The symmetry group of
$\Delta(27)\times U(1)$ is broken to
$\mathbb{Z}_{3}^{a^{2}a^{\prime}}\times\mathbb{Z}_{3}^{a^{\prime}}$, The
irreducible decompositions are given as

$\displaystyle\mathbf{3}=$
$\displaystyle\mathbf{1}_{20}\oplus\mathbf{1}_{21}\oplus\mathbf{1}_{22},$
$\displaystyle\mathbf{1}_{4}=$ $\displaystyle\mathbf{1}_{02},$
$\displaystyle\mathbf{1}_{8}=$ $\displaystyle\mathbf{1}_{01},$
$\displaystyle\mathbf{1}_{1}=$ $\displaystyle\mathbf{1}_{00},$
$\displaystyle\mathbf{1}_{2}=$ $\displaystyle\mathbf{1}_{00},$

where the left-hand side corresponds to the irreducible representations of
$\Delta(27)$, and the right-hand side
$\mathbb{Z}_{3}^{a^{2}a^{\prime}}\times\mathbb{Z}_{3}^{a^{\prime}}$. The lower
index of $\mathbf{1}_{ij}$ denotes the charges of
$\mathbb{Z}_{3}^{a^{2}a^{\prime}}\times\mathbb{Z}_{3}^{a^{\prime}}$,

$\displaystyle a^{2}a^{\prime}:$
$\displaystyle\mathbf{1}_{ij}\to\omega^{i}\mathbf{1}_{ij},$ $\displaystyle
a^{\prime}:$ $\displaystyle\mathbf{1}_{ij}\to\omega^{j}\mathbf{1}_{ij}.$

From Eq. (4.3), the automorphism of
$\mathbb{Z}_{3}^{a^{2}a^{\prime}}\times\mathbb{Z}_{3}^{a^{\prime}}$ is given
as

$\displaystyle
u^{\mathbb{Z}_{3}^{a^{2}a^{\prime}}\times\mathbb{Z}_{3}^{a^{\prime}}}:(a^{2}a^{\prime},a^{\prime})\mapsto(aa^{\prime
2},a^{\prime}).$

Thus it is CP-like transformation rather than physical (proper) CP, although
$\mathbb{Z}_{3}^{a^{2}a^{\prime}}\times\mathbb{Z}_{3}^{a^{\prime}}$ has a CIA.

If $\braket{R_{i}}$ and $\braket{T_{i}}$ develops VEVs at the same time, the
internal symmetry is broken further. Suppose the vacuum is given by

$\displaystyle\braket{R_{1}}$ $\displaystyle\in\mathbb{R},$
$\displaystyle\braket{R_{2}}$ $\displaystyle=0,$ $\displaystyle\braket{T_{1}}$
$\displaystyle=\braket{T_{2}}^{*}\neq 0.$

This vacuum does not violate the CP-like symmetry, but break $\Delta(27)\times
U(1)$ to $\mathbb{Z}_{3}^{a^{2}a^{\prime}}$, which is the center of
$\Delta(27)$. The induced automorphism is $a^{2}a^{\prime}$ to $aa^{\prime
2}=(a^{2}a^{\prime})^{-1}$. Hence it becomes to a physical (proper) CP
transformation.

####  4.1.4 Other breaking patterns of $\Delta(27)\times U(1)$

If there is an additional triplet scalar $U$ and it develops the VEV given by

$\displaystyle\braket{U}\propto(\omega,0,0)^{T},$

$\Delta(27)$ is broken to $\mathbb{Z}_{3}$ generated by $a^{\prime}$. This
vacuum is CP(-like) invariant. Since $a^{\prime}$ is fixed under $u_{CP}$, CP-
like symmetry is still CP-like at the level of $\mathbb{Z}_{3}^{a^{\prime}}$,
and physical CP is violated. Thus we can realize $\mathbb{Z}_{3}$ symmetric
model with CP-like symmetry.

If the internal symmetry is completely broken down the CP-like transformation
must turn into a physical CP transformation due to a multiplet merging, since
all the fields should belong to the same representation of the trivial
singlet.

We summarize the symmetry breaking patterns of $\Delta(54)\times U(1)$ model
obtained aforementioned processes in Fig. 4. We find that various non-standard
CP violation/restoration scenarios appear. It is shown that both the proper CP
and CP-like transformations can be mutually converted due to the mechanisms of
the multiplet splitting and merging, which depends on which subgroup remains
after the SSB. We will study its cosmological and phenomenological
implications [78].

Figure 4: The relationship between symmetry breaking of the internal symmetry
and the resultant property of the CP transformation, We only consider
CP(-like) invariant vacuum. The gray node represents a CP violated (with CP-
like symmetric) vacuum. The white node represents a physical CP symmetric
vacuum, where the CP transformation induces a CIA on the internal symmetry
group. The broken lined node represents a physical CP (CP-like) symmetric
vacuum. We only show a part of the breaking patterns for $\Delta(54)\times
U(1)$, and there are other possible breaking patterns. We also show the
breaking pattern of $SU(3)$ and $SU(2)\times U(1)$ models which are studied in
the following subsections.

###  4.2 $SU(3)$ Model

Here we show another example in a model with the proper CP symmetry for
continuous symmetry group $SU(3)$. Since the methods for calculations of CP
asymmetry and analyses of the scalar potentials are the same as in the
previous subsection, we only discuss a possible breaking pattern and their
symmetries based on the group structures for simplicity.

We start with a model with symmetry group $G=SU(3)$ and a proper CP
invariance, where the proper CP transformation is defined as
$\mathbf{3}\to\mathbf{3}^{*}$, where $\mathbf{3}$ is the triplet irreducible
representation of $SU(3)$.

####  4.2.1 Spontaneous CP violation : $SU(3)\to\Delta(54)$

If $SU(3)$ is broken down to $\Delta(54)$, the proper CP transformation
spontaneously changes to a CP-like transformation. Since we can identify the
triplet representation $\mathbf{3}$ as $\mathbf{3}_{2}$ in $\Delta(54)$ (see
Appendix A). the induced automorphism of $\Delta(54)$ is determined as

$\displaystyle u^{\Delta(54)}~{}:~{}(a,a^{\prime},b,c)\mapsto(a^{2},a^{\prime
2},b,c).$ (4.10)

From the explicit matrix representations for four doublet fields
$\phi_{\mathbf{2}_{1,2,3,4}}$, we immediately see that the corresponding CP
transformation matrix is given a CP-like form for the multiplet
$\Phi_{\mathbf{2}}=(\phi_{\mathbf{2}_{1}},\phi_{\mathbf{2}_{2}},\phi_{\mathbf{2}_{3}},\phi_{\mathbf{2}_{4}})^{T}$,

$\displaystyle\textrm{CP-like}:\Phi_{\mathbf{2}}\to
U_{\Phi_{\mathbf{2}}}\Phi_{\mathbf{2}}^{*},\quad\quad
U_{\Phi_{\mathbf{2}}}=\begin{pmatrix}\mathbf{1}&\mathbf{0}&\mathbf{0}&\mathbf{0}\\\
\mathbf{0}&\mathbf{0}&\mathbf{1}&\mathbf{0}\\\
\mathbf{0}&\mathbf{1}&\mathbf{0}&\mathbf{0}\\\
\mathbf{0}&\mathbf{0}&\mathbf{0}&\mathbf{1}\\\ \end{pmatrix},$ (4.11)

While the triplets $\mathbf{3}_{1,2}$, and two singlets $\mathbf{1}_{0,1}$
properly transform into their class-inverting representations under the CP
transformation. From the fact that irreducible decompositions of the adjoint
representation $\mathbf{8}$ of $SU(3)$ are given as
$\mathbf{8}=\mathbf{2}_{1}\oplus\mathbf{2}_{2}\oplus\mathbf{2}_{3}\oplus\mathbf{2}_{4}$,
we also see that the original CP transformation exchanges two different
representations of $\phi_{\mathbf{2}_{2}},\phi_{\mathbf{2}_{3}}$ in
$\Delta(54)$. Therefore the CP transformation spontaneously changes a CP-like
transformation due to a multiplet splitting.

####  4.2.2 Spontaneous CP restoration : $\Delta(54)\to
S_{3}\times\mathbb{Z}_{3}$

Let us consider the spontaneous breaking of $\Delta(54)\to
S_{3}\times\mathbb{Z}_{3}$. All group elements of $S_{3}$ and $\mathbb{Z}_{3}$
are given as $\\{e,b,c,b^{2},bc,b^{2}c\\}$, and $\\{e,aa^{\prime
2},a^{2}a^{\prime}\\}$, respectively. There are three conjugacy classes in
$S_{3}$:

$\displaystyle C_{1}:$ $\displaystyle\\{e\\},$ $\displaystyle C_{2}:$
$\displaystyle\\{b,b^{2}\\},$ $\displaystyle C_{3}:$
$\displaystyle\\{c,bc,b^{2}c\\},$

and there are two singlets, and one doublets. We denote all the irreducible
representations of $S_{3}\times\mathbb{Z}_{3}$ as the three doublets
$\mathbf{2}_{i}$ and six singlets $\mathbf{1}_{i}$, $\mathbf{1}^{\prime}_{i}$,
where $i=0,1,2$ represents a $\mathbb{Z}_{3}$ charge. From Eq. (4.10), we see
that the original CP transformation becomes to a CIA of
$S_{3}\times\mathbb{Z}_{3}$. Let us also see the irreducible decompositions;

$\displaystyle\mathbf{1}_{0}$ $\displaystyle=\mathbf{1}_{0},$
$\displaystyle\mathbf{1}_{1}$ $\displaystyle=\mathbf{1}^{\prime}_{0},$
$\displaystyle\mathbf{2}_{1}$ $\displaystyle=\mathbf{2}_{0},$
$\displaystyle\mathbf{2}_{2}$ $\displaystyle=\mathbf{2}_{0},$
$\displaystyle\mathbf{2}_{3}$ $\displaystyle=\mathbf{2}_{0},$
$\displaystyle\mathbf{2}_{4}$
$\displaystyle=\mathbf{1}_{0}\oplus\mathbf{1}_{0}^{\prime},$
$\displaystyle\mathbf{3}_{1i}$
$\displaystyle=\mathbf{1}_{i}\oplus\mathbf{2}_{i},$
$\displaystyle\mathbf{3}_{2i}$
$\displaystyle=\mathbf{1}^{\prime}_{i}\oplus\mathbf{2}_{i},$

where the left-hand side corresponds to the irreducible representations of
$\Delta(54)$, and the right-hand side $S_{3}\times\mathbb{Z}_{3}$. This result
is also consistent with the proper CP transformation. Since two mutually
exchanged fields $\phi_{\mathbf{2}_{2}}$ and $\phi_{\mathbf{2}_{3}}$ by the
CP-like transformation in $\Delta(54)$ eventually belong to the same
representation of $\mathbf{2}_{0}$ in $S_{3}\times\mathbb{Z}_{3}$. Thus the
physical CP symmetry can spontaneously emerge from the CP-like symmetry.

####  4.2.3 Spontaneous CP violation :
$S_{3}\times\mathbb{Z}_{3}\to\mathbb{Z}_{3}$

Next, we consider the breaking of
$S_{3}\times\mathbb{Z}_{3}\to\mathbb{Z}_{3}$. $S_{3}\times\mathbb{Z}_{3}$ has
two different subgroups of $\mathbb{Z}_{3}:\\{e,aa^{\prime
2},a^{2}a^{\prime}\\}$, and $\mathbb{Z}^{\prime}_{3}:\\{e,b,b^{2}\\}$. It is
important to notice that while these two subgroups are identical
($\mathbb{Z}_{3}$), the automorphisms induced by the original CP
transformation are different depending on the generators of each subgroup.

As for $\mathbb{Z}_{3}:\\{e,aa^{\prime 2},a^{2}a^{\prime}\\}$, the
automorphism in Eq. (4.10) correspond to the (class-inverting) outer
automorphism, so that the corresponding CP transformation is the proper CP
transformation.

On the other hand, in the case of $\mathbb{Z}^{\prime}_{3}:\\{e,b,b^{2}\\}$,
the automorphism in Eq. (4.10) correspond to the identity map (non CIA), so
that the corresponding CP transformation is the CP-like transformation. Let us
also see the irreducible decompositions of
$S_{3}\times\mathbb{Z}_{3}\to\mathbb{Z}^{\prime}_{3}$;

$\displaystyle\mathbf{1}_{i}$ $\displaystyle=\mathbf{1}_{0},$
$\displaystyle\mathbf{1}^{\prime}_{i}$ $\displaystyle=\mathbf{1}_{0},$
$\displaystyle\mathbf{2}_{i}$
$\displaystyle=\mathbf{1}_{1}\oplus\mathbf{1}_{2},\ \ \ (i=0,1,2)$

where the left-hand side corresponds to the irreducible representations of
$S_{3}\times\mathbb{Z}_{3}$, and the right-hand side
$\mathbb{Z}^{\prime}_{3}$. Thus two components of a double, which are
connected by the CP transformation, split into two fields of
$\phi_{\mathbf{1}_{1}}$ and $\phi_{\mathbf{1}_{2}}$. Again, the model has CP-
like symmetry. The symmetry breaking patterns of $SU(3)$ and CP are summarized
in Fig. 4.

###  4.3 $SU(2)\times U(1)$

We show yet another simple example in which a CP-like symmetric model with
continuous internal symmetry group can be derived from a proper CP symmetric
model. As an example, we consider $SU(2)\times U(1)$ symmetric model, and
introduce a field $\Phi^{q}=(\psi^{q}_{1},\psi^{q}_{2},\psi^{q}_{3})^{T}$ in
the triplet representation of $SU(2)$, where $q$ is $U(1)$ charge. A group
action on a triplet field $\Phi^{q}$ is represented as

$\displaystyle\Phi\to\Phi^{\prime}=$
$\displaystyle\rho_{\Phi}(\vec{c},\theta)\Phi,\quad\quad\rho_{\Phi}(\vec{c},\theta)=e^{i\vec{c}\cdot\vec{\sigma}}e^{i\theta
q},$ (4.12) $\displaystyle\vec{c}=$
$\displaystyle(c_{1},c_{2},c_{3}),\quad\quad\theta,c_{i}\in\mathbb{R}.$

where $\vec{\sigma}=(\sigma_{1},\sigma_{2},\sigma_{3})$ is the Pauli vector in
three dimensional representation, which is given in terms of the
$\mathfrak{su}(2)$ generators,

$\displaystyle\sigma_{1}$ $\displaystyle=\frac{1}{2}\begin{pmatrix}0&1&0\\\
1&0&1\\\ 0&1&0\\\ \end{pmatrix},$ $\displaystyle\sigma_{2}$
$\displaystyle=\frac{1}{2}\begin{pmatrix}0&-i&0\\\ i&0&-i\\\ 0&i&0\\\
\end{pmatrix},$ $\displaystyle\sigma_{3}$
$\displaystyle=\frac{1}{2}\begin{pmatrix}1&0&0\\\ 0&0&0\\\ 0&0&-1\\\
\end{pmatrix}.$

A proper CP transformation of $\Phi$ is given as

$\displaystyle\Phi(x)\to\Phi^{*}(\tilde{x}).$ (4.13)

The induced automorphism of $SU(2)\times U(1)$ is represented as a direct
product of the two automorphisms as $u_{CP}^{SU(2)\times
U(1)}=u_{CP}^{SU(2)}\otimes u_{CP}^{U(1)}$,

$\displaystyle
u_{CP}^{SU(2)}:\sigma_{1}\mapsto\sigma_{1}^{*}=\sigma_{1},\quad\quad\sigma_{2}\mapsto\sigma_{2}^{*}=-\sigma_{2},\quad\quad\sigma_{3}\mapsto\sigma_{3}^{*}=\sigma_{3}.$
(4.14)

We notice that while $\sigma_{1}$ and $\sigma_{3}$ are invariant, $\sigma_{2}$
is mapped to $-\sigma_{2}$ under the proper CP transformation.

####  4.3.1 CP-like Symmetry: $SU(2)\times U(1)\to U(1)\times U(1)$

The group $SU(2)$ is spontaneously broken to $U(1)$ by a nonzero VEV for a
($U(1)$ neutral) triplet scalar $S=\Phi^{0}$. Let us assume that the VEV is
given as

$\displaystyle\braket{S}=(v,0,v)^{T}\ \ (v\in\mathbb{R}),$ (4.15)

the vacuum is invariant under the CP transformation. We refer to the unbroken
symmetry as $U(1)_{Y}$ whose generator is given as $\sigma_{2}$. From the
consistency condition

$\displaystyle CP^{-1}\circ(e^{ic_{2}\sigma_{2}})^{*}\circ
CP=e^{ic_{2}\sigma_{2}},$ (4.16)

we immediately see that the proper CP transformation at the level of $SU(2)$
becomes to a CP-like transformation at the level of $U(1)_{Y}\in SU(2)$. The
triplet field are decomposed into the eigenstates for the remaining symmetry
of $U(1)_{Y}\times U(1)$,

$\displaystyle\phi_{+}^{q}$
$\displaystyle=-\frac{i}{2}\phi_{1}^{q}+\frac{1}{\sqrt{2}}\phi_{2}^{q}+\frac{i}{2}\phi_{3}^{q},$
$\displaystyle\phi_{-}^{q}$
$\displaystyle=\frac{i}{2}\phi_{1}^{q}+\frac{1}{\sqrt{2}}\phi_{2}^{q}-\frac{i}{2}\phi_{3}^{q},$
$\displaystyle\phi_{0}^{q}$
$\displaystyle=\frac{\phi_{1}^{q}+\phi_{3}^{q}}{\sqrt{2}}.$ (4.17)

The lower index indicates the $U(1)_{Y}$ charge. For a multiplet field of
these eigenstates $\tilde{\Phi}=(\phi_{+}^{q},\phi_{-}^{q},\phi_{0}^{q})^{T}$,
the original proper CP transformation (Eq. (4.13)) can be represented as a CP-
like form,

$\displaystyle\textrm{CP-like}:\tilde{\Phi}\to
U_{\tilde{\Phi}}\tilde{\Phi}^{*},\quad\quad
U_{\tilde{\Phi}}=\begin{pmatrix}0&1&0\\\ 1&0&0\\\ 0&0&1\end{pmatrix},$ (4.18)

which indicates that the induced automorphism of the generators for
$U(1)_{Y}\times U(1)$ is give as

$\displaystyle u^{U(1)_{Y}\oplus U(1)}:(q_{Y},q)\mapsto(q_{Y},-q).$ (4.19)

It should be noted that this automorphism is consistent with the CP-like
transformation defined in Eq. (3.21).

As for the vacuum stability of the model, one can investigate the following
general renormalizable scalar potential,

$\displaystyle\mathcal{V}(S)=$ $\displaystyle\frac{m^{2}}{2}S^{\dagger}\cdot
S+\frac{\mu^{2}}{2}(S\otimes S)_{0}$
$\displaystyle+\frac{\lambda_{1}}{4}\left[(S\otimes
S)_{0}\right]^{2}+\frac{\lambda_{2}}{4}(S^{\dagger}\cdot S)(S\otimes S)_{0}$
$\displaystyle+\frac{\lambda_{3}}{4}(S^{\dagger}\cdot
S)^{2}+\frac{\lambda_{4}}{4}(S^{*}\otimes S^{*})_{0}(S\otimes S)_{0}+(h.c.),$
(4.20)

where $X^{\dagger}\cdot Y$ and $(X\otimes Y)_{0}$ are the two $SU(2)$
invariant tensor products

$\displaystyle X^{\dagger}\cdot
Y=x_{1}^{*}y_{1}+x_{2}^{*}y_{2}+x_{3}^{*}y_{3},\quad\quad\quad(X\otimes
Y)_{0}=x_{1}y_{3}-x_{2}y_{2}+x_{3}y_{1},$

for two triplets of $X=(x_{1},x_{2},x_{3})^{T}$ and
$Y=(y_{1},y_{2},y_{3})^{T}$. As a result, there exists a parameter region
where the CP-like symmetric vacuum given in Eq. (4.15) is stable. Therefore we
show that a CP-like symmetric model with continuous internal symmetry can be
naturally derived from a proper CP symmetric model. Needless to say, there is
another stable vacuum with $\braket{S}=(0,v,0),\ v\in\mathbb{R}$, where the
unbroken generator is given as $\sigma_{3}$, so that a proper CP symmetry
remains after the SSB. The symmetry breaking patterns of $SU(2)\times U(1)$
and CP are summarized in Fig. 4.

##  5 Conclusion

In this paper we have extensively discussed the general CP transformations in
quantum field theory with internal global symmetry including continuous
groups. A special attention has been paid to the CP-like transformation. Since
the CP-like transformation exchanges a particle and a different antiparticles,
the physical CP symmetry is in general violated even if the theory has a CP-
like symmetry [69]. Despite its interesting property, the phenomenological
aspects of the CP-like symmetric models have not been fully investigated so
far. To address physical consequences of the CP-like symmetry we have computed
the various scattering amplitudes. As a result it has been found that physical
scattering processes that exhibit a physical CP asymmetry can also generate
the matter/antimatter asymmetry. We have classified these CP-violating
scattering amplitudes based on the representations of the asymptotic states
under the CP-like transformation, by which a required condition for generating
particle number asymmetry can be easily obtained. One striking feature of the
CP-like symmetric model is that, while there exist physical processes that can
exhibit particle number generations, the resultant particle number asymmetries
are constrained by the CP-like symmetry. Assuming that the two different
particles of $\psi_{i}$ and $\bar{\psi}_{j}$ are related to each other via the
CP-like transformation, we have explicitly calculated particle number
generations of both particles $N_{\psi_{i}}$ and $N_{\psi_{j}}$ from a CP-
violating scattering process, where it has been shown that the following
relation about the number violations should hold in general,

$\displaystyle N_{\psi_{i}}+N_{\psi_{j}}=0.$

This relation is analogous to other particle number generation processes such
as the GUT baryogenesis [72] and the sphaleron process [73]. We also have
found CP-like eigenstate which has no CP-like partner, and hence, its particle
number is not related to other antiparticles nor conserved. Existence of these
two classes of states are crucial for CP violation in CP-like symmetric
models. The mechanism for generating particle number asymmetry might be useful
for phenomenological purposes, e.g. for baryogenesis and asymmetric DM, which
we will study in the future work.

We also have discussed a dynamical origin of the CP-like symmetry and the fate
of the CP-like symmetry. It has turned out that the a proper CP and the CP-
like transformations can be mutually converted through the SSB, where it is
found that the mechanisms of the multiplet splitting/merging play an important
role. As we have pointed out that the emergence of CP-like symmetry is not
directly related to the absence of the CIA, CP-like symmetric models with a
wider class of internal symmetry groups can be obtained from a proper CP
symmetric models through the SSB. Various examples of spontaneous CP
violation/restoration have been explicitly shown, in which we have provided a
simple way to construct a CP-like symmetric model with continuous symmetries
from an underlying proper CP symmetric theory via the SSB. Our results suggest
that some quantum charge under a continuous symmetry may be defined to be
invariant under the CP-like transformation. The CP-like symmetric model may
shed light on the origin of the physical CP violation and simultaneously
provides us with novel frameworks for new physics.

Our analyses can be extended to other symmetries such as local (gauge)
symmetry and the modular symmetry. The modular symmetry includes C and P
transformation in its nature, which also acts on the flavor symmetry as an
automorphism. It would be also possible to generalize the non-CP
automorphisms. The above generalizations and applications to phenomenology may
be interesting and will be studied elsewhere.

## Acknowledgements

H. O. is supported in part by JSPS KAKENHI Grants No. 21K03554 and No.
22H00138. S. U is supported in part by JSPS KAKENHI Grant No. 22K14039. The
authors thank the Yukawa Institute for Theoretical Physics at Kyoto
University. Discussions during the YITP workshop YITP-W-23-10 on ”Progress in
Particle Physics 2023” were useful to complete this work.

##  Appendix A Group Property of $\Delta(54)$

The discrete group $\Delta(54)$ is isomorphic to
$\Delta(27)\rtimes\mathbb{Z}_{2}\cong((\mathbb{Z}_{3}\times\mathbb{Z}_{3})\rtimes\mathbb{Z}_{3})\rtimes\mathbb{Z}_{2}$.
It means all elements of $\Delta(54)$ can be represented by $a^{i}a^{\prime
j}b^{k}c^{l}$ ($i,j,k=0,1,2$ and $l=0,1$), where $a,a^{\prime},b$ corresponds
to the elements of $\Delta(27)$, and $c$ corresponds to the element of
$\mathbb{Z}_{2}$. The commutation relations including $c$ are given as
$ca=aa^{\prime}c$, $ca^{\prime}=a^{\prime 2}c$, and $cb=b^{2}c$, and those for
$\Delta(27)$ are given in Sec. 3.1. There are 10 conjugacy classes in
$\Delta(54)$:

$\displaystyle C_{1a}:$ $\displaystyle\\{e\\},~{}~{}C_{3e}:\\{aa^{\prime
2}\\},~{}~{}~{}~{}C_{3f}:\\{a^{2}a^{\prime}\\},$ $\displaystyle C_{3a}:$
$\displaystyle\\{b,b^{2},a^{2}a^{\prime}b,aa^{\prime 2}b,aa^{\prime
2}b^{2},a^{2}a^{\prime}b^{2}\\},$ $\displaystyle C_{3b}:$
$\displaystyle\\{a,a^{\prime},a^{2},aa^{\prime},a^{\prime 2},a^{2}a^{\prime
2}\\},$ $\displaystyle C_{3c}:$ $\displaystyle\\{a^{2}b,a^{\prime
2}b,aa^{\prime}b,ab^{2},a^{\prime}b^{2},a^{2}a^{\prime 2}b^{2}\\},$
$\displaystyle C_{3d}:$ $\displaystyle\\{ab,a^{\prime}b,a^{2}a^{\prime
2}b,a^{2}b^{2},aa^{\prime}b^{2},a^{\prime 2}b^{2}\\},$ $\displaystyle C_{2a}:$
$\displaystyle\\{c,a^{\prime}c,bc,a^{\prime
2}c,b^{2}c,ab^{2}c,aa^{\prime}bc,a^{2}b^{2}c,a^{2}a^{\prime 2}bc\\},$
$\displaystyle C_{6a}:$ $\displaystyle\\{ac,aa^{\prime}c,aa^{\prime
2}c,a^{\prime}bc,a^{2}bc,a^{\prime 2}b^{2}c,aa^{\prime 2}bc,aa^{\prime
2}b^{2}c,a^{2}a^{\prime 2}b^{2}c\\},$ $\displaystyle C_{6b}:$
$\displaystyle\\{a^{2}c,abc,a^{2}a^{\prime}c,a^{\prime}b^{2}c,a^{\prime
2}bc,a^{2}a^{\prime
2}c,aa^{\prime}b^{2}c,a^{2}a^{\prime}bc,a^{2}a^{\prime}b^{2}c\\}.$

Therefore $\Delta(54)$ has 10 irreducible representations of two singlets
$\mathbf{1}_{0,1}$, four doublets $\mathbf{2}_{1,2,3,4}$ and four triplets
$\mathbf{3}_{1,2},\mathbf{3}^{*}_{1,2}$. The character table of $\Delta(54)$
is summarized in Tab. 8.

| $C_{1a}$ | $C_{3a}$ | $C_{3b}$ | $C_{3c}$ | $C_{3d}$ | $C_{2a}$ | $C_{6a}$ | $C_{6b}$ | $C_{3e}$ | $C_{3f}$
---|---|---|---|---|---|---|---|---|---|---
degeneracy | $1$ | $6$ | $6$ | $6$ | $6$ | $9$ | $9$ | $9$ | $1$ | $1$
$\Delta(54)$ | $e$ | $b$ | $a^{\prime}$ | $ba^{\prime}b$ | $ba^{\prime}$ | $c$ | $ac$ | $a^{2}c$ | $aa^{\prime 2}$ | $a^{2}a^{\prime}$
$\mathbf{1}_{0}$ | $1$ | $1$ | $1$ | $1$ | $1$ | $1$ | $1$ | $1$ | $1$ | $1$
$\mathbf{1}_{1}$ | $1$ | $1$ | $1$ | $1$ | $1$ | $-1$ | $-1$ | $-1$ | $1$ | $1$
$\mathbf{2}_{1}$ | $2$ | $-1$ | $2$ | $-1$ | $-1$ | $0$ | $0$ | $0$ | $2$ | $2$
$\mathbf{2}_{2}$ | $2$ | $-1$ | $-1$ | $-1$ | $2$ | $0$ | $0$ | $0$ | $2$ | $2$
$\mathbf{2}_{3}$ | $2$ | $-1$ | $-1$ | $2$ | $-1$ | $0$ | $0$ | $0$ | $2$ | $2$
$\mathbf{2}_{4}$ | $2$ | $2$ | $-1$ | $-1$ | $-1$ | $0$ | $0$ | $0$ | $2$ | $2$
$\mathbf{3}_{1}$ | $3$ | $0$ | $0$ | $0$ | $0$ | $1$ | $\omega$ | $\omega^{2}$ | $3\omega$ | $3\omega^{2}$
$\mathbf{3}^{*}_{1}$ | $3$ | $0$ | $0$ | $0$ | $0$ | $1$ | $\omega^{2}$ | $\omega$ | $3\omega^{2}$ | $3\omega$
$\mathbf{3}_{2}$ | $3$ | $0$ | $0$ | $0$ | $0$ | $-1$ | $-\omega$ | $-\omega^{2}$ | $3\omega$ | $3\omega^{2}$
$\mathbf{3}^{*}_{2}$ | $3$ | $0$ | $0$ | $0$ | $0$ | $-1$ | $-\omega^{2}$ | $-\omega$ | $3\omega^{2}$ | $3\omega$
Table 8: Character table of $\Delta(54)$.

The faithful representation is given by the triplets. A matrix representation
of the triplets is given by

$\displaystyle\rho_{\mathbf{3}_{1}}(a)$
$\displaystyle=\rho_{\mathbf{3}_{2}}(a)=\begin{pmatrix}\omega&0&0\\\
0&\omega^{2}&0\\\ 0&0&1\\\ \end{pmatrix},~{}~{}$
$\displaystyle\rho_{\mathbf{3}_{1}}(a^{\prime})$
$\displaystyle=\rho_{\mathbf{3}_{2}}(a^{\prime})=\begin{pmatrix}1&0&0\\\
0&\omega&0\\\ 0&0&\omega^{2}\\\ \end{pmatrix},$
$\displaystyle\rho_{\mathbf{3}_{1}}(b)$
$\displaystyle=\rho_{\mathbf{3}_{2}}(b)=\begin{pmatrix}0&1&0\\\ 0&0&1\\\
1&0&0\\\ \end{pmatrix},~{}~{}$ $\displaystyle\rho_{\mathbf{3}_{1}}(c)$
$\displaystyle=-\rho_{\mathbf{3}_{2}}(c)=\begin{pmatrix}1&0&0\\\ 0&0&1\\\
0&1&0\\\ \end{pmatrix}.$

$\mathbf{3}^{*}_{i}$ is the complex conjugate representation of
$\mathbf{3}_{i}$. The matrix representations of the doublets are then given by

$\displaystyle\rho_{\mathbf{2}_{1}}(a)$
$\displaystyle=\rho_{\mathbf{2}_{1}}(a^{\prime})=\begin{pmatrix}1&0\\\
0&1\end{pmatrix},$ $\displaystyle\rho_{\mathbf{2}_{1}}(b)=$
$\displaystyle\begin{pmatrix}\omega&0\\\ 0&\omega^{2}\end{pmatrix},$
$\displaystyle\rho_{\mathbf{2}_{1}}(c)=$ $\displaystyle\begin{pmatrix}0&1\\\
1&0\end{pmatrix},$ (A.1) $\displaystyle\rho_{\mathbf{2}_{2}}(a)$
$\displaystyle=\rho_{\mathbf{2}_{2}}(a^{\prime})=\begin{pmatrix}\omega^{2}&0\\\
0&\omega\end{pmatrix},$ $\displaystyle\rho_{\mathbf{2}_{2}}(b)=$
$\displaystyle\begin{pmatrix}\omega&0\\\ 0&\omega^{2}\end{pmatrix},$
$\displaystyle\rho_{\mathbf{2}_{2}}(c)=$ $\displaystyle\begin{pmatrix}0&1\\\
1&0\end{pmatrix},$ (A.2) $\displaystyle\rho_{\mathbf{2}_{3}}(a)$
$\displaystyle=\rho_{\mathbf{2}_{3}}(a^{\prime})=\begin{pmatrix}\omega&0\\\
0&\omega^{2}\end{pmatrix},$ $\displaystyle\rho_{\mathbf{2}_{3}}(b)=$
$\displaystyle\begin{pmatrix}\omega&0\\\ 0&\omega^{2}\end{pmatrix},$
$\displaystyle\rho_{\mathbf{2}_{3}}(c)=$ $\displaystyle\begin{pmatrix}0&1\\\
1&0\end{pmatrix},$ (A.3) $\displaystyle\rho_{\mathbf{2}_{4}}(a)$
$\displaystyle=\rho_{\mathbf{2}_{4}}(a^{\prime})=\begin{pmatrix}\omega&0\\\
0&\omega^{2}\end{pmatrix},$ $\displaystyle\rho_{\mathbf{2}_{4}}(b)=$
$\displaystyle\begin{pmatrix}1&0\\\ 0&1\end{pmatrix},$
$\displaystyle\rho_{\mathbf{2}_{4}}(c)=$ $\displaystyle\begin{pmatrix}0&1\\\
1&0\end{pmatrix}.$ (A.4)

Since
$\rho_{\mathbf{2}_{i}}(a^{(\prime)})^{*}=S_{2}^{\dagger}\rho_{\mathbf{2}_{i}}(a^{(\prime)})S_{2}$,
and
$\rho_{\mathbf{2}_{i}}(b)^{*}=S_{2}^{\dagger}\rho_{\mathbf{2}_{i}}(b)S_{2}$,
where $S_{2}$ is the matrix given in Eq. (4.2). These doublets are pseudoreal
representations. We also have the trivial singlet $\mathbf{1}_{0}$ and a
nontrivial singlet $\mathbf{1}_{1}$,
$\rho_{\mathbf{1}_{1}}(a)=\rho_{\mathbf{1}_{1}}(a^{\prime})=\rho_{\mathbf{1}_{1}}(b)=1$
and $\rho_{\mathbf{1}_{1}}(c)=-1$.

The outer automorphism group of $\Delta(54)$ is isomorphic to $S_{4}$, which
is equivalent to the permutation of 4 doublets. It follows that
$\mathbf{2}_{i}\to\mathbf{2}_{i}$ corresponds to the identity element of the
outer automorphism, and we can not introduce consistent CP transformation
which induces $\mathbf{3}_{i}\to\mathbf{3}_{i}^{*}$ and
$\mathbf{2}_{i}\to\mathbf{2}_{i}$ simultaneously.

##  Appendix B Clebsch-Gordon Coefficients of $\Delta(27)$ and $\Delta(54)$

In this Appendix, we note CG coefficients of $\Delta(27)$ and $\Delta(54)$.
There are various conventions of the irreducible representations of discrete
groups. We note tensor products relative to our calculations rather than