# CP-Violating 2HDMs Emerging from 3-3-1 Models

Zhiyi Fan<EMAIL_ADDRESS>Kei Yagyu<EMAIL_ADDRESS>Department of Physics, Osaka University, Toyonaka, Osaka 560-0043, Japan

###### Abstract

We investigate CP violating 2 Higgs doublet models as an effective theory
emerging from models with an $SU(3)_{L}\otimes U(1)_{X}$ gauge symmetry.
Because of the extension of the electroweak symmetry, a characteristic
structure of Yukawa interactions appears with new CP violating phases at the
electroweak scale. In this scenario, the charged Higgs boson loop provides
dominant and sizable contributions to the neutron electric dipole moment
(nEDM) at one-loop level. We find that the prediction of the nEDM can be
slightly smaller than the current upper limit, ${\cal
O}(10^{-26})\,e\,\text{cm}$, with the mass of the charged Higgs boson and new
CP violating phases to be a few hundred GeV and $\mathcal{O}(1)$,
respectively, under the constraints from the meson mixings, the $B\to
X_{s}\gamma$ decay and current LHC data.

††preprint: OU-HET-1129

##  I Introduction

The three-generation structure of matter fields assumed in the standard model
(SM) has been established by the discovery of top quarks at Tevatron Abe  _et
al._ (1995); Abachi  _et al._ (1995). In addition, its consistency has
precisely been tested by various experiments such as $B$ factories Bevan  _et
al._ (2014) for the quark sector and LEP for the lepton sector Schael  _et
al._ (2006). Furthermore, CP violation (CPV) is naturally induced from the
phase of the Cabibbo-Kobayashi-Maskawa (CKM) matrix, which is required to
generate non-zero baryon asymmetry of the Universe. In spite of such
phenomenological successes, there is no theoretical basis for the existence of
the three generations in the SM, and it has been well known that the amount of
CPV from the CKM matrix is not sufficient to explain the observed baryon
asymmetry Huet and Sather (1995). These problems are expected to be solved in
new physics scenarios beyond the SM.

Models with an extended electroweak (EW) gauge symmetry $SU(3)_{C}\otimes
SU(3)_{L}\otimes U(1)_{X}$, the so-called 3-3-1 models Singer  _et al._
(1980); Valle and Singer (1983); Frampton (1992a), can simultaneously explain
the origin of three generations and additional sources of CPV. The former is
deduced from the condition for the gauge anomaly cancellation, by which the
number of generations has to be proportional to the color degrees of freedom
Frampton (1992a). The extension of the EW symmetry naturally requires an
extension of the Higgs sector, minimally containing three $SU(3)_{L}$ triplet
Higgs fields, in order to realize the spontaneous symmetry breaking
$SU(3)_{L}\otimes U(1)_{X}\to U(1)_{\rm em}$ and make all charged fermions
massive. Such an extended Higgs sector leads to a richer structure of Yukawa
interactions and the Higgs potential with additional sources of CPV.

In this paper, we discuss an effective theory described by the
$SU(2)_{L}\otimes U(1)_{Y}$ symmetry which is deduced from various 3-3-1
models with the minimal Higgs sector. All the non-SM fermions and gauge bosons
are decoupled from the theory by taking the vacuum expectation value (VEV) of
$SU(3)_{L}\otimes U(1)_{X}\to SU(2)_{L}\otimes U(1)_{Y}$ to be infinity. Even
in this limit, the Higgs sector can be non-minimal including two $SU(2)_{L}$
doublet fields, namely the effective theory corresponds to a two Higgs doublet
model (2HDM) Okada  _et al._ (2016). This 2HDM is categorized into that with a
discrete $Z_{2}$ symmetry whose charge for SM quarks are flavor dependent, not
the flavor universal one known as the Type-I, Type-II, Type-X and Type-Y 2HDMs
Grossman (1994); Barger  _et al._ (2009); Aoki  _et al._ (2009). Because of
the flavor dependent structure, Higgs boson couplings with quarks include new
sources of flavor violation and CPV phases other than those from the CKM
matrix, which can give sizable contributions to $B$ physics observables and
electric dipole moments (EDMs). We show that the one-loop diagram with charged
Higgs boson and top quark loops can give a significant contribution to the
neutron EDM (nEDM), by which a portion of the parameter space is excluded by
the current measurement or explored by future experiments such as the n2EDM
Ayres  _et al._ (2021) 111CPV effects on EDMs in 3-3-1 models have also been
discussed in Refs. De Conto and Pleitez (2015, 2016), in which effects of
extra fermions and gauge bosons are kept by taking their masses to be of order
1 TeV. . This result is quite different from that in the four types of 2HDMs
mentioned above Abe  _et al._ (2014); Cheung  _et al._ (2020); Altmannshofer
_et al._ (2020) and the so-called aligned 2HDMs Pich and Tuzon (2009); Jung
and Pich (2014); Kanemura  _et al._ (2020), where one-loop contributions are
negligibly smaller than the two-loop Barr-Zee type contributions Barr and Zee
(1990) due to chiral suppressions. We also find that the new phases and the
mass of the charged Higgs boson can be of order one and a few hundred GeV,
respectively, under the constraints from the meson mixings
($B^{0}$-$\bar{B}^{0}$, $D^{0}$-$\bar{D}^{0}$ and $K^{0}$-$\bar{K}^{0}$), the
$B\to X_{s}\gamma$ decay, EDMs and current LHC data.

This paper is organized as follows. In Sec. II, we classify various 3-3-1
models, and show that their effective theory can be a 2HDM. In Sec. III, we
discuss flavor constraints such as the meson mixings, the $B\to X_{s}\gamma$
decay and EDMs. The numerical evaluations for the nEDM are given in Sec. IV
under the constraint from the flavor observables. Conclusions are given in
Sec. V. In Appendix A, we present formulae for the masses of Higgs bosons and
those at the large VEV limit. In Appendix B, details of the other 3-3-1 models
are presented. In Appendix C, expressions for decay rates of the additional
Higgs bosons are explicitly shown.

##  II Models

In this section, we first classify 3-3-1 models by the difference of the
definition of the electric charge $Q$. We then show that their effective
theory corresponds to a 2HDM.

###  II.1 Classification of 3-3-1 models

Without loss of generality, $Q$ is defined as

$\displaystyle Q\equiv T_{3}+Y,~{}~{}\text{with}~{}~{}Y\equiv\zeta T_{8}+X,$
(1)

where $T_{3}$ and $T_{8}$ are the diagonal Gell-Mann matrices:

$\displaystyle T_{3}=\frac{1}{2}\text{diag}(1,-1,0),\quad
T_{8}=\frac{1}{2\sqrt{3}}\text{diag}(1,1,-2).$ (2)

As shown in Eq. (1), the hypercharge $Y$ is given as the linear combination of
$T_{8}$ and $X$ with $X$ being the $U(1)_{X}$ charge. In order to make $Y$ a
rational number, the $\zeta$ parameter should be proportional to $1/\sqrt{3}$.
We thus express $\zeta=n/\sqrt{3}$ with $n$ being an integer. The electric
charge for $SU(3)_{L}$ triplets $\bm{3}$ is then expressed as

$\displaystyle Q(\bm{3})=\begin{pmatrix}\frac{1}{2}+\frac{n}{6}+X\\\
-\frac{1}{2}+\frac{n}{6}+X\\\ -\frac{n}{3}+X\end{pmatrix}.$ (3)

The electric charge for the third component field $Q_{3}$ is expressed by that
of the first component $Q_{1}$ as $Q_{3}=Q_{1}-(n+1)/2$, so that $n$ should be
an odd number to make $Q_{3}$ integer, by which we can avoid the appearance of
exotic lepton fields with $Q$ being half integer.

The magnitude of $n$ is constrained by taking into account the relation among
gauge coupling constants Fonseca and Hirsch (2016a). Suppose that the
$SU(3)_{L}$ symmetry is spontaneously broken down to $SU(2)_{L}\otimes
U(1)_{Y}$ by a VEV of a triplet Higgs field $\Phi_{3}$ with the $X$ charge of
$n/3$, i.e., $\langle\Phi_{3}\rangle=(0,0,v_{3}/\sqrt{2})^{T}$222For $n=\pm
1$, the first or second component of $\Phi_{3}$ is electrically neutral, so
that it can generally develop a non-zero VEV. . After the breaking of
$SU(3)_{L}\to SU(2)_{L}\otimes U(1)_{Y}$, the 8th component of the $SU(3)_{L}$
gauge boson $A_{8}^{\mu}$ is mixed with the $U(1)_{X}$ gauge boson $X^{\mu}$
as

$\displaystyle\begin{pmatrix}A_{\mu}^{8}\\\
X_{\mu}\end{pmatrix}=\begin{pmatrix}\cos\theta_{331}&\sin\theta_{331}\\\
-\sin\theta_{331}&\cos\theta_{331}\end{pmatrix}\begin{pmatrix}X_{\mu}^{\prime}\\\
B_{\mu}\end{pmatrix},$ (4)

where $B_{\mu}$ ($X_{\mu}^{\prime}$) is the hypercharge (an additional neutral
massive) gauge boson. The mixing angle $\theta_{331}$ is given by

$\displaystyle\sin\theta_{331}=\frac{\zeta
g_{X}}{\sqrt{g^{2}+\zeta^{2}g_{X}^{2}}},\quad\cos\theta_{331}=\frac{g}{\sqrt{g^{2}+\zeta^{2}g_{X}^{2}}}.$
(5)

The hypercharge coupling $g_{Y}$ is given to be
$g_{Y}=g\sin\theta_{331}=g_{X}\cos\theta_{331}$. Using Eq. (5), this can also
be expressed as

$\displaystyle\frac{1}{g_{Y}^{2}}=\frac{\zeta^{2}}{g^{2}}+\frac{1}{g_{X}^{2}}.$
(6)

This relation tells us the upper limit on $|\zeta|$ which corresponds to the
case with $g_{X}\to\infty$ as

$\displaystyle\zeta^{2}<\frac{g^{2}}{g^{2}_{Y}}=\frac{1}{\tan^{2}\theta_{W}}\simeq(1.87)^{2}.$
(7)

Therefore, possible choices for $n$ are $n=\pm 1$ and $\pm 3$, i.e.,
$\zeta=\pm 1/\sqrt{3}$ and $\pm\sqrt{3}$.

3-3-1 models can also be classified into two categories depending on the way
how three generations of left-handed quarks $Q_{L}$ are embedded into the
$SU(3)_{L}$ representation. In the first category, denoting Class-I, first two
generations $Q_{L}^{1,2}$ are assigned to be $SU(3)_{L}$ anti-triplet, while
the third generation $Q_{L}^{3}$ and three left-handed leptons $L_{L}^{1,2,3}$
are assigned to be triplet. The gauge anomaly for $[SU(3)_{L}]^{3}$ is then
cancelled between six triplets from $Q_{L}^{3}$ and $L_{L}^{1,2,3}$ and six
anti-triplets from $Q_{L}^{1,2}$. In Class-I, we can further decompose models
into those with $\zeta=-\sqrt{3}$ Frampton (1992b); Pisano and Pleitez (1992),
$\zeta=-1/\sqrt{3}$ Singer (1980), $\zeta=1/\sqrt{3}$ Pleitez (1996); Özer
(1996) and $\zeta=\sqrt{3}$ Fonseca and Hirsch (2016a). In another category,
denoting Class-II, all three generations of $Q_{L}$ are embedded into anti-
triplets, while the lepton sector has to be extended to include nine
generations of left-handed lepton triplets, the so-called $E_{6}$ inspired
model Sanchez  _et al._ (2001), or to include one sextet (contributing to
$[SU(3)_{L}]^{3}$ as much as the seven triplets) and two triplets, the so-
called flipped model Fonseca and Hirsch (2016b). We note that in Class-II, the
$\zeta$ parameter is uniquely determined to be $-1/\sqrt{3}$ for the $E_{6}$
inspired model and $1/\sqrt{3}$ for the flipped model.

In the following, we mainly focus on Class-I models with $\zeta=-1/\sqrt{3}$
and its effective theory which is obtained by taking a large limit of the
$SU(3)_{L}$ breaking VEV. We then comment on models with the other $\zeta$
values and Class-II models.

###  II.2 Model with $\zeta=-1/\sqrt{3}$

Fields |  $SU(3)_{C}\otimes SU(3)_{L}\otimes U(1)_{X}\otimes U(1)^{\prime}$ |  $Z_{2}^{\rm rem}$ | Components |
---|---|---|---|---
$Q^{a}_{L}$ | $\left(\bm{3},\overline{\bm{3}},0,0\right)$ | $(+,+,-)$ |  $\left((q_{L}^{a})^{T},D^{a}_{L}\right)^{T}$, $q_{L}^{a}\equiv\left(d^{a}_{L},-u^{a}_{L}\right)^{T}$ |
$Q^{3}_{L}$ | $\left(\bm{3},\bm{3},+1/3,0\right)$ | $(+,+,-)$ | $\left((q_{L}^{3})^{T},U_{L}\right)^{T},~{}q_{L}^{3}\equiv\left(t_{L},b_{L}\right)^{T}$ |
$u^{i}_{R}$ | $\left(\bm{3},\bm{1},+2/3,q\right)$ | $+$ | $u^{i}_{R}$ |
$d^{i}_{R}$ | $\left(\bm{3},\bm{1},-1/3,-q\right)$ | $+$ | $d^{i}_{R}$ |
$U_{R}$ | $\left(\bm{3},\bm{1},+2/3,2q\right)$ | $-$ | $U_{R}$ |
$D^{a}_{R}$ | $\left(\bm{3},\bm{1},-1/3,-2q\right)$ | $-$ | $D^{a}_{R}$ |
$L^{i}_{L}$ | $\left(\bm{1},\bm{3},-1/3,0\right)$ | $(+,+,-)$ | $\left((\ell_{L}^{i})^{T},N_{R}^{ic}\right)^{T},~{}\ell_{L}^{i}\equiv\left(\nu^{i}_{L},e^{i}_{L}\right)^{T}$ |
$e^{i}_{R}$ | $\left(\bm{1},\bm{1},-1,-q\right)$ | $+$ | $e^{i}_{R}$ |
$N^{i}_{R}$ | $\left(\bm{1},\bm{1},0,+2q\right)$ | $-$ | $N^{i}_{R}$ |
$\Phi_{1}$ | $\left(\bm{1},\bm{3},+2/3,q\right)$ | $(+,+,-)$ | $\left(\phi_{1}^{T},\eta_{1}^{+}\right)^{T},~{}\phi_{1}\equiv\left(\phi_{1}^{+},\phi_{1}^{0}\right)^{T}$ |
$\Phi_{2}$ | $\left(\bm{1},\bm{3},-1/3,-q\right)$ | $(+,+,-)$ | $\left(\phi_{2}^{T},\eta_{2}^{0}\right),~{}\phi_{2}\equiv\left(\phi_{2}^{0},\phi_{2}^{-}\right)^{T}$ |
$\Phi_{3}$ | $\left(\bm{1},\bm{3},-1/3,-2q\right)$ | $(-,-,+)$ | $\left(\eta_{3}^{T},\phi_{3}^{0}\right)^{T},~{}\eta_{3}\equiv\left(\eta_{3}^{0},\eta_{3}^{-}\right)^{T}$ |
Table 1: Particle content of the model with $\zeta=-1/\sqrt{3}$, where
$U(1)^{\prime}$ is a softly-broken global symmetry and $Z_{2}^{\rm rem}$ is a
remnant symmetry after the spontaneous breaking of $SU(3)_{L}\otimes
U(1)_{X}$. The charge of $Z_{2}^{\rm rem}$ can be defined as
$(-1)^{|Q^{\prime}/q|+2\sqrt{3}T_{8}+2s}$ with $s$ being the spin and
$Q^{\prime}$ being $U(1)^{\prime}$ charge. Flavor indices $i$ and $a$ run over
1-3 and 1-2, respectively. In the last column, the component fields of
$SU(3)_{L}$ triplets are shown, where the first two components are written as
the doublet form.

In order to discuss the effective theory of 3-3-1 models, we consider the
model with $\zeta=-1/\sqrt{3}$ in Class-I as a representative one. The
particle content of this model is given in Table 1. In addition to the gauge
symmetry, we introduce a global $U(1)^{\prime}$ symmetry to avoid mixings
between SM fermions and extra fermions at tree level. The scalar sector is
composed of three $SU(3)_{L}$ triplets, which corresponds to the minimal form
to break $SU(3)_{L}\otimes U(1)_{X}$ into $U(1)_{\rm em}$ and to give all the
masses of fermions except for active neutrinos $\nu_{L}^{i}$.

The Higgs potential is generally written under the $SU(3)_{L}\otimes
U(1)_{X}\otimes U(1)^{\prime}$ symmetry as

$\displaystyle V$
$\displaystyle=\sum_{i=1,3}m_{i}^{2}|\Phi_{i}|^{2}+\left(m_{23}^{2}\Phi_{2}^{\dagger}\Phi_{3}+\mu\epsilon_{\alpha\beta\gamma}\Phi_{1}^{\alpha}\Phi_{2}^{\beta}\Phi_{3}^{\gamma}+\text{h.c.}\right)$
$\displaystyle+\sum_{i=1,3}\frac{\lambda_{i}}{2}|\Phi_{i}|^{4}+\sum_{i,j=1,3}^{j>i}\lambda_{ij}|\Phi_{i}|^{2}|\Phi_{j}|^{2}+\sum_{i,j=1,3}^{j>i}\rho_{ij}|\Phi_{i}^{\dagger}\Phi_{j}|^{2},$
(8)

where $m_{23}^{2}$ and $\mu$ terms softly break the $U(1)^{\prime}$ symmetry,
and the subscripts $\alpha,\beta,\gamma$ $(=1,2,3)$ denote the index for the
fundamental representation of $SU(3)_{L}$. By rephasing the scalar fields, all
the parameters in the potential are taken to be real without loss of
generality. We take $m_{23}^{2}=0$ such that the configuration of the VEVs
$\langle\phi^{0}_{i}\rangle=v_{i}/\sqrt{2}\neq 0$ and
$\langle\eta^{0}_{j}\rangle=0$ is realized from the tadpole conditions, see
Appendix A. In this case, the symmetry breaking is realized by the following
two steps:

$\displaystyle SU(3)_{L}\otimes U(1)_{X}\xrightarrow[v_{3}]{}SU(2)_{L}\otimes
U(1)_{Y}\xrightarrow[v]{}U(1)_{\rm em},$ (9)

where $v\equiv\sqrt{v_{1}^{2}+v_{2}^{2}}=(\sqrt{2}G_{F})^{-1/2}$ with $G_{F}$
being the Fermi constant. The ratio of these VEVs is parameterized as
$\tan\beta\equiv v_{2}/v_{1}$. In this configuration, a remnant unbroken
$Z_{2}^{\rm rem}$ symmetry appears whose charges are given in Table 1, by
which the component fields $\phi$ and $\eta$ do not mix with each other. In
addition, the lightest neutral $Z_{2}^{\rm rem}$-odd particle can be a
candidate of dark matter.

There are 18 real scalar fields in the three triplets $\Phi_{i}$, but 8 of 18
fields are absorbed into longitudinal components of the massive gauge bosons.
Thus, 10 real scalar fields remain as the physical degrees of freedom. These
can be classified into two pairs of singly-charged Higgs bosons ($H^{\pm}$ and
$\eta^{\pm}$), two “CP-odd” Higgs bosons ($A$, $\eta_{I}$) and four “CP-even”
Higgs bosons ($h$, $H$, $H_{S}$, $\eta_{R}$) with $h$ being identified with
the SM-like Higgs boson. 333Here, “CP-even” and “CP-odd” indicate the definite
CP-states when CPV phases in the Yukawa interaction are taken to be zero. In
reality, we cannot define such a CP-state, because the Higgs potential
receives CPV effects from Yukawa interactions via radiative corrections.  See
Appendix A for expressions of the scalar boson masses and the relation between
the mass eigenbases and original ones shown in Table 1.

All the $Z_{2}^{\rm rem}$-odd scalar fields and the $H_{S}$ state are
decoupled by taking the $SU(3)_{L}$ breaking VEV $v_{3}$ to be infinity. We
note that extra five gauge bosons (a pair of charged, a complex and a real
neutral gauge bosons) are also decoupled by taking this limit, see e.g., Ref.
Okada  _et al._ (2016). On the other hand, the remaining five Higgs bosons,
i.e., $H^{\pm}$, $A$, $H$ and $h$ can be of order the EW scale as long as the
$M^{2}\equiv\mu v_{3}/(\sqrt{2}\cos\beta\sin\beta)$ parameter is kept to be
(EW scale)2. Thus, the case with $M\sim{\cal O}(v)$ the Higgs sector of this
model is effectively regarded as a 2HDM. The Higgs potential of such an
effective 2HDM can be written as

$\displaystyle V_{\rm eff}$
$\displaystyle=\mu_{1}^{2}|\phi_{1}|^{2}+\mu_{2}^{2}|\phi_{2}|^{2}+\mu_{12}^{2}[\phi_{1}^{T}(i\tau_{2})\phi_{2}+\text{h.c.}]$
$\displaystyle+\frac{\lambda_{1}}{2}|\phi_{1}|^{4}+\frac{\lambda_{2}}{2}|\phi_{2}|^{4}+\lambda_{12}|\phi_{1}|^{2}|\phi_{2}|^{2}+\rho_{12}|\phi_{1}^{\dagger}\phi_{2}|^{2},$
(10)

where $\phi_{1,2}$ are the $SU(2)_{L}$ doublet Higgs fields defined in Table
1. The $\mu_{12}^{2}$ parameter is real, since the original potential does not
contain physical CPV phases. One can check that the same mass formulae in the
limit $v_{3}\to\infty$ given in Appendix A can be reproduced by starting from
the above potential with the identification
$M^{2}=\mu_{12}^{2}/(s_{\beta}c_{\beta})$.

The Yukawa interaction is given by

$\displaystyle\mathcal{L}_{Y}=$
$\displaystyle-(Y_{e})_{ij}\overline{L_{L}^{i}}\,\Phi_{1}\,e_{R}^{j}-(Y_{N})_{ij}\overline{L_{L}^{i}}\,\Phi_{3}\,N_{R}^{j}+\text{h.c.}$
$\displaystyle+(Y_{u}^{1})_{ai}\overline{Q_{L}^{a}}\,\Phi_{1}^{*}\,u_{R}^{i}-(Y_{u}^{2})_{i}\overline{Q_{L}^{3}}\,\Phi_{2}\,u_{R}^{i}-Y_{U}\overline{Q_{L}^{3}}\,\Phi_{3}\,U_{R}+\text{h.c.}$
$\displaystyle-(Y_{d}^{1})_{i}\overline{Q_{L}^{3}}\,\Phi_{1}\,d_{R}^{i}-(Y_{d}^{2})_{ai}\overline{Q_{L}^{a}}\,\Phi_{2}^{*}\,d_{R}^{i}-(Y_{D})_{ab}\overline{Q_{L}^{a}}\,\Phi_{3}^{*}\,D_{R}^{b}+\text{h.c.}$
(11)

Due to the $U(1)^{\prime}$ symmetry, mixings between SM fermions and extra
fermions ($U$ and $D^{a}$) do not appear. In addition, the new lepton Yukawa
term
$\epsilon_{\alpha\beta\gamma}(Y_{L})_{ij}(\overline{L_{L}^{ic}})^{\alpha}(L_{L}^{j})^{\beta}(\Phi_{1})^{\gamma}$
is forbidden. We see that the extra fermion masses are given by the large VEV
$v_{3}$, while those of the SM fermions are given by the smaller VEVs $v_{1}$
and $v_{2}$.

At the large $v_{3}$ limit, we obtain the low energy effective Lagrangian as
follows:

$\displaystyle{\cal L}_{\rm eff}$
$\displaystyle=-(Y_{e})_{ij}\ell_{L}^{i}\phi_{1}e_{R}^{j}+\text{h.c.}$
$\displaystyle+(Y_{u}^{1})_{ai}\bar{q}_{L}^{a}\phi_{1}^{*}u_{R}^{i}-(Y_{u}^{2})_{i}\bar{q}_{L}^{3}\phi_{2}u_{R}^{i}-(Y_{d}^{1})_{i}\bar{q}_{L}^{3}\phi_{1}d_{R}^{i}-(Y_{d}^{2})_{ai}\bar{q}_{L}^{a}\phi_{2}^{*}d_{R}^{i}+\text{h.c.},$
(12)

where $\ell_{L}^{i}$, $q_{L}^{a}$, $q_{L}^{3}$, $\phi_{1}$ and $\phi_{2}$ are
the $SU(2)_{L}$ doublet fields defined in Table 1. The Yukawa interaction
given in Eq. (12) is invariant under a flavor dependent $Z_{2}$
transformation, denoting $Z_{2}^{F}$, whose charge can be defined by
$(-1)^{3X}$, by which $\\{\phi_{2},q_{L}^{3},\ell_{L},d_{R},e_{R}\\}$ are odd
under $Z_{2}^{F}$, while the other SM fermions and $\phi_{1}$ are even. The
$Z_{2}^{F}$ symmetry is softly-broken by the $\mu_{12}^{2}$ term in the
potential, see Eq. (10). Due to the $Z_{2}^{F}$ symmetry, a characteristic
flavor dependence in the quark Yukawa interactions appears, which is not seen
in the 2HDMs with a usual (flavor independent) $Z_{2}$ symmetry. From Eq.
(12), we see that the mass matrices for quarks are composed of two Yukawa
matrices as

$\displaystyle
M_{u}=\frac{1}{\sqrt{2}}\begin{pmatrix}v_{1}(Y_{u}^{1})_{11}&v_{1}(Y_{u}^{1})_{12}&v_{1}(Y_{u}^{1})_{13}\\\
v_{1}(Y_{u}^{1})_{21}&v_{1}(Y_{u}^{1})_{22}&v_{1}(Y_{u}^{1})_{23}\\\
v_{2}(Y_{u}^{2})_{1}&v_{2}(Y_{u}^{2})_{2}&v_{2}(Y_{u}^{2})_{3}\\\
\end{pmatrix},\quad
M_{d}=\frac{1}{\sqrt{2}}\begin{pmatrix}v_{2}(Y_{d}^{2})_{11}&v_{2}(Y_{d}^{2})_{12}&v_{2}(Y_{d}^{2})_{13}\\\
v_{2}(Y_{d}^{2})_{21}&v_{2}(Y_{d}^{2})_{22}&v_{2}(Y_{d}^{2})_{23}\\\
v_{1}(Y_{d}^{2})_{1}&v_{1}(Y_{d}^{2})_{2}&v_{1}(Y_{d}^{2})_{3}\\\
\end{pmatrix}.$

Therefore, after diagonalizing these mass matrices, flavor dependent
structures of the quark Yukawa interactions appear. On the other hand, the
mass matrix for charged leptons is given by the single Yukawa matrix $Y_{e}$
as in the SM. Performing the following bi-unitary transformations:

$\displaystyle f_{L}=V_{f}f_{L}^{\prime},\quad
f_{R}=U_{f}f_{R}^{\prime},~{}~{}(f=u,d,e),$ (13)

we can diagonalize these mass matrices. We then can extract the interaction
terms in the mass eigenbases for fermions as follows:

$\displaystyle{\cal L}_{\rm eff}$
$\displaystyle=-\frac{1}{v}\bar{e}^{\prime}M_{e}^{\rm
diag}[(s_{\beta-\alpha}-t_{\beta}c_{\beta-\alpha})h+(c_{\beta-\alpha}+t_{\beta}s_{\beta-\alpha})H+it_{\beta}\gamma_{5}A]e^{\prime}$
$\displaystyle-\frac{1}{v}\sum_{\varphi=h,H,A}\sum_{q=u,d}p_{\varphi}^{q}\bar{q}^{\prime}\Gamma_{\varphi}^{q}M_{q}^{\rm
diag}P_{R}q^{\prime}\varphi+\text{h.c.}$
$\displaystyle-\frac{\sqrt{2}}{v}\left[\bar{e}^{\prime}M_{e}^{\rm
diag}t_{\beta}P_{L}\nu^{\prime}+\bar{d}^{\prime}\left(M_{L}P_{L}+M_{R}P_{R}\right)u^{\prime}\right]H^{-}+\text{h.c.},$
(14)

where $c_{X}=\cos X$, $s_{X}=\sin X$, $t_{X}=\tan X$,
$p_{h}^{q}=p_{H}^{q}=1,~{}p_{A}^{q}=2iI_{q}$ with $I_{q}=1/2(-1/2)$ for
$q=u(d)$ and

$\displaystyle M_{L}=-M_{d}^{\rm diag}\Gamma_{A}^{d}V_{\rm
CKM}^{\dagger},\quad M_{R}=V_{\rm CKM}^{\dagger}\Gamma_{A}^{u}M_{u}^{\rm
diag}.$ (15)

In Eq. (14), $P_{L,R}$ are the projection operator, and $V_{\rm CKM}\equiv
V_{u}^{\dagger}V_{d}$ is the CKM matrix. The interaction matrices are given by

$\displaystyle\begin{split}\Gamma_{A}^{u}&=V_{u}^{\dagger}\text{diag}(-t_{\beta},-t_{\beta},t_{\beta}^{-1})V_{u},\quad\Gamma_{A}^{d}=V_{d}^{\dagger}\text{diag}(t_{\beta}^{-1},t_{\beta}^{-1},-t_{\beta})V_{d},\\\
\Gamma_{H}^{u}&=V_{u}^{\dagger}\text{diag}\left(c_{\beta-\alpha}+t_{\beta}s_{\beta-\alpha},c_{\beta-\alpha}+t_{\beta}s_{\beta-\alpha},c_{\beta-\alpha}-t_{\beta}^{-1}s_{\beta-\alpha}\right)V_{u},\\\
\Gamma_{H}^{d}&=V_{d}^{\dagger}\text{diag}\left(c_{\beta-\alpha}-t_{\beta}^{-1}s_{\beta-\alpha},c_{\beta-\alpha}-t_{\beta}^{-1}s_{\beta-\alpha},c_{\beta-\alpha}+t_{\beta}s_{\beta-\alpha}\right)V_{d},\\\
\Gamma_{h}^{u}&=V_{u}^{\dagger}\text{diag}\left(s_{\beta-\alpha}-t_{\beta}c_{\beta-\alpha},s_{\beta-\alpha}-t_{\beta}c_{\beta-\alpha},s_{\beta-\alpha}+t_{\beta}^{-1}c_{\beta-\alpha}\right)V_{u},\\\
\Gamma_{h}^{d}&=V_{d}^{\dagger}\text{diag}\left(s_{\beta-\alpha}+t_{\beta}^{-1}c_{\beta-\alpha},s_{\beta-\alpha}+t_{\beta}^{-1}c_{\beta-\alpha},s_{\beta-\alpha}-t_{\beta}c_{\beta-\alpha}\right)V_{d}.\end{split}$
(16)

All these $\Gamma_{\varphi}^{q}$ matrices are hermitian which turn out to be
important for the discussion of the EDMs in the next section. In the so-called
alignment limit, i.e., $s_{\beta-\alpha}\to 1$, these matrices take the simple
form as

$\displaystyle\Gamma_{H}^{q}$
$\displaystyle=-\Gamma_{A}^{q},\quad\Gamma_{h}^{q}=1~{}~{}(q=u,d).$ (17)

Thus, the Yukawa couplings for the 125 GeV Higgs boson $h$ are the same as
those of the SM values at tree level. On the other hand, the Yukawa couplings
for extra Higgs bosons are non-diagonal form, because of the flavor dependent
structure of the matrices $\text{diag}(-t_{\beta},-t_{\beta},t_{\beta}^{-1})$
and $\text{diag}(t_{\beta}^{-1},t_{\beta}^{-1},-t_{\beta})$ as aforementioned.
We note that in the 2HDMs with a flavor independent $Z_{2}$ symmetry, the
diagonal matrices, e.g., $\text{diag}(-t_{\beta},-t_{\beta},t_{\beta}^{-1})$
are replaced by the $3\times 3$ identity matrix $I_{3}$ times a constant, so
that the above $\Gamma_{\varphi}^{q}$ matrices are also proportional to
$I_{3}$ due to the unitarity of the $V_{q}$ matrices.

In order to parameterize the $\Gamma_{\varphi}^{q}$ matrices, we express the
unitary matrix $V_{u}$ as follows:

$\displaystyle
V_{u}=\text{diag}(e^{i\delta_{1}},e^{i\delta_{2}},e^{i\delta_{3}})R_{12}(\phi)R_{13}(\theta)R_{23}(\psi)\text{diag}(e^{i\delta_{4}},e^{i\delta_{5}},e^{i\delta_{6}}),$
(18)

where

$\displaystyle R_{12}(X)=\begin{pmatrix}c_{X}&-s_{X}&0\\\ s_{X}&c_{X}&0\\\
0&0&1\end{pmatrix},~{}R_{13}(X)=\begin{pmatrix}c_{X}&0&-s_{X}\\\ 0&1&0\\\
s_{X}&0&c_{X}\end{pmatrix},~{}R_{23}(X)=\begin{pmatrix}1&0&0\\\
0&c_{X}&-s_{X}\\\ 0&s_{X}&c_{X}\end{pmatrix}.$ (19)

The unitary matrix for down-type quarks $V_{d}$ is then obtained from
$V_{d}=V_{u}V_{\rm CKM}$. In this parameterization, the $\Gamma_{\varphi}^{q}$
matrices do not depend on the $(\delta_{1},\delta_{2},\delta_{3},\phi)$
parameters.

$\zeta$ | Quarks | Leptons | Scalars
---|---|---|---
$-1/\sqrt{3}$ | $Q_{-1/3}(2)$ | $Q_{2/3}(1)$ | $L_{0}(3)$ | $H_{S}$ | $\eta^{0}$ | $\eta^{\pm}$
$+1/\sqrt{3}$ | $Q_{-1/3}(1)$ | $Q_{2/3}(2)$ | $L_{-1}(3)$ | $H_{S}$ | $\eta^{0}$ | $\eta^{\pm}$
$-\sqrt{3}$ | $Q_{-4/3}(2)$ | $Q_{5/3}(1)$ | - | $H_{S}$ | $\eta^{\pm}$ | $\eta^{\pm\pm}$
$+\sqrt{3}$ | $Q_{-4/3}(1)$ | $Q_{5/3}(2)$ | $L_{-2}(3)$ | $H_{S}$ | $\eta^{\pm}$ | $\eta^{\pm\pm}$
Table 2: Content of heavy extra degrees of freedom in the Class-I models with
four $\zeta$ values. The subscript with quarks ($Q$) and leptons ($L$)
represents the electric charge, and the number inside parentheses denotes the
number of flavors. For scalar particles, $H_{S}$ represents a $Z_{2}^{\rm
rem}$-even real scalar boson, while $\eta^{0}$, $\eta^{\pm}$ and
$\eta^{\pm\pm}$ are $Z_{2}^{\rm rem}$-odd complex neutral, singly-charged and
doubly-charged scalar bosons, respectively.

Before closing this section, let us comment on the other Class-I models with
$\zeta=+1/\sqrt{3}$ and $\zeta=\pm\sqrt{3}$. In these models, heavy extra
degrees of freedom, which are decoupled by taking the large $v_{3}$ limit, are
different from those in the model with $\zeta=-1/\sqrt{3}$. In Table 2, we
show the content of extra fermions and scalar bosons decoupled from the
theory. See also Appendix B for details of the particle content. The important
thing here is that when we take the large $v_{3}$ limit, the effective theory
of these three models is the same as that of the model with
$\zeta=-1/\sqrt{3}$. For the Class-II models, however, the effective theory is
different from that discussed in this section, because all the three
generations of quarks are embedded into the same $SU(3)_{L}$ representation,
which do not lead to the flavor dependent structure as shown in Eq. (12). In
addition, the lepton sector involves more complicated structures including a
sextet representation in the flipped model and nice triplets in the $E_{6}$
model, so that the lepton sector in their effective theory can be richer than
that discussed in this section.

##  III Flavor Constraints

As we have seen in Sec. II, 3-3-1 models introduce a 2HDM with a
characteristic flavor structure in quark Yukawa interactions at the EW scale.
We thus take into account flavor constraints on the parameter space,
particularly those from neutral meson mixings, $B\to X_{s}\gamma$ and EDMs.
There are the other flavor constraints associated with leptons such as
$M^{\pm}\to\ell^{\pm}\nu$, $M^{0}\to\ell^{+}\ell^{-}$ and $\tau^{\pm}\to
M^{\pm}\nu$ with $M^{\pm}$ and $M^{0}$ being charged and neutral mesons,
respectively. These processes typically become important in the case where
leptonic Yukawa interactions are largely enhanced, see e.g., Refs. Crivellin
_et al._ (2013); Arbey  _et al._ (2018); Enomoto and Watanabe (2016). In our
model, such an enhancement can be realized for large $\tan\beta$ cases which
are highly constrained by the processes considered in this section. We thus do
not discuss such processes in details.

###  III.1 Meson mixings

We first consider the constraint from meson mixings which happen at tree level
via neutral Higgs boson exchanges, because of the flavor violating
interactions given in Eq. (16). According to Ref. Gabbiani  _et al._ (1996),
new contributions to the $B_{q}^{0}$–$\bar{B}_{q}^{0}$ ($q=d,s$) mixing
$\Delta m_{B_{q}}$ are given by

$\displaystyle\Delta m_{B_{q}}$
$\displaystyle=2\text{Re}\left[C_{LR}^{B_{q}}\left(\frac{1}{24}+\frac{m_{B_{q}}^{2}}{4(m_{b}+m_{q})^{2}}\right)-\frac{5}{24}\left(C_{LL}^{B_{q}}+C_{RR}^{B_{q}}\right)\frac{m_{B_{q}}^{2}}{(m_{b}+m_{q})^{2}}\right]m_{B_{q}}f_{B_{q}}^{2},$
(20)

where $m_{B_{q}}$ and $f_{B_{q}}$ are the mass of the $B_{q}^{0}$ meson and
the decay constant, respectively. 444The coefficients $C_{LL}$, $C_{RR}$ and
$C_{LR}$ respectively correspond to those for the effective operators $Q_{2}$,
$\tilde{Q}_{2}$ and $Q_{4}$ defined in Ref. Gabbiani  _et al._ (1996).  The
expressions for the $D^{0}$–$\bar{D}^{0}$ and $K^{0}$–$\bar{K}^{0}$ mixings
are also obtained by the replacement of
$(m_{B_{q}},f_{B_{q}},m_{b},m_{q},C_{ij}^{B_{q}})\to(m_{D},f_{D},m_{u},m_{c},C_{ij}^{K})$
and
$(m_{B_{q}},f_{B_{q}},m_{b},m_{q},C_{ij}^{B_{q}})\to(m_{K},f_{K},m_{d},m_{s},C_{ij}^{K})$,
respectively. The coefficients $C_{ij}^{M}$ $(M=B_{q},D,K)$ are expressed as:

$\displaystyle\begin{split}C_{LL}^{B_{q}}&=-\sum_{\varphi}\frac{\big{[}p_{\varphi}^{d}m_{q}(\Gamma^{d*}_{\varphi})_{3q}\big{]}^{2}}{2m^{2}_{\varphi}v^{2}},~{}C_{RR}^{B_{q}}=-\sum_{\varphi}\frac{\big{[}p_{\varphi}^{d}m_{b}(\Gamma^{d}_{\varphi})_{q3}\big{]}^{2}}{2m^{2}_{\varphi}v^{2}},~{}C_{LR}^{B_{q}}=-\sum_{\varphi}\frac{m_{q}m_{b}[(\Gamma^{d}_{\varphi})_{q3}]^{2}}{m^{2}_{\varphi}v^{2}},\\\
C_{LL}^{D}&=-\sum_{\varphi}\frac{\big{[}p_{\varphi}^{u}m_{u}(\Gamma^{u*}_{\varphi})_{21}\big{]}^{2}}{2m^{2}_{\varphi}v^{2}},~{}C_{RR}^{D}=-\sum_{\varphi}\frac{\big{[}p_{\varphi}^{u}m_{c}(\Gamma^{u}_{\varphi})_{12}\big{]}^{2}}{2m^{2}_{\varphi}v^{2}},~{}C_{LR}^{D}=-\sum_{\varphi}\frac{m_{u}m_{c}[(\Gamma^{u}_{\varphi})_{12}]^{2}}{m^{2}_{\varphi}v^{2}},\\\
C_{LL}^{K}&=-\sum_{\varphi}\frac{\big{[}p_{\varphi}^{d}m_{d}(\Gamma^{d*}_{\varphi})_{21}\big{]}^{2}}{2m^{2}_{\varphi}v^{2}},~{}C_{RR}^{K}=-\sum_{\varphi}\frac{\big{[}p_{\varphi}^{d}m_{s}(\Gamma^{d}_{\varphi})_{12}\big{]}^{2}}{2m^{2}_{\varphi}v^{2}},~{}C_{LR}^{K}=-\sum_{\varphi}\frac{m_{d}m_{s}[(\Gamma^{d}_{\varphi})_{12}]^{2}}{m^{2}_{\varphi}v^{2}},\end{split}$
(21)

where the summation takes over $\varphi=\\{h,H,A\\}$. For $M=B_{q}$,
$(\Gamma^{d*}_{\varphi})_{3q}$ represents $(\Gamma^{d*}_{\varphi})_{31}$ and
$(\Gamma^{d*}_{\varphi})_{32}$ for $q=d$ and $q=s$, respectively, and the same
thing applies to $(\Gamma^{d}_{\varphi})_{q3}$. We checked that these
expressions are consistent with those given in Refs. Crivellin  _et al._
(2013); Jana  _et al._ (2021).

At the alignment limit, i.e., $s_{\beta-\alpha}=1$, the contribution from the
$h$ mediation vanishes, because the Yukawa coupling of $h$ takes a diagonal
form. When we take $m_{H}=m_{A}$ in addition to $s_{\beta-\alpha}=1$, $C_{LL}$
and $C_{RR}$ vanish, because the cancellation happens between the
contributions from the $H$ and $A$ mediations. In these limits, the expression
for $\Delta m_{M}$ ($M=B,D,K$) takes a simple form as

$\displaystyle\begin{split}\Delta
m_{K}&=m_{K}\frac{m_{d}m_{s}f_{B}^{2}}{m^{2}_{A}v^{2}}\left[\frac{1}{6}+\frac{m_{K}^{2}}{(m_{d}+m_{s})^{2}}\right]\text{Re}\left\\{[(\Gamma^{d}_{A})_{12}]^{2}\right\\},\\\
\Delta
m_{D}&=m_{D}\frac{m_{u}m_{c}f_{D}^{2}}{m^{2}_{A}v^{2}}\left[\frac{1}{6}+\frac{m_{D}^{2}}{(m_{u}+m_{c})^{2}}\right]\text{Re}\left\\{[(\Gamma^{u}_{A})_{12}]^{2}\right\\},\\\
\Delta
m_{B_{q}}&=m_{B_{q}}\frac{m_{q}m_{b}f_{B}^{2}}{m^{2}_{A}v^{2}}\left[\frac{1}{6}+\frac{m_{B_{q}}^{2}}{(m_{q}+m_{b})^{2}}\right]\text{Re}\left\\{[(\Gamma^{d}_{A})_{q3}]^{2}\right\\}.\end{split}$
(22)

The current measured values of $\Delta m_{M}$ are Zyla  _et al._ (2020a)

$\displaystyle\begin{split}\Delta m_{K}^{\rm exp}&=(3.484\pm 0.006)\times
10^{-12}~{}~{}\text{MeV},\\\ \Delta m_{D}^{\rm exp}&=(6.251\pm 2.70)\times
10^{-12}~{}~{}\text{MeV},\\\ \Delta m_{B_{d}}^{\rm exp}&=(3.334\pm
0.013)\times 10^{-10}~{}~{}\text{MeV},\\\ \Delta m_{B_{s}}^{\rm
exp}&=(1.1677\pm 0.0013)\times 10^{-8}~{}~{}\text{MeV}.\end{split}$ (23)

By comparing Eqs. (22) and (23), we can constrain $m_{A}(=m_{H})$, $t_{\beta}$
and the mixing/phase parameters ($\theta$, $\psi$, $\delta_{4,5,6}$). In the
numerical evaluation, we simply require that the absolute values of the new
contribution given in Eq. (22) do not exceed the central value of the
measurement.

###  III.2 $B\to X_{s}\gamma$

The mass of the charged Higgs boson and its Yukawa coupling are constrained by
the $B\to X_{s}\gamma$ process. The current measured value of the branching
ratio is given by Amhis  _et al._ (2021)

$\displaystyle{\cal B}(B\to X_{s}\gamma)=(3.32\pm 0.15)\times 10^{-4}.$ (24)

The bound on $m_{H^{\pm}}$ has been studied in the 2HDMs with a softly-broken
$Z_{2}$ symmetry at the next-to-leading order (NLO) Ciuchini  _et al._ (1998);
Kagan and Neubert (1999); Borzumati and Greub (1998, 1999) and at the next-to-
NLO (NNLO) Misiak  _et al._ (2007); Misiak and Steinhauser (2017); Misiak  _et
al._ (2020) in QCD, where $m_{H^{\pm}}\gtrsim$ 800 GeV has been taken in the
Type-II 2HDM with $t_{\beta}\gtrsim 2$ Misiak  _et al._ (2020).

In our effective 2HDM, the situation is similar to the Type-II 2HDM if the
mixing parameters $\theta$, $\psi$ and the phases $\delta_{4,5,6}$ are taken
to be small. However, when these parameters get larger values, the constraint
should be drastically different from the Type-II case. In fact, we will see in
Sec. IV that the bound from the $B\to X_{s}\gamma$ decay can be relaxed when
the mixing parameters are switched on.

For the numerical evaluation of the branching ratio of $B\to X_{s}\gamma$, we
employ the expression given in Ref. Borzumati and Greub (1998), in which QCD
and QED corrections are implemented at NLO. In Ref. Borzumati and Greub
(1998), the branching ratio is given in terms of the $X$ and $Y$ parameters
which are the coefficients of the Yukawa coupling for the charged Higgs
boson555In the Type-I (Type-II) 2HDM, $X=-\cot\beta~{}(\tan\beta)$ and
$Y=\cot\beta~{}(\cot\beta)$ Borzumati and Greub (1998). , and they appear as
combinations of $XY^{*}$ and $|Y|^{2}$. In our model, $XY^{*}$ and $|Y|^{2}$
are expressed as666We only consider the dominant top loop contribution to the
decay rate.

$\displaystyle XY^{*}$
$\displaystyle=\frac{(M_{L}^{*})_{33}(M_{R})_{23}}{m_{t}m_{b}(V_{\rm
CKM})_{33}(V_{\rm
CKM}^{*})_{32}},\quad|Y|^{2}=\frac{(M_{R}^{*})_{33}(M_{R})_{23}}{m_{t}^{2}(V_{\rm
CKM})_{33}(V_{\rm CKM}^{*})_{32}}.$ (25)

###  III.3 Electric Dipole Moments

Figure 1: One-loop contributions to $d_{d}$.

Generally, new CPV phases in the Yukawa interaction can induce new
contributions to EDMs. Effects of CPV can be described by the following
effective Lagrangian up to dimension six as

$\displaystyle\mathcal{L}_{\text{eff}}$
$\displaystyle=-\frac{id_{f}}{2}\overline{\psi}_{f}\sigma^{\mu\nu}\gamma_{5}\psi_{f}F_{\mu\nu}-\frac{id^{C}_{f}}{2}\overline{\psi}_{f}\sigma^{\mu\nu}\gamma_{5}T^{A}\psi_{f}G^{A}_{\mu\nu}$
$\displaystyle+\frac{1}{3}C_{W}f_{ABC}G_{\mu\nu}^{A}\tilde{G}^{B\,\nu\rho}G_{\rho}^{C\,\mu}+C_{ff^{\prime}}(\bar{f}f)(\bar{f}^{\prime}i\gamma_{5}f^{\prime}).$
(26)

In the first line, $d_{f}$ $(d_{f}^{C})$ denotes the coefficients of the EDM
and the Chromo EDM (CEDM), where
$\sigma^{\mu\nu}=\frac{i}{2}[\gamma^{\mu},\gamma^{\nu}]$, $F_{\mu\nu}$ and
$G_{\mu\nu}^{A}$ are respectively the field strength tensor for the photon and
gluon with $A(=1,\dots,8)$ being the index for the $SU(3)_{C}$ adjoint
representation. In the second line, $C_{W}$ ($C_{ff^{\prime}}$) represents the
coefficients of the Weinberg operator Weinberg (1989) (four-fermi
interaction), where $f_{ABC}$ is the structure constant of the $SU(3)$ group
and $\tilde{G}^{A}_{\mu\nu}=\epsilon_{\mu\nu\rho\sigma}G^{A\,\rho\sigma}$.

We first consider one-loop contributions to the (C)EDM. The contributions from
the neutral Higgs boson $\varphi=\\{h,H,A\\}$ loops are calculated to be zero
shown as

$\displaystyle
d_{q}\propto\text{Im}[\Gamma_{\varphi}^{q}\Gamma_{\varphi}^{q}]_{11}=\text{Im}[(\Gamma_{\varphi}^{q})_{1i}(\Gamma_{\varphi}^{q})_{i1}]=\text{Im}[(\Gamma_{\varphi}^{q})_{1i}(\Gamma_{\varphi}^{q*})_{1i}]=\text{Im}[|(\Gamma_{\varphi}^{q})_{1i}|^{2}]=0.$
(27)

Here, we used the hermiticity of the matrices $\Gamma_{\varphi}^{q}$. The
contribution from the charged Higgs boson loop, shown in Fig. 1, is expressed
as

$\displaystyle
d_{d}^{\text{1-loop}}=\frac{em_{d}}{8\pi^{2}v^{2}}\text{Im}\left[Q_{u}\left(V_{\rm
CKM}^{\dagger}\Gamma_{A}^{u}\,G_{1}^{u}\,\bar{\Gamma}_{A}^{u}V_{\rm
CKM}\right)_{11}+\left(V_{\rm
CKM}^{\dagger}\Gamma_{A}^{u}\,G_{2}^{u}\,\bar{\Gamma}_{A}^{u}V_{\rm
CKM}\right)_{11}\right],$ (28) $\displaystyle
d_{u}^{\text{1-loop}}=\frac{em_{u}}{8\pi^{2}v^{2}}\text{Im}\left[Q_{d}\left(\bar{\Gamma}_{A}^{u}V_{\rm
CKM}G_{1}^{d}V_{\rm
CKM}^{\dagger}\Gamma_{A}^{u}\right)_{11}-\left(\bar{\Gamma}_{A}^{u}V_{\rm
CKM}\,G_{2}^{d}\,V_{\rm CKM}^{\dagger}\Gamma_{A}^{u}\right)_{11}\right],$ (29)

where $Q_{u}~{}(Q_{d})=2/3~{}(-1/3)$, and we use $\Gamma_{A}^{d}=V_{\rm
CKM}^{\dagger}\bar{\Gamma}_{A}^{u}V_{\rm CKM}$ with
$\bar{\Gamma}_{A}^{u}\equiv\Gamma_{A}^{u}|_{\text{diag}(-t_{\beta},-t_{\beta},t_{\beta}^{-1})\to\text{diag}(t_{\beta}^{-1},t_{\beta}^{-1},-t_{\beta})}$.
The matrices $G_{1,2}^{q}$ are defined as

$\displaystyle G_{i}^{u}$
$\displaystyle=\text{diag}\left[G_{i}\left(\frac{m_{H^{\pm}}^{2}}{m_{u}^{2}}\right),G_{i}\left(\frac{m_{H^{\pm}}^{2}}{m_{c}^{2}}\right),G_{i}\left(\frac{m_{H^{\pm}}^{2}}{m_{t}^{2}}\right)\right],$
(30) $\displaystyle G_{i}^{d}$
$\displaystyle=\text{diag}\left[G_{i}\left(\frac{m_{H^{\pm}}^{2}}{m_{d}^{2}}\right),G_{i}\left(\frac{m_{H^{\pm}}^{2}}{m_{s}^{2}}\right),G_{i}\left(\frac{m_{H^{\pm}}^{2}}{m_{b}^{2}}\right)\right]~{}~{}(i=1,2),$
(31)

with

$\displaystyle G_{1}(x)=\frac{1-4x+3x^{2}-2x^{2}\ln x}{2(1-x)^{3}},\quad
G_{2}(x)=\frac{1-x^{2}+2x\ln x}{2(1-x)^{3}}.$ (32)

Similarly, the contribution to the CEDM is obtained by replacing the external
photon with the external gluon in the left diagram of Fig. 1. We obtain

$\displaystyle d_{d}^{C\text{1-loop}}$
$\displaystyle=\frac{g_{s}m_{d}}{8\pi^{2}v^{2}}\text{Im}\left[V_{\rm
CKM}^{\dagger}\Gamma_{A}^{u}\,G_{1}^{u}\,\bar{\Gamma}_{A}^{u}V_{\rm
CKM}\right]_{11},~{}d_{u}^{C\text{1-loop}}=\frac{g_{s}m_{u}}{8\pi^{2}v^{2}}\text{Im}\left[\bar{\Gamma}_{A}^{u}V_{\rm
CKM}\,G_{1}^{d}\,V_{\rm CKM}^{\dagger}\Gamma_{A}^{u}\right]_{11},$ (33)

where $g_{s}$ is the $SU(3)_{C}$ gauge coupling constant.

The magnitude of $d_{d}^{(C)\text{1-loop}}$ is mainly determined by the top-
loop diagram, because the other contributions are suppressed by
$m_{q}^{2}/m_{H^{\pm}}^{2}$ ($q\neq t$). Thus, the one-loop contribution can
approximately be expressed as

$\displaystyle d_{d}^{\text{1-loop}}$
$\displaystyle\simeq\frac{em_{d}}{24\pi^{2}v^{2}}\left[2G_{1}\left(\frac{m^{2}_{H^{\pm}}}{m^{2}_{t}}\right)+3G_{2}\left(\frac{m^{2}_{H^{\pm}}}{m^{2}_{t}}\right)\right]\text{Im}\left[\left(V_{\rm
CKM}^{\dagger}\Gamma_{A}^{u}\right)_{13}\left(\bar{\Gamma}_{A}^{u}V_{\rm
CKM}\right)_{31}\right],$ (34) $\displaystyle d_{d}^{C\text{1-loop}}$
$\displaystyle\simeq\frac{g_{s}m_{d}}{8\pi^{2}v^{2}}G_{1}\left(\frac{m^{2}_{H^{\pm}}}{m^{2}_{t}}\right)\text{Im}\left[\left(V_{\rm
CKM}^{\dagger}\Gamma_{A}^{u}\right)_{13}\left(\bar{\Gamma}_{A}^{u}V_{\rm
CKM}\right)_{31}\right].$ (35)

We note that $d_{u}^{(C)\text{1-loop}}$ is negligibly smaller than
$d_{d}^{(C)\text{1-loop}}$ because of the absence of such heavy quark loops.

It is important to emphasize here that one-loop contributions to $d_{q}$ are
negligibly smaller than two-loop Barr-Zee type contributions in models with
flavor conservation at tree level such as the four types of the 2HDMs and
Yukawa aligned 2HDMs. This is because one-loop contributions from charged
Higgs boson loops to $d_{q}$ are significantly suppressed by the facotr of
$m_{d}\times m_{U}^{2}/m_{H^{\pm}}^{2}|V_{\rm CKM}^{Ud}|^{2}$ ($U=u,c,t$) for
$q=d$ and $m_{u}\times m_{D}^{2}/m_{H^{\pm}}^{2}|V_{\rm CKM}^{uD}|^{2}$
($D=d,s,b$) for $q=u$, while those from neutral Higgs boson loops vanish. On
the other hand, the two-loop Barr-Zee contribution with a neutral Higgs and a
photon exchange is only linearly suppressed by a small fermion mass and not
suppressed by a tiny CKM matrix element, so that it usually dominates with
respect to the one-loop contribution. The above statement does not hold in
models without flavor conservation at tree level as in our present model.

Figure 2: Contributions to $d_{d}$ from two-loop Barr-Zee type diagrams.

In order to show the dominance of the one-loop contribution, we consider
contributions from two-loop Barr-Zee type diagrams Barr (1990) to $d_{q}$.
Similar to the one-loop case, diagrams with neutral Higgs boson exchanges do
not contribute to the EDM, because of the hermiticity of the matrices
$\Gamma_{\varphi}^{q}$. Thus, the diagram with the charged Higgs boson
exchange, shown in Fig. 2, contributes to the EDM. In the alignment limit
$s_{\beta-\alpha}\to 1$, contributions from $W$-$W$-$h$ and $W$-$W$-$H$ loops
vanish, and the scalar-loop $d_{d}^{\text{BZ}}(S)$ and the fermion-loop
$d_{d}^{\text{BZ}}(F)$ give non-zero contributions. These are calculated as

$\displaystyle d_{d}^{\text{BZ}}(S)$
$\displaystyle=-\left(\frac{1}{16\pi^{2}}\right)^{2}\frac{eg^{2}}{2v}\lambda_{H^{+}H^{-}H}\sum_{k}\text{Im}(M_{L}^{1k}V_{\rm
CKM}^{k1})$ $\displaystyle\times\int_{0}^{1}dxx\left[C_{0}^{\rm
BZ}(\bar{m}_{H^{\pm}}^{2}+\tilde{m}_{H}^{2},m_{W}^{2},m_{H^{\pm}}^{2})+m_{u_{k}}^{2}D_{0}^{\rm
BZ}(\bar{m}_{H^{\pm}}^{2}+\tilde{m}_{H}^{2},m_{W}^{2},m_{u_{k}}^{2},m_{H^{\pm}}^{2})\right],$
(36) $\displaystyle d_{d}^{\text{BZ}}(F)$
$\displaystyle=-\left(\frac{1}{16\pi^{2}}\right)^{2}\frac{eg^{2}}{v^{2}}N_{c}\int_{0}^{1}dx\,\left(Q_{d}+\frac{1-x}{x}Q_{u}\right)$
$\displaystyle\times\sum_{i,j,k}\text{Im}\left[M_{L}^{1k}V_{\rm
CKM}^{k1}\left(x\,M_{L}^{*}V_{\rm CKM}^{*}M_{d}^{\rm
diag}+(x-2)\,M_{R}^{*}M_{u}^{\rm diag}V_{\rm CKM}^{*}\right)_{ij}\right]$
$\displaystyle\times\left[C_{0}^{\rm
BZ}(\bar{m}_{d_{i}}^{2}+\tilde{m}_{u_{j}}^{2},m_{W}^{2},m_{H^{\pm}}^{2})+m_{u_{k}}^{2}D_{0}^{\rm
BZ}(\bar{m}_{d_{i}}^{2}+\tilde{m}_{u_{j}}^{2},m_{W}^{2},m_{u_{k}}^{2},m_{H^{\pm}}^{2})\right],$
(37)

where $\bar{m}_{X}^{2}=m_{X}^{2}/(1-x)$, $\tilde{m}_{X}^{2}=m_{X}^{2}/x$, and
$M_{L,R}$ are given in Eq. (15). The scalar trilinear coupling
$\lambda_{H^{+}H^{-}H}$ is defined by the coefficient of the vertex
$H^{+}H^{-}H$ in the Lagrangian, which is extracted to be

$\displaystyle\lambda_{H^{+}H^{-}H}=\frac{2}{v}(m_{H}^{2}-m_{A}^{2})\cot
2\beta,$ (38)

at the alignment limit. The loop functions $C_{0}^{\rm BZ}$ and $D_{0}^{\rm
BZ}$ are given by

$\displaystyle C_{0}^{\rm BZ}(a,b,c)$ $\displaystyle=-\frac{a\ln
a}{(a-b)(a-c)}+\text{(cyclic)},$ (39) $\displaystyle D_{0}^{\rm BZ}(a,b,c,d)$
$\displaystyle=\frac{a\ln a}{(a-b)(a-c)(a-d)}+\text{(cyclic)},$ (40)

where (cyclic) denotes the other terms obtained by replacing the first term in
the cyclic way, e.g., $(a,b,c)\to(b,c,a)$. Similarly, we obtain

$\displaystyle d_{u}^{\text{BZ}}(S)$
$\displaystyle=\left(\frac{1}{16\pi^{2}}\right)^{2}\frac{eg^{2}}{2v}\lambda_{H^{+}H^{-}H}\sum_{k}\text{Im}(V_{\rm
CKM}^{1k}M_{R}^{k1})$ $\displaystyle\times\int_{0}^{1}dxx\left[C_{0}^{\rm
BZ}(\bar{m}_{H^{\pm}}^{2}+\tilde{m}_{H}^{2},m_{W}^{2},m_{H^{\pm}}^{2})+m_{d_{k}}^{2}D_{0}^{\rm
BZ}(\bar{m}_{H^{\pm}}^{2}+\tilde{m}_{H}^{2},m_{W}^{2},m_{d_{k}}^{2},m_{H^{\pm}}^{2})\right],$
(41) $\displaystyle d_{u}^{\text{BZ}}(F)$
$\displaystyle=-\left(\frac{1}{16\pi^{2}}\right)^{2}\frac{eg^{2}}{v^{2}}N_{c}\sum_{i,j,k}\int_{0}^{1}dx\left(Q_{u}+\frac{x}{1-x}Q_{d}\right)$
$\displaystyle\times\sum_{i,j,k}\text{Im}\left[V_{\rm
CKM}^{1k}M_{R}^{k1}\left((1+x)M_{L}^{*}V_{\rm CKM}^{*}M_{d}^{\rm
diag}-(1-x)M_{R}^{*}M_{u}^{\rm diag}V_{\rm CKM}^{*}\right)_{ij}\right]$
$\displaystyle\times\left[C_{0}^{\rm
BZ}(\bar{m}_{d_{i}}^{2}+\tilde{m}_{u_{j}}^{2},m_{W}^{2},m_{H^{\pm}}^{2})+m_{d_{k}}^{2}D_{0}^{\rm
BZ}(\bar{m}_{d_{i}}^{2}+\tilde{m}_{u_{j}}^{2},m_{W}^{2},m_{d_{k}}^{2},m_{H^{\pm}}^{2})\right].$
(42)

We note that the diagrams which are obtained by replacing the external photon
by the gluon in Fig. 2 vanish, because the color index is not closed in the
loop, so that the Barr-Zee type diagram does not contribute to $d_{q}^{C}$.

The nEDM $d_{n}$ can be calculated by a linear combination of the EDM and CEDM
by using the QCD sum rule, which are expressed as Abe  _et al._ (2014)

$\displaystyle
d_{n}=0.79d_{d}-0.20d_{u}+\frac{e}{g_{s}}(0.59d_{d}^{C}+0.30d_{u}^{C}).$ (43)

In our calculation, we identify
$d_{q}=d_{q}^{\text{1-loop}}+d_{q}^{\text{BZ}}$ and
$d_{q}^{C}=d_{q}^{C\text{1-loop}}$. The current upper limit on the nEDM
$d_{n}$ has been given by

$\displaystyle|d_{n}|<1.8\times
10^{-26}~{}e\,\text{cm}~{}~{}(90\%~{}~{}\text{CL}),$ (44)

from the nEDM experiment Abel  _et al._ (2020). In future, the bound on
$|d_{n}|$ will be improved to be $1\times 10^{-27}e\,\text{cm}$ by the n2EDM
experiment with its designed performance and to be of order
$10^{-28}~{}e\,\text{cm}$ by its possible modifications Ayres  _et al._
(2021).

We here comment on the other contributions to $d_{n}$ from the Weinberg
operator $C_{W}$ and the four-fermi interaction $C_{ff}$. It has been known
that the contribution from the charged Higgs boson exchange in two-loop
diagrams to $C_{W}$ vanishes Jung and Pich (2014); Logan  _et al._ (2021). In
addition, the contribution from the neutral Higgs boson exchanges to $C_{W}$
in two-loop diagrams and that to $C_{ff}$ in tree level diagrams vanish due to
the hermicity of the $\Gamma_{\varphi}^{q}$ matrices.

Let us also comment on the constraint from the electron EDM $d_{e}$ whose
current upper limit is given to be $|d_{e}|<1.1\times 10^{-29}e\,\text{cm}$ at
90% CL at the ACME experiment Andreev  _et al._ (2018). 777The ACME experiment
has been performed with thorium monoxide (ThO), which has a sensitivity to a
combination of $d_{e}$ and $C_{S}$ with the latter being related to the
coefficient of the electron-nucleon interaction
$(\bar{N}N)(\bar{e}i\gamma_{5}e)$. The value of $C_{S}$ can be estimated from
$C_{ff^{\prime}}$ which are zero at tree level in our model as mentioned
above, while the bound $|d_{e}|<1.1\times 10^{-29}e\,\text{cm}$ has been
obtained by taking $C_{S}$ to be zero Andreev  _et al._ (2018).  888In
addition to the EDM measured through paramagnetic systems such as ThO, that
from diamagnetic atoms, e.g., Hg Griffith  _et al._ (2009) and Xe Inoue  _et
al._ (2013) also provide a sensitivity to the coefficients given in Eq. (26).
It is, however, quite difficult to obtain robust constraints on the
coefficients from them due to huge theoretical uncertainties Jung and Pich
(2014).  The dominant contribution to $d_{e}$ comes from Barr-Zee diagrams
which are obtained by replacing the external down quark $d$ (internal up-type
quarks $u_{k}$) with the electron (neutrinos) in Fig. 2. Since there is no CPV
phase in the lepton Yukawa couplings for $H^{\pm}$ in the limit of massless
neutrinos, only the diagram with quark loops (right panel of Fig. 2)
contributes to $d_{e}$. Thus, $d_{e}$ can be expressed as

$\displaystyle d_{e}$
$\displaystyle\simeq-\left(\frac{1}{16\pi^{2}}\right)^{2}\frac{eg^{2}m_{e}}{v^{2}}t_{\beta}N_{c}\int_{0}^{1}dx\,\left(Q_{d}+\frac{1-x}{x}Q_{u}\right)$
$\displaystyle\times\sum_{i,j,k}\text{Im}\left[\left(x\,M_{L}^{*}V_{\rm
CKM}^{*}M_{d}^{\rm diag}+(x-2)\,M_{R}^{*}M_{u}^{\rm diag}V_{\rm
CKM}^{*}\right)_{ij}\right]C_{0}^{\rm
BZ}(\bar{m}_{d_{i}}^{2}+\tilde{m}_{u_{j}}^{2},m_{W}^{2},m_{H^{\pm}}^{2}).$
(45)

We confirm that the value of $|d_{e}|$ can be maximally about $7\times
10^{-31}\,e$ cm for $m_{H^{\pm}}=600$ GeV and $t_{\beta}=3$ by scanning the
parameters $(\theta,\psi,\delta_{4,5,6})$, so that we can safely avoid the
current upper limit. For $t_{\beta}>10$, $|d_{e}|$ can exceed the upper limit,
but such a large $t_{\beta}$ value is also excluded by the constraints from
$d_{n}$ as well as the flavor experiments as we will see in the next section.

##  IV Numerical evaluations

$\alpha_{\rm em}^{-1}$ Zyla  _et al._ (2020a) |  $m_{u}$ Xing  _et al._ (2008) |  $m_{c}$ Xing  _et al._ (2008) |  $m_{d}$ Xing  _et al._ (2008) |  $m_{s}$ Xing  _et al._ (2008) |  $m_{b}$ Xing  _et al._ (2008) |  $m_{B}$ Zyla  _et al._ (2020a)
---|---|---|---|---|---|---
137.036 | 1.9 MeV | 0.901 GeV | 4.22 MeV | 80 MeV |  $4.2$ GeV | 5.279 GeV
$m_{B_{s}}$ Zyla  _et al._ (2020a) |  $m_{D}$ Zyla  _et al._ (2020a) |  $m_{K}$ Zyla  _et al._ (2020a) |  $f_{B}$ Aoki  _et al._ (2020) |  $f_{B_{s}}$ Aoki  _et al._ (2020) |  $f_{D}$ Aoki  _et al._ (2020) |  $f_{K}$ Aoki  _et al._ (2020)
5.366 GeV | 1.865 GeV | 0.498 GeV | 0.190 GeV | 0.230 GeV | 0.212 GeV | 0.156 GeV

Table 3: SM input parameters for the calculation of the meson mixings and
${\cal B}(B\to X_{s}\gamma$), where we take $m_{B_{d}}=m_{B}$ and
$f_{B_{d}}=f_{B}$.

$\alpha_{\rm em}^{-1}$ Zyla  _et al._ (2020a) |  $\alpha_{s}$ Aoki  _et al._ (2020) |  $m_{u}$ Xing  _et al._ (2008) |  $m_{c}$ Xing  _et al._ (2008) |  $m_{d}$ Xing  _et al._ (2008) |  $m_{s}$ Xing  _et al._ (2008) |  $m_{b}$ Xing  _et al._ (2008)
---|---|---|---|---|---|---
127.952 | 0.1182 |  $1.27$ MeV |  $0.619$ GeV |  $2.93$ MeV |  $55$ MeV |  $2.89$ GeV

Table 4: SM input parameters at the $m_{Z}$ scale for the calculation of the
EDM constraints.

$m_{t}$ Xing  _et al._ (2008) |  $m_{Z}$ Zyla  _et al._ (2020a) |  $m_{W}$ Zyla  _et al._ (2020a) |  $A$ Charles  _et al._ (2015) |  $\lambda$ Charles  _et al._ (2015) |  $\overline{\rho}$ Charles  _et al._ (2015) |  $\overline{\eta}$ Charles  _et al._ (2015)
---|---|---|---|---|---|---
$171.7$ GeV | 91.1876 GeV | 80.379 GeV | 0.810 | 0.22548 | 0.145 | 0.343

Table 5: Common SM input parameters.

In this section, we numerically evaluate the constraint on the parameter space
from the meson mixings, the $B\to X_{s}\gamma$ decay and the nEDM $d_{n}$
discussed in Sec. III. We then study decays of the additional Higgs bosons in
a few benchmark points which are allowed by all the constraints considered in
this section.

We use the SM input parameters summarized in Tables 5, 5 and 5. In Table 5,
$A$, $\lambda$, $\bar{\rho}$ and $\bar{\eta}$ denote the Wolfenstein
parameters for the CKM matrix defined as Zyla  _et al._ (2020b) :

$\displaystyle
V_{\text{CKM}}=\left(\begin{matrix}1-\lambda^{2}/2&\lambda&A\lambda^{3}(\bar{\rho}-i\bar{\eta})\\\
-\lambda&1-\lambda^{2}/2&A\lambda^{2}\\\
A\lambda^{3}(1-\bar{\rho}-i\bar{\eta})&-A\lambda^{2}&1\\\
\end{matrix}\right)+\mathcal{O}(\lambda^{4}).$ (46)

In the following, we particularly focus on the case with the alignment limit
$s_{\beta-\alpha}\to 1$ for simplicity, in which the $h$ state has the same
couplings as those of the SM Higgs boson at tree level, although a small
deviation from the alignment limit is allowed by the current LHC data ATL
(2021); CMS (2020), e.g., $|c_{\beta-\alpha}|\lesssim 0.1$ at $\tan\beta\simeq
1$ in the Type-II 2HDM. Such a deviation can mainly affect on the constraint
from the meson mixings as discussed in Sec. III.1, while its influence on the
EDMs can be negligible, because the hermiticity of the $\Gamma_{\varphi}^{q}$
matrices also works for the non-alignment case. For the impact of the
deviation from the alignment limit on the meson mixings, see Ref. Okada  _et
al._ (2016).

We also would like to mention constraints on the masses of the additional
Higgs bosons from experiments other than flavor ones. First, they can be
constrained from the EW oblique parameters such as the $S$, $T$ and $U$
parameters Peskin and Takeuchi (1990). In particular, the $T$ parameter
severely constrains the mass difference between charged and neutral Higgs
bosons, because it breaks the custodial $SU(2)$ symmetry Pomarol and Vega
(1994). It has been known that new contributions to the $T$ parameters vanish
in the case with $m_{A}=m_{H^{\pm}}$, and also those to $S$ disappear for
$m_{H}=m_{A}=m_{H^{\pm}}$, see e.g., Refs. Kanemura  _et al._ (2011); Blasi
_et al._ (2017). We thus take the degenerate mass case
($m_{H}=m_{A}=m_{H^{\pm}}$) in what follows, in which the constraints from the
meson mixings also become quite milder than the case with $m_{H}\neq m_{A}$.
The mass of the additional Higgs bosons can also be constrained from direct
searches for additional Higgs bosons at LHC. As we will see in Sec. IV.3,
their decay products typically include top quarks e.g., $H/A\to
t\bar{u},~{}t\bar{c}$, $t\bar{t}$ and $H^{\pm}\to tb$. The current LHC data do
not exclude scenarios with additional Higgs bosons which dominantly decay into
the above modes Hou and Modak (2021).

###  IV.1 Case without new phases

We first discuss the case without the new CPV phases, i.e.,
$\delta_{4,5,6}=0$, in which the CPV effect purely comes from the CKM phase.
As we will see at the end of this subsection, the contribution from the two-
loop Barr-Zee type diagrams to $d_{n}$ is negligibly smaller than the one-loop
contribution. Thus, we focus on the one-loop contribution.

Using the Wolfenstein parameterization given in Eq. (46), the approximate
formulae for $d_{d}^{(C)\text{1--loop}}$ given in Eqs. (34) and (35) are
expressed up to ${\cal O}(\lambda^{5})$ as

$\displaystyle\frac{d_{d}^{\text{1--loop}}}{e}$
$\displaystyle\simeq\frac{A\bar{\eta}\lambda^{3}}{24\pi^{2}}\frac{m_{d}(1-t_{\beta}^{2})}{v^{2}s_{\beta}^{2}}c_{\theta}c_{\psi}\left(s_{\theta}-\lambda
c_{\theta}s_{\psi}-\frac{\lambda^{2}}{2}s_{\theta}\right)\left[2G_{1}\left(\frac{m^{2}_{H^{\pm}}}{m^{2}_{t}}\right)+3G_{2}\left(\frac{m^{2}_{H^{\pm}}}{m^{2}_{t}}\right)\right],$
(47) $\displaystyle\frac{d^{C\text{1--loop}}_{d}}{g_{s}}$
$\displaystyle\simeq\frac{A\bar{\eta}\lambda^{3}}{8\pi^{2}}\frac{m_{d}(1-t_{\beta}^{2})}{v^{2}s_{\beta}^{2}}c_{\theta}c_{\psi}\left(s_{\theta}-\lambda
c_{\theta}s_{\psi}-\frac{\lambda^{2}}{2}s_{\theta}\right)G_{1}\left(\frac{m^{2}_{H^{\pm}}}{m^{2}_{t}}\right).$
(48)

As we can see that the dominant contribution is proportional to $\lambda^{3}$,
because one has to pick up the CPV parameter $\bar{\eta}$ in the element
$V_{\rm CKM}^{13}$ or $V_{\rm CKM}^{31}$ at least once. Before numerical
evaluations, let us give a few remarks on the above approximate formulae as
follows:

  1. 1.

For $\psi=\pi/2+n\pi$ $(n\in\mathbb{Z})$, $d_{d}^{(C)\text{1--loop}}$ vanishes
up to ${\cal O}(\lambda^{5})$ as it is proportional to $c_{\psi}$. In this
case, the matrices $\Gamma_{A}^{u}$ and $\bar{\Gamma}_{A}^{u}$ are a block
diagonal form with $2\times 2$ and $1\times 1$ blocks, so that the product of
the combinations $(V_{\rm CKM}^{\dagger}\Gamma_{A}^{u})_{13}[=(V_{\rm
CKM}^{*})_{31}(\Gamma_{A}^{u})_{33}]$ and $(\bar{\Gamma}_{A}^{u}V_{\rm
CKM})_{31}[=(\bar{\Gamma}_{A}^{u})_{33}(V_{\rm CKM})_{31}]$ gives a real value
at order $\lambda^{6}$.

  2. 2.

For $\theta=\pi/2+n\pi$ $(n\in\mathbb{Z})$, $d_{d}^{(C)\text{1--loop}}$
vanishes at all orders of $\lambda$. In this case, the matrices
$\Gamma_{A}^{u}$ and $\bar{\Gamma}_{A}^{u}$ are diagonal forms, and the
combinations $(V_{\rm
CKM}^{\dagger}\Gamma_{A}^{u}\,G_{i}^{u}\,\bar{\Gamma}_{A}^{u}V_{\rm CKM})$ and
$(\bar{\Gamma}_{A}^{u}V_{\rm CKM}G_{1}^{d}V_{\rm
CKM}^{\dagger}\Gamma_{A}^{u})$ appearing in Eqs. (28) and (29) are hermitian
matrices.

  3. 3.

For $t_{\beta}=1$, $d_{d}^{(C)\text{1--loop}}$ vanishes at all orders of
$\lambda$. In this case, $\bar{\Gamma}_{A}^{u}=-\Gamma_{A}^{u}$ holds, so that
$(V_{\rm CKM}^{\dagger}\Gamma_{A}^{u}\,G_{i}^{u}\,\bar{\Gamma}_{A}^{u}V_{\rm
CKM})$ and $(\bar{\Gamma}_{A}^{u}V_{\rm CKM}G_{1}^{d}V_{\rm
CKM}^{\dagger}\Gamma_{A}^{u})$ are hermitian matrices.

  4. 4.

$|d_{d}^{(C)\text{1--loop}}|$ is centrosymmetric about
$(\theta,\psi)=(n\pi/2,m\pi)$ ($n,m\in\mathbb{Z}$) and symmetric about the
lines with $\psi=\pi/2+n\pi$ $(n\in\mathbb{Z})$.

Figure 3: Comparison of the value of $|d_{n}/e|$ for $\theta=\psi=\pi/3$ and
$\delta_{4,5,6}=0$ by using the approximate formulae given in Eqs. (47) and
(48) up to $\mathcal{O}(\lambda^{3})$, $\mathcal{O}(\lambda^{4})$ and
$\mathcal{O}(\lambda^{5})$ and by using the exact formulae given in Eqs. (28),
(29) and (33). We fix $m_{H^{\pm}}=400$ GeV (left) and $\tan\beta=3$ (right).

In Fig. 3, we show the comparison of the numerical values of $d_{n}$ at one-
loop level by using the exact formula given in Eqs. (28), (29) and (33) and
the approximate one given in Eqs. (47) and (48) as a function of $t_{\beta}$
(left panel) and $m_{H^{\pm}}$ (right panel). For the approximate formula, we
use the expression up to $\mathcal{O}(\lambda^{3})$,
$\mathcal{O}(\lambda^{4})$ and $\mathcal{O}(\lambda^{5})$. We here fix the
mixing angles $\theta=\psi=\pi/3$ as an example, but we confirm that the
validity of the approximation is not spoiled depending on values of the mixing
angles except for the case with $\psi=\pi/2+n\pi$ ($n\in\mathbb{Z}$) 999In
this case, the approximation is no longer good, because the contribution from
${\cal O}(\lambda^{6})$ turns out to be dominant. It, however, gives
negligibly smaller values of $|d_{n}|$ as compared with the current upper
limit, so that we do not need to care about such breakdown of the
approximation. . It is clear that the approximation with the
$\mathcal{O}(\lambda^{5})$ term is quite good agreement with the exact one. In
addition, we see that $|d_{n}|$ increases by $t_{\beta}^{2}$ for
$t_{\beta}>1$, and decreases by $m_{t}^{2}/m_{H^{\pm}}^{2}$ for
$m_{H^{\pm}}\gg m_{t}$ as they can be read from Eqs. (47) and (48).

Figure 4: Contour plots of $|d_{n}|$ on the $\theta$–$\psi$ plane with
$t_{\beta}=2$ and $m_{H^{\pm}}=600$ GeV (left panel) and
$t_{\beta}$–$m_{H^{\pm}}$ plane with $\theta=-7\pi/16$ and $\psi=\pi/8$ (right
panel). We take the new phases to be zero ($\delta_{4,5,6}=0$) and degenerate
masses as $m_{A}=m_{H}=m_{H^{\pm}}$. The shaded regions are excluded by the
constraint from the $B^{0}$-$\bar{B}^{0}$ (red), $D^{0}$-$\bar{D}^{0}$ (blue),
$K^{0}$-$\bar{K}^{0}$ (brown) and $B\to X_{s}\gamma$ (black) data,
respectively. The green-solid, blue-dashed and yellow-dotted contours denote
$|d_{n}|=1.8\times 10^{-26}\,e\,\text{cm}$, $9.0\times 10^{-27}$ and
$1.0\times 10^{-27}\,e\,\text{cm}$, respectively.

Next in Fig. 4, we show various parameter dependences, e.g., $\theta$, $\psi$,
$t_{\beta}$ and $m_{H^{\pm}}$, of $|d_{n}|$ under the constraints from the
meson mixings and $B\to X_{s}\gamma$ data. We numerically find that $|d_{n}|$
takes a maximum value at $(\theta,\psi)\simeq(-0.79,0.18)$, and these values
almost do not depend on the choice of $t_{\beta}$ and $m_{H^{\pm}}$, as it is
expected from Eqs. (47) and (48). From the left panel of Fig. 4, we see that
the wide region of the parameter space is excluded by the flavor constraints,
while the allowed region typically predicts values of $|d_{n}|$ with one order
of magnitude smaller than the current upper limit, which can be explored at
the n2EDM experiment. The right panel shows the constrained region on the
$t_{\beta}$–$m_{H^{\pm}}$ plane with the fixed values of the mixing angles
$\theta=-7\pi/16$ and $\psi=\pi/8$, which are allowed by all the three
constraints for $t_{\beta}=2$ and $m_{H^{\pm}}=600$ GeV, see the left panel.
We can extract the upper limit on $t_{\beta}$ to be about 3 (5) for
$m_{H^{\pm}}=400$ (800) GeV from the meson mixings ($|d_{n}|$). Remarkably,
the light charged Higgs boson with a mass of about 250 GeV is allowed
depending on $\theta$, $\psi$ and $t_{\beta}$ by the $B\to X_{s}\gamma$ data.
Such a light charged Higgs boson scenario has been excluded in the Type-II
2HDM.

Figure 5: Contour plots of the contribution from two-loop Barr-Zee diagrams to
$|d_{n}|$ on the $\theta$–$\psi$ plane with $t_{\beta}=2$ and
$m_{H^{\pm}}=600$ GeV (left panel) and $t_{\beta}$–$m_{H^{\pm}}$ plane with
$\theta=-7\pi/16$ and $\psi=\pi/8$ (right panel). We take the new phases to be
zero ($\delta_{4,5,6}=0$) and degenerate masses as $m_{A}=m_{H}=m_{H^{\pm}}$.
The solid, dashed and dotted contours denote $|d_{n}|=1.0\times
10^{-28}\,e\,\text{cm}$, $1.0\times 10^{-29}$ and $1.0\times
10^{-30}\,e\,\text{cm}$, respectively.

As aforementioned, the contributions from the Barr-Zee diagram are negligibly
smaller than the one-loop contribution which is shown in Fig. 4. In order to
numerically show this, we exhibit the value of $|d_{n}|$ from the Barr-Zee
diagram only in Fig. 5 with the same configuration used in Fig. 4. From Fig.
5, we see that the contribution to $|d_{n}|$ is typically of order $10^{-29}$
or smaller, which is roughly two or more than two orders of magnitude smaller
than the one-loop contribution. This behavior does not significantly change
for the case with non-zero phases. We thus can safely neglect the Barr-Zee
diagrams.

###  IV.2 Non-zero phases

Figure 6: Contour plots of $|d_{n}|$ with non-zero CPV phases $\delta_{4,5,6}$
on the $\delta_{4}$–$\delta_{6}$ plane with $t_{\beta}=2$,
$m_{H}=m_{A}=m_{H^{\pm}}$, $\theta=-7\pi/16$ and $\psi=\pi/8$. We take
$m_{H^{\pm}}=400$ GeV for the upper panels and 600 GeV for the lower panels.
The remaining phase $\delta_{5}$ is taken to be $\delta_{5}=0$ (left panel),
$\delta_{5}=1$ (center panel) and $\delta_{5}=-1$ (right panel). The regions
shaded by black color are excluded by the constraint from $B\to X_{s}\gamma$
data. The dashed and dotted contours denote $|d_{n}|=9.0\times 10^{-27}$ and
$1.0\times 10^{-27}\,e\,\text{cm}$, respectively.

Next, we discuss the case with non-zero phases $\delta_{4,5,6}$. In Fig. 6, we
show the region constrained from the meson mixings, $B\to X_{s}\gamma$ data
and the nEDM on the $\delta_{4}$–$\delta_{6}$ plane for $m_{H^{\pm}}=400$ GeV
(upper panels) and 600 GeV (lower panels). The other parameters are fixed as
explicitly written in the caption. From the left panels ($\delta_{5}=0$), we
see that at the point of the no phase limit $(\delta_{4},\delta_{6})=(0,0)$,
the value of $|d_{n}|$ is given of order $10^{-27}$, while the region with
$\delta_{4,6}\neq 0$ the value of $|d_{n}|$ can quickly glow up to $10^{-26}$.
We also show the similar plots for $\delta_{5}=1$ (center) and $\delta_{5}=-1$
(right), where the pattern of the prediction for $|d_{n}|$ is similar to the
case with $\delta_{5}=0$, but the region allowed by the $B\to X_{s}\gamma$
data is shifted to upper and lower regions, respectively.

To conclude, we clarify that the new CPV phases $\delta_{4,5,6}$ with
$\mathcal{O}(1)$ are allowed by the current $B$ physics observables and the
EDM data even for the case with a relatively light charged Higgs boson with
the mass of a few hundred GeV. It should be emphasized that our scenario
becomes quite similar to the Type-II 2HDM in the no-mixing limit, i.e.,
$\theta=\psi=0$, so that the constraint from the $B\to X_{s}\gamma$ decay is
severe, i.e., $m_{H^{\pm}}\gtrsim 600$ GeV. This, however, can drastically be
changed for the case with non-zero mixing and/or non-zero additional CPV
phases, where the constraint from the $B\to X_{s}\gamma$ decay is
significantly relaxed which makes a light charged Higgs boson scenario
possible. Such a scenario is also possible in the Type-I or Type-X 2HDM, but
our scenario can be distinguished from them by measuring flavor violating
decays of the additional Higgs bosons, e.g., $H/A\to tc$ and/or flavor
dependent deviations in the $h$ couplings from the SM predictions which do not
appear in the $Z_{2}$ symmetric 2HDMs, see Ref. Okada  _et al._ (2016). In
addition, our scenario can be probed by future EDM experiments, e.g., n2EDM,
and also can directly be tested at collider experiments by looking at decays
of additional Higgs bosons, see e.g., Kanemura  _et al._ (2021) for the probe
of CPV phases at electron-positron colliders.

###  IV.3 Decays of the additional Higgs bosons

| $\theta$ | $\psi$ | $\delta_{4}$ | $~{}\delta_{5}~{}$ | $\delta_{6}$ | $\tan\beta$ |  $m_{H}(=m_{A}=m_{H^{\pm}})$ [GeV]
---|---|---|---|---|---|---|---
BP1 | $-0.438\pi$ | $0.125\pi$ | $-0.250\pi$ | 0 | $-0.250\pi$ | 1.5 | 300
BP2 | $0.223\pi$ | $-0.446\pi$ | $-0.250\pi$ | 0 | $-0.250\pi$ | 2.5 | 300
BP3 | $-0.438\pi$ | $0.125\pi$ | $-0.250\pi$ | 0 | $-0.250\pi$ | 3 | 500
BP4 | $-0.125\pi$ | $0.0625\pi$ | $-0.271\pi$ | 0 | $-0.0637\pi$ | 3 | 500

Table 6: Input parameters for each benchmark point.

|  $|\Delta m_{K}|$ [MeV] |  $|\Delta m_{D}|$ [MeV] |  $|\Delta m_{B_{d}}|$ [MeV] |  $|\Delta m_{B_{s}}|$ [MeV] | ${\cal B}(B\to X_{s}\gamma)$ |  $|d_{n}|/e$ [cm]
---|---|---|---|---|---|---
BP1 | $3.17\times 10^{-12}$ | $\simeq 0$ | $1.47\times 10^{-10}$ | $1.22\times 10^{-10}$ | $3.06\times 10^{-4}$ | $4.24\times 10^{-27}$
BP2 | $3.82\times 10^{-13}$ | $\simeq 0$ | $4.50\times 10^{-11}$ | $8.52\times 10^{-10}$ | $3.26\times 10^{-4}$ | $1.05\times 10^{-26}$
BP3 | $2.70\times 10^{-12}$ | $\simeq 0$ | $1.26\times 10^{-10}$ | $1.05\times 10^{-10}$ | $3.29\times 10^{-4}$ | $1.25\times 10^{-26}$
BP4 | $1.87\times 10^{-13}$ | $2.82\times 10^{-13}$ | $2.13\times 10^{-10}$ | $3.06\times 10^{-10}$ | $3.57\times 10^{-4}$ | $1.31\times 10^{-26}$

Table 7: Predictions of flavor observables for each benchmark point.

|  ${\cal B}(H)$ [%] |  ${\cal B}(A)$ [%] |  ${\cal B}(H^{\pm})$ [%]
---|---|---|---
BP1 | 98.3 ($tu$) | 0.8 ($gg$) | 0.6 ($tc$) | 98.8 ($tu$) | 0.6 ($tc$) | 0.3 ($gg$) | 93.3 ($tb$) | 6.6 ($td$) | 0.04 ($ts$)
BP2 | 56.0 ($tc$) | 41.0 ($tu$) | 2.4 ($gg$) | 57.0 ($tc$) | 41.7 ($tu$) | 0.8 ($gg$) | 97.7 ($tb$) | 1.3 ($ts$) | 0.9 ($td$)
BP3 | 92.0 ($tt$) | 7.6 ($tu$) | 0.3 ($gg$) | 95.8 ($tt$) | 4.2 ($tu$) | 0.02 ($tc$) | 96.0 ($tb$) | 3.9 ($td$) | 0.02 ($ts$)
BP4 | 80.0 ($tu$) | 17.7 ($tc$) | 2.0 ($tt$) | 78.6 ($tu$) | 17.4 ($tc$) | 3.8 ($tt$) | 78.4 ($td$) | 17.4 ($ts$) | 4.2 ($tb$)

Table 8: Predictions of the biggest three decay branching ratios of $H$, $A$
and $H^{\pm}$ for each benchmark point. The decay mode is written in the
parentheses. For the branching ratios of flavor violating modes of $H$ and
$A$, we take the sum of possible two final states, e.g., $H/A\to t\bar{c}$ and
$H/A\to c\bar{t}$.

We briefly discuss decays of the additional Higgs bosons. In the case with
$s_{\beta-\alpha}=1$ and $m_{H}=m_{A}=m_{H^{\pm}}$, the additional Higgs
bosons can mainly decay into a fermion pair and sub-dominantly decay into two
gluons, where the latter is induced at one-loop level for $H$ and $A$. The
other loop induced decays $H/A\to\gamma\gamma/Z\gamma$ and $H^{\pm}\to
W^{\pm}\gamma/W^{\pm}Z$ are negligible Aiko  _et al._ (2021). The analytic
expressions for the relevant decay rates are given in Appendix C.

In Table 8, we give four benchmark points (BP1-BP4) all of which are allowed
by the constraints from various flavor experiments as shown in Table 8. In
Table 8, we show the predictions of the branching ratios of $H$, $A$ and
$H^{\pm}$ for each benchmark point. At the BP1 and BP2, the masses of $H$ and
$A$ are taken to be 300 GeV, so that they cannot decay into a top-quark pair.
In such a case, these additional Higgs bosons can typically decay into
$b\bar{b}$ or $\tau^{+}\tau^{-}$ in the four types of the 2HDMs Aoki  _et al._
(2009), but they dominantly decay into $ts$ or $tc$ via the flavor violating
interaction in our scenario. In these benchmark points, $H^{\pm}$ dominantly
decay into $tb$ and sub-dominantly decay into $td$ and $ts$. At the BP3 and
BP4, the masses of the additional Higgs bosons are taken to be 500 GeV, so
that they can decay into a top-quark pair. We find that even in such a heavier
case, $H$ and $A$ ($H^{\pm}$) mainly decay into $tu$ and $tc$ ($td$ and $ts$)
at the BP4. Therefore, the flavor violating decays of the additional Higgs
bosons can be important to test our scenario. Recently in Ref. Hou and Modak
(2021), LHC phenomenology for flavor violating Higgs bosons has been discussed
in the 2HDM without a $Z_{2}$ symmetry. It has been shown that the case with
the magnitude of a parameter $\rho_{tc}$, corresponding to our
$\sqrt{2}m_{t}(\Gamma_{A}^{u})_{32}/v$, to be larger than about 0.5 and with
the mass range of $350\lesssim m_{H/A}\lesssim 570$ GeV can be proved with
5$\sigma$ level from the three top process assuming 14 TeV for the collision
energy and 3000 fb-1 for the integrated luminosity Hou and Modak (2021). We
note that $|\rho_{tc}|$ is given to be 0.03, 0.28, 0.044 and 0.55 at the BP1,
BP2, BP3 and BP4, respectively. Thus, our scenario, especially BP4, can be
proved at LHC.

##  V Conclusions

We have discussed a CPV 2HDM emerging from 3-3-1 models at the EW scale, which
are motivated to explain the three generation structure of chiral fermions.
All the non-SM fermions and gauge bosons are decoupled from the theory in our
approach. We have shown that 3-3-1 models categorized as Class-I deduce the
same structure of the 2HDM, in which flavor violating Higgs-quark couplings
appear, and they contain additional CPV phases. The Yukawa couplings for
quarks can be parameterized not only by $\tan\beta$ but also the mixing
parameters $\theta$, $\psi$ and the new phases $\delta_{4,5,6}$ which arise
from the bi-unitary transformation of up-type and down-type quarks. These new
parameters do not appear in the softly-broken $Z_{2}$ symmetric 2HDMs, i.e.,
the Type-I, Type-II, Type-X and Type-Y 2HDMs, because of the flavor universal
structure of the Yukawa interaction. If we take additional mixing and phase
parameters to be zero, our scenario becomes Type-II 2HDM like as long as we
neglect Yukawa interactions with the first and second generation fermions.

We then have studied constraints on the parameter space from flavor
observables such as the $B^{0}$-$\bar{B}^{0}$, $D^{0}$-$\bar{D}^{0}$,
$K^{0}$-$\bar{K}^{0}$ mixings, the $B\to X_{s}\gamma$ decay and the EDMs
particularly for the nEDM $d_{n}$. Because of the characteristic flavor
structure, the neutral Higgs bosons do not contribute to the EDMs at one-loop
and two-loop Barr-Zee diagrams. It has been clarified that the dominant
contribution comes from charged Higgs boson loops at one-loop level. We have
found that even in the case without new phases, i.e., $\delta_{4,5,6}=0$ the
charged Higgs boson contribution to $d_{n}$ can be sizable due to the CKM
phase and the non-zero mixing angles $\theta$ and $\psi$, which can be of
order $10^{-27}$ in the allowed parameter region. We also have found that the
constraint from the $B\to X_{s}\gamma$ decay can significantly be relaxed as
compared with the Type-II 2HDM in the case with $\theta,~{}\psi\neq 0$, and
have shown that a light charged Higgs boson with a few hundred GeV is
possible. For the case with non-zero CPV phases, i.e., $\delta_{4,5,6}\neq 0$,
we see that the effect of these phases on $d_{n}$ is constructive in a portion
of the parameter space which are already excluded by the current measurement
of $d_{n}$. On the other hand, there are regions where the new phases give a
destructive effect, in which new phases of order one can be allowed under the
flavor constraints. Such sizable CPV phases can be indirectly be tested at
future EDM experiments such as the n2EDM experiment, and can directly be
tested at collider experiments such as the high-luminosity LHC and future
electron-positron colliders.

###### Acknowledgements.

The work of KY was supported in part by Grant-in-Aid for Early-Career
Scientists, No. 19K14714.

##  Appendix A Masses of Higgs bosons

We discuss details of the mass spectrum for scalar bosons in the model with
$\zeta=-1/\sqrt{3}$ whose particle content is given in Table 1.

There are five neutral components in the three Higgs triplets, which can be
parameterized as

$\displaystyle\phi_{i}^{0}=\frac{1}{\sqrt{2}}(v_{i}+h_{i}+ia_{i})~{}~{}(i=1,2,3),\quad\eta_{j}^{0}=\frac{1}{\sqrt{2}}(v_{j}^{\prime}+\eta_{j}^{R}+i\eta_{j}^{I})~{}~{}(j=2,3).$
(49)

We here introduce the VEVs for the $\eta_{j}^{0}$ fields, but either
$v_{2}^{\prime}$ and $v_{3}^{\prime}$ can be taken to be zero by the field
redefinition of $\Phi_{2}$ and $\Phi_{3}$ without loss of generality. We thus
set $v_{2}^{\prime}=0$. From the tadpole condition for $\eta_{2}^{R}$, we
obtain

$\displaystyle
m_{23}^{2}v_{3}+\frac{v_{3}^{\prime}s_{\beta}}{2}\left(vv_{3}\rho_{23}+\frac{vM^{2}}{v_{3}}\right)=0.$
(50)

For the case with $m_{23}^{2}=0$101010This can be regarded by the choice of
the soft-breaking term of the $U(1)^{\prime}$ symmetry such that the subgroup
$\tilde{Z}_{2}$ survives, where its charge is defined by
$(-1)^{|Q^{\prime}|/q}$ Das  _et al._ (2020). , this can be satisfied by
taking $v_{3}^{\prime}=0$ for arbitrary values of $v$ and $v_{3}$. We take
this configuration throughout the paper. In this setup, the tadpole condition
for $\eta_{3}^{R}$ is automatically satisfied, while those for $h_{1,2,3}$
states are expressed under the assumption with $v,v_{3}\neq 0$ as

$\displaystyle
m_{1}^{2}-M^{2}s_{\beta}^{2}+\frac{\lambda_{1}}{2}v^{2}c_{\beta}^{2}+\frac{\lambda_{12}}{2}v^{2}s_{\beta}^{2}+\frac{\lambda_{13}}{2}v_{3}^{2}=0,$
(51) $\displaystyle
m_{2}^{2}-M^{2}c_{\beta}^{2}+\frac{\lambda_{12}}{2}v^{2}c_{\beta}^{2}+\frac{\lambda_{2}}{2}v^{2}s_{\beta}^{2}+\frac{\lambda_{23}}{2}v_{3}^{2}=0,$
(52) $\displaystyle
m_{3}^{2}-M^{2}\frac{v^{2}}{v_{3}^{2}}c_{\beta}^{2}s_{\beta}^{2}+\frac{\lambda_{13}}{2}v^{2}c_{\beta}^{2}+\frac{\lambda_{23}}{2}v^{2}s_{\beta}^{2}+\frac{\lambda_{3}}{2}v_{3}^{2}=0,$
(53)

where $M^{2}\equiv\mu v_{3}/(\sqrt{2}c_{\beta}s_{\beta})$.

The mass eigenstates for the $Z_{2}^{\rm rem}$-even scalars can be defined as

$\displaystyle\begin{pmatrix}\phi_{1}^{\pm}\\\ \phi_{2}^{\pm}\\\
\end{pmatrix}$ $\displaystyle=R(\beta)\begin{pmatrix}G^{\pm}\\\ H^{\pm}\\\
\end{pmatrix},~{}R(x)=\begin{pmatrix}\cos x&\sin x\\\ -\sin x&\cos
x\end{pmatrix}.$ (54) $\displaystyle\begin{pmatrix}a_{1}\\\ a_{2}\\\
a_{3}\end{pmatrix}$
$\displaystyle=\begin{pmatrix}-\frac{vc_{\beta}}{\sqrt{v_{3}^{2}+v^{2}c_{\beta}^{2}}}&-c_{\beta}&\frac{v_{3}s_{\beta}}{\sqrt{v_{3}^{2}+s_{\beta}^{2}c_{\beta}^{2}v^{2}}}\\\
0&s_{\beta}&\frac{v_{3}c_{\beta}}{\sqrt{v_{3}^{2}+s_{\beta}^{2}c_{\beta}^{2}v^{2}}}\\\
\frac{v_{3}}{\sqrt{v_{3}^{2}+v^{2}c_{\beta}^{2}}}&0&\frac{vs_{\beta}c_{\beta}}{\sqrt{v_{3}^{2}+s_{\beta}^{2}c_{\beta}^{2}v^{2}}}\end{pmatrix}\begin{pmatrix}G_{1}^{0}\\\
G_{2}^{0}\\\ A\\\ \end{pmatrix},~{}\begin{pmatrix}h_{1}\\\ h_{2}\\\
h_{3}\end{pmatrix}=R_{H}\begin{pmatrix}H\\\ h\\\ H_{S}\end{pmatrix}.$ (55)

where $G^{\pm}$, $G_{1}^{0}$ and $G_{2}^{0}$ are the Nambu-Goldstone (NG)
bosons, and $h$ can be identified with the discovered Higgs boson with a mass
of 125 GeV. The matrix $R_{H}$ is the $3\times 3$ orthogonal matrix. We note
that the eigenvectors for $G_{1}^{0}$ and $G_{2}^{0}$, corresponding to the
first and second column vectors of the $3\times 3$ matrix in Eq. (55),
respectively, are not orthogonal, so that the orthogonal NG states are
obtained by taking an appropriate basis transformation of
$(G_{1}^{0},G_{2}^{0})$. The masses of physical scalar states are calculated
as

$\displaystyle m_{H^{\pm}}^{2}$
$\displaystyle=M^{2}+\frac{\rho_{12}}{2}v^{2},\quad
m_{A}^{2}=M^{2}\left(1+\frac{v^{2}}{v_{3}^{2}}s_{\beta}^{2}c_{\beta}^{2}\right),\quad
m_{H_{i}}^{2}=(R_{H}^{T}M_{H}^{2}R_{H})_{ii},$ (56)

where $M_{H}^{2}$ is the squared mass matrix for CP-even Higgs bosons in the
basis of $(h_{1},h_{2},h_{3})$ given as

$\displaystyle
M_{H}^{2}=\begin{pmatrix}v^{2}c_{\beta}^{2}\lambda_{1}+M^{2}s_{\beta}^{2}&(v^{2}\lambda_{12}-M^{2})s_{\beta}c_{\beta}&\frac{v}{v_{3}}c_{\beta}(v_{3}^{2}\lambda_{13}-M^{2}s_{\beta}^{2})\\\
&v^{2}s_{\beta}^{2}\lambda_{2}+M^{2}c_{\beta}^{2}&\frac{v}{v_{3}}s_{\beta}(v_{3}^{2}\lambda_{23}-M^{2}c_{\beta}^{2})\\\
&&v_{3}^{2}\lambda_{3}+\frac{v^{2}}{v_{3}^{2}}M^{2}s_{\beta}^{2}c_{\beta}^{2}\end{pmatrix}.$
(57)

For the $Z_{2}^{\text{rem}}$-odd fields, the mass eigenstates are defined as
follows:

$\displaystyle\left(\begin{matrix}\eta^{\pm}_{3}\\\
\eta_{1}^{\pm}\end{matrix}\right)=R(\gamma)\left(\begin{matrix}G^{\pm}_{\eta}\\\
\eta^{\pm}\end{matrix}\right),~{}\left(\begin{matrix}\eta_{3}^{0}\\\
\eta_{2}^{0}\end{matrix}\right)=R(\delta)\left(\begin{matrix}G_{\eta}^{0}\\\
\eta^{0}\end{matrix}\right)$ (58)

where $G_{\eta}^{\pm}$ and $G_{\eta}^{0}$ are the NG bosons, $\tan\gamma\equiv
v_{1}/v_{3}$ and $\tan\delta\equiv v_{2}/v_{3}$. The masses of these scalar
bosons are given by

$\displaystyle
m_{\eta^{\pm}}^{2}=\left(1+\frac{v^{2}}{v_{3}^{2}}c_{\beta}^{2}\right)\left(M^{2}s_{\beta}^{2}+\frac{v_{3}^{2}}{2}\rho_{13}\right),\quad
m_{\eta^{0}}^{2}=\left(1+\frac{v^{2}}{v_{3}^{2}}s_{\beta}^{2}\right)\left(M^{2}c_{\beta}^{2}+\frac{v_{3}^{2}}{2}\rho_{23}\right).$
(59)

Let us take the large VEV limit, i.e., $v_{3}\gg v$ with keeping the $M^{2}$
parameter to be the size of $v^{2}$. In this case, the $\eta^{\pm}$ and
$\eta^{0}$ are decoupled as their masses contain the $v_{3}^{3}$ term. On the
other hand, the matrix $M_{H}^{2}$ takes a block diagonal form with small
corrections proportional to $v^{2}/v_{3}^{2}$ or $M^{2}/v_{3}^{2}$ as

$\displaystyle
M_{H}^{2}=\begin{pmatrix}v^{2}c_{\beta}^{2}\lambda_{1}+M^{2}s_{\beta}^{2}&(v^{2}\lambda_{12}-M^{2})s_{\beta}c_{\beta}&0\\\
&v^{2}s_{\beta}^{2}\lambda_{2}+M^{2}c_{\beta}^{2}&0\\\
0&0&v_{3}^{2}\lambda_{3}+\frac{v^{2}}{v_{3}^{2}}M^{2}s_{\beta}^{2}c_{\beta}^{2}\end{pmatrix}+{\cal
O}\left(\frac{v^{2}}{v_{3}^{2}},\frac{M^{2}}{v_{3}^{2}}\right).$ (60)

The orthogonal matrix $R_{H}$ is then approximately expressed as

$\displaystyle R_{H}=\begin{pmatrix}\cos\alpha+{\cal
O}(\epsilon^{2})&-\sin\alpha+{\cal O}(\epsilon^{2})&{\cal O}(\epsilon)\\\
\sin\alpha+{\cal O}(\epsilon^{2})&\cos\alpha+{\cal O}(\epsilon^{2})&{\cal
O}(\epsilon)\\\ {\cal O}(\epsilon)&{\cal O}(\epsilon)&1+{\cal
O}(\epsilon^{2})\end{pmatrix}$ (61)

with $\epsilon=v/v_{3}$. Neglecting the ${\cal O}(\epsilon)$ term, the masses
of CP-even Higgs bosons are expressed as

$\displaystyle m^{2}_{H}$ $\displaystyle=(M_{H}^{\prime
2})_{11}c_{\beta-\alpha}^{2}-2(M_{H}^{\prime
2})_{12}c_{\beta-\alpha}s_{\beta-\alpha}+(M_{H}^{\prime
2})_{22}s_{\beta-\alpha}^{2},$ (62) $\displaystyle m^{2}_{h}$
$\displaystyle=(M_{H}^{\prime 2})_{11}c_{\beta-\alpha}^{2}+2(M_{H}^{\prime
2})_{12}c_{\beta-\alpha}s_{\beta-\alpha}+(M_{H}^{\prime
2})_{22}s_{\beta-\alpha}^{2},$ (63) $\displaystyle m^{2}_{H_{S}}$
$\displaystyle=v_{3}^{2}\lambda_{3},$ (64)

and the mixing angle $\beta-\alpha$ is expressed as

$\displaystyle\tan 2(\beta-\alpha)=\frac{-2(M_{H}^{\prime
2})_{12}}{(M_{H}^{\prime 2})_{11}-(M_{H}^{\prime 2})_{22}}.$ (65)

In the above expressions, $M_{H}^{\prime 2}$ is the $2\times 2$ mass matrix in
the basis of $R(-\beta)(h_{1},h_{2})^{T}$:

$\displaystyle(M^{\prime 2}_{H})_{11}$
$\displaystyle=v^{2}\left(\lambda_{1}c_{\beta}^{4}+\lambda_{2}s_{\beta}^{4}+2\lambda_{12}c_{\beta}^{2}s_{\beta}^{2}\right),$
(66) $\displaystyle(M^{\prime 2}_{H})_{12}$
$\displaystyle=v^{2}c_{\beta}s_{\beta}\left(\lambda_{2}s^{2}_{\beta}-\lambda_{1}c^{2}_{\beta}+\lambda_{12}c_{2\beta}\right),$
(67) $\displaystyle(M^{\prime 2}_{H})_{22}$
$\displaystyle=M^{2}+c_{\beta}^{2}s_{\beta}^{2}v^{2}\left(\lambda_{1}+\lambda_{2}-2\lambda_{12}\right).$
(68)

We see that at the large $v_{3}$ limit, the particle content of the Higgs
sector coincides with that of the 2HDM, i.e., $H^{\pm}$, $A$, $H$ and $h$.

##  Appendix B Other Class-I models

Fields |  $SU(3)_{C}\otimes SU(3)_{L}\otimes U(1)_{X}\otimes U(1)^{\prime}$ |  $Z_{2}^{\rm rem}$ | Components |
---|---|---|---|---
$Q^{a}_{L}$ | $\left(\bm{3},\overline{\bm{3}},1/3,0\right)$ | $(+,+,-)$ | $\left(d^{a}_{L},-u^{a}_{L},U^{a}_{L}\right)^{T}$ |
$Q^{3}_{L}$ | $\left(\bm{3},\bm{3},0,0\right)$ | $(+,+,-)$ | $\left(t_{L},b_{L},D_{L}\right)^{T}$ |
$u^{i}_{R}$ | $\left(\bm{3},\bm{1},+2/3,q\right)$ | $+$ | $u^{i}_{R}$ |
$d^{i}_{R}$ | $\left(\bm{3},\bm{1},-1/3,-q\right)$ | $+$ | $d^{i}_{R}$ |
$U_{R}^{a}$ | $\left(\bm{3},\bm{1},+2/3,-2q\right)$ | $-$ | $U_{R}^{a}$ |
$D_{R}$ | $\left(\bm{3},\bm{1},-1/3,2q\right)$ | $-$ | $D_{R}$ |
$L^{i}_{L}$ | $\left(\bm{1},\bm{3},-2/3,0\right)$ | $(+,+,-)$ | $\left(\nu^{i}_{L},e^{i}_{L},E_{L}^{i}\right)^{T}$ |
$e^{i}_{R}$ | $\left(\bm{1},\bm{1},-1,-q\right)$ | $+$ | $e^{i}_{R}$ |
$E^{i}_{R}$ | $\left(\bm{1},\bm{1},-1,2q\right)$ | $-$ | $E^{i}_{R}$ |
$\Phi_{1}$ | $\left(\bm{1},\bm{3},-2/3,-q\right)$ | $(+,+,-)$ | $\left(\phi_{1}^{0},\phi_{1}^{-},\eta_{1}^{-}\right)$ |
$\Phi_{2}$ | $\left(\bm{1},\bm{3},+1/3,q\right)$ | $(+,+,-)$ | $\left(\phi_{2}^{+},\phi_{2}^{0},\eta_{2}^{0}\right)$ |
$\Phi_{3}$ | $\left(\bm{1},\bm{3},+1/3,-2q\right)$ | $(-,-,+)$ | $\left(\eta_{3}^{+},\eta_{3}^{0},\phi_{3}^{0}\right)$ |
Table 9: Particle content of the model with $\zeta=+1/\sqrt{3}$, where $E^{i}$ are the extra lepton with the electric charge of $-1$.  Fields |  $SU(3)_{c}\otimes SU(3)_{L}\otimes U(1)_{X}$ |  $Z_{2}^{\rm rem}$ | Components |
---|---|---|---|---
$Q^{a}_{L}$ | $\left(\bm{3},\overline{\bm{3}},-1/3\right)$ | $(+,+,-)$ | $\left(d^{a}_{L},-u^{a}_{L},J^{a}_{L}\right)^{T}$ |
$Q^{3}_{L}$ | $\left(\bm{3},\bm{3},+2/3\right)$ | $(+,+,-)$ | $\left(t_{L},b_{L},K_{L}\right)^{T}$ |
$u^{i}_{R}$ | $\left(\bm{3},\bm{1},+2/3\right)$ | $+$ | $u^{i}_{R}$ |
$d^{i}_{R}$ | $\left(\bm{3},\bm{1},-1/3\right)$ | $+$ | $d^{i}_{R}$ |
$K_{R}$ | $\left(\bm{3},\bm{1},+5/3\right)$ | $-$ | $K_{R}$ |
$J^{a}_{R}$ | $\left(\bm{3},\bm{1},-4/3\right)$ | $-$ | $J^{a}_{R}$ |
$L^{i}_{L}$ | $\left(\bm{1},\bm{3},0\right)$ | $(+,+,-)$ | $\left(\nu^{i}_{L},e^{i}_{L},(e_{R}^{i})^{c}\right)^{T}$ |
$\Phi_{1}$ | $\left(\bm{1},\bm{3},0\right)$ | $(+,+,-)$ | $\left(\phi_{1}^{0},\phi_{1}^{-},\eta_{1}^{+}\right)$ |
$\Phi_{2}$ | $\left(\bm{1},\bm{3},1\right)$ | $(+,+,-)$ | $\left(\phi_{2}^{+},\phi_{2}^{0},\eta_{2}^{++}\right)$ |
$\Phi_{3}$ | $\left(\bm{1},\bm{3},-1\right)$ | $(-,-,+)$ | $\left(\eta_{3}^{-},\eta_{3}^{--},\phi_{3}^{0}\right)$ |
Table 10: Particle content for the model with $\zeta=-\sqrt{3}$, where $J^{a}$ and $K$ are exotic quarks with the electric charge of $-4/3$ and $+5/3$, respectively.  Fields |  $SU(3)_{c}\otimes SU(3)_{L}\otimes U(1)_{X}$ |  $Z_{2}^{\rm rem}$ | Components |
---|---|---|---|---
$Q^{a}_{L}$ | $\left(\bm{3},\overline{\bm{3}},2/3\right)$ | $(+,+,-)$ | $\left(d^{a}_{L},-u^{a}_{L},K^{a}_{L}\right)^{T}$ |
$Q^{3}_{L}$ | $\left(\bm{3},\bm{3},-1/3\right)$ | $(+,+,-)$ | $\left(t_{L},b_{L},J_{L}\right)^{T}$ |
$u^{i}_{R}$ | $\left(\bm{3},\bm{1},+2/3\right)$ | $+$ | $u^{i}_{R}$ |
$d^{i}_{R}$ | $\left(\bm{3},\bm{1},-1/3\right)$ | $+$ | $d^{i}_{R}$ |
$K_{R}$ | $\left(\bm{3},\bm{1},+5/3\right)$ | $-$ | $K_{R}^{a}$ |
$J^{a}_{R}$ | $\left(\bm{3},\bm{1},-4/3\right)$ | $-$ | $J_{R}$ |
$L^{i}_{L}$ | $\left(\bm{1},\bm{3},-1\right)$ | $(+,+,-)$ | $\left(\nu^{i}_{L},e^{i}_{L},F_{L}^{i}\right)^{T}$ |
$e^{i}_{R}$ | $\left(\bm{1},\bm{1},-1\right)$ | $+$ | $e^{i}_{R}$ |
$F^{i}_{R}$ | $\left(\bm{1},\bm{1},-2\right)$ | $-$ | $F^{i}_{R}$ |
$\Phi_{1}$ | $\left(\bm{1},\bm{3},-1\right)$ | $(+,+,-)$ | $\left(\phi_{1}^{0},\phi_{1}^{-},\eta_{1}^{--}\right)$ |
$\Phi_{2}$ | $\left(\bm{1},\bm{3},0\right)$ | $(+,+,-)$ | $\left(\phi_{2}^{+},\phi_{2}^{0},\eta_{2}^{-}\right)$ |
$\Phi_{3}$ | $\left(\bm{1},\bm{3},1\right)$ | $(-,-,+)$ | $\left(\eta_{3}^{++},\eta_{3}^{+},\phi_{3}^{0}\right)$ |
Table 11: Particle content for the model with $\zeta=+\sqrt{3}$, where $J$ and
$K$ ($F$) are exotic quarks (leptons) with the electric charge of $-4/3$ and
$+5/3$ ($-2$), respectively.

The particle content of the other Class-I models with $\zeta=+1/\sqrt{3}$,
$-\sqrt{3}$ and $+\sqrt{3}$ are shown in Tables 9, 10 and 11, respectively.
The indices $i$ and $a$ run over 1-3 and 1-2, respectively. For the models
with $\zeta=\pm\sqrt{3}$, we do not need to impose the $U(1)^{\prime}$
symmetry, because all the extra fermions have different electric charges from
that of SM fermions, and thus there is no mixing among SM and extra fermions.
In the model with $\zeta=-\sqrt{3}$, the remnant $Z_{2}^{\rm rem}$ symmetry
does not appear, because the SM right-handed charged leptons are embedded into
the third component of the lepton triplet. In other words, the anti-symmetric
Yukawa interaction $\overline{L_{L}^{c}}L_{L}\Phi_{1}$ breaks the remnant
symmetry, by which $\eta$ particles can decay into SM particles. On the other
hand, in the model with $\zeta=+\sqrt{3}$, the remnant $Z_{2}^{\rm rem}$
symmetry appears, and its charges can be extracted from interaction terms of
component fields, which are shown in Table 11. However, it has to be
explicitly broken, because there is no neutral $Z_{2}^{\rm rem}$-odd particle,
i.e., it gives rise to a stable charged particle which has been excluded by
cosmological observations. A simple way to break the $Z_{2}^{\rm rem}$
symmetry is to introduce doubly-charged singlet scalars
$S^{++}\sim({\bm{1}},{\bm{1}},+2)$, by which
$\Phi_{3}^{\dagger}\Phi_{1}S^{++}$ and $\overline{e_{R}^{c}}e_{R}S^{++}$ terms
can be constructed, and these allow $Z_{2}^{\rm rem}$-odd particles to decay
into SM particles.

##  Appendix C Decay rates of the Higgs bosons

The decays rates into a pair of fermions are given by

$\displaystyle\begin{split}\Gamma(H\to\bar{q}_{i}q_{j})&=\frac{N_{c}}{8\pi}\frac{m_{H}^{3}}{v^{2}}\left|(\Gamma_{H}^{q})_{ij}\right|^{2}\left[\frac{r_{H}^{q_{i}}+r_{H}^{q_{j}}}{2}(1-r_{H}^{q_{i}}-r_{H}^{q_{j}})-2r_{H}^{q_{i}}r_{H}^{q_{j}}\right]\sqrt{F_{\text{2-body}}(r_{H}^{q_{i}},r_{H}^{q_{j}})},\\\
\Gamma(H\to\bar{\ell}\ell)&=\frac{1}{8\pi}\frac{m_{H}^{3}}{v^{2}}r_{H}^{\ell}(c_{\beta-\alpha}+\tan\beta
s_{\beta-\alpha})^{2}\sqrt{F_{\text{2-body}}(r_{H}^{\ell},r_{H}^{\ell})}^{3},\\\
\Gamma(A\to\bar{q}_{i}q_{j})&=\frac{N_{c}}{8\pi}\frac{m_{A}^{3}}{v^{2}}\left|(\Gamma_{A}^{q})_{ij}\right|^{2}\left[\frac{r_{A}^{q_{i}}+r_{A}^{q_{j}}}{2}(1-r_{A}^{q_{i}}-r_{A}^{q_{j}})+2r_{A}^{q_{i}}r_{A}^{q_{j}}\right]\sqrt{F_{\text{2-body}}(r_{A}^{q_{i}},r_{A}^{q_{j}})},\\\
\Gamma(A\to\bar{\ell}\ell)&=\frac{1}{8\pi}\frac{m_{A}^{3}}{v^{2}}r_{A}^{\ell}(c_{\beta-\alpha}+\tan\beta
s_{\beta-\alpha})^{2}\sqrt{F_{\text{2-body}}(r_{A}^{\ell},r_{A}^{\ell})},\\\
\Gamma(H^{+}\to\bar{d}_{i}u_{j})&=\frac{N_{c}}{8\pi}\frac{m_{H^{\pm}}^{3}}{v^{2}}\Bigg{\\{}[|(\Gamma_{A}^{d}V_{\rm
CKM}^{\dagger})_{ij}|^{2}r_{H^{\pm}}^{d_{i}}+|(V_{\rm
CKM}^{\dagger}\Gamma_{A}^{u})_{ij}|^{2}r_{H^{\pm}}^{u_{j}}](1-r_{H^{\pm}}^{d_{i}}-r_{H^{\pm}}^{u_{j}})\\\
&+4r_{H^{\pm}}^{d_{i}}r_{H^{\pm}}^{u_{j}}\text{Re}[(\Gamma_{A}^{d}V_{\rm
CKM}^{\dagger})_{ij}^{*}(V_{\rm
CKM}^{\dagger}\Gamma_{A}^{u})_{ij}]\Bigg{\\}}\sqrt{F_{\text{2-body}}(r_{H^{\pm}}^{d_{i}},r_{H^{\pm}}^{u_{j}})},\\\
\Gamma(H^{+}\to\bar{\ell}\nu)&=\frac{1}{8\pi}\frac{m_{H^{\pm}}^{3}}{v^{2}}r_{H^{\pm}}^{\ell}\tan^{2}\beta\sqrt{F_{\text{2-body}}(r_{H^{\pm}}^{\ell},0)},\end{split}$
(69)

where $r_{X}^{Y}=m_{Y}^{2}/m_{X}^{2}$ and $N_{c}=3$. In Eq. (69),
$F_{\text{2-body}}$ is the 2-body phase space function given as

$\displaystyle
F_{\text{2-body}}(x,y)=[(1-x-y)^{2}-4xy]\Theta(1-\sqrt{x}-\sqrt{y}),$ (70)

with $\Theta$ being the Heaviside step function. The neutral Higgs bosons $H$
and $A$ can also decay into two gluons at one-loop level, and its decay rate
is given by

$\displaystyle\begin{split}\Gamma(H\to
gg)&=\frac{\alpha_{s}^{2}}{2\pi^{3}}\frac{m_{H}^{3}}{v^{2}}\left|r_{H}^{t}(\Gamma_{H}^{u})_{33}I_{t}+r_{H}^{b}(\Gamma_{H}^{d})_{33}I_{b}\right|^{2},\\\
\Gamma(A\to
gg)&=\frac{\alpha_{s}^{2}}{2\pi^{3}}\frac{m_{A}^{3}}{v^{2}}\left|r_{A}^{t}(\Gamma_{A}^{u})_{33}J_{t}+r_{A}^{b}(\Gamma_{A}^{d})_{33}J_{b}\right|^{2},\end{split}$
(71)

where

$\displaystyle I_{q}$ $\displaystyle=1+(1-4r_{H}^{q})f(r_{q}),\quad
J_{q}=r_{A}^{q}f(r_{q}),$ (72)

with

$\displaystyle f(x)$
$\displaystyle=\arcsin^{2}\left[\frac{1}{2\sqrt{x}}\right],~{}\text{for}~{}x\geq\frac{1}{4},$
(73) $\displaystyle f(x)$
$\displaystyle=-\frac{1}{4}\left[\ln\left(\frac{1+\sqrt{1-4x}}{1-\sqrt{1-4x}}\right)-i\pi\right]^{2},~{}\text{for}~{}x<\frac{1}{4}.$
(74)

## References

  * Abe  _et al._ (1995) F. Abe _et al._ (CDF), Phys. Rev. Lett. 74, 2626 (1995), arXiv:hep-ex/9503002 .
  * Abachi  _et al._ (1995) S. Abachi _et al._ (D0), Phys. Rev. Lett. 74, 2632 (1995), arXiv:hep-ex/9503003 .
  * Bevan  _et al._ (2014) A. J. Bevan _et al._ (BaBar, Belle), Eur. Phys. J. C 74, 3026 (2014), arXiv:1406.6311 [hep-ex] .
  * Schael  _et al._ (2006) S. Schael _et al._ (ALEPH, DELPHI, L3, OPAL, SLD, LEP Electroweak Working Group, SLD Electroweak Group, SLD Heavy Flavour Group), Phys. Rept. 427, 257 (2006), arXiv:hep-ex/0509008 .
  * Huet and Sather (1995) P. Huet and E. Sather, Phys. Rev. D 51, 379 (1995), arXiv:hep-ph/9404302 .
  * Singer  _et al._ (1980) M. Singer, J. W. F. Valle, and J. Schechter, Phys. Rev. D22, 738 (1980).
  * Valle and Singer (1983) J. W. F. Valle and M. Singer,  _11th International Symposium on Lepton and Photon Interactions at High Energies Ithaca, New York, August 4-9, 1983_ , Phys. Rev. D28, 540 (1983).
  * Frampton (1992a) P. H. Frampton, Phys. Rev. Lett. 69, 2889 (1992a).
  * Okada  _et al._ (2016) H. Okada, N. Okada, Y. Orikasa, and K. Yagyu, Phys. Rev. D 94, 015002 (2016), arXiv:1604.01948 [hep-ph] .
  * Grossman (1994) Y. Grossman, Nucl. Phys. B426, 355 (1994), arXiv:hep-ph/9401311 [hep-ph] .
  * Barger  _et al._ (2009) V. Barger, H. E. Logan, and G. Shaughnessy, Phys. Rev. D 79, 115018 (2009), arXiv:0902.0170 [hep-ph] .
  * Aoki  _et al._ (2009) M. Aoki, S. Kanemura, K. Tsumura, and K. Yagyu, Phys. Rev. D80, 015017 (2009), arXiv:0902.4665 [hep-ph] .
  * Ayres  _et al._ (2021) N. J. Ayres _et al._ (n2EDM), Eur. Phys. J. C 81, 512 (2021), arXiv:2101.08730 [physics.ins-det] .
  * De Conto and Pleitez (2015) G. De Conto and V. Pleitez, Phys. Rev. D 91, 015006 (2015), arXiv:1408.6551 [hep-ph] .
  * De Conto and Pleitez (2016) G. De Conto and V. Pleitez, (2016), arXiv:1606.01747 [hep-ph] .
  * Abe  _et al._ (2014) T. Abe, J. Hisano, T. Kitahara, and K. Tobioka, JHEP 01, 106 (2014), [Erratum: JHEP 04, 161 (2016)], arXiv:1311.4704 [hep-ph] .
  * Cheung  _et al._ (2020) K. Cheung, A. Jueid, Y.-N. Mao, and S. Moretti, Phys. Rev. D 102, 075029 (2020), arXiv:2003.04178 [hep-ph] .
  * Altmannshofer  _et al._ (2020) W. Altmannshofer, S. Gori, N. Hamer, and H. H. Patel, Phys. Rev. D 102, 115042 (2020), arXiv:2009.01258 [hep-ph] .
  * Pich and Tuzon (2009) A. Pich and P. Tuzon, Phys. Rev. D 80, 091702 (2009), arXiv:0908.1554 [hep-ph] .
  * Jung and Pich (2014) M. Jung and A. Pich, JHEP 04, 076 (2014), arXiv:1308.6283 [hep-ph] .
  * Kanemura  _et al._ (2020) S. Kanemura, M. Kubota, and K. Yagyu, JHEP 08, 026 (2020), arXiv:2004.03943 [hep-ph] .
  * Barr and Zee (1990) S. M. Barr and A. Zee, Phys. Rev. Lett. 65, 21 (1990), [Erratum: Phys.Rev.Lett. 65, 2920 (1990)].
  * Fonseca and Hirsch (2016a) R. M. Fonseca and M. Hirsch, Phys. Rev. D 94, 115003 (2016a), arXiv:1607.06328 [hep-ph] .
  * Frampton (1992b) P. H. Frampton, Physical Review Letters 69, 2889 (1992b).
  * Pisano and Pleitez (1992) F. Pisano and V. Pleitez, Physical Review D 46, 410 (1992).
  * Singer (1980) M. Singer, Physical Review D 22, 738 (1980).
  * Pleitez (1996) V. Pleitez, Physical Review D 53, 514 (1996).
  * Özer (1996) M. Özer, Physical Review D 54, 1143 (1996).
  * Sanchez  _et al._ (2001) L. A. Sanchez, W. A. Ponce, and R. Martinez, Phys. Rev. D 64, 075013 (2001), arXiv:hep-ph/0103244 .
  * Fonseca and Hirsch (2016b) R. M. Fonseca and M. Hirsch, Journal of High Energy Physics 2016 (2016b), 10.1007/jhep08(2016)003.
  * Crivellin  _et al._ (2013) A. Crivellin, A. Kokulu, and C. Greub, Phys. Rev. D 87, 094031 (2013), arXiv:1303.5877 [hep-ph] .
  * Arbey  _et al._ (2018) A. Arbey, F. Mahmoudi, O. Stal, and T. Stefaniak, Eur. Phys. J. C 78, 182 (2018), arXiv:1706.07414 [hep-ph] .
  * Enomoto and Watanabe (2016) T. Enomoto and R. Watanabe, JHEP 05, 002 (2016), arXiv:1511.05066 [hep-ph] .
  * Gabbiani  _et al._ (1996) F. Gabbiani, E. Gabrielli, A. Masiero, and L. Silvestrini, Nucl. Phys. B 477, 321 (1996), arXiv:hep-ph/9604387 .
  * Jana  _et al._ (2021) S. Jana, S. Klett, and M. Lindner, (2021), arXiv:2112.09155 [hep-ph] .
  * Zyla  _et al._ (2020a) P. Zyla _et al._ (Particle Data Group), PTEP 2020, 083C01 (2020a).
  * Amhis  _et al._ (2021) Y. S. Amhis _et al._ (HFLAV), Eur. Phys. J. C 81, 226 (2021), arXiv:1909.12524 [hep-ex] .
  * Ciuchini  _et al._ (1998) M. Ciuchini, G. Degrassi, P. Gambino, and G. F. Giudice, Nucl. Phys. B 527, 21 (1998), arXiv:hep-ph/9710335 .
  * Kagan and Neubert (1999) A. L. Kagan and M. Neubert, Eur. Phys. J. C 7, 5 (1999), arXiv:hep-ph/9805303 .
  * Borzumati and Greub (1998) F. Borzumati and C. Greub, Phys. Rev. D 58, 074004 (1998), arXiv:hep-ph/9802391 .
  * Borzumati and Greub (1999) F. Borzumati and C. Greub, Phys. Rev. D 59, 057501 (1999), arXiv:hep-ph/9809438 .
  * Misiak  _et al._ (2007) M. Misiak _et al._ , Phys. Rev. Lett. 98, 022002 (2007), arXiv:hep-ph/0609232 .
  * Misiak and Steinhauser (2017) M. Misiak and M. Steinhauser, Eur. Phys. J. C 77, 201 (2017), arXiv:1702.04571 [hep-ph] .
  * Misiak  _et al._ (2020) M. Misiak, A. Rehman, and M. Steinhauser, JHEP 06, 175 (2020), arXiv:2002.01548 [hep-ph] .
  * Weinberg (1989) S. Weinberg, Physical Review Letters 63, 2333 (1989).
  * Barr (1990) S. M. Barr, Physical Review Letters 65, 21 (1990).
  * Abel  _et al._ (2020) C. Abel _et al._ (nEDM), Phys. Rev. Lett. 124, 081803 (2020), arXiv:2001.11966 [hep-ex] .
  * Logan  _et al._ (2021) H. E. Logan, S. Moretti, D. Rojas-Ciofalo, and M. Song, JHEP 07, 158 (2021), arXiv:2012.08846 [hep-ph] .
  * Andreev  _et al._ (2018) V. Andreev _et al._ (ACME), Nature 562, 355 (2018).
  * Griffith  _et al._ (2009) W. C. Griffith, M. D. Swallows, T. H. Loftus, M. V. Romalis, B. R. Heckel, and E. N. Fortson, Phys. Rev. Lett. 102, 101601 (2009), arXiv:0901.2328 [physics.atom-ph] .
  * Inoue  _et al._ (2013) T. Inoue _et al._ , Hyperfine Interact. 220, 59 (2013).
  * Xing  _et al._ (2008) Z.-z. Xing, H. Zhang, and S. Zhou, Phys. Rev. D 77, 113016 (2008), arXiv:0712.1419 [hep-ph] .
  * Aoki  _et al._ (2020) S. Aoki _et al._ (Flavour Lattice Averaging Group), Eur. Phys. J. C 80, 113 (2020), arXiv:1902.08191 [hep-lat] .
  * Charles  _et al._ (2015) J. Charles _et al._ , Phys. Rev. D 91, 073007 (2015), arXiv:1501.05013 [hep-ph] .
  * Zyla  _et al._ (2020b) P. A. Zyla _et al._ (Particle Data Group), PTEP 2020, 083C01 (2020b).
  * ATL (2021) (ATLAS Collaboration) ATLAS-CONF-2021-053 (2021).
  * CMS (2020) (CMS Collaboration) CMS-PAS-HIG-19-005 (2020).
  * Peskin and Takeuchi (1990) M. E. Peskin and T. Takeuchi, Phys. Rev. Lett. 65, 964 (1990).
  * Pomarol and Vega (1994) A. Pomarol and R. Vega, Nucl. Phys. B413, 3 (1994), arXiv:hep-ph/9305272 [hep-ph] .
  * Kanemura  _et al._ (2011) S. Kanemura, Y. Okada, H. Taniguchi, and K. Tsumura, Phys. Lett. B 704, 303 (2011), arXiv:1108.3297 [hep-ph] .
  * Blasi  _et al._ (2017) S. Blasi, S. De Curtis, and K. Yagyu, JHEP 09, 091 (2017), arXiv:1707.05273 [hep-ph] .
  * Hou and Modak (2021) W.-S. Hou and T. Modak, Mod. Phys. Lett. A 36, 2130006 (2021), arXiv:2012.05735 [hep-ph] .
  * Kanemura  _et al._ (2021) S. Kanemura, M. Kubota, and K. Yagyu, JHEP 04, 144 (2021), arXiv:2101.03702 [hep-ph] .
  * Aiko  _et al._ (2021) M. Aiko, S. Kanemura, M. Kikuchi, K. Mawatari, K. Sakurai, and K. Yagyu, Nucl. Phys. B 966, 115375 (2021), arXiv:2010.15057 [hep-ph] .
  * Das  _et al._ (2020) A. Das, K. Enomoto, S. Kanemura, and K. Yagyu, Phys. Rev. D 101, 095007 (2020), arXiv:2003.05857 [hep-ph] .