IPMU21-0032

KANAZAWA-21-07

Vanishing or non-vanishing rainbow?
Reduction formulas of electric dipole moment

Motoko<EMAIL_ADDRESS>, Junji
<EMAIL_ADDRESS>, Takashi
<EMAIL_ADDRESS>
1Department of Physics, Nagoya University,
Furo-cho Chikusa-ku, Nagoya, 464-8602 Japan
2Kobayashi-Maskawa Institute for the Origin of Particles and the Universe,
Nagoya University,Furo-cho Chikusa-ku, Nagoya, 464-8602 Japan
3Kavli IPMU (WPI), UTIAS, University of Tokyo, Kashiwa, 277-8584, Japan
4Institute of Liberal Arts and Science, Kanazawa University,
Kakuma-machi, Kanazawa, 920-1192 Japan
5Institute for Theoretical Physics, Kanazawa University, Kanazawa, 920-1192
Japan

##  1 Introduction

The CP violation is the key to understand the nature of the universe dominated
by baryons. In the Standard Model (SM), the possible CP violating source is
the Cabbibo-Kobayashi-Maskawa (CKM) matrix that appears in the processes
mediated by $W$ bosons between up-type and down-type quarks. Such CP violating
processes are suppressed by the small Jarlskog invariant in the electroweak
theory [1, 2].

The electric dipole moment (EDM) is one of the observables connected with CP
violation, and is being explored by various experiments. The current upper
bound on the neutron EDM is given by the nEDM Collaboration while the bound on
the electron EDM is given by the ACME Collaboration, which are $|d_{n}|\leq
1.8\times 10^{-26}~{}e\hskip 2.84544pt\mathrm{cm}$ [3] and $|d_{e}|\leq
1.1\times 10^{-29}~{}e\hskip 2.84544pt\mathrm{cm}$ [4] at $90\%$ confidence
level, respectively. For future prospects, the TUCAN EDM experiment aims to
measure the neutron EDM at $10^{-27}~{}e\hskip 2.84544pt\mathrm{cm}$ [5] and
the ACME Collaboration will update the bound for electron EDM at
$\mathcal{O}(10^{-30})~{}e\hskip 2.84544pt\mathrm{cm}$ [6, 7]. Theoretically,
the EDM is induced from the flavor diagonal effective Hamiltonian induced by
quantum corrections. For the neutron EDM in the SM, a leading order diagram is
naively expected to be the two-loop level that is proportional to the fourth
order of the CKM matrix elements. Figure 1 shows the two-loop “rainbow-type”
diagram, a diagram with a irreducible structure in the outer loop, in the SM
where the photon couples to arbitrary charged particles.

However, this naive expectation is failed as shown by E. P. Shabalin [8]. The
author has performed the explicit calculations in the SM and shown that the
neutron EDM induced from these two-loop diagrams totally cancels out.
Therefore, the leading order contribution to the neutron EDM is induced by the
three-loop level.111It is rather induced from the bound state effect that is
estimated to be $|d_{n}|\sim 10^{-31}~{}e\hskip 2.84544pt\mathrm{cm}$ [9, 10].
For the charged leptons in the SM, the EDMs are further suppressed and induced
at four-loop order because the CP violation has to be brought from the quark
sector [11, 12]. In addition to the SM discussion, the similar cancellation of
charged lepton EDMs in Type-I seesaw models has also been discussed in Refs.
[13, 14, 15].

According to Ref. [8], this non-trivial EDM cancellation between the diagrams
shown in Fig. 1 can be attributed to the electromagnetic gauge symmetry. These
rainbow-type diagrams have the structures of one-loop sub-diagrams that induce
the fermion self-energy and the vertex correction. These contributions are
related to each other through the Ward-Takahashi identity of the gauge
symmetry. This is one of the most essential points to understand the neutron
EDM cancellation at the two-loop level in the SM. We also expect the same
technique using the Ward-Takahashi identity should be useful to derive the EDM
formulas not only in the SM but also in a wider class of models.

Figure 1: Two-loop diagram with $W$ boson propagation in the SM. The on-shell
photon can couple to arbitrary charged particles.

In this paper, we generalize Shabalin’s explicit calculation which can be
applied to rainbow-type diagrams with a more general setup. Namely, we
introduce the effective off-shell couplings for fermions that are read out
from the self-energy and the vertex correction. These terms have non-trivial
relations through the Ward-Takahashi identity. We also introduce the form
factors to express the fermion self-energy and the vertex correction, and
explicitly show which terms induce non-zero EDM contributions. With this
general setup, we finally derive reduced formulas of the EDM that can be
applicable to a larger class of models including the SM. The exact
cancellation of the EDM between rainbow-type diagrams in the SM is recovered
from the derived formulas. Furthermore, we discuss the other possible
generalization of the EDM calculation for the models with multi-scalars.

The remaining parts of the paper are organized as follows. In Sec. 2, we
derive the reduction formulas for EDMs induced by two-loop or higher order
diagrams that include sub-diagram structures. In Sec. 3, we apply our formulas
of the EDM to some specific models such as the SM, the scotogenic model and
the singlet-triplet extended model. Our reduction formulas are general enough
to not only reproduce the EDM cancellation between the rainbow-type diagrams
in the SM but also provide the predicted EDMs in a larger class of models
beyond the SM. In Sec. 4, we also consider the other possible generalizations
of the EDM calculation for the models with multi-scalars. Our conclusions are
given in Sec. 5.

##  2 Reduction formulas of EDM

In this section, we derive reduction formulas for EDM that are induced from
the diagrams with sub-diagram structures. First, we briefly review the SM
calculation of the neutron EDM diagrams at the two-loop level as discussed in
Ref. [8]. Second, we summarize the essential points in the SM calculation and
derive the EDM reduction formulas with a more general setup.

###  2.1 EDM cancellation in the Standard Model

We focus on the electroweak interactions between up-type and down-type quarks
as the CP violation sources in the SM, and calculate the EDM based on the
following Lagrangian

$\displaystyle{\mathcal{L}}$
$\displaystyle=-\Bigg{(}\frac{g_{2}}{\sqrt{2}}V_{i\alpha}W_{\mu}^{+}\overline{u}_{i}\gamma^{\mu}P_{L}d_{\alpha}+\mathrm{h.c.}\Bigg{)}-eQ_{u}A_{\mu}\overline{u}_{i}\gamma^{\mu}u_{i},~{}~{}~{}~{}(i,\alpha=1,2,3),$
(2.1)

where $Q_{u}$ is the electric charge for the up-type quarks, $g_{2}$ is the
SU(2)L gauge coupling and $V_{i\alpha}$ denotes the CKM matrix elements. For
the later convenience, we express the generation indices as roman characters
($i,j$) for the up-type quarks, and greek characters ($\alpha,\beta$) for the
down-type quarks, respectively. In the EDM calculation, it suffices to focus
on the CP violating terms that are up to the first order of photon momentum as
pointed out in Ref. [8]. One can naively expect that the leading order CP
violating terms arise from the amplitude at ${\mathcal{O}}(\alpha_{2}^{2})$
$(\alpha_{2}=g_{2}^{2}/4\pi)$ as shown in Fig. 1.222The discussion for the
diagrams with the external up-type quarks is parallel.  The on-shell photon
can couple to arbitrary charged particles in the diagram. These amplitudes are
proportional to the four CKM matrix elements, and the non-zero CP phases arise
from their imaginary parts,

$\displaystyle\mathrm{Im}\left[V^{*}_{j\alpha}V_{j\beta}V^{*}_{i\beta}V_{i\alpha}\right],$
(2.2)

which is the Jarlskog invariant [1, 2]. If we exchange the internal fermion
flavor indices $i\leftrightarrow j$, this Jarlskog invariant acquires $(-1)$
factor as shown below.

$\displaystyle\mathrm{Im}\left[V^{*}_{j\alpha}V_{j\beta}V^{*}_{i\beta}V_{i\alpha}\right]$
$\displaystyle\xrightarrow[]{i\leftrightarrow
j}-\mathrm{Im}\left[V^{*}_{j\alpha}V_{j\beta}V^{*}_{i\beta}V_{i\alpha}\right].$
(2.3)

Thanks to this anti-symmetricity of the Jarlskog invariant, we can further
narrow down the diagrams that may induce non-zero EDM contributions. If the
photon couples to the outer $W$ boson loop, the expression of the amplitude is
anti-symmetric under $i\leftrightarrow j$. Therefore, this contribution
cancels out after summing up all the flavor indices.

Figure 2: Variations of diagrams that have a chance to induce non-zero EDM. We
find the self-energy structure for $u_{i}$ and $u_{j}$ from the diagrams in
the upper row. On the other hand, we find vertex correction from the diagrams
in the lower row.

The remaining diagrams are shown in Fig. 2. We need to have $i\neq j$ in these
diagrams to pick up non-zero CP phases. There are four variations of the
diagrams that a photon can attach to the charged particles. These individual
diagrams induce non-zero EDM contributions. However, in Ref. [8], it was
pointed out that a non-trivial cancellation occurs between these diagrams. The
point is to find the sub-diagram structures of the self-energy and vertex
correction for the up-type quarks, $u_{i}$ and $u_{j}$. These self-energy and
vertex correction are related to each other through the Ward-Takahashi
identity of the electromagnetic gauge symmetry. This is one of the most
crucial points to understand the EDM cancellation in these diagrams. At the
same time, we expect the same reduction technique using the Ward-Takahashi
identity can be applied for a more general setup other than the SM. In the
succeeding section, we summarize the points in the SM calculation and show how
to generalize our calculation setup.

###  2.2 Derivation of EDM reduction formulas

The important points for neutron EDM cancellation at the two-loop level in the
SM are summarized below [8].

  * •

The amplitude is expanded by the photon momentum up to the first order, and
the EDM contributions arise from the first order term of the on-shell photon
momentum.

  * •

The overall CKM factors (Jarlskog invariant) have anti-symmetricity under the
exchange of the internal fermion flavor indices ($i\leftrightarrow j$).

  * •

The Ward-Takahashi identity works between the self-energy and the vertex
correction for internal fermions.

Note that the second point in the above is rather model-dependent and can be
adapted only for a narrow class of models involving the SM. In the following
generalization of the EDM calculation, it can be reinterpreted as a result of
imposing the hermiticity of the self-energy and the vertex correction in the
effective Lagrangian.

| $f_{\alpha}$ | $\psi_{i}$ | $\phi$
---|---|---|---
Spin | $1/2$ | $1/2$ | $0$
electric charge | $Q_{f}$ | $Q_{\psi}$ | $Q_{\phi}$
Table 1: Particle contents of the general model we consider where the electric
charges $Q_{f}$, $Q_{\psi}$ and $Q_{\phi}$ can be arbitrary values.

To generalize our calculation setup without loss of the essence in the SM
calculation, we introduce the following effective Lagrangian

$\displaystyle\mathcal{L}_{\mathrm{eff}}=$
$\displaystyle-\Big{(}y_{i\alpha}\phi^{*}\overline{\psi_{i}}P_{L}f_{\alpha}+\mathrm{H.c.}\Big{)}-\overline{\psi_{i}}\Sigma_{ij}\psi_{j}-eA_{\mu}\overline{\psi_{i}}\Lambda_{ij}^{\mu}\psi_{j}$
$\displaystyle-
eQ_{\psi}A_{\mu}\overline{\psi_{i}}\gamma^{\mu}\psi_{i}+ieQ_{\phi}A_{\mu}\big{(}\phi\partial^{\mu}\raisebox{9.38945pt}{\tiny{$\leftrightarrow$}\hskip
2.84544pt}\phi^{*}\big{)},$ (2.4)

where $\phi\partial^{\mu}\raisebox{9.38945pt}{\tiny{$\leftrightarrow$}\hskip
2.84544pt}\phi^{*}=\phi\partial^{\mu}\phi^{*}-\phi^{*}\partial^{\mu}\phi$ and

$\displaystyle\Lambda_{ij}^{\mu}=\sum_{p}Q_{p}\big{(}\Lambda_{p}\big{)}_{ij}^{\mu}.$
(2.5)

The $W$ boson, up-type and down-type quarks in the SM are replaced by a scalar
$\phi$, internal fermion $\psi_{i}$ and external fermion $f_{\alpha}$,
respectively. The spin and electric charges of these particles are summarized
in Tab. 1. The Yukawa couplings $y_{i\alpha}$ are possible sources of a CP
violation. The symbols $\Sigma_{ij}$ and $\Lambda_{ij}^{\mu}$ are the self-
energy and the vertex correction induced at the one-loop or higher loop level.
The subscript $p$ in the vertex correction denotes the particle attaching a
photon in the loop. All the possible vertex corrections are summed up with the
electric charge of the particle $p$ ($Q_{p}$). From this effective Lagrangian,
one can draw the diagrams which potentially induce non-zero EDMs of the
fermion $f_{\alpha}$ as shown in Fig. 3.333In Fig. 3, we also show the diagram
with photon that couples to the outer scalar loop, which corresponds to the
vector loop diagram not to be shown in Fig. 2. We explicitly check the
cancellation of EDM from this diagram in the view from the hermiticity of the
self-energy and the vertex correction.

Figure 3: Feynman diagrams potentially inducing non-zero EDMs of the fermion
$f_{\alpha}$ with the multi-fermions $\psi_{i}$ and a scalar $\phi$.

One can also consider the diagrams with a vector boson $X$ in the outer loop
instead of the scalar $\phi$. However the calculation is essentially the same
with the scalar case, and the derived formula for the vector boson case will
be rather redundant. Thus we will give our calculation for the scalar loop
below based on the Lagrangian in Eq. (2.4), and we provide only the final
result for the vector boson in Sec. 2.2.5, which is obtained by merely
replacing the corresponding factors in the scalar case. Although we have
considered the left chirality of the fermion $f_{\alpha}$ in Eq. (2.4), one
can consider right-chirality instead. In this case, the following calculations
simply change with opposite chirality. Note that if both chiralities are
involved at tree-level, non-zero EDM of the fermions $f_{\alpha}$ are induced
at one-loop levels, thus our reduction of the EDM calculation which will be
shown in the following is spoiled.

The amplitude of each diagram in Fig. 3 can be given by

$\displaystyle i\mathcal{M}_{\mathrm{A1}}=$
$\displaystyle~{}eQ_{\psi}\int\\!\frac{d^{4}k}{(2\pi)^{4}}\overline{u}(p_{2})P_{R}\left(\not{k}_{2}+m_{j}\right)\tilde{\Sigma}_{ji}(\not{k}_{2})\left(\not{k}_{2}+m_{i}\right)\not{\epsilon}\left(\not{k}_{1}+m_{i}\right)P_{L}u(p_{1})$
$\displaystyle\times\frac{1}{(k-p)^{2}-m_{\phi}^{2}}\frac{1}{k_{1}^{2}-m_{i}^{2}}\frac{1}{k_{2}^{2}-m_{i}^{2}}\frac{1}{k_{2}^{2}-m_{j}^{2}},$
(2.6) $\displaystyle i\mathcal{M}_{\mathrm{A2}}=$
$\displaystyle~{}eQ_{\psi}\int\\!\frac{d^{4}k}{(2\pi)^{4}}\overline{u}(p_{2})P_{R}\left(\not{k}_{2}+m_{j}\right)\not{\epsilon}\left(\not{k}_{1}+m_{j}\right)\tilde{\Sigma}_{ji}(\not{k}_{1})\not{\epsilon}\left(\not{k}_{1}+m_{i}\right)P_{L}u(p_{1})$
$\displaystyle\times\frac{1}{(k-p)^{2}-m_{\phi}^{2}}\frac{1}{k_{1}^{2}-m_{i}^{2}}\frac{1}{k_{1}^{2}-m_{j}^{2}}\frac{1}{k_{2}^{2}-m_{j}^{2}},$
(2.7) $\displaystyle i\mathcal{M}_{\mathrm{A3}}=$
$\displaystyle~{}e\epsilon_{\mu}\int\\!\frac{d^{4}k}{(2\pi)^{4}}\overline{u}(p_{2})P_{R}\left(\not{k}_{2}+m_{j}\right)\tilde{\Lambda}_{ji}^{\mu}(k_{1},k_{2})\left(\not{k}_{1}+m_{i}\right)P_{L}u(p_{1})$
$\displaystyle\times\frac{1}{(k-p)^{2}-m_{\phi}^{2}}\frac{1}{k_{1}^{2}-m_{i}^{2}}\frac{1}{k_{2}^{2}-m_{j}^{2}},$
(2.8) $\displaystyle i\mathcal{M}_{\mathrm{A4}}=$
$\displaystyle~{}eQ_{\phi}\epsilon_{\mu}\int\\!\frac{d^{4}k}{(2\pi)^{4}}\overline{u}(p_{2})P_{R}\left(\not{k}+m_{j}\right)\tilde{\Sigma}_{ji}(\not{k})\left(\not{k}+m_{i}\right)P_{L}u(p_{1})$
$\displaystyle\times\frac{1}{k^{2}-m_{i}^{2}}\frac{1}{k^{2}-m_{j}^{2}}\frac{1}{(k-p_{1})^{2}-m_{\phi}^{2}}\frac{1}{(k-p_{2})^{2}-m_{\phi}^{2}}\left(p_{1}+p_{2}-2k\right)^{\mu},$
(2.9)

where $\tilde{\Sigma}_{ji}(\not{k})$ and
$\tilde{\Lambda}_{ji}^{\mu}(k_{1},k_{2})$ are defined by

$\displaystyle\tilde{\Sigma}_{ji}(\not{k})\equiv
y_{j\alpha}^{*}\Sigma_{ji}(\not{k})y_{i\alpha},\qquad\tilde{\Lambda}_{ji}^{\mu}(k_{1},k_{2})\equiv
y_{j\alpha}^{*}\Lambda_{ji}^{\mu}(k_{1},k_{2})y_{i\alpha},$ (2.10)

and the momenta in the above equations are defined by $q\equiv
k_{2}-k_{1}(=p_{2}-p_{1})$, $k_{1}\equiv k-q/2$, $k_{2}\equiv k+q/2$ and
$p\equiv(p_{1}+p_{2})/2$ as shown in Fig. 4. The photon polarization vector is
given by $\epsilon_{\mu}$. The fermion self-energy $\Sigma_{ji}(\not{k})$ can
be expressed by

$\displaystyle\Sigma_{ji}(\not{k})=A_{ji}^{L}(k^{2})\not{k}P_{L}+A_{ji}^{R}(k^{2})\not{k}P_{R}+B_{ji}^{L}(k^{2})P_{L}+B_{ji}^{R}(k^{2})P_{R},$
(2.11)

with the form factors $A_{ji}^{L/R}(k^{2})$ and $B_{ji}^{L/R}(k^{2})$ for each
chirality. All of these form factors are Lorentz scalars as a function of
$k^{2}$. This expansion is always possible without loss of generality at any
order of perturbative expansions. We impose the hermiticity on the self-energy
since the absorptive part of the self-energy has nothing to do with our EDM
calculation.444This fact can be found by considering whether or not all the
intermediate states can satisfy the on-shell conditions in the EDM loop
diagrams. We focus on the situation where the total mass of loop propagating
particles are heavier than the mass of the external fermion mass. As long as
this condition is satisfied, all the intermediate states are not allowed to be
on-shell.  Namely, the condition

$\displaystyle\big{(}\Sigma^{\dagger}\big{)}_{ji}\gamma^{0}=\gamma^{0}\Sigma_{ij},$
(2.12)

is imposed, and thus we obtain the relations between the form factors.

$\displaystyle A_{ji}^{L}=A_{ij}^{L*},\qquad A_{ji}^{R}=A_{ij}^{R*},\qquad
B_{ji}^{L}=B_{ij}^{R*},\qquad B_{ji}^{R}=B_{ij}^{L*}.$ (2.13)

In the following calculation, we express the amplitude in terms of
$A_{ji}^{L}$, $A_{ji}^{R}$ and $B_{ji}^{L}(\equiv B_{ji})$.

Figure 4: Assignment of the momenta for the self-energy and vertex correction.

####  2.2.1 The amplitude (A1)+(A2)

As already stated, it is sufficient to expand the amplitude up to
$\mathcal{O}(q)$ in order to extract the term of static EDMs [8]. The CP
violating part of the partial amplitude $i\mathcal{M}_{\mathrm{A1+A2}}\equiv
i\mathcal{M}_{\mathrm{A1}}+i\mathcal{M}_{\mathrm{A2}}$ relevant to the EDM of
the fermion $f_{\alpha}$ can be evaluated as

$\displaystyle i\mathcal{M}_{\mathrm{A1+A2}}^{\mathrm{CP}}\approx$
$\displaystyle~{}2ieQ_{\psi}\int\\!\frac{d^{4}k}{(2\pi)^{4}}\overline{u}(p_{2})\not{\epsilon}P_{L}u(p_{1})\frac{\left(k\\!\cdot\\!q\right)m_{i}}{(k-p)^{2}-m_{\phi}^{2}}\frac{1}{\left(k^{2}-m_{i}^{2}\right)^{2}}\frac{1}{\left(k^{2}-m_{j}^{2}\right)^{2}}$
$\displaystyle\times\mathrm{Im}\Bigg{[}k^{4}m_{i}\frac{d\tilde{A}_{ji}^{R}}{dk^{2}}-m_{j}^{3}\bigg{(}\tilde{A}_{ji}^{L}+k^{2}\frac{d\tilde{A}_{ji}^{L}}{dk^{2}}\bigg{)}+\left(k^{4}-2k^{2}m_{j}^{2}+m_{i}^{2}m_{j}^{2}\right)\frac{d\tilde{B}_{ji}}{dk^{2}}\Bigg{]}$
$\displaystyle-2ieQ_{\psi}\int\\!\frac{d^{4}k}{(2\pi)^{4}}\overline{u}(p_{2})\not{k}\not{\epsilon}\not{k}P_{L}u(p_{1})\frac{\left(k\\!\cdot\\!q\right)m_{j}^{2}}{(k-p)^{2}-m_{\phi}^{2}}\frac{1}{\left(k^{2}-m_{i}^{2}\right)^{2}}\frac{1}{\left(k^{2}-m_{j}^{2}\right)^{2}}$
$\displaystyle\times\mathrm{Im}\Bigg{[}\tilde{A}_{ji}^{R}+k^{2}\frac{d\tilde{A}_{ji}^{R}}{dk^{2}}+m_{i}m_{j}\frac{d\tilde{A}_{ji}^{L}}{dk^{2}}+2m_{i}\frac{d\tilde{B}_{ji}}{dk^{2}}\Bigg{]},$
(2.14)

where the hermiticity condition in Eq. (2.13) is used to reduce the equation.
We introduce the following notation in the same manner as defined in Eq.
(2.10).

$\displaystyle\tilde{A}_{ji}^{L/R}(k^{2})\equiv
y_{j\alpha}^{*}A_{ji}^{L/R}(k^{2})y_{i\alpha},\qquad\tilde{B}_{ji}(k^{2})\equiv
y_{j\alpha}^{*}B_{ji}(k^{2})y_{i\alpha}.$ (2.15)

####  2.2.2 The amplitude (A3)

For the amplitude $i\mathcal{M}_{\mathrm{A3}}$, the vertex correction
$\Lambda^{\mu}(k_{1},k_{2})$ obeys the Ward-Takahashi identity

$\displaystyle
q_{\mu}\Lambda^{\mu}(k_{1},k_{2})=\Sigma\left(\not{k}_{1}\right)-\Sigma\left(\not{k}_{2}\right).$
(2.16)

The full vertex function $\Gamma^{\mu}$ involving the tree-level contribution
is related with the correction $\Lambda^{\mu}$ as
$\Gamma^{\mu}=\gamma^{\mu}+\Lambda^{\mu}$. The vertex correction
$\Lambda^{\mu}(k_{1},k_{2})$ can be decomposed into the longitudinal part
$\Lambda_{\mathrm{L}}^{\mu}(k_{1},k_{2})$ and the transverse part
$\Lambda_{\mathrm{T}}^{\mu}(k_{1},k_{2})$, which is orthogonal to the photon
momentum [16], namely

$\displaystyle\Lambda^{\mu}(k_{1},k_{2})=\Lambda_{\mathrm{L}}^{\mu}(k_{1},k_{2})+\Lambda_{\mathrm{T}}^{\mu}(k_{1},k_{2}),$
(2.17)

where $\Lambda_{\mathrm{L}}^{\mu}(k_{1},k_{2})$ and
$\Lambda_{\mathrm{T}}^{\mu}(k_{1},k_{2})$ obey

$\displaystyle
q_{\mu}\Lambda_{\mathrm{L}}^{\mu}(k_{1},k_{2})=\Sigma\left(\not{k}_{1}\right)-\Sigma\left(\not{k}_{2}\right),\qquad
q_{\mu}\Lambda_{\mathrm{T}}^{\mu}(k_{1},k_{2})=0.$ (2.18)

In general, the vertex correction $\Lambda^{\mu}(k_{1},k_{2})$ can be expanded
by 12 independent vectors (24 vectors taking into account chirality operators
$P_{L}$ and $P_{R}$ for any chiral models) [17, 16]. However, 4 of these
vectors (8 for chiral models) can be eliminated by the Ward-Takahashi identity
shown in Eq. (2.16).555The detailed calculation is given in Appendix A. As a
result, the longitudinal part is written as

$\displaystyle\Lambda_{\mathrm{L}}^{\mu}(k_{1},k_{2})=$
$\displaystyle~{}\Bigg{[}\frac{A^{L}(k_{1}^{2})-A^{L}(k_{2}^{2})}{\left(k\\!\cdot\\!q\right)}k^{\mu}\not{k}-\frac{A^{L}(k_{1}^{2})+A^{L}(k_{2}^{2})}{2}\gamma^{\mu}+\frac{B(k_{1}^{2})-B(k_{2}^{2})}{\left(k\\!\cdot\\!q\right)}k^{\mu}\Bigg{]}P_{L}$
$\displaystyle~{}+\Bigg{[}\frac{A^{R}(k_{1}^{2})-A^{R}(k_{2}^{2})}{\left(k\\!\cdot\\!q\right)}k^{\mu}\not{k}-\frac{A^{R}(k_{1}^{2})+A^{R}(k_{2}^{2})}{2}\gamma^{\mu}+\frac{B^{{\dagger}}(k_{1}^{2})-B^{{\dagger}}(k_{2}^{2})}{\left(k\\!\cdot\\!q\right)}k^{\mu}\Bigg{]}P_{R},$
(2.19)

with the form factors in Eq. (2.11). On the other hand, the transverse part is
given by the remaining 8 independent vectors $V_{a}^{\mu}~{}(a=1-8)$ as

$\displaystyle\Lambda_{\mathrm{T}}^{\mu}(k_{1},k_{2})=$
$\displaystyle~{}\sum_{a=1}^{8}\Big{[}C_{a}^{L}(k_{1},k_{2})V_{a}^{\mu}P_{L}+C_{a}^{R}(k_{1},k_{2})V_{a}^{\mu}P_{R}\Big{]},$
(2.20)

where we take the independent vectors $V_{a}^{\mu}$ as follows

$\displaystyle V_{1}^{\mu}=$
$\displaystyle~{}\left(k\\!\cdot\\!q\right)q^{\mu}-q^{2}k^{\mu},\hskip
38.41139ptV_{2}^{\mu}=\not{k}\left[\left(k\\!\cdot\\!q\right)q^{\mu}-q^{2}k^{\mu}\right],$
$\displaystyle V_{3}^{\mu}=$
$\displaystyle~{}\not{k}\not{q}\left[\left(k\\!\cdot\\!q\right)q^{\mu}-q^{2}k^{\mu}\right],\hskip
14.22636ptV_{4}^{\mu}=\not{q}q^{\mu}-q^{2}\gamma^{\mu},$ $\displaystyle
V_{5}^{\mu}=$
$\displaystyle~{}\left(k\\!\cdot\\!q\right)\gamma^{\mu}-\not{q}k^{\mu},\hskip
42.67912ptV_{6}^{\mu}=\not{k}\left[\left(k\\!\cdot\\!q\right)\gamma^{\mu}-\not{q}k^{\mu}\right],$
$\displaystyle V_{7}^{\mu}=$ $\displaystyle~{}i\sigma^{\mu\nu}q_{\nu},\hskip
85.35826ptV_{8}^{\mu}=i\epsilon^{\mu\nu\rho\sigma}\gamma_{5}\gamma_{\nu}k_{\rho}q_{\sigma},$
(2.21)

with $\sigma^{\mu\nu}\equiv
i\left(\gamma^{\mu}\gamma^{\nu}-\gamma^{\nu}\gamma^{\mu}\right)/2$ and the
anti-symmetric tensor $\epsilon^{\mu\nu\rho\sigma}~{}(\epsilon^{0123}=+1)$.
Note that the form factors $C_{a}^{L}$ and $C_{a}^{R}$ can be regarded as a
function of only $k^{2}$, namely $C_{a}^{L}(k^{2})$ and $C_{a}^{R}(k^{2})$, as
long as the $\mathcal{O}(q)$ amplitude is considered because the vectors
$V_{a}^{\mu}$ are already $\mathcal{O}(q)$ at least. One can explicitly show
that Eqs. (2.19) and (2.20) obey Eq. (2.18). Furthermore, the following
relations are satisfied,

$\displaystyle\frac{\partial\Lambda_{\mathrm{L}}^{\mu}}{\partial
q_{\nu}}\Big{|}_{q=0}=0,\qquad\Lambda_{\mathrm{T}}^{\mu}(k,k)=0.$ (2.22)

From the above argument, the vertex correction expanded up to $\mathcal{O}(q)$
can be separated into the derivative of the self-energy and the transverse
part as follows [18],

$\displaystyle\Lambda^{\mu}(k_{1},k_{2})\approx$
$\displaystyle~{}\Lambda^{\mu}\Big{|}_{q=0}+q_{\nu}\frac{\partial\Lambda^{\mu}}{\partial
q_{\nu}}\Big{|}_{q=0}$ $\displaystyle=$
$\displaystyle~{}\Lambda_{\mathrm{L}}^{\mu}\Big{|}_{q=0}+q_{\nu}\frac{\partial\Lambda_{\mathrm{T}}^{\mu}}{\partial
q_{\nu}}\Big{|}_{q=0}=-\frac{d\Sigma}{dk_{\mu}}+q_{\nu}\frac{\partial\Lambda_{\mathrm{T}}^{\mu}}{\partial
q_{\nu}}\Big{|}_{q=0}.$ (2.23)

Using Eq. (2.23), the amplitude $i\mathcal{M}_{\mathrm{A3}}$ can be evaluated
at $\mathcal{O}(q)$. The CP violating part coming from the first term of the
last expression in Eq. (2.23), which corresponds to the longitudinal part of
the vertex correction, is calculated as

$\displaystyle i\mathcal{M}_{\mathrm{A3L}}^{\mathrm{CP}}\approx$
$\displaystyle-2ieQ_{\psi}\int\\!\frac{d^{4}k}{(2\pi)^{4}}\overline{u}(p_{2})\not{\epsilon}P_{L}u(p_{1})\frac{\left(k\\!\cdot\\!q\right)m_{i}}{(k-p)^{2}-m_{\phi}^{2}}\frac{1}{\left(k^{2}-m_{i}^{2}\right)^{2}}\frac{1}{\left(k^{2}-m_{j}^{2}\right)^{2}}$
$\displaystyle\times\mathrm{Im}\Bigg{[}k^{4}m_{i}\frac{d\tilde{A}_{ji}^{R}}{dk^{2}}-m_{j}^{3}\Bigg{(}\tilde{A}_{ji}^{L}+k^{2}\frac{d\tilde{A}_{ji}^{L}}{dk^{2}}\Bigg{)}+k^{2}\left(m_{i}^{2}-m_{j}^{2}\right)\frac{d\tilde{B}_{ji}}{dk^{2}}\Bigg{]}$
$\displaystyle+2ieQ_{\psi}\int\\!\frac{d^{4}k}{(2\pi)^{4}}\overline{u}(p_{2})\not{k}\not{\epsilon}\not{k}P_{L}u(p_{1})\frac{\left(k\\!\cdot\\!q\right)m_{j}^{2}}{(k-p)^{2}-m_{\phi}^{2}}\frac{1}{\left(k^{2}-m_{i}^{2}\right)^{2}}\frac{1}{\left(k^{2}-m_{j}^{2}\right)^{2}}$
$\displaystyle\times\mathrm{Im}\Bigg{[}\tilde{A}_{ji}^{R}+k^{2}\frac{d\tilde{A}_{ji}^{R}}{dk^{2}}+m_{i}m_{j}\frac{d\tilde{A}_{ji}^{L}}{dk^{2}}+2m_{i}\frac{d\tilde{B}_{ji}}{dk^{2}}\Bigg{]}$
$\displaystyle-2ieQ_{\psi}\int\\!\frac{d^{4}k}{(2\pi)^{4}}\overline{u}(p_{2})\not{q}P_{L}u(p_{1})\frac{\left(\epsilon\\!\cdot\\!k\right)m_{i}}{(k-p)^{2}-m_{\phi}^{2}}\frac{1}{k^{2}-m_{i}^{2}}\frac{1}{k^{2}-m_{j}^{2}}\mathrm{Im}\Bigg{[}\frac{d\tilde{B}_{ji}}{dk^{2}}\Bigg{]},$
(2.24)

where the hermiticity condition given in Eq. (2.13) and the conservation of
the electromagnetic charges $\sum_{p}Q_{p}=Q_{\psi}$ are used.

The remaining transverse part of the amplitude $i\mathcal{M}_{\mathrm{A3}}$,
which corresponds to the second term of the last expression in Eq. (2.23), can
be evaluated. First, since the vectors $V_{a}^{\mu}~{}(a=1-4)$ are
$\mathcal{O}(q^{2})$, we can ignore these terms in the EDM calculation.
Second, one can find that the terms for the vectors $V_{5}^{\mu}$ and
$V_{8}^{\mu}$ have no CP violation as discussed below. The following relations
are derived from the hermiticity of the vertex correction
$\Lambda_{\mathrm{T}}^{\mu}(k_{1},k_{2})$

$\displaystyle\left(C_{5}^{L}\right)_{ji}=\left(C_{5}^{L}\right)_{ij}^{*},\qquad\left(C_{5}^{R}\right)_{ji}=\left(C_{5}^{R}\right)_{ij}^{*},$
$\displaystyle\left(C_{6}^{L}\right)_{ji}=\left(C_{6}^{R}\right)_{ij}^{*},\qquad\left(C_{6}^{R}\right)_{ji}=\left(C_{6}^{L}\right)_{ij}^{*},$
$\displaystyle\left(C_{7}^{L}\right)_{ji}=\left(C_{7}^{R}\right)_{ij}^{*},\qquad\left(C_{7}^{R}\right)_{ji}=\left(C_{7}^{L}\right)_{ij}^{*},$
$\displaystyle\left(C_{8}^{L}\right)_{ji}=\left(C_{8}^{L}\right)_{ij}^{*},\qquad\left(C_{8}^{R}\right)_{ji}=\left(C_{8}^{R}\right)_{ij}^{*}.$

Although each term is complex, multiplying the Yukawa coupling
$y_{j\alpha}^{*}$ and $y_{i\alpha}$, and summing up $i$ and $j$, it can be
found that only the real part remains and the imaginary part involving CP
violation vanishes for the vectors $V_{5}^{\mu}$ and $V_{8}^{\mu}$. On the
other hand, the terms for the vectors $V_{6}^{\mu}$ and $V_{7}^{\mu}$
involving even numbers of gamma matrices violate the CP. From the above
consideration, one can extract the CP violating term from the transverse part
as

$\displaystyle i\mathcal{M}_{\mathrm{A3T}}^{\mathrm{CP}}=$
$\displaystyle~{}2iem_{i}\int\\!\frac{d^{4}k}{(2\pi)^{4}}\overline{u}(p_{2})\Big{[}\left(k\\!\cdot\\!q\right)\not{\epsilon}-\left(\epsilon\\!\cdot\\!k\right)\not{q}\Big{]}P_{L}u(p_{1})$
$\displaystyle\times\frac{1}{(k-p)^{2}-m_{\phi}^{2}}\frac{1}{k^{2}-m_{i}^{2}}\frac{1}{k^{2}-m_{j}^{2}}\mathrm{Im}\bigg{[}k^{2}\tilde{C}_{ji}(k^{2})+\tilde{D}_{ji}(k^{2})\bigg{]},$
(2.25)

where $\tilde{C}_{ji}(k^{2})$ and $\tilde{D}_{ji}(k^{2})$ are defined by the
sum of all the vertex corrections for the left-chirality

$\displaystyle\tilde{C}_{ji}(k^{2})\equiv\sum_{p}Q_{p}\Big{[}y_{j\alpha}^{*}\big{(}C_{p6}^{L}(k^{2})\big{)}_{ji}y_{i\alpha}\Big{]},\quad\tilde{D}_{ji}(k^{2})\equiv\sum_{p}Q_{p}\Big{[}y_{j\alpha}^{*}\big{(}C_{p7}^{L}(k^{2})\big{)}_{ji}y_{i\alpha}\Big{]}.$
(2.26)

####  2.2.3 The amplitude (A4)

The amplitude $i\mathcal{M}_{\mathrm{A4}}$ can be evaluated by directly
substituting the self-energy of Eq. (2.11) into Eq. (2.9)

$\displaystyle i\mathcal{M}_{\mathrm{A4}}=$
$\displaystyle~{}eQ_{\phi}\int\\!\frac{d^{4}k}{(2\pi)^{4}}\overline{u}(p_{2})\not{k}P_{L}u(p_{1})\Big{[}\epsilon\\!\cdot\\!\left(p_{1}+p_{2}-2k\right)\Big{]}$
$\displaystyle~{}\times\Big{[}m_{j}m_{i}\tilde{A}_{ji}^{L}(k^{2})+k^{2}\tilde{A}_{ji}^{R}(k^{2})+m_{i}\tilde{B}_{ji}(k^{2})+m_{j}\tilde{B}_{ij}^{*}(k^{2})\Big{]}$
$\displaystyle~{}\times\frac{1}{k^{2}-m_{i}^{2}}\frac{1}{k^{2}-m_{j}^{2}}\frac{1}{(k-p_{1})^{2}-m_{\phi}^{2}}\frac{1}{(k-p_{2})^{2}-m_{\phi}^{2}}.$
(2.27)

Note that the indices $i$ and $j$ are summed up. One can find that the CP
violating part is eventually zero due to the hermiticity of the self-energy,
and thus there is no contribution to the EDM in the amplitude.

####  2.2.4 The total amplitude

Combining Eqs. (2.14), (2.24) and (2.25), the CP violating term of the total
amplitude in Fig. 3 is summarized as

$\displaystyle i\mathcal{M}_{\mathrm{scalar}}^{\mathrm{CP}}=$
$\displaystyle~{}2iem_{i}\int\\!\frac{d^{4}k}{(2\pi)^{4}}\overline{u}(p_{2})\left[\left(k\\!\cdot\\!q\right)\not{\epsilon}-\left(\epsilon\\!\cdot\\!k\right)\not{q}\right]P_{L}u(p_{1})\frac{1}{\left(k-p\right)^{2}-m_{\phi}^{2}}\frac{1}{k^{2}-m_{i}^{2}}\frac{1}{k^{2}-m_{j}^{2}}$
$\displaystyle\times\mathrm{Im}\left[Q_{\psi}\frac{d\tilde{B}_{ji}}{dk^{2}}+k^{2}\tilde{C}_{ji}(k^{2})+\tilde{D}_{ji}(k^{2})\right].$
(2.28)

This is our main result of the reduction of the EDM calculation. Note that we
do not use any approximation to derive Eq. (2.28) except for the
$\mathcal{O}(q)$ expansion, which does not affect the EDM calculation.
Therefore, the above calculation for the EDM is exact.

The formula in Eq. (2.28) does not directly give the EDM of the fermion
$f_{\alpha}$, and we have to assume a specific model to further proceed with
the calculation to obtain a concrete formula of EDMs. However, once a model is
fixed, using the Gordon identity, the EDM coefficient $d_{\alpha}$ can
straightforwardly be extracted from

$\displaystyle
i\mathcal{M}_{\mathrm{total}}^{\mathrm{CP}}=d_{\alpha}\epsilon_{\mu}\overline{u}(p_{2})i\sigma^{\mu\nu}q_{\nu}\gamma_{5}u(p_{1}),$
(2.29)

where $q=p_{2}-p_{1}$. In particular, if $p^{2}=m_{\alpha}^{2}\ll
m_{\phi}^{2}$, one can expand the $\phi$ propagator in Eq. (2.28) as

$\displaystyle\frac{1}{(k-p)^{2}-m_{\phi}^{2}}$
$\displaystyle\approx\frac{1}{k^{2}-m_{\phi}^{2}}+\frac{2\left(k\\!\cdot\\!p\right)}{\left(k^{2}-m_{\phi}^{2}\right)^{2}}.$
(2.30)

Then it can be found that the contribution from the first term in the right-
hand side of Eqs. (2.30) vanishes because the resultant integrand is an odd
function in terms of the loop momentum $k$. Thus the second term in Eq. (2.30)
gives a leading contribution to the EDM. One can derive the approximate EDM
formula for $m_{\alpha}^{2}\ll m_{\phi}^{2}$,

$\displaystyle\frac{d_{\alpha}}{e}\approx$
$\displaystyle-\frac{im_{i}m_{\alpha}}{2}\int\\!\frac{d^{4}k}{(2\pi)^{4}}\frac{k^{2}}{\left(k^{2}-m_{\phi}^{2}\right)^{2}}\frac{1}{k^{2}-m_{i}^{2}}\frac{1}{k^{2}-m_{j}^{2}}$
$\displaystyle\times\mathrm{Im}\left[Q_{\psi}\frac{d\tilde{B}_{ji}}{dk^{2}}+k^{2}\tilde{C}_{ji}(k^{2})+\tilde{D}_{ji}(k^{2})\right],$
(2.31)

where $\tilde{B}_{ji}(k^{2})$ is the form factor of the self-energy, and
$\tilde{C}_{ji}(k^{2})$ and $\tilde{D}_{ji}(k^{2})$ are defined in Eq.
(2.26).666We can read out the higher dimensional operator that induces the EDM
with equations of motion. Here, we introduce the following loop function.
$\displaystyle\int\\!\frac{d^{4}k}{(2\pi)^{4}}\frac{k_{\mu}}{\left[\left(k-p\right)^{2}-m_{\phi}^{2}\right]\left(k^{2}-m_{i}^{2}\right)\left(k^{2}-m_{j}^{2}\right)}\mathrm{Im}\left[Q_{\psi}\frac{d\tilde{B}_{ji}}{dk^{2}}+k^{2}\tilde{C}_{ji}(k^{2})+\tilde{D}_{ji}(k^{2})\right]\equiv
iC^{\rm CP}p_{\mu},$ where $C^{\rm CP}$ is the scalar function. After
performing the loop integral of Eq. (2.28), we can read out the dimension six
operator shown below.  $\displaystyle{\mathcal{L}}_{\rm eff}=-em_{i}C^{\rm
CP}\left[\overline{f}_{\alpha}i{D^{\mu}\raisebox{9.38945pt}{\tiny{$\leftrightarrow$}\hskip
2.84544pt}}~{}\gamma^{\nu}P_{L}f_{\alpha}\right]F_{\mu\nu},$ where $D_{\mu}$
is the covariant derivative for $f_{\alpha}$.

In the next section, some examples of the EDM calculation using Eq. (2.28)
will be shown in specific models.

Figure 5: Feynman diagrams potentially inducing non-zero EDMs of the fermion
$f_{\alpha}$ with the multi-fermions $\psi_{i}$ and a vector boson $X$.

####  2.2.5 EDM reduction for diagrams with vector boson in outer loop

For the case that a vector boson propagates instead of the scalar $\phi$ as
shown in Fig. 5, the calculation is essentially the same as the scalar case.
The Lagrangian for this case is given by

$\displaystyle\mathcal{L}=$
$\displaystyle-\Big{(}g_{i\alpha}X_{\mu}^{*}\overline{\psi_{i}}\gamma^{\mu}P_{L}f_{\alpha}+\mathrm{H.c.}\Big{)}-\overline{\psi}_{i}\Sigma_{ij}\psi_{j}-eA_{\mu}\overline{\psi_{i}}\Lambda_{ij}^{\mu}\psi_{j}$
$\displaystyle-
Q_{\psi}A_{\mu}\overline{\psi_{i}}\gamma^{\mu}\psi_{i}+ieQ_{X}\Big{(}A_{\mu}\big{(}X_{\nu}^{*}\partial^{\mu}\raisebox{9.38945pt}{\tiny{$\leftrightarrow$}\hskip
2.84544pt}X^{\nu}\big{)}+X_{\mu}\big{(}A_{\nu}\partial^{\mu}\raisebox{9.38945pt}{\tiny{$\leftrightarrow$}\hskip
2.84544pt}X^{*\nu}\big{)}+X_{\mu}^{*}\big{(}X_{\nu}\partial^{\mu}\raisebox{9.38945pt}{\tiny{$\leftrightarrow$}\hskip
2.84544pt}A^{\nu}\big{)}\Big{)}.$ (2.32)

There is no contribution to EDM from the diagram (B4), and the CP violating
term induced by the other diagrams is summarized as

$\displaystyle i\mathcal{M}_{\mathrm{vector}}^{\mathrm{CP}}=$
$\displaystyle~{}-2iem_{i}\int\\!\frac{d^{4}k}{(2\pi)^{4}}\overline{u}(p_{2})\gamma^{\rho}\Big{[}\left(k\\!\cdot\\!q\right)\not{\epsilon}-\left(\epsilon\\!\cdot\\!k\right)\not{q}\Big{]}P_{L}\gamma^{\sigma}u(p_{1})$
$\displaystyle~{}\times\frac{1}{\left(k-p\right)^{2}-m_{X}^{2}}\frac{1}{k^{2}-m_{i}^{2}}\frac{1}{k^{2}-m_{j}^{2}}\left(g_{\rho\sigma}-\frac{\left(k-p\right)_{\rho}\left(k-p\right)_{\sigma}}{m_{X}^{2}}\right)$
$\displaystyle~{}\times\mathrm{Im}\left[Q_{\psi}\frac{d\tilde{B}_{ji}}{dk^{2}}+k^{2}\tilde{C}_{ji}(k^{2})+\tilde{D}_{ji}(k^{2})\right],$
(2.33)

where $p\equiv(p_{1}+p_{2})/2$ as same as the scalar case, and we chose the
unitary gauge. The form factors $\tilde{B}(k^{2})$, $\tilde{C}(k^{2})$ and
$\tilde{D}(k^{2})$ are defined as same as Eqs. (2.10) and (2.26) with the
gauge coupling $g_{j\alpha}^{*}$, $g_{i\alpha}$ instead of the Yukawa coupling
$y_{j\alpha}^{*}$, $y_{i\alpha}$.

With the approximation $p^{2}=m_{\alpha}^{2}\ll m_{X}^{2}$, the EDM formula
can be extracted from Eq. (2.29), which is given by777To read out the higher
dimensional operators, we introduce the following loop functions.
$\displaystyle\int\\!\frac{d^{4}k}{(2\pi)^{4}}\frac{k_{\mu}\left(g_{\rho\sigma}-\frac{\left(k-p\right)_{\rho}\left(k-p\right)_{\sigma}}{m_{X}^{2}}\right)}{\left[\left(k-p\right)^{2}-m_{X}^{2}\right]\left(k^{2}-m_{i}^{2}\right)\left(k^{2}-m_{j}^{2}\right)}\mathrm{Im}\left[Q_{\psi}\frac{d\tilde{B}_{ji}}{dk^{2}}+k^{2}\tilde{C}_{ji}(k^{2})+\tilde{D}_{ji}(k^{2})\right]$
$\displaystyle\equiv$ $\displaystyle~{}iC^{\rm
CP}_{1}p_{\mu}g_{\rho\sigma}+iC^{\rm
CP}_{2}\left(g_{\mu\rho}p_{\sigma}+g_{\mu\sigma}p_{\rho}\right)+iC^{\rm
CP}_{3}p_{\mu}p_{\rho}p_{\sigma},$ where $C^{\rm CP}_{i}(i=1,2,3)$ are the
scalar functions. After performing the loop integral of Eq. (2.33), we can
read out the dimension six and dimension eight operators shown below.
$\displaystyle{\mathcal{L}}_{\rm eff}=$ $\displaystyle-2em_{i}\left(C^{\rm
CP}_{1}+2C^{\rm
CP}_{2}\right)\left[\overline{f}_{\alpha}i{D^{\mu}\raisebox{9.38945pt}{\tiny{$\leftrightarrow$}\hskip
2.84544pt}}~{}\gamma^{\nu}P_{R}f_{\alpha}\right]F_{\mu\nu}+\frac{em_{i}}{4}C^{\rm
CP}_{3}\left[\overline{f}_{\alpha}i{D^{\mu}\raisebox{9.38945pt}{\tiny{$\leftrightarrow$}\hskip
2.84544pt}}~{}\gamma^{\nu}\bigl{(}i{D\raisebox{9.38945pt}{\tiny{$\leftrightarrow$}\hskip
2.84544pt}}\bigr{)}^{2}P_{R}f_{\alpha}\right]F_{\mu\nu}.$ These operators
induce the EDMs with equations of motion.

$\displaystyle\frac{d_{\alpha}}{e}\approx$
$\displaystyle~{}\frac{3im_{i}m_{\alpha}}{4}\int\\!\frac{d^{4}k}{(2\pi)^{4}}\frac{k^{2}}{m_{X}^{2}}\frac{k^{2}}{\left(k^{2}-m_{X}^{2}\right)^{2}}\frac{1}{k^{2}-m_{i}^{2}}\frac{1}{k^{2}-m_{j}^{2}}$
$\displaystyle\times\mathrm{Im}\left[Q_{\psi}\frac{d\tilde{B}_{ji}}{dk^{2}}+k^{2}\tilde{C}_{ji}(k^{2})+\tilde{D}_{ji}(k^{2})\right].$
(2.34)

As shown in our reduction formulas, Eq. (2.28) and Eq. (2.33), it is essential
to pick up the chirality flipping effects from the internal loop in order to
obtain the non-zero EDM contributions. We immediately conclude that EDM
cancellation happens as long as we consider the setup where no chirality
flipping effects are induced in the self-energy and the vertex correction of
the internal fermion.

###  2.3 Reduction formulas for Chromo EDM

We can apply our reduction scheme to the calculation of the chromo-EDM. The
chromo-EDM arises from the diagrams with the external gluon instead of the
external photon. For the non-Abelian gauge theory, we have the Slavnov-Taylor
identity imposed by the Becchi-Rouet-Stora-Tyutin (BRST) symmetry, which has a
more complicated form than the Ward-Takahashi identity in the Abelian gauge
theory. However, the Ward-Takahashi identity holds for the background gluon
vertices, which is shown using the background gauging method [20, 19]. We can
treat gluons as the background fields in the chromo-EDM calculation. We also
note that the diagrams that the gluon couples to the outer loop do not induce
the chromo-EDM for the same reason as the diagram (A4) and diagram (B4) in
Fig. 3 and Fig. 5, respectively. The remaining diagrams have the same group
factors of SU(3)c symmetry, which are automatically determined once we fix the
concrete model, and thus the same reduction mechanism works. Consequently, we
obtain the same EDM reduction formulas for the chromo-EDM except for the
overall group factor.

##  3 Application to specific models

In this section, we apply our reduction formulas to several models including
the SM. As we clarify the conditions to get non-zero EDMs, we understand when
and why we have exact EDM cancellation in rainbow-type diagrams. We also show
the setup that we can apply our formulas and obtain the non-zero values of the
EDM prediction.

###  3.1 The Standard Model

The well-known result of the SM, in which the neutron EDM induced from the
two-loop $W$ boson diagrams totally cancels out, can be reproduced from our
formula in Eq. (2.33). Recall that only the left-handed fermions couple to the
$W$ boson in the SM as in Eq. (2.1). At the one-loop level, the self-energy
for the up-type quarks is explicitly given by

$\displaystyle\Sigma_{ji}(\not{k})$
$\displaystyle=-\frac{ig_{2}^{2}}{2}V_{j\alpha}V_{i\alpha}^{*}\int\\!\frac{d^{4}\ell}{(2\pi)^{4}}\gamma^{\rho}P_{L}\frac{\not{k}-\not{\ell}+m_{\beta}}{(k-\ell)^{2}-m_{\beta}^{2}}P_{R}\gamma^{\sigma}\frac{1}{\ell^{2}-m_{W}^{2}}\left(g_{\rho\sigma}-\frac{\ell_{\rho}\ell_{\sigma}}{m_{W}^{2}}\right).$
(3.35)

Due to the projection operators $P_{L}$ and $P_{R}$, we have no chirality
flipping term. In this case, therefore, we only have $A^{L}_{ji}(k^{2})$ in
terms of the notation introduced in Eq. (2.11). Thus one can find that there
is no EDM contribution from the self-energy.

Regarding the contribution from the vertex correction $\Lambda_{ji}^{\mu}$,
there are two possibilities that the photon can be attached to the down-type
quark and the $W$ boson, which are explicitly given by

$\displaystyle\left(\Lambda_{d}\right)_{ji}^{\mu}(k_{1},k_{2})=$
$\displaystyle-\frac{ig_{2}^{2}}{2}V_{j\beta}V_{i\beta}^{*}\int\\!\frac{d^{4}\ell}{(2\pi)^{4}}\gamma^{\rho}P_{L}\frac{\left(\not{\ell}_{2}+m_{\beta}\right)}{\ell_{2}^{2}-m_{\beta}^{2}}\gamma^{\mu}\frac{\left(\not{\ell}_{1}+m_{\beta}\right)}{\ell_{1}^{2}-m_{\beta}^{2}}P_{R}\gamma^{\sigma}$
$\displaystyle~{}\times\frac{1}{\left(k-\ell\right)^{2}-m_{W}^{2}}\left(g_{\rho\sigma}-\frac{\left(k-\ell\right)_{\rho}\left(k-\ell\right)_{\sigma}}{m_{W}^{2}}\right),$
(3.36) $\displaystyle\left(\Lambda_{W^{+}}\right)_{ji}^{\mu}(k_{1},k_{2})=$
$\displaystyle-\frac{ig_{2}^{2}}{2}V_{j\beta}V_{i\beta}^{*}\int\\!\frac{d^{4}\ell}{(2\pi)^{4}}\gamma^{\rho}P_{L}\frac{\left(\not{k}-\not{\ell}+m_{\beta}\right)}{\left(k-\ell\right)^{2}-m_{\beta}^{2}}P_{R}\gamma^{\sigma}\frac{1}{\ell_{1}^{2}-m_{W}^{2}}\frac{1}{\ell_{2}^{2}-m_{W}^{2}}$
$\displaystyle~{}\times\Bigg{[}+\left(g_{\sigma}^{~{}\nu}-\frac{\ell_{1\sigma}\ell_{1}^{\nu}}{m_{W}^{2}}\right)\left(g_{\nu\rho}-\frac{\ell_{2\nu}\ell_{2\rho}}{m_{W}^{2}}\right)\left(\ell_{1}+\ell_{2}\right)^{\mu}$
$\displaystyle~{}\hskip
21.33955pt-\left(g_{\sigma}^{~{}\mu}-\frac{\ell_{1\sigma}\ell_{1}^{\mu}}{m_{W}^{2}}\right)\left(g_{\nu\rho}-\frac{\ell_{2\nu}\ell_{2\rho}}{m_{W}^{2}}\right)\left(\ell_{1}-q\right)^{\nu}$
$\displaystyle~{}\hskip
21.33955pt-\left(g_{\sigma\nu}-\frac{\ell_{1\sigma}\ell_{1\nu}}{m_{W}^{2}}\right)\left(g^{\mu}_{~{}\rho}-\frac{\ell_{2}^{\mu}\ell_{2\rho}}{m_{W}^{2}}\right)\left(\ell_{2}+q\right)^{\nu}\Bigg{]},$
(3.37)

where $\ell_{1}=\ell-q/2$ and $\ell_{2}=\ell+q/2$. From these expressions, one
can easily find that there is no contribution to the form factors
$\tilde{C}_{ji}(k^{2})$ and $\tilde{D}_{ji}(k^{2})$ shown in Eq. (2.33)
because only the odd numbers of the gamma matrices are involved in the vertex
corrections. Therefore the neutron EDM induced from the weak interactions in
the SM at the two-loop level exactly vanishes, which is consistent with the
result in Ref. [8].

###  3.2 The scotogenic model

In the scotogenic model proposed by E. Ma [21], it is known that the charged
lepton EDMs induced from a subset of the diagrams exactly cancel [22, 23]. As
in the SM, we have no chirality flipping term in the self-energy due to the
chirality projection operators at the one-loop level. In addition, there is no
$\tilde{C}_{ji}(k^{2})$ and $\tilde{D}_{ji}(k^{2})$ terms in Eq. (2.28) coming
from the vertex correction. Therefore, the EDM cancellations, which have been
found in the previous analysis [22, 23], can be understood from our
generalized EDM formula. Moreover, our derivation using form factors can be
applicable for both Dirac and Majorana fermions.

On the other hand in this model, there are some additional two-loop diagrams
that are not categorized as the diagrams as shown in Fig. 3.888See diagrams
shown in Fig. 2 of Ref. [23], for example. We can not find the sub-diagram
structures from these diagrams.  Our calculation is not applicable for these
diagrams, and indeed a non-zero contribution to charged lepton EDMs arises.

###  3.3 The singlet-triplet extended model

The singlet-triplet model which is named the ST model in Ref. [23] has been
considered in the previous work. It has turned out that the charged lepton EDM
arising from the rainbow-type diagrams also vanishes in this model. Here we
further extend the model introducing a triplet scalar $\Phi$ to demonstrate
how we can obtain non-zero EDM from the rainbow-type diagrams using our
formulas. The particle contents are given in Tab. 2, and the Lagrangian
relevant to the charged lepton EDMs is given by

$\displaystyle\mathcal{L}=$
$\displaystyle~{}\left(D_{\mu}\eta\right)^{{\dagger}}\left(D^{\mu}\eta\right)+\frac{1}{2}\left(D_{\mu}\Phi\right)\left(D^{\mu}\Phi\right)+\frac{1}{2}\overline{\Psi_{i}}\left(i\not{D}-m_{i}\right)\Psi_{i}$
$\displaystyle-\left(y_{i\alpha}\eta\overline{\Psi_{i}}P_{L}L_{\alpha}+\frac{\lambda_{ij}}{2}\Phi\overline{\Psi_{i}^{c}}P_{R}\Psi_{j}+\mathrm{h.c.}\right),$
(3.38)

where $\Psi_{1}\equiv N$ (singlet) and $\Psi_{2}\equiv\Sigma$ (triplet) and
$i,j=1,2$. The new coupling $\lambda_{ij}$ which is not included in the
original singlet-triplet model plays an important role in inducing a non-zero
EDM due to chirality flipping effects. Note that the cubic term
$\Phi|H|_{\bm{3}}^{2}+\mathrm{h.c.}$ in the scalar potential induces the
mixing between $H$ and $\Phi$ where $|H|_{\bm{3}}^{2}$ represents the
$SU(2)_{L}$ triplet. However, this is ignored in the following calculation for
simplicity.

| $\eta$ | $\Phi$ | $N$ | $\Sigma$
---|---|---|---|---
$SU(2)_{L}$ | $\bm{2}$ | $\bm{3}$ | $\bm{1}$ | $\bm{3}$
$U(1)_{Y}$ | $1/2$ | $0$ | $0$ | $0$
$\mathbb{Z}_{2}$ | $-1$ | $+1$ | $-1$ | $-1$
Spin | $0$ | $0$ | $1/2$ | $1/2$
Table 2: Contents of the new particle in the singlet-triplet fermion model.
Figure 6: Feynman diagrams inducing non-zero charged lepton EDM in the
singlet-triplet extended model where $\eta^{+}$, $H$ and $A$ are the charged,
CP-even and CP-odd neutral components of the $SU(2)_{L}$ doublet $\eta$,
respectively.

We can use our formula Eq. (2.28) to extract the charged lepton EDM. The
relevant Feynman diagrams are shown in Fig. 6. For the three diagrams in the
top, the EDM contribution comes from the self-energy correction (or equivalent
to the longitudinal part of the vertex correction). The form factor
$B_{ji}(k^{2})$ of the self-energy can be evaluated as

$\displaystyle
B_{22}(k^{2})=-\frac{\lambda^{*2}m_{1}}{(4\pi)^{2}}\int_{0}^{1}\\!dxdy\delta(x+y-1)\lim_{\epsilon\to
0}\Gamma\left(\frac{\epsilon}{2}\right)\left(\frac{4\pi}{\Delta_{1}(k^{2})}\right)^{\epsilon/2},$
(3.39)

where $\Delta_{1}(k^{2})\equiv xm_{1}^{2}+ym_{\phi^{+}}^{2}-xyk^{2}$,
$\lambda\equiv\left(\lambda_{12}+\lambda_{21}\right)/2$ and $\epsilon\equiv
4-d$ is the parameter of the dimensional regularization. There is no
contribution from the transverse part of the vertex correction since it does
not include terms with two gamma matrices.

For the two diagrams at the bottom of Fig. 6, the EDM contribution is induced
from the vertex correction of the right diagram, which is evaluated as

$\displaystyle
D_{11}(k^{2})=\frac{\lambda^{*2}}{(4\pi)^{2}}\int_{0}^{1}dxdy\delta(x+y-1)\frac{xm_{2}}{\Delta_{2}(k^{2})},$
(3.40)

where $\Delta_{2}(k^{2})=xm_{2}^{2}+ym_{\phi^{+}}^{2}-xyk^{2}$. Substituting
the above expressions into the formula of Eq. (2.28) and using the Gordon
identity, we can extract the EDM from Eq. (2.29). The result is summarized as

$\displaystyle\frac{d_{\alpha}}{e}=$
$\displaystyle~{}\frac{\mathrm{Im}\lambda^{2}}{2(4\pi)^{4}}\frac{m_{\alpha}}{m_{\eta^{+}}^{2}}\sqrt{\xi_{1}\xi_{2}}\int_{0}^{1}\\!dxdy\delta(x+y-1)\int_{0}^{1}\\!dsdtdu\delta(s+t+u-1)$
$\displaystyle\times
xy^{2}su\Bigg{[}\frac{x|y_{2\alpha}|^{2}}{\Delta^{2}\left(\xi_{2},\xi_{1},\xi_{H},\xi_{\phi^{+}},\xi_{\alpha}\right)}+\frac{x|y_{2\alpha}|^{2}}{\Delta^{2}\left(\xi_{2},\xi_{1},\xi_{A},\xi_{\phi^{+}},\xi_{\alpha}\right)}-\frac{2|y_{1\alpha}|^{2}}{\Delta^{2}\left(\xi_{1},\xi_{2},1,\xi_{\phi^{+}},\xi_{\alpha}\right)}\Bigg{]},$
(3.41)

where $\Delta(\xi_{i},\xi_{j},\xi_{\varphi},\xi_{\phi^{+}},\xi_{\alpha})\equiv
xy\left(s\xi_{i}+u\xi_{\varphi}\right)+xt\xi_{\phi^{+}}+yt\xi_{j}-xyu(1-u)\xi_{\alpha}$
with the dimensionless parameter $\xi_{a}\equiv m_{a}^{2}/m_{\eta^{+}}^{2}$.
Note that we have taken into account only the rainbow-type diagrams here, and
some additional diagrams which are topologically different from the rainbow-
type ones may give another contribution comparable or even dominant over Eq.
(3.41).

The magnitude of the electron EDM is simply estimated assuming the same order
of the fermion masses ($m_{e}\ll m_{1}\sim m_{2}$), the scalar masses
($m_{e}\ll m_{\eta^{+}}\sim m_{\phi^{+}}\sim m_{H}\sim m_{A}$) and the Yukawa
coupling ($|y_{1e}|\sim|y_{2e}|$). In this case, we have numerically checked
that the loop functions are approximated by

$\displaystyle~{}\int_{0}^{1}\\!dxdy\delta(x+y-1)\int_{0}^{1}\\!dsdtdu\delta(s+y+u-1)\frac{x^{2}y^{2}su}{\Delta^{2}\left(\xi_{1},\xi_{1},1,1,0\right)},$
$\displaystyle\approx$
$\displaystyle\left\\{\begin{array}[]{ccc}0.15&\text{for}&\xi_{1}\ll 1\\\
0.15\xi_{1}^{-2}&\text{for}&\xi_{1}\gg 1\end{array}\right.,$ (3.44)
$\displaystyle~{}\int_{0}^{1}\\!dxdy\delta(x+y-1)\int_{0}^{1}\\!dsdtdu\delta(s+y+u-1)\frac{xy^{2}su}{\Delta^{2}\left(\xi_{1},\xi_{1},1,1,0\right)},$
$\displaystyle\approx$
$\displaystyle\left\\{\begin{array}[]{ccc}0.25\log\xi_{1}^{-1}&\text{for}&\xi_{1}\ll
1\\\ 0.30\xi_{1}^{-2}&\text{for}&\xi_{1}\gg 1\end{array}\right.,$ (3.47)

therefore the electron EDM is

$\displaystyle\frac{d_{e}}{e}\approx-\frac{\mathrm{Im}\lambda^{2}|y_{1e}|^{2}}{(4\pi)^{4}}\frac{m_{e}}{m_{\eta^{+}}^{2}}\xi_{1}\times\left\\{\begin{array}[]{ccc}0.25\log\xi_{1}^{-1}&\text{for}&\xi_{1}\ll
1\\\ 0.30\xi_{1}^{-2}&\text{for}&\xi_{1}\gg 1\end{array}\right..$ (3.50)

Fixing the mass ratio $\xi_{1}=0.1$ and $10$ for examples, the magnitude of
the electron EDM is estimated as

$\displaystyle\frac{|d_{e}|}{e}=\left\\{\begin{array}[]{ccc}\displaystyle
1.7\times
10^{-30}~{}\mathrm{cm}\left(\frac{\mathrm{Im}\lambda^{2}}{0.3}\right)\left(\frac{|y_{1e}|}{0.5}\right)^{2}\left(\frac{1~{}\mathrm{TeV}}{m_{\eta^{+}}}\right)^{2}&\text{for}&\xi_{1}=0.1\\\
\displaystyle 9.1\times
10^{-30}~{}\mathrm{cm}\left(\frac{\mathrm{Im}\lambda^{2}}{0.3}\right)\left(\frac{|y_{1e}|}{0.5}\right)^{2}\left(\frac{1~{}\mathrm{TeV}}{m_{1}}\right)^{2}&\text{for}&\xi_{1}=10\end{array}\right..$
(3.53)

These values are comparable to the future prospect of the ACME Collaboration
that is $\mathcal{O}(10^{-30})~{}\mathrm{cm}$ [6, 7].

##  4 The other generalizations

Another generalization of the EDM reduction is possible in models with multi-
scalars $\phi_{i}$ and a fermion $\psi$ instead of multi-fermions $\psi_{i}$
and a scalar $\phi$, or a vector boson $X$. The Feynman diagrams relevant to
the EDM calculation are shown in Fig. 7. The calculation for this case is
essentially the same as Sec. 2. We take the effective Lagrangian given by

$\displaystyle\mathcal{L}_{\mathrm{eff}}=$
$\displaystyle-\Big{(}y_{i\alpha}\phi^{*}_{i}\overline{\psi}P_{L}f_{\alpha}+\mathrm{h.c.}\Big{)}-\phi_{i}^{*}M^{2}_{ij}\phi_{j}-eA_{\mu}\phi_{i}^{*}\Lambda_{ij}^{\mu}\phi_{j},$
$\displaystyle-
eQ_{\psi}A_{\mu}\overline{\psi}\gamma^{\mu}\psi+ieQ_{\phi}A_{\mu}\big{(}\phi_{i}\partial^{\mu}\raisebox{9.38945pt}{\tiny{$\leftrightarrow$}\hskip
2.84544pt}\phi_{i}^{*}\big{)},$ (4.54)

where $M_{ij}^{2}$ is the self-energy of the scalar particles and
$\Lambda_{ij}^{\mu}$ is the vertex correction induced at the loop level

$\displaystyle\Lambda_{ij}^{\mu}(k_{1},k_{2})=\sum_{p}Q_{p}\big{(}\Lambda_{p}\big{)}_{ij}^{\mu}(k_{1},k_{2}).$
(4.55)

For the self-energy, the hermiticity condition

$\displaystyle M_{ji}^{2}(k^{2})=M_{ij}^{*2}(k^{2}),$ (4.56)

is imposed as same as Sec. 2.

Figure 7: Feynman diagrams which may contribute to EDMs of the fermions
$f_{\alpha}$ with multi-scalars $\phi_{i}$ and a fermion $\psi$.

The amplitude for each diagram in Fig. 7 can be evaluated as

$\displaystyle i\mathcal{M}_{\mathrm{C1}}=$
$\displaystyle~{}eQ_{\phi}\int\\!\frac{d^{4}k}{(2\pi)^{4}}\tilde{M}_{ji}^{2}(k_{2}^{2})\overline{u}(p_{2})P_{R}\left(\not{k}-\not{p}+m_{\psi}\right)P_{L}u(p_{1})\epsilon_{\mu}\left(k_{1}+k_{2}\right)^{\mu}$
$\displaystyle\times\frac{1}{(k-p)^{2}-m_{\psi}^{2}}\frac{1}{k_{1}^{2}-m_{i}^{2}}\frac{1}{k_{2}^{2}-m_{i}^{2}}\frac{1}{k_{2}^{2}-m_{j}^{2}},$
(4.57) $\displaystyle i\mathcal{M}_{\mathrm{C2}}=$
$\displaystyle~{}eQ_{\phi}\int\\!\frac{d^{4}k}{(2\pi)^{4}}\tilde{M}_{ji}^{2}(k_{1}^{2})\overline{u}(p_{2})P_{R}\left(\not{k}-\not{p}+m_{\psi}\right)P_{L}u(p_{1})\epsilon_{\mu}\left(k_{1}+k_{2}\right)^{\mu}$
$\displaystyle\times\frac{1}{(k-p)^{2}-m_{\psi}^{2}}\frac{1}{k_{1}^{2}-m_{i}^{2}}\frac{1}{k_{1}^{2}-m_{j}^{2}}\frac{1}{k_{2}^{2}-m_{j}^{2}},$
(4.58) $\displaystyle i\mathcal{M}_{\mathrm{C3}}=$
$\displaystyle~{}e\int\\!\frac{d^{4}k}{(2\pi)^{4}}\epsilon_{\mu}\tilde{\Lambda}_{ji}^{\mu}(k_{1},k_{2})\overline{u}(p_{2})P_{R}\left(\not{k}-\not{p}+m_{\psi}\right)P_{L}u(p_{1})\frac{1}{k_{1}^{2}-m_{i}^{2}}\frac{1}{k_{2}^{2}-m_{j}^{2}},$
(4.59) $\displaystyle i\mathcal{M}_{\mathrm{C4}}=$
$\displaystyle~{}eQ_{\psi}\int\\!\frac{d^{4}k}{(2\pi)^{4}}\tilde{M}_{ji}^{2}(k^{2})\overline{u}(p_{2})P_{R}\big{(}\not{p}_{2}-\not{k}+m_{\psi}\big{)}\gamma^{\mu}\big{(}\not{p}_{1}-\not{k}+m_{\psi}\big{)}P_{L}u(p_{1})$
$\displaystyle\times\frac{1}{k^{2}-m_{i}^{2}}\frac{1}{k^{2}-m_{j}^{2}}\frac{1}{(p_{1}-k)^{2}-m_{\psi}^{2}}\frac{1}{(p_{2}-k)^{2}-m_{\psi}^{2}},$
(4.60)

where $\tilde{M}_{ji}^{2}(k^{2})$ and
$\tilde{\Lambda}_{ji}^{\mu}(k_{1},k_{2})$ are defined by

$\displaystyle\tilde{M}_{ji}^{2}(k^{2})\equiv
y_{j\alpha}^{*}M_{ij}^{2}(k^{2})y_{i\alpha},\qquad\Lambda_{ji}^{\mu}(k_{1},k_{2})\equiv
y_{j\alpha}^{*}\Lambda_{ji}^{\mu}(k_{1},k_{2})y_{i\alpha}.$ (4.61)

As in the previous calculation, the scalar vertex correction
$\Lambda^{\mu}(k_{1},k_{2})$ can be decomposed into the longitudinal part and
transverse part which can be given by [16]

$\displaystyle\Lambda_{\mathrm{L}}^{\mu}(k_{1},k_{2})=$
$\displaystyle~{}\frac{M^{2}(k_{1}^{2})-M^{2}(k_{2}^{2})}{\left(k\\!\cdot\\!q\right)}k^{\mu},$
(4.62) $\displaystyle\Lambda_{\mathrm{T}}^{\mu}(k_{1},k_{2})=$
$\displaystyle~{}C(k_{1},k_{2})\Big{[}\left(k\\!\cdot\\!q\right)q^{\mu}-q^{2}k^{\mu}\Big{]},$
(4.63)

where $k_{1}=k-q/2$ and $k_{2}=k+q/2$. These correspond to Eqs. (2.19) and
(2.20) in the multi-fermion case. Note that the vertex correction can be
expanded by two vectors $k^{\mu}$ and $q^{\mu}$. Then similarly to Sec. 2, one
vector is removed by the condition of the Ward-Takahashi identity. As a
result, we obtain the above decomposition. One can explicitly find that the
longitudinal and transverse parts of the vertex correction obey

$\displaystyle q_{\mu}\Lambda_{\mathrm{L}}^{\mu}(k_{1},k_{2})=$
$\displaystyle~{}M^{2}(k_{1}^{2})-M^{2}(k_{2}^{2}),$ (4.64) $\displaystyle
q_{\mu}\Lambda_{\mathrm{T}}^{\mu}(k_{1},k_{2})=$ $\displaystyle~{}0$ (4.65)

and thus the Ward-Takahashi identity

$\displaystyle q_{\mu}\Lambda^{\mu}(k_{1},k_{2})=$
$\displaystyle~{}M^{2}(k_{1}^{2})-M^{2}(k_{2}^{2}).$ (4.66)

From Eq. (4.63), it is clear that the transverse part
$\Lambda_{\mathrm{T}}^{\mu}(k_{1},k_{2})$ is $\mathcal{O}(q^{2})$, and does
not contribute to EDMs. Thus expanding the full vertex correction
$\Lambda^{\mu}(k_{1},k_{2})$ up to $\mathcal{O}(q)$, one obtains

$\displaystyle\Lambda^{\mu}(k_{1},k_{2})\approx\Lambda_{\mathrm{L}}^{\mu}\Big{|}_{q=0}+q_{\nu}\frac{\partial\Lambda_{\mathrm{T}}^{\mu}}{\partial
q_{\nu}}\Big{|}_{q=0}=-\frac{dM^{2}}{dk_{\mu}}.$ (4.67)

The CP violating part of the amplitude arising from (C1) and (C2) diagrams in
Fig. 7 can be evaluated as

$\displaystyle i\mathcal{M}_{\mathrm{C1+C2}}^{\mathrm{CP}}=$
$\displaystyle-2eQ_{\phi}m_{j}^{2}\int\\!\frac{d^{4}k}{(2\pi)^{4}}\left(k\\!\cdot\\!q\right)\mathrm{Im}\frac{d\tilde{M}_{ji}^{2}}{dk^{2}}\overline{u}(p_{2})P_{R}\left(\not{k}-\not{p}+m_{\psi}\right)P_{L}u(p_{1})2\left(\epsilon\\!\cdot\\!k\right)$
$\displaystyle\times\frac{1}{(k-p)^{2}-m_{\psi}^{2}}\frac{1}{\big{(}k^{2}-m_{i}^{2}\big{)}^{2}}\frac{1}{\left(k^{2}-m_{j}^{2}\right)^{2}},$
(4.68)

at $\mathcal{O}(q)$. For the amplitude of the diagram (C3), it is exactly the
same as Eq. (4.68) except for the opposite sign. The amplitude of the diagram
(C4) is totally real and does not give CP violation due to the hermiticity
condition of Eq. (4.56). Therefore summarizing the calculation, it turns out
that the CP violating part of the total amplitude in Fig. 7 exactly vanishes.

One can also consider multi-vector bosons $X_{i}$ instead of multi-scalars
$\phi_{i}$, and derive the reduced EDM formula. However, since there are no
well-motivated models involving the multi-vector bosons, we do not consider
this case.

##  5 Conclusions

In this paper, we have derived the reduced formulas for the EDM calculation at
two-loop level or higher introducing the form factors of the self-energy and
vertex correction that are related through the Ward-Takahashi identity. The
formulas indicate that one may extract only a part of the self-energy and
vertex correction in order to evaluate the fermion EDM arising from the
corresponding Feynman diagrams. The formulas can be employed in a larger class
of models with multi-fermions and a scalar (or a vector boson), and
significantly reduces the EDM calculation compared to the usual way. We find
it is essential to pick up the chirality flipping effects from the internal
loop in order to obtain the non-zero EDM contributions as shown in our
reduction formulas. From this fact, we immediately find that EDMs exactly
cancel if we consider a model with no chirality flipping effects in the self-
energy and the vertex correction for the internal fermion.

As for an application of our EDM formulas to specific models, first we have
revisited the EDM calculation in the SM and the scotogenic model. It has been
easily verified that the neutron EDM induced by the $W$ bosons vanishes at the
two-loop level in the SM. Similarly, in the scotogenic model, the charged
lepton EDM induced by the relevant rainbow-type diagrams also vanishes. In
addition, we have considered the singlet-triplet model extended with a triplet
scalar to demonstrate how we can obtain a non-zero EDM from our reduction
formulas.

Although the derived EDM formulas can be adapted in a large class of models,
one should note that additional diagrams of different topology which are not
treated in this paper may exist. For example, in the scotogenic model [23],
different topological diagrams exist and give a leading contribution to the
EDM.

## Acknowledgments

MF would like to thank Teppei Kitahara for the useful discussion. This work
was supported by JSPS Grant-in-Aid for Scientific Research KAKENHI Grant No.
JP20J12392 (MF), JP20H01895 (JH), JP21K03572 (JH), and JP20K22349 (TT). The
work of J.H. was supported by Grant-in-Aid for Scientific research from the
Ministry of Education, Science, Sports, and Culture (MEXT), Japan (Grant
Numbers 16H06492). The work of J.H. was also supported by JSPS Core-to-Core
Program (Grant Numbers JPJSCCA20200002), and World Premier International
Research Center Initiative (WPI Initiative), MEXT, Japan.

##  Appendix A Decomposition of vertex correction

One can take the following 12 independent vectors to construct the vertex
correction $\Lambda^{\mu}(k_{1},k_{2})$ [17],

$\displaystyle\mathcal{V}_{1}^{\mu}=$ $\displaystyle~{}k^{\mu},\quad\hskip
19.0633pt\mathcal{V}_{2}^{\mu}=q^{\mu},\quad\hskip
63.16515pt\mathcal{V}_{3}^{\mu}=\gamma^{\mu},\quad\hskip
14.22636pt\mathcal{V}_{4}^{\mu}=i\sigma^{\mu\nu}k_{\nu},$
$\displaystyle\mathcal{V}_{5}^{\mu}=$
$\displaystyle~{}i\sigma^{\mu\nu}q_{\nu},\quad\mathcal{V}_{6}^{\mu}=\not{k}q^{\mu},\quad\hskip
56.33633pt\mathcal{V}_{7}^{\mu}=\not{q}k^{\mu},\quad\hskip
8.5359pt\mathcal{V}_{8}^{\mu}=\not{k}k^{\mu},$
$\displaystyle\mathcal{V}_{9}^{\mu}=$
$\displaystyle~{}\not{q}q^{\mu},\quad\hskip
11.66573pt\mathcal{V}_{10}^{\mu}=i\epsilon^{\mu\nu\rho\sigma}\gamma_{5}\gamma_{\nu}k_{\rho}q_{\sigma},\quad\mathcal{V}_{11}^{\mu}=\not{k}\not{q}k^{\mu},\quad\mathcal{V}_{12}^{\mu}=\not{k}\not{q}q^{\mu},$
(A.69)

where $q=k_{2}-k_{1}$, $k_{1}=k-q/2$ and $k_{2}=k+q/2$. Then the vertex
correction $\Lambda^{\mu}(k_{1},k_{2})$ is given by

$\displaystyle\Lambda^{\mu}(k_{1},k_{2})=\sum_{a=1}^{12}\Big{[}\mathcal{C}_{a}^{L}(k_{1},k_{2})\mathcal{V}_{a}^{\mu}P_{L}+\mathcal{C}_{a}^{R}(k_{1},k_{2})\mathcal{V}_{a}^{\mu}P_{R}\Big{]}.$
(A.70)

Multiplying the photon momentum $q_{\mu}$ to Eq. (A.70), it should satisfy the
Ward-Takahashi identity shown in Eq. (2.16). Using the explicit form of the
self-energy in Eq. (2.11), and comparing the coefficients of the terms
proportional to $\not{k}\not{q}$, $\not{k}$, $\not{q}$ and $1$, one can obtain
the following 4 equations for the left chirality

$\displaystyle\mathcal{C}_{4}^{L}+\mathcal{C}_{11}^{L}\left(k\\!\cdot\\!q\right)+\mathcal{C}_{12}^{L}q^{2}=$
$\displaystyle~{}0,$
$\displaystyle\mathcal{C}_{6}^{L}q^{2}+\mathcal{C}_{8}^{L}\left(k\\!\cdot\\!q\right)=$
$\displaystyle~{}A^{L}(k_{1}^{2})-A^{L}(k_{2}^{2}),$
$\displaystyle\mathcal{C}_{3}^{L}+\mathcal{C}_{7}^{L}\left(k\\!\cdot\\!q\right)+\mathcal{C}_{9}^{L}q^{2}=$
$\displaystyle-\frac{A^{L}(k_{1}^{2})+A^{L}(k_{2}^{2})}{2},$
$\displaystyle\mathcal{C}_{1}^{L}\left(k\\!\cdot\\!q\right)+\mathcal{C}_{2}^{L}q^{2}-\mathcal{C}_{4}^{L}\left(k\\!\cdot\\!q\right)=$
$\displaystyle~{}B^{L}(k_{1}^{2})-B^{L}(k_{2}^{2}),$

and the same equations for the other chirality are also obtained. Note that
for the vectors $\mathcal{V}_{5}^{\mu}$ and $\mathcal{V}_{10}^{\mu}$, it gives
identically zero when $q_{\mu}$ is multiplied. From these equations, 4
coefficients for each chirality can be eliminated, which are chosen as
$\mathcal{C}_{1}^{L/R}$, $\mathcal{C}_{3}^{L/R}$, $\mathcal{C}_{8}^{L/R}$ and
$\mathcal{C}_{11}^{L/R}$ here. As a result, one can decompose the vertex
correction into the longitudinal part

$\displaystyle\Lambda_{\mathrm{L}}^{\mu}(k_{1},k_{2})=$
$\displaystyle~{}\Bigg{[}\frac{A^{L}(k_{1}^{2})-A^{L}(k_{2}^{2})}{\left(k\\!\cdot\\!q\right)}k^{\mu}\not{k}-\frac{A^{L}(k_{1}^{2})+A^{L}(k_{2}^{2})}{2}\gamma^{\mu}+\frac{B^{L}(k_{1}^{2})-B^{L}(k_{2}^{2})}{\left(k\\!\cdot\\!q\right)}k^{\mu}\Bigg{]}P_{L}$
$\displaystyle~{}+\left(L\leftrightarrow R\right),$ (A.71)

which corresponds to Eq. (2.19), and the transverse part

$\displaystyle\Lambda_{\mathrm{T}}^{\mu}(k_{1},k_{2})=$
$\displaystyle~{}\frac{\mathcal{C}_{2}^{L}}{\left(k\\!\cdot\\!q\right)}\big{[}\left(k\\!\cdot\\!q\right)q^{\mu}-q^{2}k^{\mu}\big{]}P_{L}+\frac{\mathcal{C}_{4}^{L}}{\left(k\\!\cdot\\!q\right)}\not{k}\big{[}\left(k\\!\cdot\\!q\right)\gamma^{\mu}-\not{q}k^{\mu}\big{]}P_{L}+\mathcal{C}_{5}^{L}i\sigma^{\mu\nu}q_{\nu}P_{L}$
$\displaystyle~{}+\frac{\mathcal{C}_{6}^{L}}{\left(k\\!\cdot\\!q\right)}\not{k}\big{[}\left(k\\!\cdot\\!q\right)q^{\mu}-q^{2}k^{\mu}\big{]}P_{L}-\mathcal{C}_{7}^{L}\big{[}\left(k\\!\cdot\\!q\right)\gamma^{\mu}-\not{q}k^{\mu}\big{]}P_{L}+\mathcal{C}_{9}^{L}\big{[}\not{q}q^{\mu}-q^{2}\gamma^{\mu}\big{]}P_{L}$
$\displaystyle~{}+\mathcal{C}_{10}^{L}\epsilon^{\mu\nu\rho\sigma}\gamma_{5}\gamma_{\nu}k_{\rho}q_{\sigma}P_{L}+\frac{\mathcal{C}_{12}^{L}}{\left(k\\!\cdot\\!q\right)}\not{k}\not{q}\big{[}\left(k\\!\cdot\\!q\right)q^{\mu}-q^{2}k^{\mu}\big{]}P_{L}+\left(L\rightarrow
R\right).$ (A.72)

Finally, replacing the coefficients

$\displaystyle\frac{\mathcal{C}_{2}^{L/R}}{\left(k\\!\cdot\\!q\right)}$
$\displaystyle\to C_{1}^{L/R},\quad\hskip
11.38092pt\frac{\mathcal{C}_{4}^{L/R}}{\left(k\\!\cdot\\!q\right)}\to
C_{6}^{L/R},\quad\mathcal{C}_{5}^{L/R}\to
C_{7}^{L/R},\quad\frac{\mathcal{C}_{6}^{L/R}}{\left(k\\!\cdot\\!q\right)}\to
C_{2}^{L/R},$ $\displaystyle-\mathcal{C}_{7}^{L/R}$ $\displaystyle\to
C_{5}^{L/R},\quad~{}~{}~{}~{}\mathcal{C}_{9}^{L/R}\to
C_{4}^{L/R},\quad\mathcal{C}_{10}^{L/R}\to
C_{8}^{L/R},\quad\frac{\mathcal{C}_{12}^{L/R}}{\left(k\\!\cdot\\!q\right)}\to
C_{3}^{L/R},$ (A.73)

Eq. (A.72) eventuates in Eq. (2.20).

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