#  Quark level and hadronic contributions to the electric dipole moment of
charged leptons in the standard model

Yasuhiro Yamaguchi1,2 1Advanced Science Research Center, Japan Atomic Energy
Agency (JAEA), Tokai 319-1195, Japan  2RIKEN Nishina Center, RIKEN, Wako,
Saitama 351-0198, Japan<EMAIL_ADDRESS>Nodoka Yamanaka3,4
3Amherst Center for Fundamental Interactions, Department of Physics,
University of Massachusetts Amherst, Massachusetts 01003, USA  4Yukawa
Institute for Theoretical Physics, Kyoto University, Kitashirakawa-Oiwake,
Kyoto 606-8502, Japan<EMAIL_ADDRESS>
###### Abstract

We evaluate the electric dipole moment of charged leptons in the standard
model, where the complex phase of the Cabibbo-Kobayashi-Maskawa matrix is the
only source of CP violation. We first prove that, at the quark-gluon level, it
is suppressed by a factor of $m_{b}^{2}m_{c}^{2}m_{s}^{2}$ at all orders of
perturbation due to the Glashow-Iliopoulos-Maiani mechanism. We then calculate
the hadronic long distance contribution generated by vector mesons at one-loop
level. The $|\Delta S|=1$ weak hadronic interaction is derived using the
factorization, and the strong interaction is modeled by the hidden local
symmetry framework. We find that the electric dipole moments of charged
leptons obtained from this hadronic mechanism are much larger than the
perturbative four-loop level quark-gluon process, by several orders of
magnitude.

###### pacs:

11.30.Er,12.15.Lk,13.20.-v,13.40.Em

##  I Introduction

The electric dipole moment (EDM) He:1990qa ; Bernreuther:1990jx ; Barr:1992dq
; Jungmann:2013sga ; NaviliatCuncic:2012zza ; Khriplovich:1997ga ;
Ginges:2003qt ; Pospelov:2005pr ; Raidal:2008jk ; Fukuyama:2012np ;
Engel:2013lsa ; Yamanaka:2014 ; Roberts:2014bka ; deVries:2015gea ;
Yamanaka:2016umw ; Yamanaka:2017mef ; Chang:2017wpl ; Chupp:2017rkp ;
Safronova:2017xyt ; Orzel:2018cui ; Yamanaka:2019ifh  is a very sensitive
observable for the detection of the CP violation in many candidate models of
new physics beyond the standard model (SM) such as the supersymmetry
Ellis:1982tk ; delAguila:1983dfr ; Dugan:1984qf ; Kizukuri:1992nj ;
Fischler:1992ha ; Ibrahim:1997gj ; Ibrahim:1998je ; Chang:1998uc ;
Pokorski:1999hz ; Lebedev:2002ne ; Chang:2002ex ; Pilaftsis:2002fe ;
Demir:2003js ; ArkaniHamed:2004yi ; Ellis:2008zy ; Lee:2012wa ;
Yamanaka:2012hm ; Yamanaka:2012ia ; Yamanaka:2012ep ; Dhuria:2014fba ;
Yamanaka:2014nba ; Zhao:2014vga ; Li:2015yla ; Bian:2016zba ; Nakai:2016atk ;
Cesarotti:2018huy ; Zheng:2019knr ; Yang:2019aao ; Yang:2020ebs , extended
Higgs model Weinberg:1990me ; Barr:1990vd ; Leigh:1990kf ; Kao:1992jv ;
Hayashi:1994ha ; Barger:1996jc ; BowserChao:1997bb ; Jung:2013hka ;
Abe:2013qla ; Inoue:2014nva ; Bian:2014zka ; Chen:2017com ; Fontes:2017zfn ;
Alves:2018kjr ; Egana-Ugrinovic:2018fpy ; Panico:2018hal ; Brod:2018pli ;
Brod:2018lbf ; Okada:2018yrn ; Cirigliano:2019vfc ; Cheung:2019bkw ;
Oshimo:2019ylq ; Chun:2019oix ; Oredsson:2019mni ; Fuyuto:2019svr ;
Eeg:2019eei ; Fuchs:2020uoc ; Cheung:2020ugr ; Kanemura:2020ibp , Majorana
fermion Ng:1995cs ; Archambault:2004td ; Fukuyama:2019jiq ; Chang:2017vzi ,
and other interesting models Appelquist:2004mn ; Fuyuto:2018scm ;
Dekens:2018bci ; Panico:2018hal ; Abe:2019wku ; Okawa:2019arp ;
Altmannshofer:2020ywf ; Kirpichnikov:2020tcf ; Pan:2020qqd ;
Kirpichnikov:2020lws . Among systems in which the EDM may be measured, the
charged leptons are the most frequently studied experimentally. The electron
EDM is known to be enhanced by relativistic effect of heavy atomic and
molecular systems Carrico:1968zz ; sandars1 ; sandars2 ; Flambaum:1976vg ;
Sandars:1975zz ; Labzovskii ; Sushkovmolecule ; kelly ; Kozlov:1994zz ;
Kozlov:1995xz ; flambaumfr1 ; nayak1 ; nayak3 ; nayak2 ; Natarajrubidium ;
Mukherjeefrancium ; Dzuba:2009mw ; Nataraj:2010vn ; flambaumybftho ;
Porsev:2012zx ; Roberts:2013zra ; Chubukov:2014rba ; abe ; sunaga ;
Radziute:2015apa ; Denis ; Skripnikov ; Sunaga:2018lja ; Sunaga:2018pjn ;
Sunaga:2019pfo ; Malika:2019jhn ; Fazil:2019esp ; Talukdar:2020ban , and it is
currently the object of a massive experimental competition Sandars:1964zz ;
Weisskopf:1968zz ; Stein:1969zz ; Player:1970zz ; Murthy:1989zz ;
Abdullah:1990nh ; Commins:1994gv ; Chin:2001zz ; Regan:2002ta ; Hudson:2011zz
; Sakemi:2011zz ; Kara:2012ay ; Baron:2013eja ; Cairncross:2017fip ;
Kozyryev:2017cwq ; Andreev:2018ayy ; Andreev:2018ayy . The EDM of the muon is
directly measureable in experiments using storage rings Bennett:2008dy . That
of the $\tau$ lepton can be extracted by analyzing collider experimental data
Chen:2018cxt ; Koksal:2018env ; Koksal:2018xyi ; Koksal:2018vtt ;
Dyndal:2020yen .

In the SM, the Cabibbo-Kobayashi-Maskawa (CKM) matrix Kobayashi:1973fv  has a
CP violating complex phase, so it may generate the EDM. In the search for new
physics beyond the SM, this contribution must be assessed as the leading
background. It is known that, in most cases, it is unobservably much smaller
than the experimental sensitivity Shabalin:1978rs ; Shabalin:1980tf ;
Eeg:1982qm ; Eeg:1983mt ; Khriplovich:1985jr ; Czarnecki:1997bu ;
Pospelov:1991zt ; Booth:1993af ; Pospelov:2013sca ; Pospelov:1994uf ;
Khriplovich:1981ca ; McKellar:1987tf ; Seng:2014lea ; Yamanaka:2015ncb ;
Yamanaka:2016fjj ; Lee:2018flm . However, the hadronic contribution to the EDM
of charged leptons has never been evaluated in the past. This is just the aim
of this paper to quantify it.

In this paper, we first prove that the contribution at the quark-gluon level
is suppressed by a factor of $m_{b}^{2}m_{c}^{2}m_{s}^{2}$ at all orders of
perturbation due to the Glashow-Iliopoulos-Maiani (GIM) mechanism. Next, we
calculate the hadronic long distance contribution to the charged lepton EDM
generated by vector mesons at one-loop level. The $|\Delta S|=1$ weak hadronic
interaction is derived using the factorization, while the strong interaction
is given by the hidden local symmetry framework. Part of the results have been
briefly reported in Yamaguchi:2020eub . A complete report of our study is
given in this article.

This paper is organized as follows. In the next section, we review the quark-
gluon level calculation of the CKM contribution to the EDM of charged leptons
and prove that it is actually suppressed by factors of quark masses at all
orders of perturbation. We then describe in Sec. III the setup of the
evaluation of the hadronic contribution to the EDMs of charged leptons in the
hidden local symmetry framework, with the weak interaction derived with the
factorization. In Sec. IV, we show the result of our calculation and analyze
the theoretical uncertainty. The final section gives the summary of this work.

##  II Quark level estimation of the EDM of charged leptons and the GIM
mechanism

Let us first review the previous works on the calculation of the short
distance (quark-gluon level) effect to the EDM of charged leptons in the SM.
Since we are supposing that the CP violation is generated by the physical
complex phase of the CKM matrix, the Feynman diagrams contributing to the
lepton EDM must have at least a quark loop, with sufficient flavor changes so
as to fulfill the Jarlskog combination Jarlskog:1985ht . The Jarlskog
invariant is given by the product of four CKM matrix elements ($J={\rm
Im}[V_{us}V_{td}V^{*}_{ud}V^{*}_{ts}]=(3.18\pm 0.15)\times 10^{-5}$
Tanabashi:2018oca ), so the quark loop must have four $W$ boson-quark
vertices. By noting that the $W$ boson must also be connected to the electron,
the two-loop level diagram which has only two vertices in the quark loop does
not contribute to the EDM due to the cancellation of the complex phase.

Figure 1:  Two-loop level diagram contributing to the EDM of $W$ boson in the
SM at the quark level. The external photon field is attached to all possible
propagators. The sum of all diagrams vanishes, so the EDM of charged leptons
at the three-loop level, which is generated by attaching the two external $W$
boson lines to the lepton line, also cancels.

The first plausible contribution appears then at the three-loop order
Hoogeveen:1990cb  (two-loop level diagrams of the EDM of $W$ boson as shown in
Fig. 1, which is attached to the lepton line). However, extensive three-loop
level analyses revealed us that it exactly cancels due to the antisymmetry of
the Jarlskog invariant under the flavor exchange (also called the GIM
mechanism, a consequence of the CKM unitarity) Pospelov:1991zt ; Booth:1993af
; Pospelov:2013sca . The cancellation works as follows. If we can find two
quark propagators of the same type (up type or down type) in the diagram with
identical momenta and sandwiched by $W$ boson vertices, the sum of the direct
product of these two parts over the $d$-type quark flavors reads

$\displaystyle\sum_{D\neq D^{\prime}}{\rm
Im}[V_{U^{\prime}D}V^{*}_{UD}V_{U^{\prime}D^{\prime}}V^{*}_{UD^{\prime}}]P_{L}S_{D}\gamma^{\mu}P_{L}\otimes
P_{L}S_{D^{\prime}}\gamma^{\nu}P_{L}$ (1) $\displaystyle=$ $\displaystyle{\rm
Im}[V_{U^{\prime}d}V^{*}_{Ud}V_{U^{\prime}s}V^{*}_{Us}](P_{L}S_{d}\gamma^{\mu}P_{L}\otimes
P_{L}S_{s}\gamma^{\nu}P_{L}-P_{L}S_{s}\gamma^{\mu}P_{L}\otimes
P_{L}S_{d}\gamma^{\nu}P_{L})$ $\displaystyle+{\rm
Im}[V_{U^{\prime}s}V^{*}_{Us}V_{U^{\prime}b}V^{*}_{Ub}](P_{L}S_{s}\gamma^{\mu}P_{L}\otimes
P_{L}S_{b}\gamma^{\nu}P_{L}-P_{L}S_{b}\gamma^{\mu}P_{L}\otimes
P_{L}S_{s}\gamma^{\nu}P_{L})$ $\displaystyle+{\rm
Im}[V_{U^{\prime}b}V^{*}_{Ub}V_{U^{\prime}d}V^{*}_{Ud}](P_{L}S_{b}\gamma^{\mu}P_{L}\otimes
P_{L}S_{d}\gamma^{\nu}P_{L}-P_{L}S_{d}\gamma^{\mu}P_{L}\otimes
P_{L}S_{b}\gamma^{\nu}P_{L})$ $\displaystyle=$ $\displaystyle{\rm
Im}[V_{U^{\prime}d}V^{*}_{Ud}V_{U^{\prime}s}V^{*}_{Us}](P_{L}k/\,\gamma^{\mu}P_{L})\otimes(P_{L}k/\,\gamma^{\nu}P_{L})$
$\displaystyle\times\Biggl{\\{}\frac{1}{k^{2}-m_{s}^{2}}\cdot\frac{1}{k^{2}-m_{d}^{2}}-\frac{1}{k^{2}-m_{d}^{2}}\cdot\frac{1}{k^{2}-m_{s}^{2}}+\frac{1}{k^{2}-m_{b}^{2}}\cdot\frac{1}{k^{2}-m_{s}^{2}}-\frac{1}{k^{2}-m_{s}^{2}}\cdot\frac{1}{k^{2}-m_{b}^{2}}$
$\displaystyle\hskip
20.00003pt+\frac{1}{k^{2}-m_{d}^{2}}\cdot\frac{1}{k^{2}-m_{b}^{2}}-\frac{1}{k^{2}-m_{b}^{2}}\cdot\frac{1}{k^{2}-m_{d}^{2}}\Biggr{\\}}$
$\displaystyle=$ $\displaystyle 0,\ \ \ $

where $S_{D}\equiv\frac{i(k/\,+m_{D})}{k^{2}-m_{D}^{2}}$. The projection
$P_{L}\equiv\frac{1}{2}(1-\gamma_{5})$ comes from the $W$ boson-quark
vertices. The mass insertions of $S_{D}$ cancel since odd number of chirality
flips is not allowed when $S_{D}$ is sandwiched by $W$ boson-quark vertices.
It turns out that the pair of propagators with the same ($u$\- or $d$-) type
quarks can always be found in the two-loop level contribution to the EDM of
$W$ boson, and consequently in the three-loop level diagrams of the EDM of
charged leptons. The most trivial ones are the symmetric diagrams with two
quark propagators of the same type, but there are also diagrams which have
nonsymmetric insertions of the external photon. The latter ones can actually
be recast into the symmetric form of quark propagators by using the Ward-
Takahashi identity Pospelov:1991zt ; Booth:1993af . Similar cancellation also
occurs in the case of the quark EDM/chromo-EDM Shabalin:1978rs ;
Shabalin:1980tf ; Eeg:1982qm ; Eeg:1983mt ; Khriplovich:1985jr ;
Czarnecki:1997bu  or the Weinberg operator (gluon chromo-EDM) Pospelov:1994uf
.

Figure 2:  Example of four-loop level diagram contributing to the lepton EDM
in the SM at the quark level.  Figure 3:  Boson emissions/absorptions of the
quark loop with four flavor changing vertices respecting the Jarlskog
combination. The gluon, photon, and the neutral Higgs boson are denoted by the
wiggly, wavy, and dashed lines, respectively, and the ellipses means that they
each may be of arbitrary number. The sum of the quark flavors removes the
contribution without flip of chirality due to the GIM mechanism. The emitted
bosons have $O(m_{W})\approx O(m_{t})$ momenta, and they may also form loops,
or be connected to other fermion loops, which are not interfering with the
flavor structure of the one considered in this figure.

The first nonvanishing contribution which avoids the above symmetric
cancellation appears at the four-loop level (see Fig. 2). Although the four-
loop level contribution has never completely been calculated, it is possible
to estimate its size by symmetry consideration. It is indeed possible to prove
that the GIM mechanism Glashow:1970gm ; Ellis:1976fn  always brings additional
suppression of quark mass factors $m_{q}^{2}$, independently of the order of
perturbation. Let us first consider the quark loop with several insertions of
vertices of flavor unchanging (neutral) bosons, i.e. gluons, photons, or Higgs
bosons (see Fig. 3). We focus on the direct product of the $U$ and
$U^{\prime}$ quark lines with vertex insertions of Fig. 3, which may be
expressed by the Taylor expansion in terms of the quark masses, as follows:

$\sum_{U\neq U^{\prime}}{\rm
Im}[V_{U^{\prime}D}V^{*}_{UD}V_{U^{\prime}D^{\prime}}V^{*}_{UD^{\prime}}]\sum_{n=0}a^{(1)}_{n}m_{U}^{2n}\otimes\sum_{n^{\prime}=0}a^{(2)}_{n^{\prime}}m_{U^{\prime}}^{2n^{\prime}},$
(2)

where $a^{(1)}_{n}$ and $a^{(2)}_{n^{\prime}}$ are polynomials of the electric
charge of up-type quarks, the strong coupling, the inverse of the Higgs vacuum
expectation value (appearing from the Yukawa coupling of the Higgs boson after
factoring out quark masses), and all momenta carried by the bosons attached to
$U$ and $U^{\prime}$, respectively, which depend on the diagram considered.
Here we took the direct product $\otimes$ to show that the above Taylor
expansion also works for the case where Dirac matrices are involved.

We can actually prove that the terms involving $a^{(1)}_{0}$ and $a^{(2)}_{0}$
always vanish due to the GIM mechanism. The case of $a^{(1)}_{0}\otimes
a^{(2)}_{0}$ is easy to show, since the sum is just proportional to the sum of
Jarlskog invariants, which cancel due to the antisymmetry in the flavor
exchange. The remaining possibilities are
$\sum_{n=1}a^{(1)}_{n}m_{U}^{2n}\otimes a^{(2)}_{0}$ and
$a^{(1)}_{0}\otimes\sum_{n^{\prime}=1}a^{(2)}_{n^{\prime}}m_{U^{\prime}}^{2n^{\prime}}$
which are also not difficult to treat. For the former case, we have

$\displaystyle\sum_{U\neq U^{\prime}}{\rm
Im}[V_{U^{\prime}D}V^{*}_{UD}V_{U^{\prime}D^{\prime}}V^{*}_{UD^{\prime}}]\sum_{n=0}a^{(1)}_{n}m_{U}^{2n}\otimes
a^{(2)}_{0}$ (3) $\displaystyle=$ $\displaystyle{\rm
Im}[V_{tD}V^{*}_{uD}V_{tD^{\prime}}V^{*}_{uD^{\prime}}]\sum_{n=1}a^{(1)}_{n}m_{u}^{2n}\otimes
a^{(2)}_{0}+{\rm
Im}[V_{tD}V^{*}_{cD}V_{tD^{\prime}}V^{*}_{cD^{\prime}}]\sum_{n=1}a^{(1)}_{n}m_{c}^{2n}\otimes
a^{(2)}_{0}$ $\displaystyle+{\rm
Im}[V_{uD}V^{*}_{cD}V_{uD^{\prime}}V^{*}_{cD^{\prime}}]\sum_{n=1}a^{(1)}_{n}m_{c}^{2n}\otimes
a^{(2)}_{0}+{\rm
Im}[V_{uD}V^{*}_{tD}V_{uD^{\prime}}V^{*}_{tD^{\prime}}]\sum_{n=1}a^{(1)}_{n}m_{t}^{2n}\otimes
a^{(2)}_{0}$ $\displaystyle+{\rm
Im}[V_{cD}V^{*}_{tD}V_{cD^{\prime}}V^{*}_{tD^{\prime}}]\sum_{n=1}a^{(1)}_{n}m_{t}^{2n}\otimes
a^{(2)}_{0}+{\rm
Im}[V_{cD}V^{*}_{uD}V_{cD^{\prime}}V^{*}_{uD^{\prime}}]\sum_{n=1}a^{(1)}_{n}m_{u}^{2n}\otimes
a^{(2)}_{0}$ $\displaystyle=$ $\displaystyle{\rm
Im}[V_{cD}V^{*}_{uD}V_{cD^{\prime}}V^{*}_{uD^{\prime}}]\Biggl{[}\sum_{n=1}a^{(1)}_{n}m_{u}^{2n}-\sum_{n=1}a^{(1)}_{n}m_{c}^{2n}+\sum_{n=1}a^{(1)}_{n}m_{c}^{2n}-\sum_{n=1}a^{(1)}_{n}m_{t}^{2n}+\sum_{n=1}a^{(1)}_{n}m_{t}^{2n}-\sum_{n=1}a^{(1)}_{n}m_{u}^{2n}\Biggr{]}\otimes
a^{(2)}_{0}$ $\displaystyle=$ $\displaystyle 0.$

We may repeat the same calculation to show the cancellation for the case of
$a^{(1)}_{0}\otimes\sum_{n^{\prime}=1}a^{(2)}_{n^{\prime}}m_{U^{\prime}}^{2n^{\prime}}$
as well. We thus proved that the leading order CP violation of the quark loop
is accompanied by two factors of squared mass of two different up-type quarks
to all orders of perturbation in QED, QCD, and Higgs corrections. We may also
exactly repeat the above procedure for the down-type quark contribution which
is independent of the up-type ones. The CP violating part of the quark loop is
then at least having a suppression factor of
$m_{t}^{2}m_{b}^{2}m_{c}^{2}m_{s}^{2}$, which of course persists even if some
of the neutral or $W$ bosons are contracted each other or with other quark
loops. The appearance of this factor has actually been already discussed in
the general case of the CP violation of the CKM matrix Jarlskog:1985cw , and
it also appeared in the result of the calculation of the Weinberg operator
which is also generated by a quark loop Pospelov:1994uf .

The presence of the suppression due to quark mass factors, i.e. the
cancellation of the zeroth order terms of the Taylor expansion of the quark
lines with neutral boson insertions, may also more elegantly be shown using
the unitarity of the CKM matrix. At the order of four $W$ boson-quark
vertices, the general flavor structure of the quark loop, with the sum over
the flavor taken, is expressed by the following trace

${\rm
Tr}[V^{\dagger}Q_{U}^{(1)}VR_{D}^{(1)}V^{\dagger}Q_{U}^{(2)}VR_{D}^{(2)}],$
(4)

where $V$ is the $3\times 3$ CKM matrix, and
$Q_{U}^{(k)}\equiv\sum_{n_{k}=1}a^{(k)}_{n_{k}}m_{U}^{2n_{k}}$,
$R_{D}^{(l)}\equiv\sum_{n_{l}=1}b^{(l)}_{n_{l}}m_{D}^{2n_{l}}$ ($k,l=1,2$) are
the down-type and up-type quark lines with arbitrary number of neutral boson
insertions, respectively. We note that $Q_{U}^{(k)}$ and $R_{D}^{(l)}$ are
$3\times 3$ matrices that only have diagonal components. By taking the zeroth
order term of $Q_{U}^{(1)}$, we have

$\displaystyle{\rm
Tr}[V^{\dagger}a_{0}^{(1)}VR_{D}^{(1)}V^{\dagger}Q_{U}^{(2)}VR_{D}^{(2)}]$
$\displaystyle=$ $\displaystyle a_{0}^{(1)}{\rm
Tr}[V^{\dagger}VR_{D}^{(1)}V^{\dagger}Q_{U}^{(2)}VR_{D}^{(2)}]$ (5)
$\displaystyle=$ $\displaystyle a_{0}^{(1)}{\rm
Tr}[R_{D}^{(2)}R_{D}^{(1)}V^{\dagger}Q_{U}^{(2)}V]\,=\,a_{0}^{(1)}\sum_{i,j=1}^{3}(R_{D}^{(2)})_{i}(R_{D}^{(1)})_{i}|V_{ij}|^{2}(Q_{U}^{(2)})_{j}.$

Here we used the unitarity of the CKM matrix $V^{\dagger}V=1$, the fact that
$R_{D}^{(2)},R_{D}^{(1)}$, and $Q_{U}^{(2)}$ are diagonal, and that
$a_{0}^{(1)}$ is flavor blind, i.e. proportional to the unit matrix. The above
trace is therefore purely real and the zeroth order terms of the Taylor
expansion of the quark lines with neutral boson insertions does not contribute
to the EDM. This expression is exactly equivalent with Eq. (3), and at this
order $O(V^{4})$ the imaginary part only survives when the flavors of all
quarks are different, to avoid the appearance of the squared absolute values
of the CKM matrix elements.

Next, we have to see higher order corrections with $W$ boson-quark vertices
which may be treated in a similar manner. Here again the unitarity of the CKM
matrix plays a crucial role. Let us consider the case with six $W$ boson-quark
vertices. The general flavor structure of this quark loop, with the flavor
summed, looks like

${\rm
Tr}[V^{\dagger}Q_{U}^{(1)}VR_{D}^{(1)}V^{\dagger}Q_{U}^{(2)}VR_{D}^{(2)}V^{\dagger}Q_{U}^{(3)}VR_{D}^{(3)}].$
(6)

We now show that the correction at this order ($V^{6}$) is not larger than
that of $O(V^{4})$ which has the quark mass factors
$m_{t}^{2}m_{b}^{2}m_{c}^{2}m_{s}^{2}$. A potentially large contribution may
arise from the zeroth order terms of the Taylor expansion $a^{(k)}_{0}$ and
$b^{(l)}_{0}$. For example, by considering one such insertion,

$\displaystyle{\rm
Tr}[V^{\dagger}a^{(1)}_{0}VR_{D}^{(1)}V^{\dagger}Q_{U}^{(2)}VR_{D}^{(2)}V^{\dagger}Q_{U}^{(3)}VR_{D}^{(3)}]$
$\displaystyle=$ $\displaystyle a^{(1)}_{0}{\rm
Tr}[V^{\dagger}VR_{D}^{(1)}V^{\dagger}Q_{U}^{(2)}VR_{D}^{(2)}V^{\dagger}Q_{U}^{(3)}VR_{D}^{(3)}]$
(7) $\displaystyle=$ $\displaystyle a^{(1)}_{0}{\rm
Tr}[R_{D}^{(3)}R_{D}^{(1)}V^{\dagger}Q_{U}^{(2)}VR_{D}^{(2)}V^{\dagger}Q_{U}^{(3)}V],$

where we again used the unitarity of the CKM matrix. By noting that
$R_{D}^{(3)}R_{D}^{(1)}$ is also a diagonal matrix with each component
depending only on the mass of one quark flavor, we see that the flavor
structure of this contribution is exactly the same as that of the $O(V^{4})$
process with neutral boson insertions discussed previously in this section
[Fig. 3, Eqs. (3) and (4)]. This means that the the $O(V^{6})$ quark loop
having one zeroth order term of the Taylor expansion is also having the quark
mass factor $m_{t}^{2}m_{b}^{2}m_{c}^{2}m_{s}^{2}$. We also note that the
contribution with the three up-type quarks being all top quarks, which may
potentially be larger than the $O(V^{4})$ terms, has no effect to the EDM,
since it will be proportional to three factors of the absolute values of
squared CKM matrix elements $|V_{tD}|^{2}$, i.e. at least a factor of
$m_{c}^{2}$ or $m_{u}^{2}$ is needed. This analysis may be extended to
arbitrary higher orders recursively, since the zeroth order terms
$a^{(k)}_{0}$ or $b^{(l)}_{0}$, proportional to the unit matrix, contract two
CKM matrix elements $V$ and $V^{\dagger}$ to form another unit matrix,
reducing the flavor trace of $O(V^{2N})$ to $O(V^{2N-2})$. Since the
$O(V^{4})$ contribution is having a factor of
$m_{t}^{2}m_{b}^{2}m_{c}^{2}m_{s}^{2}$, this is also so at $O(V^{6})$ and at
all other higher orders of $W$ boson-quark vertices ($V$).

We can also show with the above approach the cancellation of the quark loop at
$O(V^{2})$ and at the two-loop level in a more elegant manner. At $O(V^{2})$,
we have

${\rm
Tr}[V^{\dagger}Q_{U}^{(1)}VR_{D}^{(1)}]=\sum_{i=1}^{3}H_{ii}(R_{D}^{(1)})_{i},$
(8)

where $H\equiv V^{\dagger}Q_{U}^{(1)}V$ is an Hermitian matrix. Since
$R_{D}^{(1)}$ is diagonal and real, its trace with $H$ is taking only the
diagonal elements, which are also real. There is no room for the imaginary
part, so CP is conserved at $O(V^{2})$, even accounting for all order
corrections of neutral bosons.

At the two-loop level (of the quark loop), we previously saw that we can
always find a symmetric set of either up or down-type propagators with the
same momentum argument Pospelov:1991zt ; Booth:1993af ; Pospelov:2013sca . We
may then write the trace as

${\rm
Tr}[VS_{D}V^{\dagger}Q_{U}^{(1)}VS_{D}V^{\dagger}Q_{U}^{(2)}]=\sum_{i,j=1}^{3}H^{\prime}_{ij}(Q_{U}^{(1)})_{j}H^{\prime}_{ji}(Q_{U}^{(2)})_{i}=\sum_{i,j=1}^{3}|H^{\prime}_{ij}|^{2}(Q_{U}^{(1)})_{j}(Q_{U}^{(2)})_{i},$
(9)

where we used the Hermiticity of $H^{\prime}\equiv VS_{D}V^{\dagger}$. Due to
the absolute value, there is no CP violation, and there is thus no
contribution to the EDM of charged leptons at the three-loop level. We also
see that, if the symmetry between the two $S_{D}$ is destroyed, the two
$H^{\prime}$ will no longer be complex conjugates, and the imaginary part will
be generated.

Let us now estimate the EDM of charged leptons according to the above
discussion. The correct dimensional analysis of the four-loop level
contribution according to the above proof therefore yields

$d_{l}\sim\frac{eJ\alpha_{s}\alpha_{\rm
QED}^{3}m_{l}m_{b}^{2}m_{c}^{2}m_{s}^{2}}{\sin^{6}\theta_{W}m_{t}^{8}(4\pi)^{4}},$
(10)

which is transcribed to

$\displaystyle d_{e}$ $\displaystyle=$ $\displaystyle O(10^{-50})e\,{\rm cm},$
(11) $\displaystyle d_{\mu}$ $\displaystyle=$ $\displaystyle
O(10^{-48})e\,{\rm cm},$ (12) $\displaystyle d_{\tau}$ $\displaystyle=$
$\displaystyle O(10^{-47})e\,{\rm cm}.$ (13)

Here we did not consider the logarithmic enhancement which may enlarge the
above values by one or two orders of magnitude. Nevertheless, these results
are actually telling us that the short distance contribution is extremely
small. From this analysis, we see that the enormous suppression of the EDM of
charged leptons is not due to the fact that it appears at the four-loop level,
but rather due to the cancellation by the GIM mechanism.

We stress that this suppression mechanism does apply only when all momenta
involved are of $O(m_{W}\sim m_{t})$. In the case where nonperturbative
physics is relevant in the infrared region, the coefficients $a_{n}$,
$b_{n^{\prime}}$ of Eq. (2) may be enhanced by $O(1/\Lambda_{\rm
QCD}^{2})=O({\rm GeV}^{-2})$ factors. In the next section, we recast the soft
momentum physics into phenomenological hadron physics where the weak
interacting hard part is given by low energy constants, which are also
calculated with phenomenological models.

##  III Setup of the calculation

###  III.1 The long distance effect

The leading order contribution of the CKM matrix to the lepton EDM is
constructed with at least two $W$ boson exchanges. To avoid severe GIM
cancellation as we saw in the previous section, we have to split the short
distance flavor changing process at least into two parts at the hadron level
(the long distance effect), while keeping the Jarlskog combination of the CKM
matrix elements. The largest long distance contribution should involve
unflavored and $|S|=1$ mesons rather than the heavy flavored ($c,b$) ones.
Another important condition is that the charged lepton EDM is generated by
one-loop level diagrams involving vector mesons, because the interaction of
pseudoscalar mesons with the lepton will change the chirality, suppressing the
EDM by at least by a factor of $m_{l}^{2}$ ($l=e,\mu,\tau$) (for an example of
a one-loop level diagram with pseudoscalar mesons suppressed by chirality
flips, see Fig. 4). The charged lepton EDM is then generated by diagrams
involving a $K^{*}$ meson. The one-loop level diagrams must not have a
neutrino in the intermediate state of the long distance process, since the
small neutrino mass will not provide sufficient chirality flip required in the
generation of the EDM. Moreover, if the process contains two weak
$K^{*}$-charged lepton vertices, the chirality selection will not allow an
EDM. The $K^{*}$ meson must therefore change to an unflavored meson which in
turn becomes a photon which will be absorbed by the charged lepton. Under such
restrictions, we may draw diagrams shown in Fig. 5. We note that diagrams with
external photons attached to internal lepton propagator cancel when transposed
diagrams are summed.

Figure 4: Example of a one-loop level contribution to the EDM of the charged
lepton $l$ generated by pseudoscalar mesons ($\pi,K$). The grey and black
blobs denote the weak interaction. This naïvely leading diagram is suppressed
by the chirality flips of the pseudoscalar meson-lepton vertices (grey blobs).
Figure 5:  Long distance contribution to the EDM of charged lepton $l$
($=e,\mu,\tau$) in the SM. The diagrams $(a)$ and $(a^{\prime})$ ($(b)$ and
$(b^{\prime})$) are the contribution with the weak (strong) three vector meson
interactions. There are also diagrams with the $\bar{K}^{\ast}$ propagator,
which are not displayed. The grey blob denotes the $|\Delta S|=1$ semileptonic
effective interaction, while the black one is the $|\Delta S|=1$ (two- and
three-point) vector meson interactions which combine each other to form the
Jarlskog invariant.

###  III.2 Hidden local symmetry

Let us now give the interactions to calculate the diagrams of Fig. 5. It is
convenient to describe the $|\Delta S|=0$ vector meson interactions with the
hidden local symmetry (HLS) Bando:1984ej ; Bando:1984pw ; Fujiwara:1984mp ;
Bando:1985rf ; Bando:1987br ; Meissner:1987ge ; Kaiser:1990yf ; Harada:1992np
; Klingl:1996by ; Harada:2000kb ; Harada:2003jx . The HLS is a framework
introduced to extend the domain of applicability of chiral perturbation to
include vector meson resonances, and it is successful in phenomenology
Harada:2003jx . The effective Lagrangian for three vector mesons is given by

$\displaystyle{\cal L}_{3V}$ $\displaystyle=ig{\rm
tr}\left[\left(\partial_{\mu}V_{\nu}-\partial_{\nu}V_{\mu}\right)V^{\mu}V^{\nu}\right],$
(14)

where the vector meson matrix $V^{\mu}$ is given by

$\displaystyle V^{\mu}$
$\displaystyle=\left(\begin{array}[]{ccc}\frac{\rho^{0}}{\sqrt{2}}+\frac{\omega}{\sqrt{2}}&\rho^{+}&K^{\ast+}\\\
\rho^{-}&-\frac{\rho^{0}}{\sqrt{2}}+\frac{\omega}{\sqrt{2}}&K^{\ast 0}\\\
K^{\ast-}&\bar{K}^{\ast 0}&\phi\\\ \end{array}\right)^{\mu},$ (18)

where $g=m_{\rho}/(2f_{\pi})$ with the pion decay constant $f_{\pi}=93$ MeV.

The effective Lagrangian for vector meson and photon is given by Klingl:1996by

$\displaystyle{\cal L}_{\gamma V}$
$\displaystyle=-\sqrt{2}\frac{em^{2}_{\rho}}{g_{\gamma}}A_{\mu}{\rm
tr}(QV^{\mu})=-\frac{em^{2}_{\rho}}{g_{\gamma}}A_{\mu}\left(\rho^{0\mu}+\frac{1}{3}\omega^{\mu}-\frac{\sqrt{2}}{3}\phi^{\mu}\right),$
(19)

where $g_{\gamma}=5.7$ and

$\displaystyle Q$ $\displaystyle=\left(\begin{array}[]{ccc}\frac{2}{3}&0&0\\\
0&-\frac{1}{3}&0\\\ 0&0&-\frac{1}{3}\\\ \end{array}\right).$ (23)

###  III.3 $K^{*}$-lepton interaction

Let us now model the weak interaction at the hadron level. From Fig. 5, the
$|\Delta S|=1$ weak interaction appears in the $K^{*}$-lepton interaction and
in the interacting vertices between $K^{*}$ and other vector mesons. Since the
neutrino cannot appear in Fig. 5, the interaction between $K^{*}$ and the
charged lepton must be at least a one-loop level process at the quark level.
Then the best solution is to attribute the CKM matrix elements
$V_{cs}V_{cd}^{*}$ or $V_{ts}V_{td}^{*}$ to the $K^{*}$-lepton interaction,
and $V_{ud}V_{us}^{*}$ to the $K^{*}$-vector meson interactions. The latter
attribution will maximize the $|\Delta S|=1$ vector meson interactions, since
$V_{ud}V_{us}^{*}$ is given from the tree level $|\Delta S|=1$ four-quark
interaction.

Figure 6:  Short distance contribution to the $\Delta S=-1$ semileptonic
($K^{*}$-charged lepton) interaction. Here we have $l=e,\mu,\tau$.

The parity violating effective interaction between $K^{*}$ and the charged
lepton is given by

${\cal
L}_{K^{*}ll}=g_{K^{*}ll}K^{*}_{\mu}\bar{l}\gamma^{\mu}\gamma_{5}l+{\rm(H.c.)},$
(24)

where $K^{*}_{\mu}$ is the field operators of the $K^{*}$ meson. In the zero
momentum exchange limit, the coupling constant is given by

${\rm Im}(g_{K^{*}ll})\varepsilon^{K^{*}}_{\mu}={\rm
Im}(V_{ts}^{*}V_{td})\langle 0|\bar{s}\gamma_{\mu}d|K^{*}\rangle I_{dsll},$
(25)

where we fixed the complex phases of $V_{ud}V_{us}^{*}$ to be real. The
$K^{*}$ meson matrix element is given by

$\langle
0|\bar{s}\gamma_{\mu}d|K^{*}\rangle=m_{K^{*}}f_{K^{*}}\varepsilon^{K^{*}}_{\mu},$
(26)

where $\varepsilon^{K^{*}}_{\mu}$, $m_{K^{*}}=890$ MeV and $f_{K^{*}}=204$ MeV
Neubert:1997uc ; Grossmann:2015lea ; Straub:2015ica ; Chang:2018aut  are the
polarization vector, the mass, and the decay constant of $K^{*}$,
respectively. The quark level amplitude $I_{dsll}$ can be obtained by
calculating the one-loop level diagrams of Fig. 6. By neglecting all external
momenta [which are $O(\Lambda_{\rm QCD})$] and imposing $m_{t},m_{W}\gg
m_{c}$, the amplitude of the diagrams of Fig. 6 is given by

$\displaystyle{\rm Im}({\cal M}_{\rm(a)}^{K^{\ast}ll})$ $\displaystyle\approx$
$\displaystyle\frac{-\alpha_{\rm QED}^{2}{\rm
Im}(V_{ts}V_{td}^{*})}{16\sin^{4}\theta_{W}}\frac{m_{t}^{2}}{m_{t}^{2}-m_{W}^{2}}\Biggl{\\{}\frac{1}{m_{W}^{2}}+\frac{1}{m_{t}^{2}-m_{W}^{2}}\ln\,\Biggl{(}\frac{m_{W}^{2}}{m_{t}^{2}}\Biggr{)}\Biggr{\\}}\bar{u}_{e}\,\gamma^{\mu}\gamma_{5}u_{e}\cdot\bar{u}_{d}\,\gamma_{\mu}(1-\gamma_{5})u_{s},$
(27) $\displaystyle{\rm Im}({\cal M}_{\rm(b)}^{K^{\ast}ll})$
$\displaystyle\approx$ $\displaystyle\frac{-\alpha_{\rm QED}^{2}{\rm
Im}(V_{ts}V_{td}^{*})}{16\sin^{4}\theta_{W}\cos^{2}\theta_{W}m_{Z}^{2}}\bar{u}_{e}\gamma^{\mu}\gamma_{5}u_{e}\,\bar{u}_{d}\gamma_{\mu}(1-\gamma_{5})u_{s}$
(28)
$\displaystyle\times\frac{m_{t}^{2}}{m_{t}^{2}-m_{W}^{2}}\left\\{\Biggl{(}\frac{1}{2}-\frac{2}{3}\sin^{2}\theta_{W}\Biggr{)}\ln\Biggl{(}\frac{m_{W}^{2}}{m_{t}^{2}}\Biggr{)}-\Biggl{(}\frac{1}{2}+\frac{2}{3}\sin^{2}\theta_{W}\Biggr{)}\Biggl{[}1+\frac{m_{W}^{2}}{m_{t}^{2}-m_{W}^{2}}\ln\,\Biggl{(}\frac{m_{W}^{2}}{m_{t}^{2}}\Biggr{)}\Biggr{]}\right\\},$
$\displaystyle{\rm Im}({\cal M}_{\rm(c)}^{K^{\ast}ll})$ $\displaystyle\approx$
$\displaystyle\frac{3\alpha_{\rm QED}^{2}{\rm
Im}(V_{ts}V_{td}^{*})}{16\sin^{4}\theta_{W}m_{Z}^{2}}\frac{m_{t}^{2}}{(m_{W}^{2}-m_{t}^{2})^{2}}\Biggl{\\{}m_{W}^{2}-m_{t}^{2}\Biggl{[}1+\ln\,\Biggl{(}\frac{m_{W}^{2}}{m_{t}^{2}}\Biggr{)}\Biggr{]}\Biggr{\\}}\,\bar{u}_{e}\gamma^{\mu}\gamma_{5}u_{e}\cdot\bar{u}_{d}\,\gamma_{\mu}(1-\gamma_{5})u_{s},$
(29)

where $\sin^{2}\theta_{W}=0.23122$ Tanabashi:2018oca . The diagram (c) is the
largest, but all of them are of the same order. The numerical value of the
total $I_{dsll}$ is

$I_{dsll}=3.2\times 10^{-8}\,{\rm GeV}^{-2},$ (30)

which is quite consistent in absolute value with that of the naïve dimensional
analysis $I_{dsll}\sim\frac{\alpha_{\rm
QED}^{2}}{4\sin^{4}\theta_{W}m_{W}^{2}}\sim 4.3\times 10^{-8}\,{\rm
GeV}^{-2}$. We note that Eqs. (27), (28), and (29) all contain a factor of
$\frac{m_{t}^{2}}{m_{W}^{2}-m_{t}^{2}}$ which is due to the GIM cancellation.
This shows that if we invert the up-type and down-type quarks, the resulting
meson-charged lepton couplings will be suppressed by a factor of
$m_{D}^{2}/m_{W}^{2}$ ($D=d,s,b$).

###  III.4  $|\Delta S|=1$ vector meson transition and three-vector meson
interaction

We now model the $|\Delta S|=1$ vector meson transition and three-vector meson
interaction using the factorization. For that, we have to determine the Wilson
coefficients of the quark level $|\Delta S|=1$ processes. We chose the
$|\Delta S|=1$ case because it is the only allowed flavor change at low energy
scale. At the scale just below the $W$ boson mass ($m_{W}=80.4$ GeV), we have
the following $|\Delta S|=1$ effective Hamiltonian

$\displaystyle{\cal H}_{eff}(\mu=m_{W})$ $\displaystyle=$
$\displaystyle\frac{G_{F}}{\sqrt{2}}\Biggl{\\{}\sum_{i=1,2}C_{i}(\mu=m_{W})[V_{us}^{*}V_{ud}Q_{i}+V_{cs}^{*}V_{cd}Q_{i}^{c}]-\sum_{j=3}^{6}C_{j}(\mu=m_{W})V_{ts}^{*}V_{td}Q_{j}\Biggr{\\}}+{\rm
H.c.},\ \ \ \ $ (31)

with the Fermi constant $G_{F}=1.16637\times 10^{-5}{\rm GeV}^{-2}$
Tanabashi:2018oca . Here $Q^{q}_{1}$, $Q^{c}_{1}$, $Q^{q}_{2}$, $Q^{c}_{2}$,
and $Q_{j}$ ($j=3\sim 6$) are defined as Buras:1991jm ; Buchalla:1995vs

$\displaystyle Q_{1}$ $\displaystyle\equiv$
$\displaystyle\bar{s}_{\alpha}\gamma^{\mu}(1-\gamma_{5})u_{\beta}\,\bar{u}_{\beta}\gamma_{\mu}(1-\gamma_{5})d_{\alpha},$
(32) $\displaystyle Q_{1}^{c}$ $\displaystyle\equiv$
$\displaystyle\bar{s}_{\alpha}\gamma^{\mu}(1-\gamma_{5})c_{\beta}\,\bar{c}_{\beta}\gamma_{\mu}(1-\gamma_{5})d_{\alpha},$
(33) $\displaystyle Q_{2}$ $\displaystyle\equiv$
$\displaystyle\bar{s}_{\alpha}\gamma^{\mu}(1-\gamma_{5})u_{\alpha}\,\bar{u}_{\beta}\gamma_{\mu}(1-\gamma_{5})d_{\beta},$
(34) $\displaystyle Q_{2}^{c}$ $\displaystyle\equiv$
$\displaystyle\bar{s}_{\alpha}\gamma^{\mu}(1-\gamma_{5})c_{\alpha}\,\bar{c}_{\beta}\gamma_{\mu}(1-\gamma_{5})d_{\beta},$
(35) $\displaystyle Q_{3}$ $\displaystyle\equiv$
$\displaystyle\bar{s}_{\alpha}\gamma^{\mu}(1-\gamma_{5})d_{\alpha}\,\sum_{q}^{N_{f}}\bar{q}_{\beta}\gamma_{\mu}(1-\gamma_{5})q_{\beta},$
(36) $\displaystyle Q_{4}$ $\displaystyle\equiv$
$\displaystyle\bar{s}_{\alpha}\gamma^{\mu}(1-\gamma_{5})d_{\beta}\,\sum_{q}^{N_{f}}\bar{q}_{\beta}\gamma_{\mu}(1-\gamma_{5})q_{\alpha},$
(37) $\displaystyle Q_{5}$ $\displaystyle\equiv$
$\displaystyle\bar{s}_{\alpha}\gamma^{\mu}(1-\gamma_{5})d_{\alpha}\,\sum_{q}^{N_{f}}\bar{q}_{\beta}\gamma_{\mu}(1+\gamma_{5})q_{\beta},$
(38) $\displaystyle Q_{6}$ $\displaystyle\equiv$
$\displaystyle\bar{s}_{\alpha}\gamma^{\mu}(1-\gamma_{5})d_{\beta}\,\sum_{q}^{N_{f}}\bar{q}_{\beta}\gamma_{\mu}(1+\gamma_{5})q_{\alpha},$
(39)

where $\alpha$ and $\beta$ are the fundamental color indices, and the
summation over $N_{f}$ goes up to the allowed flavors at the given scale. The
Hamiltonian of Eq. (31) keeps the same form down to $\mu=m_{c}$, but the
Wilson coefficients run in the change of the scale.

The running is calculated in the next-to-leading order logarithmic
approximation (NLLA) Buras:1991jm ; Buchalla:1995vs ; Yamanaka:2015ncb . Below
$\mu=m_{c}$, the charm quark is integrated out. The resulting $|\Delta S|=1$
effective Hamiltonian becomes

${\cal
H}_{eff}(\mu)=\frac{G_{F}}{\sqrt{2}}V_{us}^{*}V_{ud}\sum_{i=1}^{6}z_{i}(\mu)Q_{i}(\mu)+{\rm
H.c.}.$ (40)

Here we quote the values of Refs. Yamanaka:2015ncb ; Yamanaka:2016fjj :

${\bf z}(\mu=1\,{\rm GeV})=\left(\begin{array}[]{c}-0.107\cr 1.02\cr
1.76\times 10^{-5}\cr-1.39\times 10^{-2}\cr 6.37\times 10^{-3}\cr-3.45\times
10^{-3}\cr\end{array}\right).$ (41)

We see that the Wilson coefficient of $Q_{2}$ is the largest. This is because
$Q_{2}$ is the sole tree level operator at $\mu=m_{W}$, and the others were
radiatively generated. Here we point that the coefficient of $Q_{1}$ is also
important since the contribution of $Q_{2}$ obtains a factor of $1/N_{c}$
after the Fierz rearrangement of the color (see below). The operators $Q_{i}$
($i=3,\cdots 6$) cannot be neglected either, because they generate the $\phi$
meson which is impossible with $Q_{1}$ and $Q_{2}$. We also note that $Q_{5}$
and $Q_{6}$, after Fierz transformation, couple to the chiral condensate which
may enhance the overall effect (see below).

For the calculation of the crossing symmetric contribution, it is convenient
to Fierz transform the $|\Delta S|=1$ four-quark operators $Q_{i}$
($i=1,\cdots 6$). The Fierz transform of Eqs. (32), (34), (36), (37), (38),
and (39) are

$\displaystyle Q_{1}$ $\displaystyle=$
$\displaystyle\frac{1}{3}\bar{s}\gamma^{\mu}(1-\gamma_{5})u\,\bar{u}\gamma_{\mu}(1-\gamma_{5})d+2\sum_{a}\bar{s}\gamma^{\mu}(1-\gamma_{5})t_{a}u\,\bar{u}\gamma_{\mu}(1-\gamma_{5})t_{a}d$
(42) $\displaystyle=$
$\displaystyle\bar{s}\gamma_{\mu}(1-\gamma_{5})d\,\bar{u}\gamma^{\mu}(1-\gamma_{5})u,$
$\displaystyle Q_{2}$ $\displaystyle=$
$\displaystyle\frac{1}{3}\bar{s}\gamma_{\mu}(1-\gamma_{5})d\,\bar{u}\gamma^{\mu}(1-\gamma_{5})u+2\sum_{a=1}^{8}\bar{s}\gamma_{\mu}(1-\gamma_{5})t_{a}d\,\bar{q}\gamma^{\mu}(1-\gamma_{5})t_{a}q,$
(43) $\displaystyle Q_{3}$ $\displaystyle=$
$\displaystyle\frac{1}{3}\sum_{q=u,d,s}\bar{s}\gamma_{\mu}(1-\gamma_{5})q\,\bar{q}\gamma^{\mu}(1-\gamma_{5})d+2\sum_{q=u,d,s}\sum_{a=1}^{8}\bar{s}\gamma_{\mu}(1-\gamma_{5})t_{a}q\,\bar{q}\gamma^{\mu}(1-\gamma_{5})t_{a}d,\
\ \ \ \ $ (44) $\displaystyle Q_{4}$ $\displaystyle=$
$\displaystyle\frac{1}{3}\bar{s}\gamma^{\mu}(1-\gamma_{5})d\,\sum_{q=u,d,s}\bar{q}\gamma_{\mu}(1-\gamma_{5})q+2\sum_{a}\bar{s}\gamma^{\mu}(1-\gamma_{5})t_{a}d\,\sum_{q=u,d,s}\bar{q}\gamma_{\mu}(1-\gamma_{5})t_{a}q$
(45) $\displaystyle=$
$\displaystyle\sum_{q=u,d,s}\bar{s}\gamma_{\mu}(1-\gamma_{5})q\,\bar{q}\gamma^{\mu}(1-\gamma_{5})d,$
$\displaystyle Q_{5}$ $\displaystyle=$
$\displaystyle-\frac{2}{3}\sum_{q=u,d,s}\bar{s}(1+\gamma_{5})q\,\bar{q}(1-\gamma_{5})d-4\sum_{q=u,d,s}\sum_{a}\bar{s}(1+\gamma_{5})t_{a}q\,\bar{q}(1-\gamma_{5})t_{a}d,$
(46) $\displaystyle Q_{6}$ $\displaystyle=$
$\displaystyle\frac{1}{3}\bar{s}\gamma^{\mu}(1-\gamma_{5})d\sum_{q=u,d,s}\bar{q}\gamma_{\mu}(1+\gamma_{5})q+2\sum_{a}\bar{s}\gamma^{\mu}(1-\gamma_{5})t_{a}d\sum_{q=u,d,s}\bar{q}\gamma_{\mu}(1+\gamma_{5})t_{a}q$
(47) $\displaystyle=$
$\displaystyle-2\sum_{q=u,d,s}\bar{s}(1+\gamma_{5})q\,\bar{q}(1-\gamma_{5})d,$

where $t_{a}$ is the generator of the color $SU(3)_{c}$ group. The summation
over the fundamental color indices runs inside each Dirac bilinear, so the
indices ($\alpha$ and $\beta$) have been omitted. As for Eqs. (42), (45), and
(47), we also displayed in the first equalities the Fierz rearrangement of the
fundamental color indices to form color singlet Dirac bilinears. We note that
an additional minus sign contributes due to the anticommutation of fermion
operators. This sign change is important since there may be interference with
crossing symmetric graphs.

Figure 7:  Factorization of the $|\Delta S|=1$ vector meson vertices ($|\Delta
S|=1$ meson transition), with (a) the two-quark process, (b) the one-quark
process, and (c) three-meson interaction. The double crosses with
”$\langle\bar{q}q\rangle$” denote the chiral condensate $\langle
0|\bar{q}q|0\rangle$ ($q=d,s$). The black blob denotes the $|\Delta S|=1$
four-quark interaction. There are similar diagrams with the $\rho$ meson
replaced by $\omega$ and $\phi$ mesons.

We use the standard factorization to derive the $|\Delta S|=1$ vector meson
interaction from the $|\Delta S|=1$ four-quark interaction of Eq. (40). We
first construct the $|\Delta S|=1$ meson transition in the factorization with
vacuum saturation approximation Lee:1972px ; Shrock:1978dm . It works as

$\displaystyle\langle\rho|\bar{s}\gamma^{\mu}d\,\bar{q}\gamma_{\mu}q|K^{*}\rangle$
$\displaystyle\approx$ $\displaystyle\langle
0|\bar{s}\gamma^{\mu}d|K^{*}\rangle\langle\rho|\bar{q}\gamma_{\mu}q|0\rangle,$
(48) $\displaystyle\langle\rho|\bar{s}d\,\bar{d}d|K^{*}\rangle$
$\displaystyle\approx$ $\displaystyle\langle\rho|\bar{s}d|K^{*}\rangle\langle
0|\bar{d}d|0\rangle,$ (49)

where $q=u,d$. We note that the vacuum saturation approximation gives the
leading contribution in the large $N_{c}$ expansion in the mesonic sector. The
$|\Delta S|=1$ four-quark interaction has two distinct contributions, as shown
in Figs. 7(a) and 7(b). The first contribution (a) is the factorization into
two meson tadpoles [see Eq. (48)]. It requires the decay constants of vector
mesons, as

$\displaystyle\langle 0|\bar{u}\gamma_{\mu}u|\rho\rangle$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{2}}\varepsilon_{\mu}m_{\rho}f_{\rho},$ (50)
$\displaystyle\langle 0|\bar{d}\gamma_{\mu}d|\rho\rangle$ $\displaystyle=$
$\displaystyle-\frac{1}{\sqrt{2}}\varepsilon_{\mu}m_{\rho}f_{\rho},$ (51)
$\displaystyle\langle 0|\bar{q}\gamma_{\mu}q|\omega\rangle$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{2}}\varepsilon_{\mu}m_{\omega}f_{\omega}\ \ \
(q=u,d),$ (52) $\displaystyle\langle 0|\bar{s}\gamma_{\mu}s|\phi\rangle$
$\displaystyle=$ $\displaystyle\varepsilon_{\mu}m_{\phi}f_{\phi},$ (53)

where $\varepsilon_{\mu}$ is the polarization of the vector meson, and
$m_{\rho}=770$ MeV, $f_{\rho}=216$ MeV, $m_{\omega}=783$ MeV, $f_{\omega}=197$
MeV, $m_{\phi}=1020$ MeV, and $f_{\phi}=233$ MeV Neubert:1997uc ;
Jansen:2009hr ; Grossmann:2015lea ; Straub:2015ica ; Chang:2018aut ;
Sun:2018cdr . The second contribution [Fig. 7 (b)] is the factorization into
scalar matrix elements [see Eq. (49)]. It appears from the Fierz
transformation of $Q_{5}$ and $Q_{6}$. The chiral condensates relevant in this
regard are $\langle 0|\bar{s}s|0\rangle\approx\langle
0|\bar{d}d|0\rangle\approx-\frac{m_{\pi}^{2}f_{\pi}^{2}}{m_{u}+m_{d}}\approx-(269\,{\rm
MeV})^{3}$ McNeile:2012xh . They are obtained at the appropriate
renormalization scale $\mu=1$ GeV with $m_{u}\approx 2.7$ MeV and
$m_{d}\approx 5.9$ MeV Tanabashi:2018oca , calculated in the two-loop level
renormalization group evolution Tarasov:1980au ; Gorishnii:1983zi . The scalar
matrix element of the vector meson is derived by using the result of the
calculation of the chiral extrapolation of the vector meson mass in lattice
QCD Leinweber:2001ac ; Rios:2008zr ; Bavontaweepanya:2018yds ; Guo:2018zvl ;
Molina:2020xxx . As derived in Appendix A, we obtain

$\displaystyle B_{\rho K^{\ast}}$
$\displaystyle\equiv\langle\rho^{0}|\bar{s}d|K^{\ast
0}\rangle=-1.14\,\,\text{GeV},$ (54) $\displaystyle B_{\omega K^{\ast}}$
$\displaystyle\equiv\langle\omega|\bar{s}d|K^{\ast
0}\rangle=1.88\,\,\text{GeV},$ (55) $\displaystyle B_{\phi K^{\ast}}$
$\displaystyle\equiv\langle\phi|\bar{d}s|K^{\ast
0}\rangle=2.14\,\,\text{GeV}.$ (56)

By using the above parameters, the lagrangian of the weak vector meson
transition is given by

$\displaystyle{\cal L}_{VK^{*}}$ $\displaystyle=$ $\displaystyle
V_{ud}V_{us}^{*}\sum_{V=\rho,\omega,\phi}g_{VK^{*}}{V}^{\nu}K^{*}_{\nu}+{\rm
H.c.},$ (57)

where $\rho^{\nu}$, $\omega^{\nu}$ and $\phi^{\nu}$ are the field operators of
the $\rho^{0}$, $\omega$, and $\phi$ mesons, respectively. The coupling
constants are given by

$\displaystyle g_{\rho K^{*}}$ $\displaystyle=$
$\displaystyle\frac{G_{F}}{\sqrt{2}}\biggl{[}\biggl{(}z_{1}+\frac{1}{3}z_{2}-\frac{1}{3}z_{3}-z_{4}\biggr{)}m_{K^{*}}f_{K^{*}}m_{\rho}\frac{f_{\rho}}{\sqrt{2}}-\biggl{(}\frac{2}{3}z_{5}+2z_{6}\biggr{)}B_{\rho
K^{\ast}}\langle 0|\bar{s}s+\bar{d}d|0\rangle\biggr{]}$ (58) $\displaystyle=$
$\displaystyle 4.4\times 10^{-8}{\rm GeV}^{2},$ $\displaystyle g_{\omega
K^{*}}$ $\displaystyle=$
$\displaystyle\frac{G_{F}}{\sqrt{2}}\biggl{[}\biggl{(}z_{1}+\frac{1}{3}z_{2}+\frac{7}{3}z_{3}+\frac{5}{3}z_{4}+2z_{5}+\frac{2}{3}z_{6}\biggr{)}m_{K^{*}}f_{K^{*}}m_{\omega}\frac{f_{\omega}}{\sqrt{2}}-\biggl{(}\frac{2}{3}z_{5}+2z_{6}\biggr{)}B_{\omega
K^{\ast}}\langle 0|\bar{s}s+\bar{d}d|0\rangle\biggr{]}$ (59) $\displaystyle=$
$\displaystyle 3.4\times 10^{-8}{\rm GeV}^{2},$ $\displaystyle g_{\phi K^{*}}$
$\displaystyle=$
$\displaystyle\frac{G_{F}}{\sqrt{2}}\biggl{[}\biggl{(}\frac{4}{3}z_{3}+\frac{4}{3}z_{4}+z_{5}+\frac{1}{3}z_{6}\biggr{)}m_{K^{*}}f_{K^{*}}m_{\phi}f_{\phi}-\biggl{(}\frac{2}{3}z_{5}+2z_{6}\biggr{)}B_{\phi
K^{\ast}}\langle 0|\bar{s}s+\bar{d}d|0\rangle\biggr{]}$ (60) $\displaystyle=$
$\displaystyle-6.6\times 10^{-9}{\rm GeV}^{2}.$

Let us also construct the weak three-meson interactions. Again by using the
vacuum saturation approximation, we have

$\displaystyle\langle\rho\,|\bar{q}\gamma_{\mu}q\,\bar{s}\gamma^{\mu}d|K^{*}\rho\rangle$
$\displaystyle\approx$ $\displaystyle\langle
0|\bar{q}\gamma_{\mu}q|\rho\rangle\langle\rho|\bar{s}\gamma^{\mu}d|K^{*}\rangle,\
\ \ \ $ (61)

with $q=u,d$. The weak three-vector meson interaction is then

$\displaystyle{\cal
L}^{V}_{V^{\prime}K^{*}}=V_{ud}V_{us}^{*}\sum_{V,V^{\prime}=\rho,\omega,\phi}g^{V}_{V^{\prime}K^{*}}V_{\mu}{V^{\prime}}^{\nu}i\overleftrightarrow{\partial}^{\mu}K^{*}_{\nu}+{\rm
H.c.},\ \ \ $ (62)

where $A\overleftrightarrow{\partial}^{\mu}B\equiv
A(\partial^{\mu}B)-(\partial^{\mu}A)B$. The coupling constants are given by

$\displaystyle g^{\rho}_{V^{\prime}K^{*}}$ $\displaystyle=$
$\displaystyle\frac{G_{F}}{\sqrt{2}}\biggl{[}z_{1}+\frac{1}{3}z_{2}-\frac{1}{3}z_{3}-z_{4}\biggr{]}m_{\rho}\frac{f_{\rho}}{\sqrt{2}}c_{V^{\prime}K^{\ast}}$
(63) $\displaystyle=$ $\displaystyle(2.4\times 10^{-7})\times
c_{V^{\prime}K^{\ast}},$ $\displaystyle g^{\omega}_{V^{\prime}K^{*}}$
$\displaystyle=$
$\displaystyle\frac{G_{F}}{\sqrt{2}}\biggl{[}z_{1}+\frac{1}{3}z_{2}+\frac{7}{3}z_{3}+\frac{5}{3}z_{4}+2z_{5}+\frac{2}{3}z_{6}\biggr{]}m_{\omega}\frac{f_{\omega}}{\sqrt{2}}c_{V^{\prime}K^{\ast}}$
(64) $\displaystyle=$ $\displaystyle(2.0\times 10^{-7})\times
c_{V^{\prime}K^{\ast}},$ $\displaystyle g^{\phi}_{V^{\prime}K^{*}}$
$\displaystyle=$
$\displaystyle\frac{G_{F}}{\sqrt{2}}\biggl{[}\frac{4}{3}z_{3}+\frac{4}{3}z_{4}+z_{5}+\frac{1}{3}z_{6}\biggr{]}m_{\phi}f_{\phi}c_{V^{\prime}K^{\ast}}$
(65) $\displaystyle=$ $\displaystyle(-2.6\times 10^{-8})\times
c_{V^{\prime}K^{\ast}},$

where $V^{\prime}=\rho,\omega,\phi$. The coefficients $c_{V^{\prime}K^{\ast}}$
are obtained as the relative strength of the meson transition ${\rm
Tr}[V^{\mu}T^{\dagger}V_{\mu}+V^{\mu\dagger}TV^{\dagger}_{\mu}]$, where $T$ is
the SU(3) ladder operator given by the Gell-Mann matrices $\lambda_{a}$
$(a=1,...,8)$ as $T=\frac{1}{2\sqrt{2}}(\lambda_{6}+i\lambda_{7})$. As a
result, we obtain $c_{\rho K^{\ast}}=1/\sqrt{2}$, $c_{\omega
K^{\ast}}=-1/\sqrt{2}$ and $c_{\phi K^{\ast}}=-1$. For the amplitudes of the
weak three-vector meson interaction, we used the approximate relation
$\langle\rho(p^{\prime})|\bar{s}\gamma^{\mu}d|K^{*}(p)\rangle\approx-(p^{\mu}+p^{\prime\mu})\varepsilon^{(\rho)\nu}\varepsilon^{(K^{*})*}_{\nu}$.

###  III.5 One-loop level calculation of the EDM of charged leptons

In this subsection, we perform the one-loop level calculation of the lepton
EDM which is given by the amplitudes shown in Fig. 5. The diagrams in Figs.
5(a) and 5(a′) are the contribution with the weak interaction of three vector
mesons, while the diagrams in Figs. 5 (b) and (b′) are that with the strong
interaction.

The scattering amplitudes with the weak three-meson interaction in Figs. 5 (a)
and (a′) are given by

$\displaystyle i{\cal M}^{K^{\ast}}_{(a)}=$ $\displaystyle-
ieJm_{K^{\ast}}f_{K^{\ast}}I_{dsll}\left(\frac{em^{2}_{\rho}}{g_{\gamma}}\right)^{2}\sum_{V,V^{\prime}=\rho,\omega,\phi}\frac{c_{V}c_{V^{\prime}}}{q^{2}-m^{2}_{V}}$
$\displaystyle\times\int\frac{d^{4}k}{(2\pi)^{4}}\frac{\bar{u}_{l}(p-q)\gamma_{\mu}[({p/}-{k/})+m_{l}]\left[g^{V}_{V^{\prime}K^{\ast}}(2k-q)\cdot\varepsilon\gamma^{\mu}+g^{V^{\prime}}_{VK^{\ast}}(k+q)^{\mu}{{\varepsilon}/}\right]\gamma_{5}u_{l}(p)}{(k-q)^{2}[(p-k)^{2}-m^{2}_{l}][(k-q)^{2}-m^{2}_{V^{\prime}}][k^{2}-m^{2}_{K^{\ast}}]},$
(66) $\displaystyle i{\cal M}^{K^{\ast}}_{(a^{\prime})}=$
$\displaystyle+ieJm_{K^{\ast}}f_{K^{\ast}}I_{dsll}\left(\frac{em^{2}_{\rho}}{g_{\gamma}}\right)^{2}\sum_{V,V^{\prime}=\rho,\omega,\phi}\frac{c_{V}c_{V^{\prime}}}{q^{2}-m^{2}_{V}}$
$\displaystyle\times\int\frac{d^{4}k}{(2\pi)^{4}}\frac{\bar{u}_{l}(p-q)\gamma_{\mu}[({p/}-{k/})-m_{l}]\left[g^{V}_{V^{\prime}K^{\ast}}(2k-q)\cdot\varepsilon\gamma^{\mu}+g^{V^{\prime}}_{VK^{\ast}}(k-2q)^{\mu}{{\varepsilon}/}\right]\gamma_{5}u_{l}(p)}{k^{2}[(p-k)^{2}-m^{2}_{l}][k^{2}-m^{2}_{V^{\prime}}][(k-q)^{2}-m^{2}_{K^{\ast}}]}.$
(67)

The masses of leptons ($l=e,\mu,\tau$) are given by $m_{e}=0.510998950$ MeV,
$m_{\mu}=0.105658$ GeV, and $m_{\tau}=1.77686$ GeV Tanabashi:2018oca . The
coefficients $c_{V},c_{V^{\prime}}$ are $c_{\rho}=1$, $c_{\omega}=1/3$,
$c_{\phi}=-\sqrt{2}/3$. In the soft photon limit ($q^{2}\sim 0$, $p\cdot q\sim
0$), the denominators of the integrands in Eqs. (66)-(67) are rewritten as

$\displaystyle\frac{1}{(k-q)^{2}[(p-k)^{2}-m^{2}_{l}][(k-q)^{2}-m^{2}_{V^{\prime}}][k^{2}-m^{2}_{K^{\ast}}]}=\Gamma(4)\int_{0}^{1}dz_{1}\int_{0}^{z_{1}}dz_{2}\int_{0}^{z_{2}}dz_{3}\frac{1}{[\ell^{2}_{a}-\Delta_{a}]^{4}},$
(68)
$\displaystyle\frac{1}{k^{2}[(p-k)^{2}-m^{2}_{l}][k^{2}-m^{2}_{V^{\prime}}][(k-q)^{2}-m^{2}_{K^{\ast}}]}=\Gamma(4)\int_{0}^{1}dz_{1}\int_{0}^{z_{1}}dz_{2}\int_{0}^{z_{2}}dz_{3}\frac{1}{[\ell^{2}_{a^{\prime}}-\Delta_{a}]^{4}},$
(69)

where

$\displaystyle\ell^{\mu}_{a}$
$\displaystyle=k^{\mu}-z_{3}p^{\mu}-(z_{1}-z_{3})q^{\mu},$ (70)
$\displaystyle\ell^{\mu}_{a^{\prime}}$
$\displaystyle=k^{\mu}-z_{3}p^{\mu}-(1-z_{1})q^{\mu},$ (71)
$\displaystyle\Delta_{a}$
$\displaystyle=m^{2}_{K^{\ast}}+(m^{2}_{V^{\prime}}-m^{2}_{K^{\ast}})z_{1}-m^{2}_{V^{\prime}}z_{2}+m^{2}_{l}z^{2}_{3}.$
(72)

The numerators of the integrands in Eqs. (66)-(67) are reduced to

$\displaystyle\bar{u}_{l}(p-q)\gamma_{\mu}[({p/}-{k/})+m_{l}]\left[g^{V}_{V^{\prime}K^{\ast}}(2k-q)\cdot\varepsilon\gamma^{\mu}+g^{V^{\prime}}_{VK^{\ast}}(k+q)_{\mu}{{\varepsilon}/}\right]\gamma_{5}u_{l}(p)$
$\displaystyle=\bar{u}_{l}(p-q)\left[4g^{V}_{V^{\prime}K^{\ast}}m_{l}z_{3}(3-2z_{1}+z_{3})p\cdot\varepsilon+g^{V^{\prime}}_{VK^{\ast}}m_{l}z_{3}\left(2p\cdot\varepsilon+{q/}{{\varepsilon}/}\right)\right]\gamma_{5}u_{l}(p)+\cdots$
(73)
$\displaystyle\bar{u}_{l}(p-q)\gamma_{\mu}[({p/}-{k/})-m_{l}]\left[g^{V}_{V^{\prime}K^{\ast}}(2k-q)\cdot\varepsilon\gamma^{\mu}+g^{V^{\prime}}_{VK^{\ast}}(k-2q)^{\mu}{{\varepsilon}/}\right]\gamma_{5}u_{l}(p)$
$\displaystyle=-\bar{u}_{l}(p-q)\left[4g^{V}_{V^{\prime}K^{\ast}}m_{l}z_{3}(3-2z_{1}+z_{3})p\cdot\varepsilon+g^{V^{\prime}}_{VK^{\ast}}m_{l}(3z_{3}-2z_{1}-2){q/}{{\varepsilon}/}\right]\gamma_{5}u_{l}(p)+\cdots$
(74)

where the terms which do not contribute to the EDM are suppressed. By
performing the integrals with respect to $\ell_{a}$ and $\ell_{a^{\prime}}$
for Eqs. (66)-(67), the amplitudes for Eqs. (66)-(67) are reduced to

$\displaystyle i{\cal M}^{K^{\ast}}_{(a)}=$
$\displaystyle-\frac{i}{(4\pi)^{2}}em_{l}Jm_{K^{\ast}}f_{K^{\ast}}I_{dsll}\left(\frac{em^{2}_{\rho}}{g_{\gamma}}\right)^{2}\sum_{V,V^{\prime}=\rho,\omega,\phi}\frac{c_{V}c_{V^{\prime}}}{m^{2}_{V}}\bar{u}_{l}(p-q)$
$\displaystyle\times\left[\int_{0}^{1}dz_{1}\int_{0}^{z_{1}}dz_{2}\int_{0}^{z_{2}}dz_{3}\frac{2g^{V}_{V^{\prime}K^{\ast}}z_{3}(3-2z_{1}+z_{3})+2g^{V^{\prime}}_{VK^{\ast}}z_{3}}{\Delta^{2}_{a}}\right]\sigma^{\mu\nu}q_{\nu}\varepsilon_{\mu}\gamma_{5}u_{l}(p),$
(75) $\displaystyle i{\cal M}^{K^{\ast}}_{(a^{\prime})}=$
$\displaystyle-\frac{i}{(4\pi)^{2}}em_{l}Jm_{K^{\ast}}f_{K^{\ast}}I_{dsll}\left(\frac{em^{2}_{\rho}}{g_{\gamma}}\right)^{2}\sum_{V,V^{\prime}=\rho,\omega,\phi}\frac{c_{V}c_{V^{\prime}}}{m^{2}_{V}}\bar{u}_{l}(p-q)$
$\displaystyle\times\left[\int_{0}^{1}dz_{1}\int_{0}^{z_{1}}dz_{2}\int_{0}^{z_{2}}dz_{3}\frac{2g^{V}_{V^{\prime}K^{\ast}}z_{3}(3-2z_{1}+z_{3})+g^{V^{\prime}}_{VK^{\ast}}(3z_{3}-2z_{1}-2)}{\Delta^{2}_{a}}\right]\sigma^{\mu\nu}q_{\nu}\varepsilon_{\mu}\gamma_{5}u_{l}(p),$
(76)

respectively, where we use the Gordon identity

$\displaystyle\bar{u}_{l}(p-q)\left[(2p-q)^{\mu}-i\sigma^{\mu\nu}q_{\nu}\right]\gamma_{5}u_{l}(p)=0.$
(77)

The integrals in Eqs. (75) and (76),

$\displaystyle I^{(a)}_{1}$
$\displaystyle=\int_{0}^{1}dz_{1}\int_{0}^{z_{1}}dz_{2}\int_{0}^{z_{2}}dz_{3}\frac{z_{3}(3-2z_{1}+z_{3})}{\Delta^{2}_{a}}$
(78) $\displaystyle I^{(a)}_{2}$
$\displaystyle=\int_{0}^{1}dz_{1}\int_{0}^{z_{1}}dz_{2}\int_{0}^{z_{2}}dz_{3}\frac{z_{3}}{\Delta^{2}_{a}}$
(79) $\displaystyle I^{(a^{\prime})}_{2}$
$\displaystyle=\int_{0}^{1}dz_{1}\int_{0}^{z_{1}}dz_{2}\int_{0}^{z_{2}}dz_{3}\frac{3z_{3}-2z_{1}-2}{\Delta^{2}_{a}}$
(80)

are performed numerically, with the results summarized in Table 1. For
$I^{(a)}_{2}$, the analytic form is obtained as shown in Appendix B.

Table 1:  Numerical values of the integrals in Eqs. (78)-(80) for the leptons $l=e,\mu,\tau$ and the vector mesons $V^{\prime}=\rho^{0},\omega,\phi$, given in units of GeV-4. $I^{(a)}_{1}$ | $\rho^{0}$ | $\omega$ | $\phi$ | $I^{(a)}_{2}$ | $\rho^{0}$ | $\omega$ | $\phi$ | $I^{(a^{\prime})}_{2}$ | $\rho^{0}$ | $\omega$ | $\phi$
---|---|---|---|---|---|---|---|---|---|---|---
$e$ | 23.3 | 22.6 | 13.6 | $e$ | 14.1 | 13.6 | 8.20 | $e$ | -82.1 | -79.6 | -47.6
$\mu$ | 4.82 | 4.70 | 3.03 | $\mu$ | 2.97 | 2.89 | 1.85 | $\mu$ | -25.9 | -25.2 | -15.5
$\tau$ | 0.209 | 0.206 | 0.153 | $\tau$ | 0.137 | 0.134 | 0.0972 | $\tau$ | -4.68 | -4.56 | -2.96

The amplitudes $i{\cal M}^{K^{\ast}}_{(a)}$ and $i{\cal
M}^{K^{\ast}}_{(a^{\prime})}$ are for the contributions with the $K^{\ast}$
propagator. In addition, the amplitudes with the $\bar{K}^{\ast}$ propagator,
denoted by $i{\cal M}^{\bar{K}^{\ast}}_{(a)}$ and $i{\cal
M}^{\bar{K}^{\ast}}_{(a^{\prime})}$, also contribute to the EDM. If we
restrict to the CP violation, we have $i{\cal M}^{\bar{K}^{\ast}}_{(a)}=i{\cal
M}^{K^{\ast}}_{(a)}$ and $i{\cal M}^{\bar{K}^{\ast}}_{(a^{\prime})}=i{\cal
M}^{K^{\ast}}_{(a^{\prime})}$. Thus the total scattering amplitude with the
weak three-vector meson interactions is given by

$\displaystyle i{\cal M}^{\rm SM}_{(a)}$ $\displaystyle=i{\cal M}_{(a)}+i{\cal
M}_{(a^{\prime})},$ (81)

where

$\displaystyle i{\cal M}_{(a)}$ $\displaystyle=i{\cal
M}^{K^{\ast}}_{(a)}+i{\cal M}^{\bar{K}^{\ast}}_{(a)}=2i{\cal
M}^{K^{\ast}}_{(a)},$ (82) $\displaystyle i{\cal M}_{(a^{\prime})}$
$\displaystyle=i{\cal M}^{K^{\ast}}_{(a^{\prime})}+i{\cal
M}^{\bar{K}^{\ast}}_{(a^{\prime})}=2i{\cal M}^{K^{\ast}}_{(a^{\prime})}.$ (83)

In a similar manner, the charged lepton EDM contributions with the strong
three-vector meson interactions shown in Figs. 5$(b)$ and 5($b^{\prime}$) are
also calculated. The scattering amplitudes of the diagrams $(b)$ and
$(b^{\prime})$ are obtained as

$\displaystyle i{\cal M}^{K^{\ast}}_{(b)}=i{\cal M}^{\bar{K}^{\ast}}_{(b)}=$
$\displaystyle-
ieJm_{K^{\ast}}f_{K^{\ast}}I_{dsll}\left(\frac{em^{2}_{\rho}}{g_{\gamma}}\right)^{2}\sum_{V,V^{\prime}=\rho^{0},\omega,\phi}\frac{c_{V}c_{V^{\prime}}}{q^{2}-m^{2}_{V}}g_{V}g_{V^{\prime}K^{\ast}}$
$\displaystyle\times\int\frac{d^{4}k}{(2\pi)^{4}}\frac{\bar{u}_{l}(p-q)\gamma_{\mu}[({p/}-{k/})+m_{l}]\left[(2k-q)\cdot\varepsilon\gamma^{\mu}+\varepsilon^{\mu}(2{q/}-{k/})-(k+q)^{\mu}{{\varepsilon}/}\right]\gamma_{5}u_{l}(p)}{(k-q)^{2}[(p-k)^{2}-m^{2}_{l}][(k-q)^{2}-m^{2}_{V^{\prime}}][(k-q)^{2}-m^{2}_{K^{\ast}}][k^{2}-m^{2}_{K^{\ast}}]},$
(84) $\displaystyle i{\cal M}^{K^{\ast}}_{(b^{\prime})}=i{\cal
M}^{\bar{K}^{\ast}}_{(b^{\prime})}=$ $\displaystyle
ieJm_{K^{\ast}}f_{K^{\ast}}I_{dsll}\left(\frac{em^{2}_{\rho}}{g_{\gamma}}\right)^{2}\sum_{V,V^{\prime}=\rho^{0},\omega,\phi}\frac{c_{V}c_{V^{\prime}}}{q^{2}-m^{2}_{V}}g_{V}g_{V^{\prime}K^{\ast}}$
$\displaystyle\times\int\frac{d^{4}k}{(2\pi)^{4}}\frac{\bar{u}_{l}(p-q)\gamma_{\mu}[({p/}-{k/})-m_{l}]\left[(2k-q)\cdot\varepsilon\gamma^{\mu}+\varepsilon^{\mu}(2{q/}-{k/})-(k+q)^{\mu}{{\varepsilon}/}\right]\gamma_{5}u_{l}(p)}{k^{2}[(p-k)^{2}-m^{2}_{l}][k^{2}-m^{2}_{V^{\prime}}][(k-q)^{2}-m^{2}_{K^{\ast}}][k^{2}-m^{2}_{K^{\ast}}]}.$
(85)

The coupling constant $g_{V}$ ($V=\rho^{0},\omega,\phi$) is defined as
$\sqrt{2}g_{\rho}=-\sqrt{2}g_{\omega}=g_{\phi}=g$. In the soft photon limit,
the denominators of the integrands in Eqs. (84)-(85) are rewritten as

$\displaystyle\frac{1}{(k-q)^{2}[(p-k)^{2}-m^{2}_{l}][(k-q)^{2}-m^{2}_{V^{\prime}}][(k-q)^{2}-m^{2}_{K^{\ast}}][k^{2}-m^{2}_{K^{\ast}}]}=\Gamma(5)\int_{0}^{1}dz_{1}\int_{0}^{z_{1}}dz_{2}\int_{0}^{z_{2}}dz_{3}\int_{0}^{z_{3}}dz_{4}\frac{1}{[\ell^{2}_{b}-\Delta_{b}]^{5}},$
(86)
$\displaystyle\frac{1}{k^{2}[(p-k)^{2}-m^{2}_{l}][k^{2}-m^{2}_{V^{\prime}}][(k-q)^{2}-m^{2}_{K^{\ast}}][k^{2}-m^{2}_{K^{\ast}}]}=\Gamma(5)\int_{0}^{1}dz_{1}\int_{0}^{z_{1}}dz_{2}\int_{0}^{z_{2}}dz_{3}\int_{0}^{z_{3}}dz_{4}\frac{1}{[\ell^{2}_{b^{\prime}}-\Delta_{b}]^{5}},$
(87)

where

$\displaystyle\ell^{\mu}_{b}$
$\displaystyle=k^{\mu}-z_{4}p^{\mu}-(z_{1}-z_{4})q^{\mu},$ (88)
$\displaystyle\ell^{\mu}_{b^{\prime}}$
$\displaystyle=k^{\mu}-z_{4}p^{\mu}-(z_{1}-z_{2})q^{\mu},$ (89)
$\displaystyle\Delta_{b}$
$\displaystyle=m^{2}_{K^{\ast}}+(m^{2}_{V^{\prime}}-m^{2}_{K^{\ast}})z_{2}-m^{2}_{V^{\prime}}z_{3}+m^{2}_{l}z^{2}_{4}.$
(90)

Performing the integrals with respect to $\ell_{b}$ and $\ell_{b^{\prime}}$,
Eqs. (84)-(85) are reduced to

$\displaystyle i{\cal M}^{K^{\ast}}_{(b)}=$
$\displaystyle-\frac{4i}{(4\pi)^{2}}eJm_{l}m_{K^{\ast}}f_{K^{\ast}}I_{dsll}\left(\frac{em^{2}_{\rho}}{g_{\gamma}}\right)^{2}\sum_{V,V^{\prime}=\rho^{0},\omega,\phi}\frac{c_{V}c_{V^{\prime}}}{m^{2}_{V}}g_{V}g_{V^{\prime}K^{\ast}}\bar{u}_{l}(p-q)$
$\displaystyle\times\left[\int_{0}^{1}dz_{1}\int_{0}^{z_{1}}dz_{2}\int_{0}^{z_{2}}dz_{3}\int_{0}^{z_{3}}dz_{4}\frac{(1-z_{1})(1-2z_{4})+1-z_{4}^{2}}{\Delta^{3}_{b}}\right]\sigma^{\mu\nu}q_{\nu}\varepsilon_{\mu}\gamma_{5}u_{l}(p),$
(91) $\displaystyle i{\cal M}^{K^{\ast}}_{(b^{\prime})}=$
$\displaystyle-\frac{4i}{(4\pi)^{2}}eJm_{l}m_{K^{\ast}}f_{K^{\ast}}I_{dsll}\left(\frac{em^{2}_{\rho}}{g_{\gamma}}\right)^{2}\sum_{V,V^{\prime}=\rho^{0},\omega,\phi}\frac{c_{V}c_{V^{\prime}}}{m^{2}_{V}}g_{V}g_{V^{\prime}K^{\ast}}\bar{u}_{l}(p-q)$
$\displaystyle\times\left[\int_{0}^{1}dz_{1}\int_{0}^{z_{1}}dz_{2}\int_{0}^{z_{2}}dz_{3}\int_{0}^{z_{3}}dz_{4}\frac{(z_{1}-z_{2})(1-2z_{4})+1-z_{4}^{2}}{\Delta^{3}_{b}}\right]\sigma^{\mu\nu}q_{\nu}\varepsilon_{\mu}\gamma_{5}u_{l}(p).$
(92)

The numerical results of the integrals

$\displaystyle I^{(b)}$
$\displaystyle=\int_{0}^{1}dz_{1}\int_{0}^{z_{1}}dz_{2}\int_{0}^{z_{2}}dz_{3}\int_{0}^{z_{3}}dz_{4}\frac{(1-z_{1})(1-2z_{4})+1-z_{4}^{2}}{\Delta^{3}_{b}},$
(93) $\displaystyle I^{(b^{\prime})}$
$\displaystyle=\int_{0}^{1}dz_{1}\int_{0}^{z_{1}}dz_{2}\int_{0}^{z_{2}}dz_{3}\int_{0}^{z_{3}}dz_{4}\frac{(z_{1}-z_{2})(1-2z_{4})+1-z_{4}^{2}}{\Delta^{3}_{b}},$
(94)

are summarized in Table 2.

Table 2:  Numerical values of the integrals in Eqs. (93)-(94) for the leptons $l=e,\mu,\tau$ and the vector mesons $V^{\prime}=\rho^{0},\omega,\phi$, given in units of GeV-6. $I^{(b)}$ | $\rho^{0}$ | $\omega$ | $\phi$ | $I^{(b^{\prime})}$ | $\rho^{0}$ | $\omega$ | $\phi$
---|---|---|---|---|---|---|---
$e$ | 13.7 | 13.3 | 7.93 | $e$ | 13.7 | 13.3 | 7.93
$\mu$ | 4.32 | 4.19 | 2.56 | $\mu$ | 4.32 | 4.19 | 2.56
$\tau$ | 0.714 | 0.697 | 0.447 | $\tau$ | 0.714 | 0.697 | 0.447

Finally, the total scattering amplitude with the strong three-vector meson
interactions is obtained as

$\displaystyle i{\cal M}^{\rm SM}_{(b)}$ $\displaystyle=i{\cal M}_{(b)}+i{\cal
M}_{(b^{\prime})},$ (95) $\displaystyle i{\cal M}_{(b)}$ $\displaystyle=i{\cal
M}^{K^{\ast}}_{(b)}+i{\cal M}^{\bar{K}^{\ast}}_{(b)}=2i{\cal
M}^{K^{\ast}}_{(b)},$ (96) $\displaystyle i{\cal M}_{(b^{\prime})}$
$\displaystyle=i{\cal M}^{K^{\ast}}_{(b^{\prime})}+i{\cal
M}^{\bar{K}^{\ast}}_{(b^{\prime})}=2i{\cal M}^{K^{\ast}}_{(b^{\prime})}.$ (97)

##  IV Results and analysis

###  IV.1 Numerical results

From the scattering amplitudes derived in the previous section, we obtain the
hadronic long distance contributions to the EDMs of charged leptons. From the
amplitudes $i{\cal M}_{(a)}$ and $i{\cal M}_{(a^{\prime})}$ of Eqs. (82) and
(83), we obtain the EDMs generated by the weak three-vector meson interactions
as

$\displaystyle d^{\rm SM}_{(a)e}$
$\displaystyle=d_{(a)e}+d_{(a^{\prime})e}=3.67\times 10^{-40}\,e\,{\rm cm},$
(98) $\displaystyle d^{\rm SM}_{(a)\mu}$
$\displaystyle=d_{(a)\mu}+d_{(a^{\prime})\mu}=-1.04\times 10^{-40}\,e\,{\rm
cm},$ (99) $\displaystyle d^{\rm SM}_{(a)\tau}$
$\displaystyle=d_{(a)\tau}+d_{(a^{\prime})\tau}=-1.12\times 10^{-37}\,e\,{\rm
cm}.$ (100)

Similarly, the amplitudes $i{\cal M}_{(b)}$ and $i{\cal M}_{(b^{\prime})}$ of
Eqs. (82) and (83) give the contribution from the strong three-vector meson
interaction:

$\displaystyle d^{\rm SM}_{(b)e}$ $\displaystyle=2.13\times 10^{-40}\,e\,{\rm
cm},$ (101) $\displaystyle d^{\rm SM}_{(b)\mu}$ $\displaystyle=1.39\times
10^{-38}\,e\,{\rm cm},$ (102) $\displaystyle d^{\rm SM}_{(b)\tau}$
$\displaystyle=3.89\times 10^{-38}\,e\,{\rm cm}.$ (103)

We finally obtain the EDMs of $e,\mu$, and $\tau$ generated by the hadronic
long distance contributions as

$\displaystyle d_{e}^{\rm SM}$ $\displaystyle=$ $\displaystyle d^{\rm
SM}_{(a)e}+d^{\rm SM}_{(b)e}=5.80\times 10^{-40}e\,{\rm cm},$ (104)
$\displaystyle d_{\mu}^{\rm SM}$ $\displaystyle=$ $\displaystyle d^{\rm
SM}_{(a)\mu}+d^{\rm SM}_{(b)\mu}=1.38\times 10^{-38}e\,{\rm cm},$ (105)
$\displaystyle d_{\tau}^{\rm SM}$ $\displaystyle=$ $\displaystyle d^{\rm
SM}_{(a)\tau}+d^{\rm SM}_{(b)\tau}=-7.32\times 10^{-38}e\,{\rm cm}.$ (106)

These values are much larger than the estimation at the four-loop level (11),
(12), and (13). The most important reason of this enhancement is due to the
relevance of the typical hadronic momenta in the loop. We recall that the
elementary (quark) level contribution only had a typical momentum of
$O(m_{W})\sim O(m_{t})$, and this feature, together with the GIM mechanism,
forced the EDM of charged leptons to have a suppression factor
$m_{b}^{2}m_{c}^{2}m_{s}^{2}$ due to the cancellation between terms with very
close values. We note that the GIM mechanism is also working at the hadron
level. However, the cancellation among contributions with different flavors
becomes much milder thanks to the fact that the typical momentum is replaced
by the mass of the heaviest hadrons of each diagram. This is probably a
general property of the hadronic CP violation in the SM, as similar
enhancement is also relevant for the case of the EDM of neutron
Czarnecki:1997bu ; McKellar:1987tf ; Seng:2014lea ; Pospelov:2013sca  or
nuclei Yamanaka:2015ncb . In this sense, the fact that the elementary
contribution to the EDM appears only at the four-loop level is not truly
essential in this strong suppression, but rather the GIM mechanism (or the
antisymmetry of the Jarlskog invariant) is the main cause Pospelov:2013sca .

We should also comment on the observable effect of the electron EDM in
experiments. The EDM of the electron is usually measured through the
paramagnetic atomic or molecular systems, since the relativistic effect
enhances its effect Carrico:1968zz ; sandars1 ; sandars2 ; Flambaum:1976vg ;
Sandars:1975zz ; Labzovskii ; Sushkovmolecule ; kelly ; Kozlov:1994zz ;
Kozlov:1995xz ; flambaumfr1 ; nayak1 ; nayak3 ; nayak2 ; Natarajrubidium ;
Mukherjeefrancium ; Dzuba:2009mw ; Nataraj:2010vn ; flambaumybftho ;
Porsev:2012zx ; Roberts:2013zra ; Chubukov:2014rba ; abe ; sunaga ;
Radziute:2015apa ; Denis ; Skripnikov ; Sunaga:2018lja ; Sunaga:2018pjn ;
Malika:2019jhn ; Fazil:2019esp . However, these systems also receive
contribution from other CP violating mechanisms such as the CP-odd electron-
nucleon interaction or the nuclear Schiff moment. Previously, the EDM of
charged lepton was believed to be extremely small and the CP-odd electron-
nucleon interaction was thought to be the dominant effect, with a benchmark
value equivalent to the electron EDM as $d_{e}^{(eN)}\sim(10^{-39}-10^{-37})e$
cm for paramagnetic systems Pospelov:2013sca ; Yamanaka:2015ncb ;
Yamanaka:2017mef ; PhysRevA.93.062503  . By considering the strong enhancement
at the hadronic level, we just obtained a value of the electron EDM which lies
in this range. It is then an interesting question to quantify which one,
between the electron EDM and the CP-odd electron-nucleon interaction, gives
the leading contribution at the atomic level.

###  IV.2 Error bar analysis

We now assess the uncertainty of our calculation. The first important source
of systematics is the nonperturbative effect of the renormalization of the
$|\Delta S|=1$ four-quark operators. This was quantified to be about 10%, by
looking at the variation of the Wilson coefficient of $Q_{2}$ in the NLLA in
the range of the scale $\mu=0.6$ GeV to $\mu=m_{c}=1.27$ GeV Buras:1991jm ;
Buchalla:1995vs ; Yamanaka:2015ncb . Another major systematics comes from the
factorization of the vector meson matrix elements. According to the large
$N_{c}$ analysis, the vacuum saturation approximation should work up to
$O(N_{c}^{-1})$ correction. To be conservative, we set the error bar
associated to it to 40%.

Figure 8:  Higher order contributions to the EDM of charged leptons within the
HLS. Here $V=\rho^{0},\omega,\phi$ is the neutral vector meson, and $s$ is the
would-be Nambu-Goldstone boson of the HLS.

Let us now see the uncertainty related to the use of the HLS. The important
point is that the one-loop level diagrams we evaluated are not divergent, so
that the uncertainty due to the counterterms is absent and the stability of
the coupling constants is guaranteed. However, we have to comment on the
contribution from higher order terms. The first process to be noted is the
two-loop level diagram with pseudoscalar mesons [see Fig. 8 (a)]. This
contribution is the most straightforward higher order effect of the HLS. The
second type of higher order process is the mixing of vector mesons with
photons [see Fig. 8 (b)]. In the mass eigenstate basis, it is possible to take
into account this mixing to all orders, but if we restrict the analysis to the
leading order, as done in this work, our particle basis and the mass one
coincide. The third contribution is the would-be Nambu-Goldstone mode which
may appear as scalar propagator insertions in between vector meson
propagations due to the choice of Feynman gauge [see Fig. 8 (c)]. This is
again a higher order effect in the HLS, just like the mixing between photons
and vector mesons. We also note that the explicit flavor SU(3) breaking effect
is a higher order of the HLS, which has already partially been included in our
calculations by introducing the physical meson masses and decay constants. We
also have to comment on the contribution from the other heavier hadrons which
were overlooked in this paper. Here we consider the axial vector meson
$K_{1}$(1270) which, in the viewpoint of the mass difference, should be the
most important hadron among the neglected ones, and show that its contribution
is likely to be subleading. First, the decay constant of $K_{1}$ is not
particularly enhanced, $f_{K_{1}}\sim 170$ MeV Nardulli:2005fn . Regarding the
other hadron matrix elements, the values do not exist in the literature, but
it is possible to show that they are not enhanced either. The axial vector
matrix element $\langle\rho|\bar{d}\gamma_{\mu}\gamma_{5}s|K_{1}\rangle$
corresponds to the quark spin, so there should be a suppression due to the
destructive interference generated by successive gluon emissions/absorptions
Yamanaka:2013zoa ; Yamanaka:2014lva . The pseudoscalar matrix element
$\langle\rho|\bar{d}\gamma_{5}s|K_{1}\rangle$ has also no reasons to be
enhanced, since this receives contribution from the Nambu-Goldstone boson
pole, which is suppressed by the $K$ meson mass in the present case. We can
consider that the effect of $K_{1}$(1270) and other heavier hadrons is part of
the higher order contribution of the HLS. We associate the theoretical
uncertainty coming from the entire higher order process mentioned above with
the expansion parameter, estimated to be $\sim 50\%$ Harada:1992np . In all,
we conclude that the theoretical uncertainty is $70\%$.

A potentially interesting way to quantify the hadronic contribution to the
electron EDM is to calculate the hadronic three-point correlator on lattice
and then attach it to a lepton line to form the EDM amplitude. This approach
is actually used to quantify the hadronic light-by-light scattering
contribution to the muon anomalous magnetic moment Blum:2019ugy . Of course
the calculation will not be easy since the three-point correlator must have a
$|\Delta S|=1$ four-quark operator in the intermediate state, but this kind of
analysis was already done in the context of $K\to\pi\pi$ decay Bai:2015nea ,
so it should not be impossible.

##  V Conclusion

Figure 9:  Plot of the SM predictions of the electron EDM compared with the
history of the records of the experimental upper limits Sandars:1964zz ;
Weisskopf:1968zz ; Player:1970zz ; Murthy:1989zz ; Abdullah:1990nh ;
Commins:1994gv ; Regan:2002ta ; Hudson:2011zz ; Baron:2013eja ;
Andreev:2018ayy . The pink band is denoting the uncertainty of the hadronic
contribution.

In conclusion, we evaluated for the first time the hadron level contribution
to the EDM of charged leptons in the SM, where the CP violation is generated
by the physical complex phase of the CKM matrix. As a result, we found that
this long distance effect is much larger than the previously known one, which
was estimated at the elementary level. We could also rigorously show that, in
the perturbative elementary level calculation at all orders, the EDM of
charged leptons is always suppressed by quark mass factors due to the GIM
mechanism. The main reason of the enhancement at the hadronic level is because
we could avoid additional factors of $m_{b,c,s}^{2}/m_{W,t}^{2}$ by embedding
the heavy $W$ boson or top quark contribution into the $|\Delta S|=1$ low
energy constants while keeping loop momenta of $O(\Lambda_{\rm QCD})\sim 1$
GeV. In Fig. 9, we plot the EDM of the electron in the SM compared with the
progress of the experimental accuracy. The electron EDM obtained in this work
is $d_{e}\sim 10^{-39}e$ cm, which is still well below the current sensitivity
of molecular beam experiments. The EDM experiments are however improving very
fast, and we have to be very sensitive to their progress and to proposals with
new ideas, with some of them claiming to be able to go beyond the level of
$O(10^{-35})e$ cm in statistical sensitivity Vutha:2017pej .

Our next object would be to extend this analysis to the flavor violation
disagreeing with the SM, recently suggested by the measurements of the decay
of $B$ mesons at several $B$ factory experiments Lees:2012xj ; Lees:2013uzd ;
Huschle:2015rga ; Sato:2016svk ; Hirose:2016wfn ; Hirose:2017dxl ;
Aaij:2015yra ; Aaij:2017uff ; Aaij:2017deq ; Aaij:2014ora ; Aaij:2015oid ;
Wehle:2016yoi ; Aaij:2017vbb ; Aaij:2019wad ; Hiller:2003js ;
Crivellin:2019qnh ; Ikeno:2019tkh , and that of $K$ meson decay of KOTO
experiment KOTOresult ; Kitahara:2019lws . In the analysis of the conjunction
of the EDM with the $B$ meson decay, we also have to include the effect of
heavy quarks, which has been omitted in this work.

###### Acknowledgements.

The authors thank T. Kugo for useful discussions. They also thank T. Morozumi
for pointing out the contribution of the semi-leptonic penguin diagram. A part
of the numerical computation in this work was carried out at the Yukawa
Institute Computer Facility. This work is supported in part by the Special
Postdoctoral Researcher Program (SPDR) of RIKEN (Y.Y.).

##  Appendix A Scalar matrix element of vector mesons

In this appendix, the scalar matrix element of vector mesons, $B_{V}$, are
derived. Up to the tree-level contribution, the scalar matrix elements of
vector mesons are obtained from the vector-meson masses expanded in terms of
the light quark mass as

$\displaystyle
m_{\rho}^{2}=M^{2}+B_{\rho}(m_{u}+m_{d})+...=M^{2}+b_{1}M^{2}B_{0}(m_{u}+m_{d})+...,$
(107) $\displaystyle
m_{\omega}^{2}=M^{2}+B_{\omega}(m_{u}+m_{d})+...=M^{2}+(b_{1}+2b_{3})M^{2}B_{0}(m_{u}+m_{d})+...,$
(108) $\displaystyle
m_{\phi}^{2}=M^{2}+B_{\phi}(2m_{s})+...=M^{2}+(b_{1}+b_{3})M^{2}B_{0}(2m_{s})+...,$
(109)

where $M$, $b_{1}$, and $b_{3}$ are the low-energy parameters of the effective
Lagrangians Bavontaweepanya:2018yds ; Guo:2018zvl 111We omit the term being
proportional to the parameter $b_{2}$ in Guo:2018zvl ; Bavontaweepanya:2018yds
, which does not contribute to the scalar matrix element for the vector-meson
transition because ${\rm tr}\left[\chi_{+}T\right]=0$.

$\displaystyle{\cal L}=\frac{b_{1}}{8}M^{2}{\rm
tr}\left[F^{\mu\nu}F_{\mu\nu}\chi_{+}\right]+\frac{b_{3}}{8}M^{2}{\rm
tr}\left[F_{\mu\nu}\right]{\rm tr}\left[F^{\mu\nu}\chi_{+}\right].$ (110)

$B_{0}$ is given by

$\displaystyle B_{0}=-\frac{\langle
0|\bar{u}u|0\rangle}{f^{2}}=\frac{(269\,\text{MeV})^{3}}{(93\,\text{MeV})^{2}}\sim
2.25\,\text{GeV}.$ (111)

The scalar matrix elements for the vector-meson transitions such as
$\langle\rho|\bar{s}d|K^{\ast}\rangle$ are related to
$B_{\rho}\approx\frac{1}{2}\langle\rho|\bar{u}u+\bar{d}d|\rho\rangle$ by
replacing $\chi_{+}\sim 2B_{0}diag(m_{u},m_{d},m_{s})$ in (110) with
$\chi_{+}T$, where $T$ is the SU(3) ladder operator. In the flavor SU(3)
limit, we obtain

$\displaystyle\langle\rho^{0}|\bar{s}d|K^{\ast 0}\rangle$
$\displaystyle\sim-\frac{1}{2}B_{\rho}=-\frac{1}{2}\left(b_{1}M^{2}B_{0}\right),$
(112) $\displaystyle\langle\omega|\bar{s}d|K^{\ast 0}\rangle$
$\displaystyle\sim\frac{1}{2}B_{\omega}=\frac{1}{2}\left\\{(b_{1}+2b_{3})M^{2}B_{0}\right\\},$
(113) $\displaystyle\langle\phi|\bar{d}s|K^{\ast 0}\rangle$
$\displaystyle\sim\frac{1}{\sqrt{2}}B_{\phi}=\frac{1}{\sqrt{2}}\left\\{(b_{1}+b_{3})M^{2}B_{0}\right\\}.$
(114)

The low-energy parameters $b_{1}$, $b_{3}$, $B_{0}$, and $M$ were determined
by fitting the lattice QCD data in Ref. Guo:2018zvl , where three fitting
results denoted by Fit 1, Fit 2, and Fit 3 were obtained as summarized in
Table 3. The scalar matrix elements obtained by using these parameters are
also summarized in Table 3. In this study, we employ the averaged values
obtained as $\langle\rho^{0}|\bar{s}d|K^{\ast 0}\rangle=-1.14$ GeV,
$\langle\omega|\bar{s}d|K^{\ast 0}\rangle=1.88$, GeV and
$\langle\phi|\bar{d}s|K^{\ast 0}\rangle=2.14$ GeV.

Table 3:  The low-energy parameters in Eqs. (107)-(109) obtained in Ref. Guo:2018zvl . | Fit 1 | Fit 2 | Fit 3
---|---|---|---
$M$ [MeV] | 759.3 | 758.8 | 757.0
$b_{1}$ [GeV-2] | 1.2224 | 1.3420 | 1.4009
$b_{3}$ [GeV-2] | 0.5131 | 0.3469 | 0.4151
$2B_{0}m/m^{2}_{\pi}$ | 1.141 | 1.077 | 1.106
$\langle\rho^{0}|\bar{s}d|K^{\ast 0}\rangle$ [GeV] | $-1.09$ | $-1.13$ | $-1.21$
$\langle\omega|\bar{s}d|K^{\ast 0}\rangle$ [GeV] | 2.01 | 1.72 | 1.92
$\langle\phi|\bar{d}s|K^{\ast 0}\rangle$ [GeV] | 2.19 | 2.01 | 2.21

##  Appendix B Analytic form of the Integral $I^{(a)}_{2}$

The analytic forms of the integral $I^{(a)}_{2}$ in Eq. (79) are summarized in
this appendix. For $m_{l}=m_{e},m_{\mu}$, $I^{(a)}_{2}$ is obtained as

$\displaystyle I^{(a)}_{2}=$
$\displaystyle\frac{1}{4{m_{l}^{4}}\left({m_{K^{\ast}}^{2}}-{m_{V^{\prime}}^{2}}\right)}\left[{m_{V^{\prime}}}\,\sqrt{{m_{V^{\prime}}^{2}}-4\,{m_{l}^{2}}}\,\log\left({{({m_{V^{\prime}}^{2}}-2\,{m_{l}^{2}})-{m_{V^{\prime}}}\,\sqrt{{m_{V^{\prime}}^{2}}-4\,{m_{l}^{2}}}}\over{2\,{m_{l}^{2}}}}\right)\right.$
$\displaystyle\left.+{m_{K^{\ast}}}\,\sqrt{{m_{K^{\ast}}^{2}}-4\,{m_{l}^{2}}}\,\log\left({{{m_{K^{\ast}}}+\sqrt{{m_{K^{\ast}}^{2}}-4\,{m_{l}^{2}}}}\over{{m_{K^{\ast}}}-\sqrt{{m_{K^{\ast}}^{2}}-4\,{m_{l}^{2}}}}}\right)+(m_{K^{\ast}}^{2}-2m_{l}^{2})\log\left(\frac{m_{l}^{2}}{m_{K^{\ast}}^{2}}\right)-(m_{V^{\prime}}^{2}-2m_{l}^{2})\log\left(\frac{m_{l}^{2}}{m_{V^{\prime}}^{2}}\right)\right],$
(115)

which can be expanded in terms of a small $m_{l}$ as

$\displaystyle I^{(a)}_{2}=$
$\displaystyle\frac{1}{(m_{K^{\ast}}m_{V^{\prime}})^{2}}\left[\log\left(\frac{m_{K^{\ast}}}{m_{l}}\right)-\frac{m_{K^{\ast}}^{2}}{m_{K^{\ast}}^{2}-m_{V^{\prime}}^{2}}\log\left(\frac{m_{K^{\ast}}}{m_{V^{\prime}}}\right)-\frac{1}{4}\right]$
$\displaystyle+\frac{2m_{l}^{2}}{(m_{K^{\ast}}m_{V^{\prime}})^{4}}\left[(m_{K^{\ast}}^{2}+m_{V^{\prime}}^{2})\log\left(\frac{m_{K^{\ast}}}{m_{l}}\right)-\frac{m_{K^{\ast}}^{4}}{m_{K^{\ast}}^{2}-m_{V^{\prime}}^{2}}\log\left(\frac{m_{K^{\ast}}}{m_{V^{\prime}}}\right)-\frac{26\,m_{K^{\ast}}^{4}-5\,m_{V^{\prime}}^{4}}{12(m_{K^{\ast}}^{2}-m_{V^{\prime}}^{2})}\right]+\cdots.$
(116)

On the other hand, $I^{(a)}_{2}$ for $m_{l}=m_{\tau}$ is given by the
different form as

$\displaystyle I^{(a)}_{2}=$
$\displaystyle\frac{1}{4m_{\tau}^{4}(m_{K^{\ast}}^{2}-m_{V^{\prime}}^{2})}\left[2(2m_{\tau}^{2}-m_{V^{\prime}}^{2})\log\left(\frac{m_{K^{\ast}}}{m_{V^{\prime}}}\right)+2(m_{V^{\prime}}^{2}-m_{K^{\ast}}^{2})\log\left(\frac{4m_{K^{\ast}}^{2}m_{\tau}^{2}-m_{V^{\prime}}^{4}}{4m_{V^{\prime}}^{2}m_{\tau}^{2}-m_{V^{\prime}}^{4}}\right)\right.$
$\displaystyle+2\sqrt{4m_{V^{\prime}}^{2}m_{\tau}^{2}-m_{V^{\prime}}^{4}}\left(\arctan\left(\frac{m_{V^{\prime}}^{2}}{\sqrt{4m_{V^{\prime}}^{2}m_{\tau}^{2}-m_{V^{\prime}}^{4}}}\right)+\arctan\left(\frac{2m_{\tau}^{2}-m_{V^{\prime}}^{2}}{\sqrt{4m_{V^{\prime}}^{2}m_{\tau}^{2}-m_{V^{\prime}}^{4}}}\right)\right)$
$\displaystyle+\left(m_{V^{\prime}}^{2}-m_{K^{\ast}}^{2}+im_{K^{\ast}}\sqrt{4m_{\tau}^{2}-m_{K^{\ast}}^{2}}\right)\log\left(2\frac{m_{K^{\ast}}^{2}-m_{V^{\prime}}^{2}}{m_{K^{\ast}}^{2}}\left(1-2\frac{4m_{V^{\prime}}^{2}m_{\tau}^{2}-m_{V^{\prime}}^{4}}{4m_{K^{\ast}}^{2}m_{\tau}^{2}-m_{V^{\prime}}^{4}}+i\frac{\sqrt{4m_{\tau}^{2}-m_{K^{\ast}}^{2}}}{m_{K^{\ast}}}\right)\right)$
$\displaystyle+\left(m_{V^{\prime}}^{2}-m_{K^{\ast}}^{2}-im_{K^{\ast}}\sqrt{4m_{\tau}^{2}-m_{K^{\ast}}^{2}}\right)\log\left(2\frac{m_{K^{\ast}}^{2}-m_{V^{\prime}}^{2}}{m_{K^{\ast}}^{2}}\left(1-2\frac{4m_{V^{\prime}}^{2}m_{\tau}^{2}-m_{V^{\prime}}^{4}}{4m_{K^{\ast}}^{2}m_{\tau}^{2}-m_{V^{\prime}}^{4}}-i\frac{\sqrt{4m_{\tau}^{2}-m_{K^{\ast}}^{2}}}{m_{K^{\ast}}}\right)\right)$
$\displaystyle-\left(m_{V^{\prime}}^{2}-m_{K^{\ast}}^{2}+im_{K^{\ast}}\sqrt{4m_{\tau}^{2}-m_{K^{\ast}}^{2}}\right)\log\left(2\frac{m_{K^{\ast}}^{2}-m_{V^{\prime}}^{2}}{m_{K^{\ast}}^{2}}\left(-1+2\frac{m_{\tau}^{2}}{m_{K^{\ast}}^{2}}\frac{4m_{K^{\ast}}^{2}m_{\tau}^{2}-m_{V^{\prime}}^{4}}{4m_{V^{\prime}}^{2}m_{\tau}^{2}-m_{V^{\prime}}^{4}}+i\frac{\sqrt{4m_{\tau}^{2}-m_{K^{\ast}}^{2}}}{m_{K^{\ast}}}\right)\right)$
$\displaystyle\left.-\left(m_{V^{\prime}}^{2}-m_{K^{\ast}}^{2}-im_{K^{\ast}}\sqrt{4m_{\tau}^{2}-m_{K^{\ast}}^{2}}\right)\log\left(2\frac{m_{K^{\ast}}^{2}-m_{V^{\prime}}^{2}}{m_{K^{\ast}}^{2}}\left(-1+2\frac{m_{\tau}^{2}}{m_{K^{\ast}}^{2}}\frac{4m_{K^{\ast}}^{2}m_{\tau}^{2}-m_{V^{\prime}}^{4}}{4m_{V^{\prime}}^{2}m_{\tau}^{2}-m_{V^{\prime}}^{4}}-i\frac{\sqrt{4m_{\tau}^{2}-m_{K^{\ast}}^{2}}}{m_{K^{\ast}}}\right)\right)\right].$
(117)

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