††institutetext:  Theoretical Division, Los Alamos National Laboratory, Los
Alamos, NM 87545, USA

Version of

LA-UR-24-20122

# Electric Dipole Moments in 5+3 Flavor Weak Effective Theory

Jacky Kumar  Emanuele Mereghetti<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

A fully generic treatment of electric dipole moments (EDMs) is presented in
the CP-violating and flavor-conserving weak effective field theory (WET) with
five flavors of quarks and three flavors of leptons. We systematically analyze
leading contributions to EDMs originating from QCD and QED renormalization
group running between the electroweak scale and low energy scales of about 2
GeV. We include the full one-loop anomalous dimension and a subset of two-loop
corrections, as well as threshold corrections at the bottom, charm and $\tau$
masses. This allows us to derive master formulae in the space of generic WET
for the neutron and proton EDMs, for EDMs of diamagnetic atoms, and for the
precession frequencies constrained in molecular EDM experiments, from which
bounds on the electron EDM are extracted. In particular, our master formulae
capture the contributions of WET CP-violating operators with heavy quark and
lepton flavors. As an application, we study EDM constraints on the Yukawa
couplings of the Higgs boson, in both the linear and non-linear realizations
of electroweak symmetry breaking.

##  1 Introduction

The origin of the baryon asymmetry in the Universe (BAU) is one of the most
pressing open problems in particle physics. While the Standard Model (SM)
satisfies the three Sakharov conditions Sakharov:1967dj for the dynamical
generation of an asymmetry, for a Higgs mass of $m_{h}=125$ GeV the
electroweak (EW) phase transition does not provide sufficient deviation from
thermal equilibrium to explain the observed BAU. Even if the SM were able to
induce a first-order phase transition, the violation of the symmetry under a
charge conjugation and parity (CP) transformation induced by the phase of the
Cabibbo-Kobayashi-Maskawa (CKM) matrix would be far too weak, underpredicting
the BAU by several orders of magnitude Gavela:1994ds ; Gavela:1993ts ;
Gavela:1994dt ; Huet:1994jb . The generation of the BAU thus quite generally
requires physics beyond the Standard Model (BSM) to have new sources of CP
violation. Permanent electric dipole moments (EDMs) of leptons, nucleons,
atoms, and molecules are extremely sensitive probes of flavor-diagonal CP-
violation, as they can be constrained very accurately and receive negligible
contributions from the phase of the CKM matrix. For example, the current bound
on the neutron EDM is $|d_{n}|<1.8\cdot 10^{-26}$ $e$ cm, naively probing BSM
physics at scales far above 10 TeV. On the other hand, this bound is five to
six orders of magnitude away from the contribution of the SM
Khriplovich:1985jr ; Czarnecki:1997bu ; Mannel:2012qk ; Seng:2014lea ;
Pospelov:2005pr . An observation of a neutron EDM in the next generation of
experiments would then be a clear indication of CP violation beyond the CKM
paradigm, and observations in multiple systems will be able to rule out a QCD
$\bar{\theta}$ term and demonstrate the existence of BSM CP violation. For
this reason, a rich experimental program is in place, with ongoing and planned
searches in different systems, from the muon and tau leptons to the neutron,
to atomic and molecular systems, aiming at improving current bounds by at
least one or two orders of magnitude Chupp:2017rkp ; Alarcon:2022ero .

Because of the strength of existing and upcoming constraints, it is important
to address the question of what existing EDM experiments teach us about CP
violation in generic BSM scenarios. In addition, assuming multiple
observations of non-zero EDMs in the next generation of experiments, we will
be faced with the “inverse problem” of identifying the fundamental mechanism
of CP violation from a set of low-energy measurements. Effective Field
Theories (EFT) provide the theoretical tools to answer these questions with
minimal reliance on specific models of BSM physics Buras:2020xsm . First of
all, EFTs such as the Weak EFT (WET) Jenkins:2017jig ; Jenkins:2017dyc allow
one to organize all possible flavor-diagonal ($\Delta F=0$) CP-violating
operators of a given dimension. Secondly, in WET it is possible to
systematically consider loop effects, such as renormalization group (RG)
evolution and matching corrections induced by heavy quark thresholds, and to
connect low-scale observables with physics at the electroweak scale. Here one
can match WET onto other EFTs, such as the Standard Model EFT (SMEFT)
Buchmuller:1985jz ; Grzadkowski:2010es or the Higgs EFT (HEFT)
Buchalla:2013rka ; Buchalla:2017jlu , and study collider phenomenology. This
tower of EFTs provides a general and model-independent link between EDMs,
other precision flavor observables, and collider experiments, which will be
necessary to solve the aforementioned inverse problem.

The advantages of parameterizing EDMs in terms of a minimal set of low-energy
operators have long been recognized Pospelov:2005pr ; Khriplovich:1997ga , and
there is a vast literature on EDM calculations in specific BSM models (see
Pospelov:2005pr ; Engel:2013lsa ; Chupp:2017rkp and references therein). In
EFTs, EDMs in the SMEFT at tree level and the matching onto hadronic EFTs were
considered in Ref. deVries:2012ab . Beyond the tree level, the contributions
to EDMs induced by classes of SMEFT operators, including in the Yukawa
Brod:2013cka ; Chien:2015xha ; Buras:2020xsm ; Brod:2022bww ; Bahl:2022yrs ;
Fuchs:2020uoc ; Brod:2023wsh , top Cirigliano:2016nyn ; Fuyuto:2017xup , heavy
flavor Ema:2022pmo ; Haisch:2021hcg , Higgs-gauge Dekens:2013zca ;
Cirigliano:2019vfc ; Haisch:2019xyi , right-handed charged current
Alioli:2017ces and flavor Endo:2019mxw ; Fajfer:2023gie sectors, have been
studied in a mostly piecemeal approach. A more systematic analysis of the
1-loop contributions to EDMs in SMEFT was carried out in Ref. Kley:2021yhn ,
but mostly with a focus on the electron and nucleon EDMs. This paper is a
first step towards a complete study of $\Delta F=0$ CP violation in EFT
extensions of the SM, and its correlations with other sectors of the theory
(such as the Higgs and flavor sectors). We focus here on CP-violating (CPV)
$\Delta F=0$ WET operators, with $n_{f}=5$ quark flavors and $n_{\ell}=3$
lepton flavors. The main new results of this work are:

  * •

We derive master formulae for the neutron, proton, and diamagnetic EDMs, and
for the precession frequencies in molecular EDM experiments, in terms of the
full set of $\Delta F=0$ WET operators at the electroweak scale. The master
formulae for $\omega_{\rm HfF}$, and $\omega_{\rm ThO}$, $\omega_{\rm YbF}$
are given in Tables 7 and 15, respectively. Contributions to $d_{\rm Hg}$ are
listed in Tables 8 and 9, while the neutron and proton EDMs, $d_{n}$ and
$d_{p}$, are given in Tables 10 and 16, respectively.

  * •

The expressions use up-to-date hadronic and nuclear matrix elements, which are
however still affected by large and often uncontrolled theoretical
uncertainties. For hadronic matrix elements, including the neutron and proton
EDMs and CPV pion-nucleon couplings, we rely as much as possible on
information from Lattice QCD, using either direct EDM calculations or indirect
calculations (such as Lattice QCD calculations of mesonic matrix elements
which contribute to pion-nucleon couplings). The implementation of the master
formulae in the package flavio Straub:2018kue allows for a quick
implementation of new matrix elements, as soon as they become available.

  * •

For WET operators involving only the $u$, $d$ and $s$ quark flavors
($n_{f}=3$) and the electron ($n_{\ell}=1$), our formulae have leading
logarithmic accuracy. We present master formulae in the Jenkins, Manohar, and
Stoffer (JMS) basis for WET Jenkins:2017jig ; Jenkins:2017dyc , which can be
easily connected to the SMEFT.

  * •

We consider the contributions of operators with heavy quark ($c$ and $b$) and
lepton ($\mu$ and $\tau$) flavors, where renormalization group mixing and
matching at the heavy fermion thresholds are crucial. In the case of scalar
four-quark operators, all operators with two heavy and two light quarks
contribute to EDMs at leading logarithmic accuracy. Scalar operators with four
heavy quarks do not contribute at the leading log. In this case, we identify
next-to-leading logarithmic contributions, arising from 1-loop running and
1-loop matching at the heavy flavor threshold. Since the renormalization group
mixing is mostly driven by QCD, $d_{n}$ and $d_{\rm Hg}$ provide strong
constraints even on scalar operators with only heavy quarks.

  * •

Four-quark vector operators with two heavy and two light quarks also
contribute at the next-to-leading log. In this case, the dominant
contributions arise from 2-loop running and 1-loop matching. Four-quark vector
operators with two $b$ and two $c$ quarks contribute at the next-to-next-to-
leading log. EDM contributions from scalar and vector operators with the same
flavors are comparable.

  * •

Leptonic operators with two heavy leptons and two electrons contribute to the
electron EDM at the leading log and next-to-leading log for scalar and vector
operators, respectively. We also find that all semileptonic operators with two
muons or $\tau$ leptons contribute to EDMs of diamagnetic atoms at the order
we are working. Since the mixing is driven by QED and proportional to the
lepton masses, however, some of the constraints are weak, especially for
muonic operators.

As we illustrate in Section 6, the WET master formulae can be easily adapted
to EFTs at the electroweak scale, by calculating the matching coefficients in
SMEFT, HEFT, or in any given BSM model.

The paper is organized as follows. In Section 2 we introduce the WET basis of
$\Delta F=0$ CPV operators. In Section 3 we provide expressions for the
neutron, atomic, and molecular EDMs in terms of WET operators at low energy.
In Section 4 we discuss the renormalization group evolution and threshold
effects and in Section 5 we provide master formulae for EDMs in terms of WET
coefficients at the electroweak scale. In Section 6 we discuss one example of
application of the master formulae, EDM constraints on non-standard Higgs
couplings. We conclude in Section 7.

##  2 Flavor-conserving CP-violation in WET

There are numerous sources of CP-violation in WET at the dim-4, 5, and 6
levels. Since our main goal is to study EDMs, we will restrict ourselves to
flavor-conserving WET operators. We can identify two kinds of contributions.
Operators constructed out of light fermion flavors, like $u,d$, or $s$ quarks
or electrons, generate EDMs at the tree level. As we will see, they will
receive the strongest constraints. Operators that contain heavier quark and
lepton generations can also contribute via loop effects originating either
from the renormalization group running or from threshold effects. Below we
consider the full list of WET operators with five quark and three lepton
flavors that can contribute to EDMs. Very schematically, WET Wilson
coefficients at low energy are obtained by solving renormalization group
equations (RGEs), which give rise to expressions of the form

$C(\mu)=\left[\sum_{n=0}\left(\frac{g^{2}}{(4\pi)^{2}}\right)^{n}\log^{n}\frac{\mu}{\mu_{ew}}+\sum_{n=1}\left(\frac{g^{2}}{(4\pi)^{2}}\right)^{n}\log^{n-1}\frac{\mu}{\mu_{ew}}+\ldots\right]\times
C(\mu_{ew}),$ (1)

where $g$ here stands for any coupling in WET, and $\mu_{ew}$ is a scale close
to the electroweak. The first and second terms in the bracket on the r.h.s of
Eq. (1) denote the leading logarithmic (LL) and next-to-leading logarithmic
(NLL) series. As we will discuss, the great majority of $\Delta F=0$ CPV
operators feeds into light flavor operators at LL. These contributions are
captured by the solution of the WET 1-loop RGEs, which are known
Buchalla:1995vs ; Aebischer:2015fzz ; Jenkins:2017dyc . Certain classes of
operators, including, for example, vector left-right operators, do not
contribute at LL. In these cases, we identify NLL corrections, arising from
2-loop running and 1-loop matching at the heavy quark threshold. A summary of
the order in which each WET operator contributes is given in Table 1, where
operators contributing at LL, NLL, and next-to-next-to-leading log (NNLL) are
indicated by no asterisk, or single and double asterisks, respectively. We
stress that, while the logarithmic series expansion in Eq. (1) provides useful
guidance to organize WET contributions to EDMs, and we will therefore use it
throughout the paper, the RGEs involve small couplings (e.g. the QED
coupling), for which the suppression from additional loops is not fully offset
by the large logarithms. In these cases, it is possible that higher loop
corrections to matching and running will be more important than those included
here. These can only be captured by pushing the calculations of WET anomalous
dimensions and matching corrections to higher order. For the sake of
generality, we also include the operators that cannot be generated in SMEFT at
leading order, i.e. the ones that violate the hypercharge quantum number.

In the JMS basis complete 1-loop anomalous dimension matrices (ADMs) are known
Jenkins:2017jig ; Jenkins:2017dyc . Moreover, the complete 1-loop matching
conditions of WET onto SMEFT has been recently worked out Dekens:2019ept in
the JMS basis. Given these developments, it is useful to express the EDMs in
terms of the Wilson coefficients (WCs) of the JMS basis so that their
connection to UV physics can be easily established. The translation of the JMS
WCs into several other popular WET bases or vice versa can be numerically
carried out using the wilson Aebischer:2018bkb program, which we also use to
perform the LL component of the RG running. Moreover, throughout this paper,
we will follow the WCxf conventions for WET and SMEFT Aebischer:2017ugx .

###  2.1 Operator Basis for 3+1 Flavors

At dim-$4$ in WET, CP can be violated by a complex quark mass term and by the
QCD $\bar{\theta}$ term. We choose to work in a basis in which the quark mass
matrices are diagonal and real, so that, at dim-4, the only CP-odd
interactions is

$\mathcal{L}_{\theta}=\bar{\theta}\,\frac{g_{s}^{2}}{32\pi^{2}}\widetilde{G}^{A}_{\mu\nu}G^{A\,\mu\nu},$
(2)

where $g_{s}$ is the strong coupling constant, $G^{A}_{\mu\nu}$ is the gluon
field strength and
$\widetilde{G}^{A}_{\mu\nu}=\frac{1}{2}\varepsilon_{\mu\nu\alpha\beta}G^{A\,\alpha\beta}$.
Throughout this paper, we assume that the contribution of the $\bar{\theta}$
term to EDMs is canceled by the Peccei-Quinn (PQ) mechanism Peccei:1977ur ;
Peccei:1977hh .

In a $n_{f}=3+1$ flavor theory, the dipole operators can contribute to EDMs at
the dim-5 level. At dim-6, there are contributions from the Weinberg three-
gluon operator and various four-fermion operators.

####  2.1.1 Dipole Operators

The flavor-conserving QCD and QED dipole operators are given by

$\displaystyle[{O}_{u\gamma}]_{ii}$
$\displaystyle=(\bar{u}_{i}\sigma^{\mu\nu}P_{R}u_{i})F_{\mu\nu}\,,\,\ \,\ \,\
\,\ \,\
[{O}_{d\gamma}]_{ii}=(\bar{d}_{i}\sigma^{\mu\nu}P_{R}d_{i})F_{\mu\nu}\,,$ (3)
$\displaystyle[{O}_{uG}]_{ii}$
$\displaystyle=(\bar{u}_{i}\sigma^{\mu\nu}T^{A}P_{R}u_{i})G_{\mu\nu}^{A}\,,\,\
\,\
[{O}_{dG}]_{ii}=(\bar{d}_{i}\sigma^{\mu\nu}T^{A}P_{R}d_{i})G_{\mu\nu}^{A}\,,$
$\displaystyle[{O}_{e\gamma}]_{ii}$
$\displaystyle=(\bar{e}_{i}\sigma^{\mu\nu}P_{R}e_{i})F_{\mu\nu}.$

Here $ii$ indicates the flavor index and $T^{A}$ are the $SU(3)_{c}$
generators. Down-type type operators having $ii=11,22$, and up-type quarks
with $ii=11$ directly contribute to the neutron and atomic EDMs. Dipole
operators with $b$ and $c$ quarks contribute via loops. In the leptonic
sector, $ii=11$, $22$, and $33$ contribute directly to electron, muon, and
$\tau$ EDMs, respectively. Since the limits on $d_{\mu}$ and $d_{\tau}$ are
much weaker than on the electron EDM, it is important also to consider the
matching and running of heavier leptons into leptonic and semileptonic
operators that contribute to molecular EDMs.

####  2.1.2 Weinberg Operator

The CPV Weinberg operator has the form

$[{O}_{\widetilde{G}}]=f^{ABC}\widetilde{G}_{\mu}^{A\nu}G_{\nu}^{B\rho}G_{\rho}^{C\mu}.$
(4)

Here $f^{ABC}$ are the structure constants of the $SU(3)_{c}$ group. Note that
the WC $\big{[}{C}_{\widetilde{G}}\big{]}$ is real by definition.

####  2.1.3 Four-fermion operators with 3+1 Flavors

In this section, we give a complete list of four-fermion WET operators with
$n_{f}=3+1$ quark and lepton flavors. All of these operators enter directly
into the EDM expressions, hence these are the leading operators. A four-
fermion basis in terms of scalar-pseudoscalar and tensor operators was
presented in Ref. Buhler:2023gsg , where the 1-loop matching to the gradient
flow scheme used in Lattice QCD calculations is also provided. Here, to
facilitate the renormalization group evolution and the matching to EFTs at the
electroweak scale, we adopt the JMS basis. In the first category, we consider
vector operators having $(LL)(RR)$ chiralities of the fermion currents. These
are given by

$\displaystyle\,[O_{uddu}^{V1,LR}]_{ijkl}$
$\displaystyle=(\bar{u}_{i}\gamma^{\mu}P_{L}d_{j})(\bar{d}_{k}\gamma_{\mu}P_{R}u_{l})\,,$
(5) $\displaystyle[O_{uddu}^{V8,LR}]_{ijkl}$
$\displaystyle=(\bar{u}_{i}\gamma^{\mu}P_{L}T^{A}d_{j})(\bar{d}_{k}\gamma_{\mu}P_{R}T^{A}u_{l}),$
$\displaystyle\,[O_{dd}^{V1,LR}]_{ijkl}$
$\displaystyle=(\bar{d}_{i}\gamma^{\mu}P_{L}d_{j})(\bar{d}_{k}\gamma_{\mu}P_{R}d_{l})\,,$
$\displaystyle[O_{dd}^{V8,LR}]_{ijkl}$
$\displaystyle=(\bar{d}_{i}\gamma^{\mu}P_{L}T^{A}d_{j})(\bar{d}_{k}\gamma_{\mu}P_{R}T^{A}d_{l}).$

These lead to six flavor structures that are CP-violating and directly
contribute to EDMs:

$[O_{uddu}^{V1,LR}]_{1111}\,,~{}[O_{uddu}^{V1,LR}]_{1221}\,,~{}[O_{uddu}^{V8,LR}]_{1111}\,,~{}[O_{uddu}^{V8,LR}]_{1221},\,[O_{dd}^{V1,LR}]_{1221}\,,~{}[O_{dd}^{V8,LR}]_{1221}.$
(6)

In the next category, there are various scalar operators. First we have
$(LR)(LR)$ up-type operators

$\displaystyle\,[O_{uu}^{S1,RR}]_{ijkl}$
$\displaystyle=(\bar{u}_{i}P_{R}u_{j})(\bar{u}_{k}P_{R}u_{l})\,,$ (7)
$\displaystyle[O_{uu}^{S8,RR}]_{ijkl}$
$\displaystyle=(\bar{u}_{i}P_{R}T^{A}u_{j})(\bar{u}_{k}P_{R}T^{A}u_{l}).$

The following two operators are CP-violating and enter at the tree-level

$[O_{uu}^{S1,RR}]_{1111}\,,~{}[O_{uu}^{S8,RR}]_{1111}.$ (8)

The down-type operators are given by

$\displaystyle\,[O_{dd}^{S1,RR}]_{ijkl}$
$\displaystyle=(\bar{d}_{i}P_{R}d_{j})(\bar{d}_{k}P_{R}d_{l})\,,$ (9)
$\displaystyle[O_{dd}^{S8,RR}]_{ijkl}$
$\displaystyle=(\bar{d}_{i}P_{R}T^{A}d_{j})(\bar{d}_{k}P_{R}T^{A}d_{l}).$

Here the following eight operators directly contribute

$\displaystyle[O_{dd}^{S1,RR}]_{1111}\,,~{}[O_{dd}^{S8,RR}]_{1111}\,,~{}[O_{dd}^{S1,RR}]_{2222}\,,~{}[O_{dd}^{S8,RR}]_{2222}\,,$
(10)
$\displaystyle[O_{dd}^{S1,RR}]_{1122}\,,~{}[O_{dd}^{S8,RR}]_{1122}\,,~{}[O_{dd}^{S1,RR}]_{1221}\,,~{}[O_{dd}^{S8,RR}]_{1221}.$

Then the operators containing both up and down-quark come in two sub-
categories. The first one is

$\displaystyle\,[O_{ud}^{S1,RR}]_{ijkl}$
$\displaystyle=(\bar{u}_{i}P_{R}u_{j})(\bar{d}_{k}P_{R}d_{l})\,,$ (11)
$\displaystyle[O_{ud}^{S8,RR}]_{ijkl}$
$\displaystyle=(\bar{u}_{i}P_{R}T^{A}u_{j})(\bar{d}_{k}P_{R}T^{A}d_{l}).$

The following four flavor structures can contribute

$[O_{ud}^{S1,RR}]_{1111}\,,~{}[O_{ud}^{S8,RR}]_{1111}\,,~{}[O_{ud}^{S1,RR}]_{1122}\,,~{}[O_{ud}^{S8,RR}]_{1122}.$
(12)

In the second sub-category, we have the operators

$\displaystyle\,[O_{uddu}^{S1,RR}]_{ijkl}$
$\displaystyle=(\bar{u}_{i}P_{R}d_{j})(\bar{d}_{k}P_{R}u_{l})\,,$ (13)
$\displaystyle[O_{uddu}^{S8,RR}]_{ijkl}$
$\displaystyle=(\bar{u}_{i}P_{R}T^{A}d_{j})(\bar{d}_{k}P_{R}T^{A}u_{l}).$

The following four operators contribute

$[O_{uddu}^{S1,RR}]_{1111}\,,~{}[O_{uddu}^{S8,RR}]_{1111}\,,~{}[O_{uddu}^{S1,RR}]_{1221}\,,~{}[O_{uddu}^{S8,RR}]_{1221}.$
(14)

In total, there are 24 four-quark operators with $n_{f}=3$, in agreement with
Ref. Buhler:2023gsg . Semileptonic operators contribute to molecular and
atomic EDMs at the tree level

$\displaystyle[{O}_{eu}^{S,RL}]_{ijkl}$ $\displaystyle=$
$\displaystyle(\bar{e}_{i}P_{R}e_{j})(\bar{u}_{k}P_{L}u_{l})\,,\,\ \,\ \,\ \,\
[{O}_{ed}^{S,RL}]_{ijkl}=(\bar{e}_{i}P_{R}e_{j})(\bar{d}_{k}P_{L}d_{l})\,,$
(15) $\displaystyle[{O}_{eu}^{S,RR}]_{ijkl}$ $\displaystyle=$
$\displaystyle(\bar{e}_{i}P_{R}e_{j})(\bar{u}_{k}P_{R}u_{l})\,,\,\ \,\ \,\ \,\
[{O}_{ed}^{S,RR}]_{ijkl}=(\bar{e}_{i}P_{R}e_{j})(\bar{d}_{k}P_{R}d_{l})\,,$
$\displaystyle[{O}_{eu}^{T,RR}]_{ijkl}$ $\displaystyle=$
$\displaystyle(\bar{e}_{i}\sigma^{\mu\nu}P_{R}e_{j})(\bar{u}_{k}\sigma_{\mu\nu}P_{R}u_{l})\,,\,\
\,\
[{O}_{ed}^{T,RR}]_{ijkl}=(\bar{e}_{i}\sigma^{\mu\nu}P_{R}e_{j})(\bar{d}_{k}\sigma_{\mu\nu}P_{R}d_{l}).$

Four-fermion Operators for EDMs with $n_{q}+n_{\ell}$ flavors
---
$u,d,s,e$ operators in 3+1 flavor WET
$[O_{uddu}^{V1,LR}]_{1111}$ | $[O_{uddu}^{V8,LR}]_{1111}$ | $[O_{uddu}^{V1,LR}]_{1221}$ | $[O_{uddu}^{V8,LR}]_{1221}$ | $[O_{dd}^{V1,LR}]_{1221}$ | $[O_{dd}^{V8,LR}]_{1221}$
$[O_{uu}^{S1,RR}]_{1111}$ | $[O_{uu}^{S8,RR}]_{1111}$ | $[O_{dd}^{S1,RR}]_{1111}$ | $[O_{dd}^{S8,RR}]_{1111}$ | $[O_{dd}^{S1,RR}]_{2222}$ | $[O_{dd}^{S8,RR}]_{2222}$
$[O_{dd}^{S1,RR}]_{1122}$ | $[O_{dd}^{S8,RR}]_{1122}$ | $[O_{dd}^{S1,RR}]_{1221}$ | $[O_{dd}^{S8,RR}]_{1221}$ | $[O_{ud}^{S1,RR}]_{1111}$ | $[O_{ud}^{S8,RR}]_{1111}$
$[O_{ud}^{S1,RR}]_{1122}$ | $[O_{ud}^{S8,RR}]_{1122}$ | $[O_{uddu}^{S1,RR}]_{1111}$ | $[O_{uddu}^{S8,RR}]_{1111}$ | $[O_{uddu}^{S1,RR}]_{1221}$ | $[O_{uddu}^{S8,RR}]_{1221}$
$[O_{eu}^{S,RL}]_{1111}$ | $[O_{eu}^{S,RR}]_{1111}$ | $[O_{ed}^{S,RL}]_{1111}$ | $[O_{ed}^{S,RR}]_{1111}$ | $[O_{ed}^{S,RL}]_{1122}$ | $[O_{ed}^{S,RR}]_{1122}$
$[O_{eu}^{T,RR}]_{1111}$ | $[O_{ed}^{T,RR}]_{1111}$ | $[O_{ed}^{T,RR}]_{1122}$ | $[O_{ee}^{S,RR}]_{1111}^{*}$ |  |
$c$ operators $\subset$ 4+1 flavor WET
$[O_{uu}^{S1,RR}]_{2222}^{*}$ | $[O_{uu}^{S8,RR}]_{2222}^{*}$ | $[O_{uu}^{S1,RR}]_{1122}$ | $[O_{uu}^{S8,RR}]_{1122}$ | $[O_{ud}^{S1,RR}]_{2211}$ | $[O_{ud}^{S8,RR}]_{2211}$
$[O_{ud}^{S1,RR}]_{2222}$ | $[O_{ud}^{S8,RR}]_{2222}$ | $[O_{uddu}^{S1,RR}]_{2112}$ | $[O_{uddu}^{S8,RR}]_{2112}$ | $[O_{uddu}^{S1,RR}]_{2222}$ | $[O_{uddu}^{S8,RR}]_{2222}$
$[O_{uu}^{S1,RR}]_{1221}$ | $[O_{uu}^{S8,RR}]_{1221}$ | $[O_{uddu}^{V1,LR}]_{2112}^{*}$ | $[O_{uddu}^{V8,LR}]_{2112}^{*}$ | $[O_{uddu}^{V1,LR}]_{2222}^{*}$ | $[O_{uddu}^{V8,LR}]_{2222}^{*}$
$[O_{uu}^{V1,LR}]_{1221}^{*}$ | $[O_{uu}^{V8,LR}]_{1221}^{*}$ |  |  |  |
$[O_{eu}^{S,RR}]_{1122}$ | $[O_{eu}^{T,RR}]_{1122}$ |  |  |  |
$b,\mu,\tau$ operators $\subset$ 5+3 flavor WET
$[O_{dd}^{S1,RR}]_{3333}^{*}$ | $[O_{dd}^{S8,RR}]_{3333}^{*}$ | $[O_{dd}^{S1,RR}]_{1133}$ | $[O_{dd}^{S8,RR}]_{1133}$ | $[O_{dd}^{S1,RR}]_{2233}$ | $[O_{dd}^{S8,RR}]_{2233}$
$[O_{dd}^{S1,RR}]_{1331}$ | $[O_{dd}^{S8,RR}]_{1331}$ | $[O_{dd}^{S1,RR}]_{2332}$ | $[O_{dd}^{S8,RR}]_{2332}$ | $[O_{ud}^{S1,RR}]_{1133}$ | $[O_{ud}^{S8,RR}]_{1133}$
$[O_{uddu}^{S1,RR}]_{1331}$ | $[O_{uddu}^{S8,RR}]_{1331}$ | $[O_{ud}^{S1,RR}]_{2233}^{*}$ | $[O_{ud}^{S8,RR}]_{2233}^{*}$ | $[O_{uddu}^{S1,RR}]_{2332}^{*}$ | $[O_{uddu}^{S8,RR}]_{2332}^{*}$
$[O_{uddu}^{V1,LR}]_{1331}^{*}$ | $[O_{uddu}^{V8,LR}]_{1331}^{*}$ | $[O_{dd}^{V1,LR}]_{1331}^{*}$ | $[O_{dd}^{V8,LR}]_{1331}^{*}$ | $[O_{dd}^{V1,LR}]_{2332}^{*}$ | $[O_{dd}^{V8,LR}]_{2332}^{*}$
$[O_{uddu}^{V1,LR}]_{2332}^{**}$ | $[O_{uddu}^{V8,LR}]_{2332}^{**}$ |  |  |  |
$[O^{T,RR}_{ed}]_{1133}$ | $[O^{T,RR}_{ed}]_{2211}$ | $[O_{ed}^{T,RR}]_{2222}$ | $[O_{ed}^{T,RR}]_{2233}^{*}$ | $[O_{ed}^{T,RR}]_{3311}$ | $[O_{ed}^{T,RR}]_{3322}$
$[O^{T,RR}_{ed}]_{3333}^{*}$ | $[O^{T,RR}_{eu}]_{2211}$ | $[O_{eu}^{T,RR}]_{2222}^{*}$ | $[O_{eu}^{T,RR}]_{3311}$ | $[O_{eu}^{T,RR}]_{3322}^{*}$ |
$[O^{S,RR}_{ed}]_{1133}$ | $[O^{S,RR}_{ed}]_{2211}$ | $[O_{ed}^{S,RR}]_{2222}$ | $[O_{ed}^{S,RR}]_{2233}^{*}$ | $[O_{ed}^{S,RR}]_{3311}$ | $[O_{ed}^{S,RR}]_{3322}$
$[O^{S,RR}_{ed}]_{3333}^{*}$ | $[O^{S,RR}_{eu}]_{2211}$ | $[O_{eu}^{S,RR}]_{2222}^{*}$ | $[O_{eu}^{S,RR}]_{3311}$ | $[O_{eu}^{S,RR}]_{3322}^{*}$ |
$[O_{ee}^{V,LR}]_{1221}^{*}$ | $[O_{ee}^{V,LR}]_{1331}^{*}$ | $[O_{ee}^{S,RR}]_{1221}$ | $[O_{ee}^{S,RR}]_{1331}$ | $[O_{ee}^{S,RR}]_{1122}$ | $[O_{ee}^{S,RR}]_{1133}$
Table 1: The complete list of four-fermion flavor conserving CP-violating
operators relevant for EDMs in $n_{q}+n_{\ell}$ =5+3 Flavor WET below the EW
scale. Here the LL, NLL, and NNLL operators are marked by no asterisk, single
asterisk, and double asterisk signs, respectively.

The relevant flavor structures for semi-leptonic operators are

$\displaystyle[O_{eu}^{S,RL}]_{1111}\,,~{}[O_{eu}^{S,RR}]_{1111}\,,~{}[O_{ed}^{S,RL}]_{1111}\,,~{}[O_{ed}^{S,RR}]_{1111}\,,$
(16)
$\displaystyle[O_{ed}^{S,RL}]_{1122}\,,~{}[O_{ed}^{S,RR}]_{1122}\,,[O_{eu}^{T,RR}]_{1111}\,,~{}[O_{ed}^{T,RR}]_{1111}\,,$
$\displaystyle[O_{ed}^{T,RR}]_{1122}.$

Finally, we note that there are no leptonic operators that can contribute at
the tree-level to any of the EDMs. As we will see, the purely leptonic
operator can contribute at one loop. In particular, the scalar operator

$[{O}_{ee}^{S,RR}]_{ijkl}=(\bar{e}_{i}P_{R}e_{j})(\bar{e}_{k}P_{R}e_{l})$ (17)

can mix onto the electron EDM. In the $3+1$ theory, the relevant flavor
structure is

$[O_{ee}^{S,RR}]_{1111}.$ (18)

###  2.2 Four-fermion Operators with 5+3 Flavors

In this section, we list the leading heavy flavor (5+3) four-fermion operators
that contribute up to the 2-loop level via matching and running effects. The
only new Lorentz structure appearing due to loop effects is the pure leptonic
operator

$[O_{ee}^{V,LR}]_{ijkl}=(\bar{e}_{i}\gamma^{\mu}P_{L}e_{j})(\bar{e}_{k}\gamma_{\mu}P_{R}e_{l})\,.$
(19)

This operator can break CP if $i\neq j$, so that the CP-odd coefficients are

$[O_{ee}^{V,LR}]_{1221}\,,~{}[O_{33}^{V,LR}]_{1331}.$ (20)

These two operators contribute to the electron EDM at the 2-loop QED level.
Otherwise, only new flavor structures are added to the EDM basis for the 5+3
theory. The complete list of relevant WET operators containing 3+1, 4+1, and
5+3 quark and lepton flavors ($n_{q}+n_{\ell}$) is given in Table 1. In this
table, the operators are categorized based on their contributions to the EDMs
at the LL, NLL, and NNLL order in RG improved perturbation theory. The order
is indicated by the asterisk signs.

##  3 Electric dipole moments

Electric dipole moments are an extremely sensitive probe of $\Delta F=0$ CP-
violation at low energy. The strongest constraints currently come from
measurements in ThO ACME:2018yjb , HfF Roussy:2022cmp , and YbF Kara:2012ay ,
which can be interpreted as bounds on the electron EDM, and from bounds on the
neutron EDM Abel:2020pzs and on the EDM of 199Hg Griffith:2009zz . Experiments
looking for the EDM of 129Xe Sachdeva:2019rkt and 225Ra Parker:2015yka are at
the moment weaker, but they provide constraints on independent combinations of
hadronic parameter and Wilson coefficients Chupp:2014gka . A summary of
current EDM bounds is shown in Table 2. In the future, the limits on the
electron EDM are expected to improve by at least a factor of ten, while new
molecular EDM experiments sensitive to nuclear Schiff moments could lead to a
large improvement in the constraints on hadronic CPV interactions Arrowsmith-
Kron:2023hcr . Several experimental collaborations, including n2EDM at the
Paul Scherrer Institute n2EDM:2021yah , PanEDM Wurm:2019yfj at the Institute
Laue Langevin, TUCAN EDM at Triumf Martin:2020lbx , and LANL EDM at Los Alamos
National Laboratory Ito:2017ywc , aim at improving the bound on the neutron
EDM by one order of magnitude. Finally, EDMs of light ions such as the proton,
deuteron, or 3He could be measured in storage ring experiments
Alexander:2022rmq . To estimate future EDM sensitivities, we will use the
projection on the second row of Table 2. In the following subsections, we will
express the EDMs of charged leptons, the neutron and proton EDMs, atomic EDMs,
and the CPV precession frequencies measured in molecular experiments in terms
of WET operators at low energy, while in Section 4 we will show how these
operators are generated from WET operators at the EW scale, via running and
matching effects.

| $d_{e}$ | $d_{n}$ | $d_{\rm Hg}$ | $d_{p}$ | $d_{\rm Xe}$ | $d_{\rm Ra}$
---|---|---|---|---|---|---
current limit | $4.0\cdot 10^{-30}$ | $1.8\cdot 10^{-26}$ | $6.2\cdot 10^{-30}$ | x |  $1.4\cdot 10^{-27}$ , | $4.2\cdot 10^{-22}$
expected limit | $4.0\cdot 10^{-29}$ | ${1.0\cdot 10^{-28}}$ | $6.2\cdot 10^{-30}$ | $1.0\cdot 10^{-29}$ | $1.4\cdot 10^{-28}$ | $1.0\cdot 10^{-27}$
| $\omega_{\rm ThO}$ | $\omega_{\rm HfF}$ | $\omega_{\rm YbF}$ |  |  |
current limit | 1.3 mrad/s |  $0.17$ mrad/s | 23.5 mrad/s |  |  |

Table 2: Current limits on the electron Roussy:2022cmp , neutron Abel:2020pzs
, and mercury Griffith:2009zz ; Graner:2016ses EDMs in units of $e$ cm ($90\%$
confidence level). We also show an indication of their prospective limits
Kumar:2013qya ; Chupp:2014gka ; Alarcon:2022ero as well as those of the
proton, xenon Sachdeva:2019rkt , and radium Parker:2015yka EDMs, which could
provide interesting constraints in the future. The electron EDM bound is
extracted from CPV precession frequencies in molecular EDM experiments and
thus depends on the assumptions that no other CPV operator contributes. In our
analysis, we will use directly the constraints on the precession frequencies
in HfF Roussy:2022cmp , ThO ACME:2018yjb , and YbF Kara:2012ay , which are
shown in the bottom part of the table. In the future, the precession
frequencies are expected to improve by one order.

###  3.1 Integrating out heavy flavors

In this paper, we integrate out heavy quarks and leptons at, or close to,
their mass threshold. In particular, we integrate out the $b$ quark at
$\mu_{b}=m^{\overline{\rm MS}}_{b}=4.18$ GeV, and the charm quark at
$\mu_{c}=2$ GeV. We thus do not consider nonperturbative matrix elements of
operators with $b$ and $c$ quarks between nucleons or nucleons and pions, but
take their effects into account in perturbation theory, via renormalization
group mixing or matching at the threshold. For the $b$ quark, with
$\alpha_{s}(m^{\overline{\rm MS}}_{b})$ well in the perturbative regime, this
approach is fully justified. Since the $c$ quark lies at the boundary between
the perturbative and nonperturbative regimes, in this case one could pursue
the alternative approach of not integrating out the $c$, and evaluating charm
matrix elements in Lattice QCD. While most Lattice QCD collaborations use
actions with dynamical charm quarks, calculations of nucleon charm matrix
elements are very preliminary in the case of quark bilinears
Alexandrou:2019brg , and non-existing for more complicated operators, such as
the charm quark chromoelectric dipole moment or four-fermion operators. One
would thus have to rely on models, which come with uncontrolled uncertainties.
As Lattice QCD calculations improve, it will become interesting to compare the
perturbative and nonperturbative treatment of heavy quarks.

###  3.2 Leptonic EDMs

For the electron, muon, and $\tau$ EDMs, there is a direct connection between
the WET dipole operators $C_{e\gamma}$ and the electric dipole moment. In
addition, the semileptonic tensor operators $\left[C^{T,RR}_{eq}\right]$ with
light quark flavors induce charged lepton dipole operator non-perturbatively
Dekens:2018pbu ; Aebischer:2021uvt , with a contribution proportional to the
correlator of the vector and tensor currents $\Pi_{VT}({\bf q})$
Cirigliano:2021img

$\displaystyle
d_{\ell}=-2\textrm{Im}\left[C_{e\gamma}-2i\Pi_{VT}(0)\left(q_{u}\left[C^{T,RR}_{eu}\right]_{1111}+q_{d}\left[C^{T,RR}_{ed}\right]_{1111}+q_{s}\left[C^{T,RR}_{ed}\right]_{1122}\right)\right]_{\ell\ell}$
$\displaystyle=-1.3\cdot 10^{-16}\left[C_{e\gamma}-2.5\cdot
10^{-5}\,{\rm{TeV}}\left(2\left[C^{T,RR}_{eu}\right]_{1111}-\left[C^{T,RR}_{ed}\right]_{1111}-\left[C^{T,RR}_{ed}\right]_{1122}\right)\right]_{\ell\ell}\,{\rm{TeV}}\,e\,{\rm
cm},$ (21)

with $q_{u}=2/3$ and $q_{d}=q_{s}=-1/3$. In Eq. (3.2), and in the rest of this
section, the WET couplings are understood to be evaluated at the scale of
$\mu=2$ GeV. Here we use a resonance model for $\Pi_{VT}$ and write
Knecht:2001xc ; Mateu:2007tr ; Cata:2008zc ; Cirigliano:2021img

$i\Pi_{VT}(0)=\frac{B_{0}F_{\pi}^{2}}{M_{\rho}^{2}},$ (22)

with the quark condensate $B_{0}=2.7$ GeV at the renormalization scale $\mu=2$
GeV, $F_{\pi}=92.2$ MeV and $M_{\rho}=0.770$ GeV. The above estimate is based
on large $N_{C}$ considerations, and on the truncation of the resonance
spectrum to the first state, and it is therefore affected by large theoretical
uncertainties, which were estimated to be at least 50% Knecht:2001xc ;
Mateu:2007tr ; Cata:2008zc ; Cirigliano:2021img .

The limit on the electron EDM, as extracted from HfF and ThO measurements
ACME:2018yjb ; Roussy:2022cmp , is shown in Table 2. Direct limits exist on
the muon EDM, from $g-2$ storage ring experiments Muong-2:2008ebm , and on the
EDM of the tau lepton, from the measurement of spin-momentum correlations in
$\tau$ decays at $e^{+}e^{-}$ machines Belle:2021ybo . The current limits are
Workman:2022ynf

$|d_{\mu}|<1.8\cdot 10^{-19}e\,{\rm cm},\qquad-0.185\cdot 10^{-16}e\,{\rm
cm}<d_{\tau}<0.061\cdot 10^{-16}e\,{\rm cm}.$ (23)

These are much weaker than the electron EDM bound so running effects of $\mu$
and $\tau$ flavor operators onto electron operators can be important.

###  3.3 ThO, YbF, and HfF Frequencies

The best limit on the electron EDM currently comes from experiments with polar
molecules. In this case, the observables are the parity ($P$) and time-
reversal ($T$) odd frequency shifts $\omega$, which receive contributions from
the electron EDM and semileptonic scalar operators. The frequency shifts are
expressed in terms of an isospin invariant and an isospin-breaking electron-
nucleon coupling, captured by the Lagrangian

$\mathcal{L}=-\frac{G_{F}}{\sqrt{2}}\bar{e}i\gamma_{5}e\,\bar{N}\left(C_{S}^{(0)}+\tau_{3}C^{(1)}_{S}\right)N,$
(24)

with $N=(p,n)$ the nucleon doublet and $\tau_{3}$ a Pauli matrix in isospin
space. The nucleon-level scalar couplings can be expressed in terms of WET
four-fermion operators as

$\displaystyle C_{S}^{(0)}=$
$\displaystyle-\frac{v^{2}}{2}\left(\textrm{Im}[C^{\rm
SRL}_{eu}]_{1111}+\textrm{Im}[C^{\rm SRR}_{eu}]_{1111}+\textrm{Im}[C^{\rm
SRL}_{ed}]_{1111}+\textrm{Im}[C^{\rm SRR}_{ed}]_{1111}\right)\frac{\sigma_{\pi
N}}{\bar{m}}$ $\displaystyle-v^{2}\left(\textrm{Im}[C^{\rm
SRL}_{ed}]_{1122}+\textrm{Im}[C^{\rm
SRR}_{ed}]_{1122}\right)\frac{\sigma_{s}}{m_{s}}$ (25) $\displaystyle
C_{S}^{(1)}=$ $\displaystyle-\frac{v^{2}}{2}\left(\textrm{Im}[C^{\rm
SRL}_{eu}]_{1111}+\textrm{Im}[C^{\rm SRR}_{eu}]_{1111}-\textrm{Im}[C^{\rm
SRL}_{ed}]_{1111}-\textrm{Im}[C^{\rm SRR}_{ed}]_{1111}\right)g^{u-d}_{S},$
(26)

where $2\bar{m}=m_{u}+m_{d}$, and $v=246$ GeV. For the nucleon $\sigma_{\pi
N}$ term, we use the extraction from $\pi$-nucleon scattering data
Hoferichter:2015hva

$\sigma_{\pi N}=59.0\pm 3.5\,{\rm MeV}.$ (27)

For the strange $\sigma$ term and the isovector scalar charge, we use the FLAG
lattice averages FlavourLatticeAveragingGroupFLAG:2021npn

$\sigma_{s}=52.9\pm 7.0\,{\rm MeV},\qquad g_{S}^{u-d}=1.02\pm 0.10,$ (28)

where the scalar charge is evaluated at the renormalization scale $\mu=2$ GeV.
With these definitions, we can express the frequency measured in the HfF, ThO,
and YbF experiments as Degenkolb:2024eve ; Chupp:2017rkp ; Dekens:2018bci

$\displaystyle\omega_{\rm HfF}$ $\displaystyle=\left[\left(34.9\pm
1.4\right)\left(\frac{d_{e}}{10^{-27}e\,{\rm
cm}}\right)+(32^{+1}_{-2})\left(\frac{C_{S}}{10^{-7}}\right)\right]\left({\rm
mrad/s}\right)$ (29) $\displaystyle\omega_{\rm ThO}$
$\displaystyle=-\left[\left(121^{+5}_{-39}\right)\left(\frac{d_{e}}{10^{-27}e\,{\rm
cm}}\right)+(182^{+42}_{-27})\left(\frac{C_{S}}{10^{-7}}\right)\right]\left({\rm
mrad/s}\right)$ (30) $\displaystyle\omega_{\rm YbF}$
$\displaystyle=-\left[\left(19.6\pm
1.5\right)\left(\frac{d_{e}}{10^{-27}e\,{\rm cm}}\right)+(17.6\pm
2.0)\left(\frac{C_{S}}{10^{-7}}\right)\right]\left({\rm mrad/s}\right)$ (31)

where

$C_{S}=C_{S}^{(0)}+\frac{Z-N}{Z+N}C_{S}^{(1)}.$ (32)

Here $(Z,A)=(72,106),(90,142),(70,103)$ for HfF, ThO, and YbF, respectively.

###  3.4 Nucleon EDMs

The interpretation of neutron and proton EDMs is complicated by the need to
match quark-level operators onto hadronic couplings. This matching requires
nonperturbative techniques and it has been carried out with different levels
of sophistication for different WET operators. In the case of the quark dipole
operators $[C_{u\gamma}]_{11}$ and $[C_{d\gamma}]_{11,22}$, the hadronic input
is captured by the quark tensor charges, which have been computed on the
lattice Gupta:2018lvp ; Alexandrou:2019brg ;
FlavourLatticeAveragingGroupFLAG:2021npn .

$\displaystyle d_{n}$ $\displaystyle=-1.3\cdot
10^{-16}\left(g_{T}^{d}\textrm{Im}\left[C_{u\gamma}\right]_{11}+g_{T}^{u}\textrm{Im}\left[C_{d\gamma}\right]_{11}+g_{T}^{s}\textrm{Im}\left[C_{d\gamma}\right]_{22}\right)\,{\rm{TeV}}\,e\,{\rm
cm},$ (33) $\displaystyle d_{p}$ $\displaystyle=-1.3\cdot
10^{-16}\left(g_{T}^{u}\textrm{Im}\left[C_{u\gamma}\right]_{11}+g_{T}^{d}\textrm{Im}\left[C_{d\gamma}\right]_{11}+g_{T}^{s}\textrm{Im}\left[C_{d\gamma}\right]_{22}\right)\,{\rm{TeV}}\,e\,{\rm
cm},$ (34)

with FlavourLatticeAveragingGroupFLAG:2021npn

$g_{T}^{u}=0.784\pm 0.030,\quad g_{T}^{d}=-0.204\pm 0.015,\quad
g_{T}^{s}=-0.0027\pm 0.0016$ (35)

at the renormalization scale $\mu=2$ GeV. The $u$ and $d$ tensor charges have
better than 10% uncertainties, while the $s$ tensor charge is harder to
compute and zero within two sigma. The $c$ tensor charge has been considered
in Ref. Alexandrou:2019brg , but it is even more uncertain,

$g^{c}_{T}=-\left(10\pm 19\right)\cdot 10^{-5}.$ (36)

For this reason, will not consider directly charm matrix elements, but
integrate out the charm quark.

Lattice QCD calculations of the nucleon EDM induced by the quark
chromoelectric dipole moment (qCEDM) operators $\left[C_{uG}\right]_{11}$,
$\left[C_{dG}\right]_{11,22}$ are at the moment only preliminary
Bhattacharya:2023qwf ; Abramczyk:2017oxr ; Kim:2021qae . Currently, the best
estimates come from QCD sum rules Pospelov:2000bw ; Pospelov:2005pr ;
Lebedev:2004va ; Fuyuto:2012yf . In Refs. Pospelov:2000bw ; Pospelov:2005pr ;
Lebedev:2004va , the $u$ and $d$ qCEDM contribution to the neutron and proton
EDMs was estimated to be

$\displaystyle d_{n}$ $\displaystyle=-\frac{4.3\cdot
10^{-17}}{g_{s}}\times(1\pm
0.5)\left(\textrm{Im}\left[C_{dG}\right]_{11}+0.5\textrm{Im}\left[C_{uG}\right]_{11}\right)\,{\rm{TeV}}\,e\,{\rm
cm},$ (37) $\displaystyle d_{p}$ $\displaystyle=+\frac{5.2\cdot
10^{-17}}{g_{s}}\times(1\pm
0.5)\left(0.5\textrm{Im}\left[C_{dG}\right]_{11}+\textrm{Im}\left[C_{uG}\right]_{11}\right)\,{\rm{TeV}}\,e\,{\rm
cm},$ (38)

with roughly 50% uncertainty. Here $g_{s}$ is the strong coupling constant. We
will use the estimates in Eq. (37) for our numerics. Ref. Fuyuto:2012yf finds
a larger value and a substantial contribution from the strange qCEDM

$\displaystyle d_{n}$ $\displaystyle=-\frac{9.8\cdot
10^{-17}}{g_{s}}\left(\textrm{Im}\left[C_{dG}\right]_{11}+1.2\textrm{Im}\left[C_{uG}\right]_{11}+0.2\textrm{Im}\left[C_{dG}\right]_{22}\right)\,{\rm{TeV}}\,e\,{\rm
cm}.$ (39)

Estimates based on the dominance of long-distance contributions mediated by
pion-nucleon couplings end up in a similar range Seng:2014pba ;
deVries:2016jox , though, of course, they are affected by uncontrolled
systematic errors. The above matrix elements are given in the assumption that
a PQ mechanism cancels the $\bar{\theta}$ term, including the shift in the
$\bar{\theta}$ term induced by the chromoelectric operators Pospelov:2005pr .
The scale dependence of $g_{s}$ and of the WET coefficients should in
principle be compensated by the scale dependence of the hadronic matrix
elements. In the sum rule calculations, such scale dependence is not
explicitly tracked, and thus, at the moment, the choice of $\mu$ introduces
some arbitrariness. Since the conversion between Lattice QCD schemes and
$\overline{\rm MS}$ is known Bhattacharya:2015rsa ; Mereghetti:2021nkt , this
issue will be overcome once Lattice QCD calculations will become available.
The contribution of the Weinberg three-gluon operator $C_{\tilde{G}}$ has been
estimated in Refs. Demir:2002gg ; Haisch:2019bml . The two evaluations are in
good agreement, and here we use the results of Ref. Haisch:2019bml

$\displaystyle d_{n}$ $\displaystyle=1.5\cdot 10^{-21}(1\pm
0.5)\left[C_{\tilde{G}}\right]\,{\rm{TeV}}^{2}\,e\,{\rm cm},$ (40)
$\displaystyle d_{p}$ $\displaystyle=-2.1\cdot 10^{-21}(1\pm
0.5)\left[C_{\tilde{G}}\right]\,{\rm{TeV}}^{2}\,e\,{\rm cm}.$ (41)

Also in this case, Lattice QCD calculations are preliminary
Bhattacharya:2022whc , even though progress has been achieved on the matching
between Lattice QCD and $\overline{\rm MS}$ schemes Cirigliano:2020msr ;
Crosas:2023anw . The estimates in Eq. (37), (39) and (40) clearly show that
hadronic matrix elements are affected by large theoretical uncertainties,
which need to be considered in any realistic analysis of EDM bounds.

The situation for four-fermion operators is even less well-developed. The
matching between gradient flow and $\overline{\rm MS}$ renormalization at one
loop has been derived in Ref. Buhler:2023gsg . Robust calculations of the
nucleon EDM currently do not exist. Here we will use the expression derived at
next-to-next-to-leading order (N2LO) in two-flavor chiral perturbation theory
Seng:2014pba

$\displaystyle d_{n}$
$\displaystyle=-\frac{eg_{A}}{8\pi^{2}F_{\pi}^{2}}\left[\left(\bar{g}_{0}-\frac{\bar{g}_{2}}{3}\right)\left(\ln\frac{m_{\pi}^{2}}{m_{N}^{2}}-\frac{\pi}{2}\frac{m_{\pi}}{m_{N}}\right)+\frac{\bar{g}_{1}}{4}(\kappa_{1}-\kappa_{0})\frac{m_{\pi}^{2}}{m_{N}^{2}}\ln\frac{m_{\pi}^{2}}{m_{N}^{2}}\right]+e\bar{d}_{n}$
(42) $\displaystyle d_{p}$
$\displaystyle=\frac{eg_{A}}{8\pi^{2}F_{\pi}^{2}}\left[\left(\bar{g}_{0}-\frac{\bar{g}_{2}}{3}\right)\left(\ln\frac{m_{\pi}^{2}}{m_{N}^{2}}-2\pi\frac{m_{\pi}}{m_{N}}\right)-\frac{\bar{g}_{1}}{4}\left(\frac{2\pi
m_{\pi}}{m_{N}}+\left(\frac{5}{2}+\kappa_{1}+\kappa_{0}\right)\frac{m_{\pi}^{2}}{m_{N}^{2}}\ln\frac{m_{\pi}^{2}}{m_{N}^{2}}\right)\right]$
$\displaystyle+e\bar{d}_{p}.$ (43)

with $g_{A}=1.27$. $\kappa_{1,0}$ are the nucleon isovector and isoscalar
anomalous magnetic moments, $\kappa_{1}=3.7$, $\kappa_{0}=-0.12$.
$\bar{g}_{0}$, $\bar{g}_{1}$, $\bar{d}_{n}$, and $\bar{d}_{p}$ are $PT$-odd
low-energy couplings in the nucleon EFT, which implicitly depend on WET Wilson
coefficients. In the following, we will provide a recipe to estimate
$\bar{g}_{0,1}$ in the case of four-fermion operators.

####  3.4.1 Pion nucleon couplings

$P$ and $T$-odd pion-nucleon ($\pi N$) couplings play an important role for
both nucleon and nuclear EDMs. In the case of the nucleon EDM, $\pi N$
couplings induce logarithmically-enhanced contributions Crewther:1979pi , as
can be seen in Eqs. (42) and (43). Since they are accompanied by unknown low-
energy constants, $\pi N$ couplings cannot completely determine the leading
contributions to the nucleon EDM, and thus expressions as Eq. (42) will be
eventually superseded by Lattice QCD calculations. Knowing the strength of
$\bar{g}_{0,1}$, however, can provide guidance for the chiral and momentum
extrapolation Dragos:2019oxn and for the removal of excited state
contamination Bhattacharya:2021lol in Lattice QCD calculations. Depending on
the chiral properties of the WET operators, $\pi N$ couplings furthermore
provide one of the leading contributions to the $PT$-odd potential
deVries:2012ab ; Maekawa:2011vs ; deVries:2020iea , which then feeds into the
calculation of Schiff moments and thus atomic EDMs Haxton:1983dq ;
Engel:1999np .

We focus here on the most important $\pi N$ couplings and refer for a more
detailed discussion to Ref. deVries:2020iea . We define the $PT$-odd $\pi N$
Lagrangian as

$\mathcal{L}_{\pi
N}=\bar{N}\left[\frac{\bar{g}_{0}}{F_{\pi}}\bm{\tau}\cdot\bm{\pi}+\frac{\bar{g}_{1}}{F_{\pi}}\pi_{0}+\frac{\bar{g}_{2}}{F_{\pi}}\left(\pi_{0}\tau_{3}-\frac{1}{3}\bm{\tau}\cdot\bm{\pi}\right)\right]N,$
(44)

where $\bm{\tau}$ are Pauli matrices in isospin space. The couplings in Eq.
(44) break chiral symmetry and are thus mostly important for chiral-symmetry-
breaking WET operators. These usually come in chiral multiplets, with the CP-
even components inducing corrections to the baryon spectrum while the CP-odd
components contribute to $\pi N$ couplings. At leading order in chiral
perturbation theory, chiral symmetry then provides a relation between
corrections to the baryon spectrum and $\pi N$ couplings deVries:2012ab ,
which survives higher order corrections deVries:2015una ; deVries:2016jox . It
can be shown that $\bar{g}_{2}$ only arises at higher order in isospin
breaking deVries:2012ab . The couplings $\bar{g}_{0}$ and $\bar{g}_{1}$
receive a “direct” contribution, proportional to the shift in the baryon
masses induced by dim-6 operators, and “indirect” contributions, arising from
“vacuum alignment” Baluni:1978rf , i.e. from imposing the condition that
$\pi_{0}$ and $\eta$ mesons do not disappear into the vacuum in the presence
of $PT$-odd interactions. We will thus write

$\displaystyle\bar{g}_{i}=\left.\bar{g}_{i}\right|_{\rm
dir}+\left.\bar{g}_{i}\right|_{\rm ind}.$ (45)

The indirect contributions depend on meson matrix elements, which are more
controlled.

For the qCEDM, the $\pi N$ couplings were extracted using QCD sum rules
Pospelov:2001ys ,

$\displaystyle\frac{\bar{g}_{0}}{F_{\pi}}$ $\displaystyle=-1.97\cdot
10^{-3}\frac{\left(1\pm
2\right)}{g_{s}}\left(\textrm{Im}\left[C_{uG}\right]_{11}+\textrm{Im}\left[C_{dG}\right]_{11}\right)\,{\rm{TeV}},$
(46) $\displaystyle\frac{\bar{g}_{1}}{F_{\pi}}$ $\displaystyle=-1.97\cdot
10^{-3}\frac{\left(4^{+8}_{-2}\right)}{g_{s}}\left(\textrm{Im}\left[C_{uG}\right]_{11}-\textrm{Im}\left[C_{dG}\right]_{11}\right)\,{\rm{TeV}}.$
(47)

Again, we are neglected contributions proportional to $\bar{\theta}_{\rm
ind}$, which are canceled if a PQ mechanism is active. In chiral perturbation
theory, the indirect part of the coupling is given by deVries:2016jox

$\displaystyle\left.\frac{\bar{g}_{0}}{F_{\pi}}\right|_{\rm ind}$
$\displaystyle=\frac{rg^{u-d}_{S}}{F_{\pi}g_{s}}\left(\textrm{Im}\left[C_{uG}\right]_{11}+\textrm{Im}\left[C_{dG}\right]_{11}\right)=-\frac{4.4\cdot
10^{-3}}{g_{s}}\left(\textrm{Im}\left[C_{uG}\right]_{11}+\textrm{Im}\left[C_{dG}\right]_{11}\right),$
(48) $\displaystyle\left.\frac{\bar{g}_{1}}{F_{\pi}}\right|_{\rm ind}$
$\displaystyle=\frac{r\sigma_{\pi
N}}{\bar{m}F_{\pi}g_{s}}\left(\textrm{Im}\left[C_{uG}\right]_{11}-\textrm{Im}\left[C_{dG}\right]_{11}\right)=-\frac{7.4\cdot
10^{-2}}{g_{s}}\left(\textrm{Im}\left[C_{uG}\right]_{11}-\textrm{Im}\left[C_{dG}\right]_{11}\right),$
(49)

where $r$ is the ratio of chromomagnetic and $\bar{q}q$ condensates, for which
we used Pospelov:2001ys

$r=-\frac{1}{2}\frac{\langle 0|\bar{q}g_{s}\sigma Gq|0\rangle}{\langle
0|\bar{q}q|0\rangle}=-\left(0.4\pm 0.1\right){\rm GeV}^{2}.$ (50)

The ratio between $\bar{g}_{1}/\bar{g}_{0}$ is determined by the ratio of
isoscalar and isovector charges, which, for the value of the sigma term we
adopt, is about $17$. The assumption of the dominance of the indirect piece
leads to larger values compared to the QCD sum rules calculations, especially
in the case of $\bar{g}_{1}$. For the qCEDM operators we will just use the sum
rules results, while for the four-fermion operators, where only the indirect
pieces can be estimated, we will capture these effects by assigning a large
uncertainty. Plugging Eqs. (46) and (47) or Eqs. (48) and (49) in the chiral
perturbation theory expression for the nucleon EDM we recover estimates that
agree in magnitude with Eqs. (37)–(39).

We discussed in some detail various estimates for the qCEDM operators to
justify our choices for four-fermion operators, for which complete evaluations
of the couplings, either on the Lattice or from QCD sum rules are not
available. The structure of the couplings is very similar to qCEDM, with a
direct piece that could be extracted from dim-6 corrections to the baryon
spectrum and an indirect piece determined by the ratio of vacuum matrix
elements Cirigliano:2016yhc ; Dekens:2022gha . The operators constructed in
Section 2.1.3 belong to the ${\mathbf{8}\times\mathbf{8}}$,
$\mathbf{6}\times\mathbf{\bar{6}}$ and $\mathbf{3}\times\mathbf{\bar{3}}$
representations of the $SU(3)$ flavor group Dekens:2022gha . The indirect
components of the $\pi N$ couplings are given by

$\displaystyle\left.\bar{g}_{0}\right|_{\rm ind}$
$\displaystyle=-\frac{F_{\pi}^{2}\bar{m}g_{S}^{u-d}}{m_{\pi}^{2}}\left[\bm{\mathcal{A}}_{\mathbf{8}\times\mathbf{8}}\,C^{(1,0)}_{\mathbf{8}\times\mathbf{8}}+\bm{\mathcal{\bar{A}}}_{\mathbf{8}\times\mathbf{8}}\,C^{(8,0)}_{\mathbf{8}\times\mathbf{8}}+\bm{\mathcal{A}}_{\mathbf{6}\times\mathbf{6}}C^{(1,0)}_{\mathbf{6}\times\mathbf{6}}+\bm{\mathcal{\bar{A}}}_{\mathbf{6}\times\mathbf{6}}C^{(8,0)}_{\mathbf{6}\times\mathbf{6}}\right.$
$\displaystyle+\left.\bm{\mathcal{A}}_{\mathbf{3}\times\mathbf{3}}C^{(1,0)}_{\mathbf{3}\times\mathbf{3}}+\bm{\mathcal{\bar{A}}}_{\mathbf{3}\times\mathbf{3}}C^{(8,0)}_{\mathbf{3}\times\mathbf{3}}\right],$
(51) $\displaystyle\left.\bar{g}_{1}\right|_{\rm ind}$
$\displaystyle=-\frac{F_{\pi}^{2}\sigma_{\pi
N}}{m_{\pi}^{2}}\left[\bm{\mathcal{A}}_{\mathbf{8}\times\mathbf{8}}C^{(1,1)}_{\mathbf{8}\times\mathbf{8}}+\bm{\mathcal{\bar{A}}}_{\mathbf{8}\times\mathbf{8}}C^{(8,1)}_{\mathbf{8}\times\mathbf{8}}+\bm{\mathcal{A}}_{\mathbf{6}\times\mathbf{6}}C^{(1,1)}_{\mathbf{6}\times\mathbf{6}}+\bm{\mathcal{\bar{A}}}_{\mathbf{6}\times\mathbf{6}}C^{(8,1)}_{\mathbf{6}\times\mathbf{6}}\right.$
$\displaystyle+\left.\bm{\mathcal{A}}_{\mathbf{3}\times\mathbf{3}}C^{(1,1)}_{\mathbf{3}\times\mathbf{3}}+\bm{\mathcal{\bar{A}}}_{\mathbf{3}\times\mathbf{3}}C^{(8,1)}_{\mathbf{3}\times\mathbf{3}}\right].$
(52)

The operators in the $LL\,RR$ class belong to the
${\mathbf{8}\times\mathbf{8}}$ representation, and we have

$\displaystyle C^{(1,0)}_{\mathbf{8}\times\mathbf{8}}$
$\displaystyle=\frac{1}{2}\left({\rm
Im}\left[C^{V1,\,LR}_{uddu}\right]_{1221}+{\rm
Im}\left[C^{V1,\,LR}_{dd}\right]_{1221}\right)~{},$ (53) $\displaystyle
C^{(8,0)}_{\mathbf{8}\times\mathbf{8}}$ $\displaystyle={\frac{1}{2}}\left({\rm
Im}\left[C^{V8,\,LR}_{uddu}\right]_{1221}+{\rm
Im}\left[C^{V8,\,LR}_{dd}\right]_{1221}\right)~{},$ (54) $\displaystyle
C^{(1,1)}_{\mathbf{8}\times\mathbf{8}}$ $\displaystyle={\rm
Im}\left[C^{V1,\,LR}_{uddu}\right]_{1111}+\frac{1}{2}{\rm
Im}\left[C^{V1,\,LR}_{uddu}\right]_{1221}-\frac{1}{2}{\rm
Im}\left[C^{V1,\,LR}_{dd}\right]_{1221}~{},$ (55) $\displaystyle
C^{(8,1)}_{\mathbf{8}\times\mathbf{8}}$ $\displaystyle={\rm
Im}\left[C^{V8,\,LR}_{uddu}\right]_{1111}+\frac{1}{2}{\rm
Im}\left[C^{V8,\,LR}_{uddu}\right]_{1221}-\frac{1}{2}{\rm
Im}\left[C^{V8,\,LR}_{dd}\right]_{1221}.$ (56)

The first superscript indicates whether the operators are color singlet or
octet, while the second superscript whether they are isospin 0 or isospin 1.
The $LR\,LR$ operators, on the other hand, belong to the
$\mathbf{6}\times\mathbf{\bar{6}}$ and $\mathbf{3}\times\mathbf{\bar{3}}$
representations. The $\mathbf{6}\times\mathbf{\bar{6}}$ coefficients are given
by

$\displaystyle C^{(1,0)}_{\mathbf{6}\times\mathbf{6}}$
$\displaystyle=\textrm{Im}\left[C^{S1,\,RR}_{uu}\right]_{1111}+\textrm{Im}\left[C^{S1,\,RR}_{dd}\right]_{1111}+{\frac{1}{2}}\left(\textrm{Im}\left[C^{S1,\,RR}_{uddu}\right]_{1111}+\textrm{Im}\left[C^{S1,\,RR}_{ud}\right]_{1111}\right)$
$\displaystyle+{\frac{1}{4}}\left(\textrm{Im}\left[C^{S1,\,RR}_{uddu}\right]_{1221}+\textrm{Im}\left[C^{S1,\,RR}_{ud}\right]_{1122}+\textrm{Im}\left[C^{S1,\,RR}_{dd}\right]_{1221}+\textrm{Im}\left[C^{S1,\,RR}_{dd}\right]_{1122}\right),$
(57) $\displaystyle C^{(1,1)}_{\mathbf{6}\times\mathbf{6}}$
$\displaystyle=\textrm{Im}\left[C^{S1,\,RR}_{uu}\right]_{1111}{-}\textrm{Im}\left[C^{S1,\,RR}_{dd}\right]_{1111}$
$\displaystyle+{\frac{1}{4}}\left(\textrm{Im}\left[C^{S1,\,RR}_{uddu}\right]_{1221}+\textrm{Im}\left[C^{S1,\,RR}_{ud}\right]_{1122}-\textrm{Im}\left[C^{S1,\,RR}_{dd}\right]_{1221}-\textrm{Im}\left[C^{S1,\,RR}_{dd}\right]_{1122}\right),$
(58)

with the same expressions for the color octet operators, with $S1\rightarrow
S8$. For the $\mathbf{3}\times\mathbf{\bar{3}}$ coefficients we have

$\displaystyle C^{(1,0)}_{\mathbf{3}\times\mathbf{3}}$
$\displaystyle={\frac{1}{2}}\left(-\textrm{Im}\left[C^{S1,\,RR}_{uddu}\right]_{1111}+\textrm{Im}\left[C^{S1,\,RR}_{ud}\right]_{1111}\right)$
$\displaystyle+{\frac{1}{4}}\left(-\textrm{Im}\left[C^{S1,\,RR}_{uddu}\right]_{1221}+\textrm{Im}\left[C^{S1,\,RR}_{ud}\right]_{1122}-\textrm{Im}\left[C^{S1,\,RR}_{dd}\right]_{1221}+\textrm{Im}\left[C^{S1,\,RR}_{dd}\right]_{1122}\right)$
(59) $\displaystyle C^{(1,1)}_{\mathbf{3}\times\mathbf{3}}$
$\displaystyle={\frac{1}{4}}\left(-\textrm{Im}\left[C^{S1,\,RR}_{uddu}\right]_{1221}+\textrm{Im}\left[C^{S1,\,RR}_{ud}\right]_{1122}+\textrm{Im}\left[C^{S1,\,RR}_{dd}\right]_{1221}-\textrm{Im}\left[C^{S1,\,RR}_{dd}\right]_{1122}\right),$
(60)

and again the color octet coefficients are obtained by $S1\rightarrow S8$.

The ${\mathbf{8}\times\mathbf{8}}$ and $\mathbf{6}\times\mathbf{\bar{6}}$ are
related by $SU(3)$ chiral symmetry to operators that contribute to
$K$-$\bar{K}$ oscillations, $K\rightarrow\pi\pi$ and to short-distance
contributions to neutrinoless double beta decay Cirigliano:2017ymo . Here we
determine the LECs from calculations of neutrinoless double beta decay matrix
elements Nicholson:2018mwc and write

$\displaystyle\bm{\mathcal{A}}_{\mathbf{8}\times\mathbf{8}}=-g_{4}^{\pi\pi},\qquad\bm{\mathcal{\bar{A}}}_{\mathbf{8}\times\mathbf{8}}=-\left[\frac{1}{2}g_{5}^{\pi\pi}-\frac{1}{2N_{c}}g_{4}^{\pi\pi}\right]$
(61)
$\displaystyle\bm{\mathcal{A}}_{\mathbf{6}\times\mathbf{6}}=-g_{2}^{\pi\pi},\qquad\bm{\mathcal{\bar{A}}}_{\mathbf{6}\times\mathbf{6}}=-\left[\frac{1}{2}g_{3}^{\pi\pi}-\frac{1}{2N_{c}}g_{2}^{\pi\pi}\right],$
(62)

with

$\displaystyle g_{2}^{\pi\pi}$ $\displaystyle=2.0(2)\;{\rm GeV}^{2},$
$\displaystyle g_{3}^{\pi\pi}$ $\displaystyle=-0.62(6)\;{\rm GeV}^{2},$ (63)
$\displaystyle g_{4}^{\pi\pi}$ $\displaystyle=-1.9(2)\;{\rm GeV}^{2},$
$\displaystyle g_{5}^{\pi\pi}$ $\displaystyle=-8.0(6)\;{\rm GeV}^{2},$ (64)

in the $\overline{\rm MS}$ scheme, at $\mu=2$ GeV. The LECs for
$\bm{\mathcal{A}}_{3\times 3}$ and $\bm{\mathcal{\bar{A}}}_{3\times 3}$ are at
the moment unknown. We will set them to zero but stress that this is an
uncontrolled assumption.

$\bm{\mathcal{A}}_{3\times 3}=0\;{\rm
GeV}^{2}\qquad\bm{{\mathcal{\bar{A}}}}_{3\times 3}=0\;{\rm GeV}^{2}$ (65)

###  3.5 Diamagnetic Atomic EDMs

We finally consider EDMs of diamagnetic atoms. The strongest bound is
currently on the EDM of 199Hg Graner:2016ses ; Griffith:2009zz , followed by
129Xe Sachdeva:2019rkt . We will also consider 225Ra, which has enhanced
sensitivity to the nuclear Schiff moment Engel:1999np ; Spevak:1996tu and
could provide important constraints in the future Alarcon:2022ero . We can
write the EDM of diamagnetic atoms as a term induced by the nuclear Schiff
moment, plus contributions from semileptonic operators

$d_{{}^{A}X}=d_{\rm
Schiff}+\sum_{N=n,p}\left(\alpha^{N}_{S}C^{(N)}_{S}+\alpha^{N}_{P}C^{(N)}_{P}+\alpha^{N}_{T}C^{(N)}_{T}\right).$
(66)

The Schiff moment component is sensitive to the neutron and proton EDM and to
$PT$-odd nucleon-nucleon interactions. These have been classified in
phenomenological one-meson-exchange models Barton:1961eg ; Towner:1994qe ;
Engel:1999np , and in chiral EFT and pionless EFT Maekawa:2011vs ;
deVries:2012ab ; Song:2012yh ; deVries:2020iea ; Yang:2020ges . For chiral-
symmetry breaking operators, such as the qCEDM and the four-quark operators,
$\bar{g}_{0,1}$ give the leading contribution to the $PT$-odd nucleon-nucleon
potential Maekawa:2011vs ; deVries:2012ab , even though short-range operators
in the ${}^{3}P_{0}$-${}^{1}S_{0}$ channel are required for renormalization
deVries:2020loy . For chiral-invariant operators, such as the Weinberg three-
gluon operator, $\pi N$ couplings and short-range isospin-invariant $NN$
couplings contribute at the same order Maekawa:2011vs ; deVries:2012ab . Since
little can be said about the size of the $NN$ CP-odd short-range interactions
induced by WET operators, we focus here on the contributions to the Schiff
moment from $\bar{g}_{0,1}$. With progress from Lattice QCD and nuclear
theory, the Schiff moment expressions could in the future be extended to
include additional operators.

We write

$d_{\rm Schiff}=A_{\rm
Schiff}\left(\alpha_{n}d_{n}+\alpha_{p}d_{p}+a_{0}\frac{\bar{g}_{0}}{F_{\pi}}+a_{1}\frac{\bar{g}_{1}}{F_{\pi}}+a_{2}\frac{\bar{g}_{2}}{F_{\pi}}\right).$
(67)

The prefactor $A_{\rm Schiff}$ accounts for the Schiff screening, while the
coefficients $\alpha_{n,p}$ and $a_{0,1,2}$ contain the information on the
nuclear structure of the systems of interest. We report them in Table 3. We
use the results of Ref. Hubert:2022pnl for the screening factors of Hg and Xe,
and Ref. Prasannaa:2020cjx for 225Ra. In the case of 199Hg, $\alpha_{n,p}$ are
known with good accuracy Dmitriev:2003sc , while in the case of Xe we use the
calculation of $\alpha_{n}$ from Yanase:2022atk . The contributions of the CP-
odd $\pi N$ couplings are affected by substantial uncertainties. Here we take
the ranges suggested in Refs. Engel:2013lsa ; Chupp:2017rkp , which considers
results from various nuclear structure calculations.

$A_{\rm Schiff}$ $\alpha_{n}$ $\alpha_{p}$ $a_{0}$ ($e$ fm) $a_{1}$ ($e$ fm)
$a_{2}$ ($e$ fm) 199Hg $-(2.40\pm 0.24)\cdot 10^{-4}$ $1.9\pm 0.1$ $0.20\pm
0.06$ $0.13^{+0.5}_{-0.07}$ $0.25^{+0.89}_{-0.63}$ $0.09^{+0.17}_{-0.04}$
129Xe $-(0.364\pm 0.025)\cdot 10^{-4}$ $-(0.29\pm 0.10)$ –
$0.10^{+0.53}_{-0.037}$ $0.076^{+0.55}_{-0.038}$ 225Ra $\left(6.3\pm
0.5\right)\cdot 10^{-4}$ – – $2.5\pm 7.5$ $-65\pm 40$ $14\pm 6.5$

Table 3: Nuclear and atomic theory input for the EDMs of diamagnetic atoms.

The contributions from semileptonic operators are usually expressed in terms
of nucleon-level operators. In addition to the scalar operators in Eq. (24),
one can write pseudoscalar and tensor terms. In a relativistic notation

$\mathcal{L}=-\frac{G_{F}}{\sqrt{2}}\left(\bar{e}e\bar{N}\left(C^{(0)}_{P}+\tau_{3}C^{(1)}_{P}\right)i\gamma_{5}N-\frac{1}{2}\varepsilon_{\mu\nu\alpha\beta}\,\bar{e}\sigma^{\mu\nu}e\,\bar{N}\sigma^{\alpha\beta}\left(C_{T}^{(0)}+C_{T}^{(1)}\tau_{3}\right)N\right),$
(68)

where the isoscalar/isovector couplings are related to proton/neutron
couplings by

$C^{(p)}_{X}=C^{(0)}_{X}+C^{(1)}_{X},\qquad
C^{(n)}_{X}=C^{(0)}_{X}-C^{(1)}_{X}.$ (69)

As can be more clearly seen in a non-relativistic notation Dekens:2018bci ,
the pseudoscalar operators vanish at zero momentum and are thus proportional
to powers of $Q/\Lambda_{\rm had}$, where $\Lambda_{\rm had}$ is an hadronic
scale. Here we only retain the isovector component, which is dominated by the
pion pole contribution and thus relatively large, with $\Lambda_{\rm had}\sim
m_{u}+m_{d}$.

$C^{(1)}_{P}=-\frac{v^{2}}{2}\left(\textrm{Im}[C^{\rm
SRR}_{eu}]_{1111}-\textrm{Im}[C^{\rm SRL}_{eu}]_{1111}-\textrm{Im}[C^{\rm
SRR}_{ed}]_{1111}+\textrm{Im}[C^{\rm
SRL}_{ed}]_{1111}\right)\frac{g_{A}m_{N}}{\bar{m}}.$ (70)

The isoscalar component is suppressed by the $\eta$ mass, rather than the pion
mass, and thus we neglect it. Its contribution was considered, for example, in
Ref. Degenkolb:2024eve . In terms of WET operators, the tensor coefficients
are

$\displaystyle C_{T}^{(0)}$
$\displaystyle=-\frac{v^{2}}{2}\left(\frac{g_{T}^{u}+g_{T}^{d}}{2}\left(\textrm{Im}[C^{\rm
TRR}_{eu}]_{1111}+\textrm{Im}[C^{\rm
TRR}_{ed}]_{1111}\right)+g_{T}^{s}\textrm{Im}[C^{\rm TRR}_{ed}]_{1122}\right)$
(71) $\displaystyle C_{T}^{(1)}$
$\displaystyle=-\frac{v^{2}}{2}\frac{g_{T}^{u}-g_{T}^{d}}{2}\left(\textrm{Im}[C^{\rm
TRR}_{eu}]_{1111}-\textrm{Im}[C^{\rm TRR}_{ed}]_{1111}\right).$ (72)

The coefficients $\alpha^{N}_{S}$ are given by

$\alpha_{S}^{p}=\mathcal{R}_{\rm
SP}\frac{Z}{A},\qquad\alpha_{S}^{n}=\mathcal{R}_{\rm SP}\frac{N}{A}.$ (73)

$(Z,N,A)$= (80, 121, 201), (54, 77, 131), (87, 136, 223) are the atomic
number, number of neutrons, and mass number of Hg, Xe, and Ra, respectively.
$\mathcal{R}_{\rm SP}$ depends solely on atomic theory, and, for Mercury and
Xexon, we use the values Fleig:2018bsf ; Fleig:2020obr .

$\left.\mathcal{R}_{\rm SP}\right|_{\rm Hg}=-\left(2.8\pm 0.6\right)\cdot
10^{-22}e\,{\rm cm}\qquad\left.\mathcal{R}_{\rm SP}\right|_{\rm
Xe}=-\left(0.71\pm 0.18\right)\cdot 10^{-23}e\,{\rm cm}.$ (74)

These values are somewhat smaller than in other calculations, especially in
the case of xenon Gaul:2023hdd . The tensor coefficients can be written as

$\alpha_{T}^{p}=\mathcal{R}_{T}\langle\sigma_{pz}\rangle,\qquad\alpha_{T}^{n}=\mathcal{R}_{T}\langle\sigma_{nz}\rangle,$
(75)

where $\mathcal{R}_{T}$ is calculated in atomic theory, while
$\langle\sigma_{pz}\rangle$ and $\langle\sigma_{nz}\rangle$ are nuclear spin
matrix elements. We use Fleig:2018bsf ; Fleig:2020obr

$\left.\mathcal{R}_{\rm T}\right|_{\rm Hg}=-\left(2.8\pm 0.6\right)\cdot
10^{-20}e\,{\rm cm}\qquad\left.\mathcal{R}_{\rm T}\right|_{\rm
Xe}=\left(0.520\pm 0.049\right)\cdot 10^{-20}e\,{\rm cm},$ (76)

and Yanase:2018qqq ; Menendez:2012tm

$\displaystyle\left.\langle\sigma_{nz}\rangle\right|_{\rm Hg}$
$\displaystyle=-0.377,\qquad\left.\langle\sigma_{pz}\rangle\right|_{\rm
Hg}=0.009$ (77) $\displaystyle\left.\langle\sigma_{nz}\rangle\right|_{\rm Xe}$
$\displaystyle=+0.658,\qquad\left.\langle\sigma_{pz}\rangle\right|_{\rm
Xe}=0.020.$ (78)

Finally, $\alpha^{N}_{P}$ are related to the tensor contributions by
Ginges:2003qt

$\alpha^{N}_{P}=\frac{Z\alpha_{\rm em}}{5m_{N}R}\alpha^{N}_{T}$ (79)

where $R=1.2A^{1/3}$ fm is the nuclear radius.

##  4 Renormalization group running and threshold effects

The WET operators containing heavier fermions (charm, bottom, muon, and tau)
can contribute to the electron, neutron, atomic, and molecular EDMs via
renormalization group evolution and threshold effects arising when the heavy
particles are integrated out. In this section, we discuss the following three
kinds of contributions of WET operators with heavier generations to the EDMs
observables given in Sec. 3, focusing only on the leading effects111Meaning if
an operator contributes at the tree-level its 1-loop effects are not discussed
in this section, and so on. :

  * •

Leading-log QCD+QED RG running from EW scale to $\mu=2$ GeV.

  * •

1-loop threshold effects at $\mu=m_{c}$ and $\mu=m_{b}$.

  * •

Next-to-leading-log 2-loop QCD+QED RG running from EW scale to $\mu=2$ GeV.

The logarithmic order at which each operator contributes is indicated in Table
1. In the following, we will dissect how these contributions arise.

###  4.1 Leading-log 1-loop RG running

Figure 1: Representative diagrams contributing to the 1-loop anomalous
dimension, which induces LL contributions to EDMs. Plain and double lines
denote light ($e$ and $u$, $d$, $s$ quarks) and heavy fermions ($\mu$, $\tau$
leptons and $c$, $b$ quarks), respectively. Wavy and wiggly lines denote
photons and gluons. Insertions of WET operators are denoted by squares, while
SM vertices by dots. Diagrams $(a)$ and $(b)$ contribute to the mixing of
heavy-light four-fermion operators onto photon and gluon dipole operators.
Four-quark operators of the type $\mathcal{O}_{uu}^{S1,RR}$,
$\mathcal{O}_{uu}^{S8,RR}$, $\mathcal{O}_{dd}^{S8,RR}$,
$\mathcal{O}_{dd}^{S8,RR}$, $\mathcal{O}_{uddu}^{S1,RR}$ and
$\mathcal{O}_{uddu}^{S8,RR}$, with two heavy and two light quarks,
semileptonic tensor operators $\mathcal{O}^{T,RR}_{eu}$
$\mathcal{O}^{T,RR}_{ed}$ with heavy quarks or heavy leptons, and the leptonic
scalar operator $\mathcal{O}^{S,RR}_{ee}$ with two heavy and two light leptons
generate LL EDMs via this path. Diagram $(c)$ denotes the mixing of dipole
operators. Diagram $(d)$ denotes the mixing of the semileptonic scalar
operators $\mathcal{O}^{S,RR}_{eu}$ and $\mathcal{O}^{S,RR}_{ed}$ onto tensor
operators. Combined with the running of the tensor into dipoles, these induce
corrections to the electron EDM starting at $\mathcal{O}(L_{\mu}^{2})$.

The complete 1-loop ADMs in WET including $\Delta F=0$ operators have been
calculated in Jenkins:2017dyc . In what follows, we identify all relevant
terms in the ADMs and provide the naive solutions for the corresponding RGEs.
Through diagrams such as diagram $(a)$ and $(c)$ in Figure 1, the QCD dipole
and four-quark scalar operators mix onto the photon dipole. Neglecting the
running of the electromagnetic and strong couplings, and the running of the
Wilson coefficients, we can denote the solution of the RGEs as

$\displaystyle\big{[}{C}_{f\gamma}\big{]}_{ii}$
$\displaystyle={8eg_{s}q_{i}C_{F}}\big{[}{C}_{fG}\big{]}_{ii}L_{\mu}\,,$ (80)
$\displaystyle\big{[}{C}_{f\gamma}\big{]}_{ii}$
$\displaystyle={2eq_{j}m_{j}}\big{[}{C}_{ff}^{S1,RR}\big{]}_{ijji}L_{\mu}\,,$
$\displaystyle\big{[}{C}_{f\gamma}\big{]}_{ii}$
$\displaystyle={2eq_{j}m_{j}C_{F}}\big{[}{C}_{ff}^{S8,RR}\big{]}_{ijji}L_{\mu}\,,$
$\displaystyle\big{[}{C}_{f\gamma}\big{]}_{ii}$
$\displaystyle=-8eq_{e}m_{j}\big{[}{C}_{ef}^{T,RR}\big{]}_{jjii}L_{\mu}\,,$
$\displaystyle\big{[}{C}_{u\gamma}\big{]}_{ii}$
$\displaystyle={eq_{j}m_{j}}\big{[}{C}_{uddu}^{S1,RR}\big{]}_{ijji}L_{\mu}\,,$
$\displaystyle\big{[}{C}_{u\gamma}\big{]}_{ii}$
$\displaystyle={eq_{j}m_{j}C_{F}}\big{[}{C}_{uddu}^{S8,RR}\big{]}_{ijji}L_{\mu}.$

where we define

$L_{\mu}=\frac{1}{16\pi^{2}}\log\frac{\mu_{low}}{\mu_{ew}}.$ (81)

Here $f=u$ or $d$, and $\mu_{low}$ denotes the scale to which we run, which
depends on the flavor of the particle in the loop. $e$ denotes the electric
charge, while $q_{j}$ is the charge of the quark or lepton $j$, with
$q_{e}=-1$, $q_{d}=-1/3$ and $q_{u}=2/3$. $C_{F}=(N_{c}^{2}-1)/(2N_{c})$ and
$C_{A}=N_{c}$, with $N_{c}$ the number of colors. The mixing of
$\big{[}{C}_{uddu}^{S1(S8),RR}\big{]}_{ijji}$ onto
$\big{[}{C}_{d\gamma}\big{]}$ is governed the last two equations in (80) by
replacing $q_{j},m_{j}$ by $q_{i},m_{i}$, respectively. The QCD dipoles mix
with the electromagnetic dipoles and four-quark scalar operators

$\displaystyle\big{[}{C}_{fG}\big{]}_{ii}$
$\displaystyle={2g_{s}m_{j}}\big{[}{C}_{ff}^{S1,RR}\big{]}_{ijji}L_{\mu}\,,$
(82) $\displaystyle\big{[}{C}_{fG}\big{]}_{ii}$
$\displaystyle={2g_{s}m_{j}}\left(C_{F}-{C_{A}\over
2}\right)\big{[}{C}_{ff}^{S8,RR}\big{]}_{ijji}L_{\mu}\,,$
$\displaystyle\big{[}{C}_{uG}\big{]}_{ii}$
$\displaystyle={g_{s}m_{j}}\big{[}{C}_{uddu}^{S1,RR}\big{]}_{ijji}L_{\mu}\,,$
$\displaystyle\big{[}{C}_{uG}\big{]}_{ii}$
$\displaystyle={g_{s}m_{j}}\left(C_{F}-{C_{A}\over
2}\right)\big{[}{C}_{uddu}^{S8,RR}\big{]}_{ijji}L_{\mu}.$

The mixing of $\big{[}{C}_{f\gamma}\big{]}$ onto $\big{[}{C}_{fG}\big{]}$
follows the exact same relation as the first equation in (80). Also, the
mixing of $\big{[}{C}_{uddu}^{S1(S8),RR}\big{]}$ onto $\big{[}{C}_{dG}\big{]}$
follows the same relations as the last two equations in (82). Eqs. (80) and
(82) show that all scalar operators with two heavy and two light quarks induce
light quark EDMs and CEDMs at one loop, leading, as we will see, to strong
constraints from neutron and mercury EDM experiments. For the electron EDM,
the mixing of the scalar and tensor operators onto the dipole operator is
governed by pure QED

$\displaystyle\big{[}{C}_{e\gamma}\big{]}_{ii}$
$\displaystyle={-8eq_{i}m_{j}N_{c}}\big{[}{C}_{ed}^{T,RR}\big{]}_{iijj}L_{\mu}\,,$
(83) $\displaystyle\big{[}{C}_{e\gamma}\big{]}_{ii}$
$\displaystyle={-8eq_{j}m_{j}N_{c}}\big{[}{C}_{eu}^{T,RR}\big{]}_{iijj}L_{\mu}\,,$
$\displaystyle\big{[}{C}_{e\gamma}\big{]}_{ii}$
$\displaystyle={2eq_{j}m_{j}}\big{[}{C}_{ee}^{S,RR}\big{]}_{ijji}L_{\mu}.$

The first two relations in (83) indicate that semileptonic operators with
heavier generations of up and down quarks can mix with the electromagnetic
dipole operator for the electron. Similarly, the muon and tau scalar operators
can also mix with the dipole operator as indicated by the last relation in
(83).

Finally, through diagrams such as diagram $(d)$ in Fig. 1, the leptonic
operators $\big{[}{C}_{ee}^{S,RR}\big{]}_{1122}$ and
$\big{[}{C}_{ee}^{S,RR}\big{]}_{1133}$ mix with
$\big{[}{C}_{e\gamma}^{S,RR}\big{]}_{11}$ via the
$\big{[}{C}_{ee}^{S,RR}\big{]}_{1221}$ and
$\big{[}{C}_{ee}^{S,RR}\big{]}_{1331}$ operators. Similarly, semileptonic
scalar operators mix into tensor operators, and scalar operators of the form
$\big{[}{C}_{uu}^{S1(8),RR}\big{]}_{iijj}$,
$\big{[}{C}_{dd}^{S1(8),RR}\big{]}_{iijj}$ and
$\big{[}{C}_{ud}^{S1(8),RR}\big{]}_{iijj}$ mix onto
$\big{[}{C}_{uu}^{S1(8),RR}\big{]}_{ijji}$,
$\big{[}{C}_{dd}^{S1(8),RR}\big{]}_{ijji}$ and
$\big{[}{C}_{uddu}^{S1(8),RR}\big{]}_{ijji}$, respectively. In this way, these
operators induce LL contribution to the electron and quark dipole operators,
which however start at two loops and go as $L_{\mu}^{2}$.

$\displaystyle\big{[}{C}_{e\gamma}\big{]}_{11}$
$\displaystyle={32q_{e}^{3}e^{3}m_{j}}\big{[}{C}_{ee}^{S,RR}\big{]}_{11jj}\,\frac{L^{2}_{\mu}}{2}\,,$
(84) $\displaystyle\big{[}{C}_{e\gamma}\big{]}_{11}$
$\displaystyle={16}{q_{e}q_{u}^{2}e^{3}m_{j}N_{c}}\big{[}{C}_{eu}^{S,RR}\big{]}_{11jj}\,{\frac{L^{2}_{\mu}}{2}}\,,$
$\displaystyle\big{[}{C}_{e\gamma}\big{]}_{11}$
$\displaystyle={16}{q_{e}q_{d}^{2}e^{3}m_{j}N_{c}}\big{[}{C}_{ed}^{S,RR}\big{]}_{11jj}\,{\frac{L^{2}_{\mu}}{2}}.$

In the first relation in (84), $jj=22,33$, for the second $jj=11,22$ and for
third $jj=11,22$, or $33$. Similarly, the semileptonic operators also mix with
the quark dipoles

$\displaystyle\big{[}{C}_{u\gamma}\big{]}_{ii}$
$\displaystyle={16}{q_{e}^{2}q_{u}e^{3}m_{j}}\big{[}{C}_{eu}^{S,RR}\big{]}_{jjii}{\frac{L^{2}_{\mu}}{2}}\,,$
(85) $\displaystyle\big{[}{C}_{d\gamma}\big{]}_{ii}$
$\displaystyle={16}{q_{e}^{2}q_{d}e^{3}m_{j}}\big{[}{C}_{ed}^{S,RR}\big{]}_{jjii}{\frac{L^{2}_{\mu}}{2}}.$

Here, $jj=11,22$ or $33$, however, only $33$ cases can induce significant
effects. Finally, four-quark scalar operators also generate $L^{2}_{\mu}$
contributions. In the case of $uu$ and $dd$ operators, the mixing is given by

$\displaystyle\big{[}{C}_{f\gamma}\big{]}_{ii}$
$\displaystyle=\frac{2eq_{f}m_{j}}{N_{c}}(16e^{2}q_{u}^{2}+16g_{s}^{2}C_{F})\big{[}{C}_{ff}^{S1,RR}\big{]}_{iijj}\frac{L^{2}_{\mu}}{2}\,,$
(86) $\displaystyle\big{[}{C}_{f\gamma}\big{]}_{ii}$
$\displaystyle=\frac{2eC_{F}q_{f}m_{j}}{N_{c}}\left(16e^{2}q_{u}^{2}C_{F}+2g_{s}^{2}\left(\frac{2}{N_{c}^{2}}+N_{c}^{2}-3\right)\right)\big{[}{C}_{ff}^{S8,RR}\big{]}_{iijj}\frac{L^{2}_{\mu}}{2}\,,$
$\displaystyle\big{[}{C}_{fG}\big{]}_{ii}$
$\displaystyle=\frac{2g_{s}m_{j}}{N_{c}}(16e^{2}q_{u}^{2}+16g_{s}^{2}C_{F})\big{[}{C}_{ff}^{S1,RR}\big{]}_{iijj}\frac{L^{2}_{\mu}}{2}\,,$
$\displaystyle\big{[}{C}_{fG}\big{]}_{ii}$
$\displaystyle=-\frac{2g_{s}}{N_{c}}m_{j}\left(C_{F}-{C_{A}\over
2}\right)\left(16e^{2}q_{u}^{2}-4g_{s}^{2}\left(\frac{2}{N_{c}}+N_{c}\right)\right)\big{[}{C}_{ff}^{S8,RR}\big{]}_{iijj}\frac{L^{2}_{\mu}}{2}\,,$

for $f=u$ or $d$. $\big{[}{C}_{ud}^{S1(8),RR}\big{]}$ operators induce both
QED and QCD dipoles. For the QED dipoles, one finds

$\displaystyle\big{[}{C}_{u\gamma}\big{]}_{ii}$
$\displaystyle=\frac{eq_{d}m_{j}}{N_{c}}(16e^{2}q_{u}q_{d}+16g_{s}^{2}C_{F})\big{[}{C}_{ud}^{S1,RR}\big{]}_{iijj}\frac{L_{\mu}^{2}}{2}\,,$
(87) $\displaystyle\big{[}{C}_{u\gamma}\big{]}_{ii}$
$\displaystyle=\frac{2eC_{F}q_{d}m_{j}}{N_{c}}\left(16e^{2}q_{u}q_{d}C_{F}+2g_{s}^{2}\left(\frac{2}{N_{c}^{2}}+N_{c}^{2}-3\right)\right)\big{[}{C}_{ud}^{S8,RR}\big{]}_{iijj}\frac{L_{\mu}^{2}}{2}\,,$
$\displaystyle\big{[}{C}_{d\gamma}\big{]}_{jj}$
$\displaystyle=\frac{eq_{d}m_{i}}{N_{c}}(16e^{2}q_{u}q_{d}+16g_{s}^{2}C_{F})\big{[}{C}_{ud}^{S1,RR}\big{]}_{iijj}\frac{L_{\mu}^{2}}{2}\,,$
$\displaystyle\big{[}{C}_{d\gamma}\big{]}_{jj}$
$\displaystyle=\frac{2eC_{F}q_{d}m_{i}}{N_{c}}\left(16e^{2}q_{u}q_{d}C_{F}+2g_{s}^{2}\left(\frac{2}{N_{c}^{2}}+N_{c}^{2}-3\right)\right)\big{[}{C}_{ud}^{S8,RR}\big{]}_{iijj}\frac{L_{\mu}^{2}}{2},$

and, for the QCD dipoles,

$\displaystyle\big{[}{C}_{uG}\big{]}_{ii}$
$\displaystyle=\frac{g_{s}m_{j}}{N_{c}}(16e^{2}q_{u}q_{d}+16g_{s}^{2}C_{F})\big{[}{C}_{ud}^{S1,RR}\big{]}_{iijj}\frac{L_{\mu}^{2}}{2}\,,$
(88) $\displaystyle\big{[}{C}_{uG}\big{]}_{ii}$
$\displaystyle=\frac{g_{s}}{N_{c}}m_{j}\left(C_{F}-{C_{A}\over
2}\right)\left(16e^{2}q_{u}q_{d}C_{F}+2g_{s}^{2}\left(\frac{2}{N_{c}^{2}}+N_{c}^{2}-3\right)\right)\big{[}{C}_{ud}^{S8,RR}\big{]}_{iijj}\frac{L_{\mu}^{2}}{2}\,,$
$\displaystyle\big{[}{C}_{dG}\big{]}_{jj}$
$\displaystyle=\frac{g_{s}}{N_{c}}m_{i}(16e^{2}q_{u}q_{d}+16g_{s}^{2}C_{F})\big{[}{C}_{ud}^{S1,RR}\big{]}_{iijj}\frac{L_{\mu}^{2}}{2}\,,$
$\displaystyle\big{[}{C}_{dG}\big{]}_{jj}$
$\displaystyle=\frac{g_{s}}{N_{c}}m_{j}\left(C_{F}-{C_{A}\over
2}\right)\left(16e^{2}q_{u}q_{d}C_{F}+2g_{s}^{2}\left(\frac{2}{N_{c}^{2}}+N_{c}^{2}-3\right)\right)\big{[}{C}_{ud}^{S8,RR}\big{]}_{iijj}\frac{L_{\mu}^{2}}{2}.$

This exhausts all the operators denoted as LL in Table 1.

Figure 2: Matching contributions arising from integrating out heavy particles.
Diagram $(a)$ shows contributions to the Weinberg three-gluon operator from
gluon dipole operators with heavy quarks. Diagram $(b)$ and $(c)$ illustrate
matching corrections from the four-quark operators
$\mathcal{O}^{V1,LR}_{uddu}$, $\mathcal{O}^{V8,LR}_{uddu}$,
$\mathcal{O}^{V1,LR}_{dd}$, $\mathcal{O}^{V8,LR}_{dd}$,
$\mathcal{O}^{V1,LR}_{uu}$ and $\mathcal{O}^{V8,LR}_{dd}$ and from the
leptonic vector operators $\mathcal{O}^{V,LR}_{ee}$ onto dipole operators.

###  4.2 1-loop Matching

In this subsection, we discuss the threshold effects at $m_{c}$, $m_{b}$,
$m_{\mu}$, and $m_{\tau}$ at the 1-loop level.

####  4.2.1 Matching to Weinberg operators

$\big{[}{C}_{dG}\big{]}_{33}$ mixes with $\big{[}{C}_{\widetilde{G}}\big{]}$
after integrating out $b$-quark Braaten:1990gq . After adjusting the
normalization of the operators, in the JMS basis, we find

$\big{[}{C}_{\widetilde{G}}\big{]}(\mu_{b}^{-})=\big{[}{C}_{\widetilde{G}}\big{]}(\mu_{b}^{+})+\frac{1}{3m_{b}}\frac{\alpha_{s}}{8\pi}{\rm
Im}\big{[}{C}_{dG}\big{]}_{33}(\mu_{b}^{+}).$ (89)

The coefficient of second term is $\approx 7.1\cdot 10^{-4}$. The 1-loop
Feynman diagram governing this mixing is given in Fig. 2, diagram $(a)$.
Similarly, we can integrate out the charm quark leading to

$\big{[}{C}_{\widetilde{G}}\big{]}(\mu_{c}^{-})=\big{[}{C}_{\widetilde{G}}\big{]}(\mu_{c}^{+})+\frac{1}{3m_{c}}\frac{\alpha_{s}}{8\pi}{\rm
Im}\big{[}{C}_{uG}\big{]}_{22}(\mu_{c}^{+}).$ (90)

The coefficient of second term is $\approx 5.2\cdot 10^{-3}$.

####  4.2.2 Matching to dipole operators

The vector operators with left-right chiral currents match onto dipole
operators at one loop Alioli:2017ces . For chromo-dipoles, in the JMS basis,
we find

$\displaystyle\big{[}{C}_{uG}\big{]}_{ii}(\mu^{-})$
$\displaystyle=\big{[}{C}_{uG}\big{]}_{ii}(\mu^{+})-{g_{s}m_{j}\over
16\pi^{2}}\left(\big{[}{C}_{uddu}^{V1,LR}\big{]}_{ijji}(\mu^{+})-{1\over
6}\big{[}{C}_{uddu}^{V8,LR}\big{]}_{ijji}(\mu^{+})\right)\,,$ (91)
$\displaystyle\big{[}{C}_{dG}\big{]}_{jj}(\mu^{-})$
$\displaystyle=\big{[}{C}_{dG}\big{]}_{jj}(\mu^{+})-{g_{s}m_{i}\over
16\pi^{2}}\left(\big{[}{C}_{uddu}^{V1,LR}\big{]}_{ijji}(\mu^{+})-{1\over
6}\big{[}{C}_{uddu}^{V8,LR}\big{]}_{ijji}(\mu^{+})\right)\,,$
$\displaystyle\big{[}{C}_{uG}\big{]}_{ii}(\mu^{-})$
$\displaystyle=\big{[}{C}_{uG}\big{]}_{ii}(\mu^{+})-{g_{s}m_{j}\over
16\pi^{2}}\left(\big{[}{C}_{uu}^{V1,LR}\big{]}_{ijji}(\mu^{+})-{1\over
6}\big{[}{C}_{uu}^{V8,LR}\big{]}_{ijji}(\mu^{+})\right)\,,$
$\displaystyle\big{[}{C}_{dG}\big{]}_{jj}(\mu^{-})$
$\displaystyle=\big{[}{C}_{dG}\big{]}_{jj}(\mu^{+})-{g_{s}m_{i}\over
16\pi^{2}}\left(\big{[}{C}_{dd}^{V1,LR}\big{]}_{ijji}(\mu^{+})-{1\over
6}\big{[}{C}_{dd}^{V8,LR}\big{]}_{ijji}(\mu^{+})\right)\,,$

similarly, using the last two equations one can obtain the matching for
$\big{[}{C}_{uG}\big{]}_{jj}$ and $\big{[}{C}_{dG}\big{]}_{ii}$ . For the
photon dipoles, we have

$\displaystyle\big{[}{C}_{u\gamma}\big{]}_{ii}(\mu^{-})$
$\displaystyle=\big{[}{C}_{u\gamma}\big{]}_{ii}(\mu^{+})+{eQ_{j}m_{j}\over
16\pi^{2}}\left(\big{[}{C}_{uddu}^{V1,LR}\big{]}_{ijji}(\mu^{+})-{4\over
3}\big{[}{C}_{uddu}^{V8,LR}\big{]}_{ijji}(\mu^{+})\right)\,,$ (92)
$\displaystyle\big{[}{C}_{d\gamma}\big{]}_{jj}(\mu^{-})$
$\displaystyle=\big{[}{C}_{d\gamma}\big{]}_{jj}(\mu^{+})+{eQ_{i}m_{i}\over
16\pi^{2}}\left(\big{[}{C}_{uddu}^{V1,LR}\big{]}_{ijji}(\mu^{+})-{4\over
3}\big{[}{C}_{uddu}^{V8,LR}\big{]}_{ijji}(\mu^{+})\right)\,,$
$\displaystyle\big{[}{C}_{u\gamma}\big{]}_{ii}(\mu^{-})$
$\displaystyle=\big{[}{C}_{u\gamma}\big{]}_{ii}(\mu^{+})+{eQ_{j}m_{j}\over
16\pi^{2}}\left(\big{[}{C}_{uu}^{V1,LR}\big{]}_{ijji}(\mu^{+})-{4\over
3}\big{[}{C}_{uu}^{V8,LR}\big{]}_{ijji}(\mu^{+})\right)\,,$
$\displaystyle\big{[}{C}_{d\gamma}\big{]}_{jj}(\mu^{-})$
$\displaystyle=\big{[}{C}_{d\gamma}\big{]}_{jj}(\mu^{+})+{eQ_{i}m_{i}\over
16\pi^{2}}\left(\big{[}{C}_{dd}^{V1,LR}\big{]}_{ijji}(\mu^{+})-{4\over
3}\big{[}{C}_{dd}^{V8,LR}\big{]}_{ijji}(\mu^{+})\right).$

In the above relations, the scale $\mu$ can be set to the appropriate matching
scale of either $m_{c}$ or $m_{b}$. Finally, for the leptonic dipoles, at
$\mu=m_{\tau}$, we obtain Aebischer:2021uvt

$\big{[}{C}_{e\gamma}\big{]}_{ii}(\mu^{-})=\big{[}{C}_{e\gamma}\big{]}_{ii}(\mu^{+})+{eQ_{j}m_{j}\over
16\pi^{2}}\big{[}{C}_{ee}^{V,LR}\big{]}_{ijji}(\mu^{+}).$ (93)

###  4.3 Two-step operator mixing

Figure 3: Representative diagrams that induce NLL contributions to EDMs.
Diagrams $(a)$ and $(b)$ denote the LL mixing of scalar four-fermion operators
with four heavy quarks onto heavy flavor dipoles, while diagram $(c)$ the
running of heavy flavor dipoles into themselves. At the heavy flavor
threshold, the gluonic dipole is integrated out, inducing
$\mathcal{O}((4\pi)^{-2}L_{\mu})$ contributions to $C_{\tilde{G}}$. Diagrams
$(d)$ and $(e)$ contribute to the 2-loop mixing of heavy-light vector
operators onto dipoles.

We next consider possible NLL effects, which give rise to the dominant EDM
contributions for those operators that do not contribute at LL. As shown in
(89) and (90) the QCD dipoles $\big{[}{C}_{dG}\big{]}_{33}$ and
$\big{[}{C}_{uG}\big{]}_{22}$ can mix with the three-gluon operator
$\big{[}{C}_{\widetilde{G}}\big{]}$ due to 1-loop threshold at the bottom and
charm thresholds, respectively. As illustrated in diagram $(b)$ in Fig. 3, the
scalar operators $\big{[}{C}_{uu}^{S1,RR}\big{]}_{2222}$ and
$\big{[}{C}_{uu}^{S8,RR}\big{]}_{2222}$ mix with $\big{[}{C}_{uG}\big{]}_{22}$
at 1-loop level. Therefore, $\big{[}{C}_{uu}^{S1,RR}\big{]}_{2222}$ and
$\big{[}{C}_{uu}^{S8,RR}\big{]}_{2222}$ induce a correction to the three gluon
operator of the form

$\displaystyle\big{[}{C}_{\widetilde{G}}\big{]}(m_{c})$ $\displaystyle={2\over
3}{g_{s}^{3}}\big{[}{C}_{uu}^{S1,RR}\big{]}_{2222}(\mu_{ew}){L_{\mu_{c}}\over
16\pi^{2}}\,,$ (94) $\displaystyle\big{[}{C}_{\widetilde{G}}\big{]}(m_{c})$
$\displaystyle={2\over 3}g_{s}^{3}\left(C_{F}-{C_{A}\over
2}\right)\big{[}{C}_{uu}^{S8,RR}\big{]}_{2222}(\mu_{ew}){L_{\mu_{c}}\over
16\pi^{2}},$

where, again, the approximate solution in Eq. (94) is obtained by neglecting
the running of the Wilson coefficients and of the strong coupling constant.
Similarly, for the operators containing bottom quarks, it reads

$\displaystyle\big{[}{C}_{\widetilde{G}}\big{]}(m_{b})$ $\displaystyle={2\over
3}g_{s}^{3}\big{[}{C}_{dd}^{S1,RR}\big{]}_{3333}(\mu_{ew}){L_{\mu_{b}}\over
16\pi^{2}}\,,$ (95) $\displaystyle\big{[}{C}_{\widetilde{G}}\big{]}(m_{b})$
$\displaystyle={2\over 3}g_{s}^{3}\left(C_{F}-{C_{A}\over
2}\right)\big{[}{C}_{dd}^{S8,RR}\big{]}_{3333}(\mu_{ew}){L_{\mu_{b}}\over
16\pi^{2}}.$

Here we denoted

$L_{\mu_{c}}=\frac{1}{16\pi^{2}}\log\left(\frac{m_{c}}{\mu_{ew}}\right),\qquad
L_{\mu_{b}}=\frac{1}{16\pi^{2}}\log\left(\frac{m_{b}}{\mu_{ew}}\right),$ (96)

and the appearance of one power of $L_{\mu}$ and an additional power of
$(16\pi^{2})^{-1}$ is typical of a NLL correction. Eqs. (94) and (95) are the
leading contributions of scalar operators with charm and bottom quarks
$\big{[}{C}_{uu}^{S1(S8),RR}\big{]}_{2222}$ and
$\big{[}{C}_{dd}^{S1(S8),RR}\big{]}_{3333}$ to EDMs. The operators
$\big{[}{C}_{uddu}^{S1(S8)}\big{]}_{2332}$ can also mix with the three-gluon
operator by first generating a $b$ or $c$ chromoelectric EDM

$\displaystyle\big{[}{C}_{\widetilde{G}}\big{]}(m_{c})$
$\displaystyle={m_{b}\over
3m_{c}}{g_{s}^{3}}\big{[}{C}_{uddu}^{S1,RR}\big{]}_{2332}(\mu_{ew}){L_{\mu_{c}}\over
16\pi^{2}}\,,$ (97) $\displaystyle\big{[}{C}_{\widetilde{G}}\big{]}(m_{c})$
$\displaystyle={m_{b}\over 3m_{c}}{g_{s}^{3}}\left(C_{F}-{C_{A}\over
2}\right)\big{[}{C}_{uddu}^{S8,RR}\big{]}_{2332}(\mu_{ew}){L_{\mu_{c}}\over
16\pi^{2}}\,,$ $\displaystyle\big{[}{C}_{\widetilde{G}}\big{]}(m_{b})$
$\displaystyle={m_{c}\over
3m_{b}}{g_{s}^{3}}\big{[}{C}_{uddu}^{S1,RR}\big{]}_{2332}(\mu_{ew}){L_{\mu_{b}}\over
16\pi^{2}}\,,$ $\displaystyle\big{[}{C}_{\widetilde{G}}\big{]}(m_{b})$
$\displaystyle={m_{c}\over 3m_{b}}g_{s}^{3}\left(C_{F}-{C_{A}\over
2}\right)\big{[}{C}_{uddu}^{S8,RR}\big{]}_{2332}(\mu_{ew}){L_{\mu_{b}}\over
16\pi^{2}}.$

The operators $\big{[}{C}_{ud}^{S1(8),RR}\big{]}_{2233}$ have 1-loop QCD and
QED mixing with $\big{[}{C}_{uddu}^{S1(8),RR}\big{]}_{2332}$, and thus
generate contributions to the $b$ and $c$ QED and QCD dipoles analogous to
Eqs. (87) and (88). These generate corrections to
$\big{[}{C}_{\widetilde{G}}\big{]}$ that scale as $L^{2}_{\mu}/(16\pi^{2})$.

###  4.4 2-loop anomalous dimension

We have seen that four-quark scalar operators in the $(LR)(LR)$ class
contribute to EDM at LL, in the case of operators with four light quarks or
two heavy and two light quarks, or NLL, for operators with four heavy quarks.
We next analyze vector operators in the $(LL)(RR)$ class. For heavy-light
operators, the first contribution arises at NLL. At this order, we need to
consider the 2-loop running between $\mu_{ew}$ and $m_{b}$ or $m_{c}$, and the
1-loop matching contribution discussed in Sec. 4.2.2. Both the anomalous
dimension and the matching coefficient are separately scheme-dependent, and
the scheme dependence cancels between the two Ciuchini:1993ks . The 2-loop
anomalous dimension can be obtained by generalizing the results of Misiak and
Cho Cho:1993zb . They calculated the mixing between $\Delta F=1$ vector left-
right and dipole operators in the “standard” basis. Here we generalize their
results to to the mixing of $\Delta F=0$ four-fermion and dipole operators in
the JMS basis. A typical 2-loop Feynman diagram governing this mixing is given
by diagram $(d)$ in Fig. 3. The $\Delta F=1$ effective Lagrangian used in Ref.
Cho:1993zb was

$\mathcal{L}_{\rm eff}^{\Delta
F=1}=V_{tb}V_{ts}^{*}\frac{G_{F}}{\sqrt{2}}\sum_{i\in 7,8,9,10}C_{i}Q_{i}\,,$
(98)

with

$\displaystyle Q_{7}$
$\displaystyle=\frac{e}{16\pi^{2}}m_{b}(\bar{s}_{\alpha}\sigma^{\mu\nu}P_{R}b_{\alpha})F_{\mu\nu}\,,\quad
Q_{8}=\frac{g_{s}}{16\pi^{2}}m_{b}(\bar{s}_{\alpha}\sigma^{\mu\nu}P_{R}T^{A}_{\alpha\beta}b_{\beta})G_{\mu\nu}^{A}\,,$
(99) $\displaystyle Q_{9}$ $\displaystyle={m_{b}\over
m_{c}}(\bar{s}_{\alpha}\gamma_{\mu}P_{L}u_{\beta})(\bar{u}_{\beta}\gamma^{\mu}P_{R}b_{\alpha})\,,\qquad
Q_{10}={m_{b}\over
m_{c}}(\bar{s}_{\alpha}\gamma_{\mu}P_{L}u_{\alpha})(\bar{u}_{\beta}\gamma^{\mu}P_{R}b_{\beta}).$

The RGE for corresponding WCs can be written as

$2\mu^{2}{d\over d\mu^{2}}\vec{C_{S}}(\mu)={\alpha_{s}\over
4\pi}(\hat{\gamma}_{S}^{u})^{T}\vec{C}_{S}(\mu).$ (100)

The $4\times 4$ ADM for the standard operator vector
$\vec{Q}_{S}=\\{Q_{7},Q_{8},Q_{9},Q_{10}\\}$ is given in equation (4.7) of
Ref. Cho:1993zb for $u=c$. Keeping the explicit dependence on the electric
charges of the $u$-type and $d$-type quarks, it reads222We thank M. Misiak for
providing us the expressions of the ADM for generic $q_{u}$ and $q_{d}$
charges.

${\hat{\gamma}_{S}^{u}\over 2}=\begin{pmatrix}16\over 3&0&0&0\\\ -16\over
9&14\over 3&0&0\\\ 40q_{u}&-2&-8&0\\\ 8q_{u}+{16\over 3}q_{d}&4\over
3&-3&1\end{pmatrix}.$ (101)

The superscript $u$ indicates that the $u$-quarks are closed in diagram $(d)$
in Fig. 3. Since the JMS basis uses the unit normalization for the operators,
using (103), first we go to an intermediate basis and obtain the ADM for it

$\vec{Q}^{\prime}=\\{Q_{7}^{\prime},Q_{8}^{\prime},Q_{9}^{\prime},Q_{10}^{\prime}\\}.$
(102)

The $Q_{i}^{\prime}$’s are the same as the operators given in (99) but with
unit normalization like the JMS operators and follows a JMS like RGEs for the
WCs

$16\pi^{2}\mu{d\vec{C}^{\prime}(\mu)\over
d\mu}={\hat{\gamma}}^{\prime}\vec{C}^{\prime}(\mu).$ (103)

The ADM in this basis (when closing $u$-quark) comes out to be

${\hat{\gamma}}^{\prime u}=\begin{pmatrix}{8\over 3}g_{s}^{2}&-32eg_{s}\over
9&{5eg_{s}^{2}m_{u}\over\pi^{2}}q_{u}&{eg_{s}^{2}m_{u}\over
8\pi^{2}}({16q_{d}\over 3}+8q_{u})\\\ 0&-19g_{s}^{2}\over
3&-g_{s}^{3}m_{u}\over 4\pi^{2}&g_{s}^{3}m_{u}\over 6\pi^{2}\\\
0&0&-16g_{s}^{2}&-6g_{s}^{2}\\\ 0&0&0&2g_{s}^{2}\end{pmatrix}.$ (104)

The relation between ${\hat{\gamma}}^{\prime u}$ and ${\hat{\gamma}}_{S}^{u}$
is given by

${\hat{\gamma}}^{\prime u}=16\pi^{2}\mu{dR(\mu)\over
d\mu}R(\mu)^{-1}+R(\mu)g_{s}^{2}({\hat{\gamma}}_{S}^{u})^{T}R(\mu)^{-1}.$
(105)

The matrix $R(\mu)$ is the transformation matrix defined by
$C^{\prime}(\mu)=R(\mu)C_{S}(\mu)$. The RGE for $R(\mu)$ is dictated by the
RGEs of $g_{s}(\mu)$ and $m_{b}(\mu)$

$16\pi^{2}\mu{dR(\mu)\over d\mu}=\begin{pmatrix}-8eg_{s}^{2}m_{b}&0&0&0\\\
0&{-47\over 3}g_{s}^{3}m_{b}&0&0\\\ 0&0&0&0\\\ 0&0&0&0\end{pmatrix}.$ (106)

We have used $16\pi^{2}\mu({dm_{b}/d\mu})=-6C_{F}g_{s}^{2}m_{b}$ and
$16\pi^{2}\mu({dg_{s}/d\mu})=-23g_{s}^{3}/3$ for five flavors. The
correspondence between the $Q^{\prime}$ and the JMS bases is given by

$\hat{\gamma}_{\text{JMS}}=P\hat{\gamma}^{\prime}P^{-1}.$ (107)

Here the matrix $P$ defines the transformation between the WCs in the two
bases, namely $C_{\text{JMS}}=PC^{\prime}$. In what follows we will derive
$\hat{\gamma}_{\text{JMS}}$ for specific sets of the JMS $\Delta F=0$
operators entering the EDMs.

####  4.4.1 ${C_{\\!\underset{prrp}{uddu}}^{{\tiny
V1,LR}}},{C_{\\!\underset{prrp}{uddu}}^{{\tiny V8,LR}}}$ onto dipoles

In this case, the operators ${C_{\\!\underset{prrp}{uddu}}^{{\tiny V1,LR}}}$
and ${C_{\\!\underset{prrp}{uddu}}^{{\tiny V8,LR}}}$ contribute to EDMs, here
the index $p$ can be equal or unequal to $r$. At 2-loop order, these two
operators can mix with
$\\{{C_{\\!\underset{pp}{u\gamma}}},{C_{\\!\underset{pp}{uG}}}\\}$, if one
closes the $d$ quark loop, or
$\\{{C_{\\!\underset{rr}{d\gamma}}},{C_{\\!\underset{rr}{dG}}}\\}$, by closing
the $u$ quark loop. Using the color identity $T^{A}_{ij}T^{A}_{kl}={1\over
2}\left(\delta_{il}\delta_{jk}-{1\over 3}\delta_{ij}\delta_{kl}\right)$, the
tree-level transformation matrix between the intermediate basis with properly
adjusted down-quark flavors to $\Delta F=0$ and the JMS basis
$\vec{\mathcal{C}}_{u}=\\{{C_{\\!\underset{rr}{d\gamma}}},{C_{\\!\underset{rr}{dG}}},$
${C_{\\!\underset{prrp}{uddu}}^{{\tiny
V1,LR}}},{C_{\\!\underset{prrp}{uddu}}^{{\tiny V8,LR}}}\\}$ is found to be

$P=\begin{pmatrix}1&0&0&0\\\ 0&1&0&0\\\ 0&0&1/3&1\\\ 0&0&2&0\end{pmatrix}.$
(108)

Using (101), (106) and (108) in (105), we obtain 333Note that one needs to
properly adjust the value of electric charge for the case of up-type dipoles
in contrast to the original result which is valid for the down-quark dipoles.
the ADMs for $\vec{\mathcal{C}}_{u}$ when the $u$-quark loop is closed444Note
that in the original result of Misiak and Cho the charm-quark loop was closed.

$\gamma_{\text{JMS}}^{u}=\begin{pmatrix}8g_{s}^{2}\over 3&-32eg_{s}\over
9&4eg_{s}^{2}m_{u}\over 9\pi^{2}&43eg_{s}^{2}m_{u}\over 27\pi^{2}\\\
-8eg_{s}\over 3&-19g_{s}^{2}\over 3&g_{s}^{3}m_{u}\over
6\pi^{2}&-11g_{s}^{3}m_{u}\over 72\pi^{2}\\\ 0&0&0&-8g_{s}^{2}\over 3\\\
0&0&-12g_{s}^{2}&-14g_{s}^{2}\end{pmatrix}.$ (109)

Here $u=u$ or $c$.

For down-quark loops we have a four-vector
$\vec{\mathcal{C}}_{d}=\\{{C_{\\!\underset{pp}{u\gamma}}},{C_{\\!\underset{pp}{uG}}},$
${C_{\\!\underset{prrp}{uddu}}^{{\tiny
V1,LR}}},{C_{\\!\underset{prrp}{uddu}}^{{\tiny V8,LR}}}\\}$. The
transformation matrix is the same as given in (108). However, in this case,
first, we need to make the replacements $q_{u}\to q_{d}$ and $q_{d}\to q_{u}$
in the original ADM (101). This brings us

$\gamma_{\text{JMS}}^{d}=\begin{pmatrix}8g_{s}^{2}\over 3&64eg_{s}\over
9&eg_{s}^{2}m_{d}\over 9\pi^{2}&-23eg_{s}^{2}m_{d}\over 27\pi^{2}\\\
16eg_{s}\over 3&-19g_{s}^{2}\over 3&g_{s}^{3}m_{d}\over
6\pi^{2}&-11g_{s}^{3}m_{d}\over 72\pi^{2}\\\ 0&0&0&-8g_{s}^{2}\over 3\\\
0&0&-12g_{s}^{2}&-14g_{s}^{2}\end{pmatrix}.$ (110)

Here $d$ represents $d,s$ or $b$. Obviously, only the $1-3$ and $1-4$ entries
corresponding to the mixing of vector left-right operators onto QED dipole
operators are modified as compared to $\gamma_{\text{JMS}}^{u}$.

####  4.4.2 ${C_{\\!\underset{prrp}{uu}}^{{\tiny
V1,LR}}},{C_{\\!\underset{prrp}{uu}}^{{\tiny V8,LR}}}$ onto dipoles

The operators ${C_{\\!\underset{prrp}{uu}}^{{\tiny V1,LR}}}$ and
${C_{\\!\underset{prrp}{uu}}^{{\tiny V8,LR}}}$ can contribute to EDMs if
$p\neq r$. These mix with
$\\{{C_{\\!\underset{pp}{u\gamma}}},{C_{\\!\underset{pp}{uG}}}\\}$ and
$\\{{C_{\\!\underset{rr}{u\gamma}}},{\mathcal{O}_{\\!\underset{rr}{uG}}}\\}$
depending on whether the quarks with $rr$ or $pp$ flavor indices are closed.
In this case, we have two four vectors, namely
$\vec{\mathcal{C}}_{ur}=\\{{C_{\\!\underset{pp}{u\gamma}}},{C_{\\!\underset{pp}{uG}}},$
${C_{\\!\underset{prrp}{uu}}^{{\tiny
V1,LR}}},{C_{\\!\underset{prrp}{uu}}^{{\tiny V8,LR}}}\\}$ and
$\vec{\mathcal{C}}_{up}=\\{{C_{\\!\underset{rr}{u\gamma}}},{C_{\\!\underset{rr}{uG}}},$
${C_{\\!\underset{prrp}{uu}}^{{\tiny
V1,LR}}},{C_{\\!\underset{prrp}{uu}}^{{\tiny V8,LR}}}\\}$ . Both cases involve
an up-quark loop with flavor indices $rr$ and $pp$ respectively. The
transformation matrix between the standard (after properly adjusting quark
flavors in (99)) and JMS vectors $\vec{\mathcal{C}}_{ur}$ and
$\vec{\mathcal{C}}_{up}$ is given by (108). In this case, in the original
2-loop ADM (101) one has to make replacements $q_{u}\to q_{u}$ and $q_{d}\to
q_{u}$ which lead us to

$\gamma_{\text{JMS}}^{ur}=\begin{pmatrix}8g_{s}^{2}\over 3&{64}eg_{s}\over
9&10eg_{s}^{2}m_{u}\over 9\pi^{2}&40eg_{s}^{2}m_{u}\over 27\pi^{2}\\\
16eg_{s}\over 3&-19g_{s}^{2}\over 3&g_{s}^{3}m_{u}\over
6\pi^{2}&-11g_{s}^{3}m_{u}\over 72\pi^{2}\\\ 0&0&0&-8g_{s}^{2}\over 3\\\
0&0&-12g_{s}^{2}&-14g_{s}^{2}\end{pmatrix}.$ (111)

The $\gamma_{\text{JMS}}^{up}$ is exactly same as $\gamma_{\text{JMS}}^{ur}$.

####  4.4.3 ${C_{\\!\underset{prrp}{dd}}^{{\tiny
V1,LR}}},{C_{\\!\underset{prrp}{dd}}^{{\tiny V8,LR}}}$ onto dipoles

In this case, we have two four vectors, namely
$\vec{\mathcal{C}}_{dr}=\\{{C_{\\!\underset{pp}{d\gamma}}},{C_{\\!\underset{pp}{dG}}},$
${C_{\\!\underset{prrp}{dd}}^{{\tiny
V1,LR}}},{C_{\\!\underset{prrp}{dd}}^{{\tiny V8,LR}}}\\}$ and
$\vec{\mathcal{C}}_{dp}=\\{{C_{\\!\underset{rr}{d\gamma}}},{C_{\\!\underset{rr}{dG}}},$
${C_{\\!\underset{prrp}{dd}}^{{\tiny
V1,LR}}},{C_{\\!\underset{prrp}{dd}}^{{\tiny V8,LR}}}\\}$ . In the original
2-loop ADM (101) one has to make replacements $q_{u}\to q_{d}$ and $q_{d}\to
q_{d}$. This leads us to

$\gamma_{\text{JMS}}^{dr}=\begin{pmatrix}8g_{s}^{2}\over 3&-32eg_{s}\over
9&-5eg_{s}^{2}m_{d}\over 9\pi^{2}&-20eg_{s}^{2}m_{d}\over 27\pi^{2}\\\
-8eg_{s}\over 3&-19g_{s}^{2}\over 3&g_{s}^{3}m_{d}\over
6\pi^{2}&-11g_{s}^{3}m_{d}\over 72\pi^{2}\\\ 0&0&0&-8g_{s}^{2}\over 3\\\
0&0&-12g_{s}^{2}&-14g_{s}^{2}\end{pmatrix}.$ (112)

The $\gamma_{\text{JMS}}^{dp}$ is exactly same as $\gamma_{\text{JMS}}^{dr}$.

All four-quark and dipole operators with bottom-quark in 5+3 WET
---
| $\eta_{i}^{j}(\mu_{\rm low},\mu_{\rm ew})$ | $\eta_{i}^{j}(\mu_{b},\mu_{\rm ew})$
$i\downarrow$ | $\underset{11}{C_{uG}}$ | $\underset{11}{C_{u\gamma}}$ | $\underset{11}{C_{dG}}$ | $\underset{22}{C_{dG}}$ | $\underset{11}{C_{d\gamma}}$ | $\underset{22}{C_{d\gamma}}$ | $C_{\tilde{G}}$ | $\underset{11}{C_{e\gamma}}$ | $\underset{33}{C_{dG}}$ | $\underset{33}{C_{d\gamma}}$ | $\underset{22}{C_{uG}}$ | $\underset{22}{C_{u\gamma}}$
$[C^{S1,RR}_{dd}]_{3333}$ | $4.6\cdot 10^{-8}$ | - | $9.8\cdot 10^{-8}$ | $2.0\cdot 10^{-6}$ | - | - | $-1.4\cdot 10^{-4}$ | - | $-2.6\cdot 10^{-1}$ | $1.2\cdot 10^{-2}$ | - | -
$[C^{S8,RR}_{dd}]_{3333}$ | - | - | $-2.0\cdot 10^{-8}$ | $-4.1\cdot 10^{-7}$ | - | - | $3.0\cdot 10^{-5}$ | - | $5.5\cdot 10^{-2}$ | $1.4\cdot 10^{-2}$ | - | -
$[C^{S1,RR}_{dd}]_{1133}$ | - | - | $2.6\cdot 10^{-2}$ | - | $3.6\cdot 10^{-4}$ | - | $1.3\cdot 10^{-8}$ | - | $2.4\cdot 10^{-5}$ | $1.4\cdot 10^{-7}$ | - | -
$[C^{S8,RR}_{dd}]_{1133}$ | - | - | $8.9\cdot 10^{-3}$ | - | $-1.1\cdot 10^{-3}$ | - | - | - | $8.1\cdot 10^{-6}$ | $-1.2\cdot 10^{-6}$ | - | -
$[C^{S1,RR}_{dd}]_{2233}$ | - | - | - | $2.6\cdot 10^{-2}$ | - | $3.6\cdot 10^{-4}$ | $2.6\cdot 10^{-7}$ | - | $4.8\cdot 10^{-4}$ | $2.7\cdot 10^{-6}$ | - | -
$[C^{S8,RR}_{dd}]_{2233}$ | - | - | - | $8.9\cdot 10^{-3}$ | - | $-1.1\cdot 10^{-3}$ | $8.9\cdot 10^{-8}$ | - | $1.6\cdot 10^{-4}$ | $-2.4\cdot 10^{-5}$ | - | -
$[C^{S1,RR}_{dd}]_{1331}$ | - | - | $-1.7\cdot 10^{-1}$ | - | $4.2\cdot 10^{-3}$ | - | $-8.5\cdot 10^{-8}$ | - | $-1.6\cdot 10^{-4}$ | $5.9\cdot 10^{-6}$ | - | -
$[C^{S8,RR}_{dd}]_{1331}$ | - | - | $2.2\cdot 10^{-2}$ | - | $8.2\cdot 10^{-3}$ | - | $1.1\cdot 10^{-8}$ | - | $2.0\cdot 10^{-5}$ | $8.4\cdot 10^{-6}$ | - | -
$[C^{S1,RR}_{dd}]_{2332}$ | - | - | - | $-1.7\cdot 10^{-1}$ | - | $4.2\cdot 10^{-3}$ | $-1.7\cdot 10^{-6}$ | - | $-3.2\cdot 10^{-3}$ | $1.2\cdot 10^{-4}$ | - | -
$[C^{S8,RR}_{dd}]_{2332}$ | - | - | - | $2.2\cdot 10^{-2}$ | - | $8.2\cdot 10^{-3}$ | $2.2\cdot 10^{-7}$ | - | $4.0\cdot 10^{-4}$ | $1.7\cdot 10^{-4}$ | - | -
$[C^{S1,RR}_{ud}]_{1133}$ | $2.6\cdot 10^{-2}$ | $-7.6\cdot 10^{-4}$ | - | - | - | - | - | - | $1.1\cdot 10^{-5}$ | $2.5\cdot 10^{-8}$ | - | -
$[C^{S8,RR}_{dd}]_{1133}$ | - | - | $8.9\cdot 10^{-3}$ | - | $-1.1\cdot 10^{-3}$ | - | - | - | $8.1\cdot 10^{-6}$ | $-1.2\cdot 10^{-6}$ | - | -
$[C^{S1,RR}_{uddu}]_{1331}$ | $-1.7\cdot 10^{-1}$ | $1.4\cdot 10^{-2}$ | - | - | - | - | $-4.0\cdot 10^{-8}$ | - | $-7.3\cdot 10^{-5}$ | $-8.1\cdot 10^{-6}$ | - | -
$[C^{S8,RR}_{uddu}]_{1331}$ | $2.1\cdot 10^{-2}$ | $6.9\cdot 10^{-3}$ | - | - | - | - | - | - | $9.0\cdot 10^{-6}$ | $-7.5\cdot 10^{-6}$ | - | -
$[C^{S1,RR}_{ud}]_{2233}$ | - | - | - | $-4.5\cdot 10^{-8}$ | - | - | $9.9\cdot 10^{-5}$ | - | $6.0\cdot 10^{-3}$ | $1.3\cdot 10^{-5}$ | $2.4\cdot 10^{-2}$ | $-3.2\cdot 10^{-4}$
$[C^{S8,RR}_{ud}]_{2233}$ | - | - | - | $-1.5\cdot 10^{-8}$ | - | - | $3.2\cdot 10^{-5}$ | - | $1.9\cdot 10^{-3}$ | $6.5\cdot 10^{-4}$ | $7.7\cdot 10^{-3}$ | $-1.4\cdot 10^{-3}$
$[C^{S1,RR}_{uddu}]_{2332}$ | - | - | $1.4\cdot 10^{-8}$ | $2.9\cdot 10^{-7}$ | - | - | $-6.5\cdot 10^{-4}$ | - | $-3.9\cdot 10^{-2}$ | $-4.3\cdot 10^{-3}$ | $-1.6\cdot 10^{-1}$ | $1.1\cdot 10^{-2}$
$[C^{S8,RR}_{uddu}]_{2332}$ | - | - | - | $-3.6\cdot 10^{-8}$ | - | - | $7.9\cdot 10^{-5}$ | - | $4.8\cdot 10^{-3}$ | $-4.0\cdot 10^{-3}$ | $1.9\cdot 10^{-2}$ | $7.5\cdot 10^{-3}$
$[C^{V1,LR}_{uddu}]_{1331}$ | $-5.0\cdot 10^{-2}$ | $1.1\cdot 10^{-3}$ | - | - | - | - | - | - | $-1.7\cdot 10^{-6}$ | $-2.3\cdot 10^{-6}$ | - | -
$[C^{V8,LR}_{uddu}]_{1331}$ | $1.3\cdot 10^{-2}$ | $1.2\cdot 10^{-2}$ | - | - | - | - | - | - | $2.6\cdot 10^{-6}$ | $-5.5\cdot 10^{-6}$ | - | -
$[C^{V1,LR}_{dd}]_{1331}$ | - | - | $-4.9\cdot 10^{-2}$ | - | $3.2\cdot 10^{-3}$ | - | - | - | $-3.6\cdot 10^{-6}$ | $4.1\cdot 10^{-6}$ | - | -
$[C^{V8,LR}_{dd}]_{1331}$ | - | - | $1.4\cdot 10^{-2}$ | - | $1.2\cdot 10^{-2}$ | - | - | - | $5.7\cdot 10^{-6}$ | $5.6\cdot 10^{-6}$ | - | -
$[C^{V1,LR}_{dd}]_{2332}$ | - | - | - | $-4.9\cdot 10^{-2}$ | - | $3.2\cdot 10^{-3}$ | $-3.9\cdot 10^{-8}$ | - | $-7.1\cdot 10^{-5}$ | $8.1\cdot 10^{-5}$ | - | -
$[C^{V8,LR}_{dd}]_{2332}$ | - | - | - | $1.4\cdot 10^{-2}$ | - | $1.2\cdot 10^{-2}$ | $6.1\cdot 10^{-8}$ | - | $1.1\cdot 10^{-4}$ | $1.1\cdot 10^{-4}$ | - | -
$[C^{V1,LR}_{uddu}]_{2332}$ | - | - | - | - | - | - | $-1.8\cdot 10^{-4}$ | - | $-8.6\cdot 10^{-4}$ | $-1.2\cdot 10^{-3}$ | $-4.5\cdot 10^{-2}$ | $2.9\cdot 10^{-4}$
$[C^{V8,LR}_{uddu}]_{2332}$ | - | - | - | - | - | - | $5.0\cdot 10^{-5}$ | - | $1.3\cdot 10^{-3}$ | $-2.7\cdot 10^{-3}$ | $1.3\cdot 10^{-2}$ | $1.2\cdot 10^{-2}$
$[C_{dG}]_{33}$ | $-2.2\cdot 10^{-7}$ | - | $-4.7\cdot 10^{-7}$ | $-9.5\cdot 10^{-6}$ | - | $-4.2\cdot 10^{-8}$ | $6.9\cdot 10^{-4}$ | - | $1.3\cdot 10^{0}$ | $3.2\cdot 10^{-2}$ | - | -
$[C_{d\gamma}]_{33}$ | - | - | - | $-1.9\cdot 10^{-7}$ | - | - | $1.3\cdot 10^{-5}$ | - | $2.5\cdot 10^{-2}$ | $9.0\cdot 10^{-1}$ | - | -
| $\underset{11}{C_{uG}}$ | $\underset{11}{C_{u\gamma}}$ | $\underset{11}{C_{dG}}$ | $\underset{22}{C_{dG}}$ | $\underset{11}{C_{d\gamma}}$ | $\underset{22}{C_{d\gamma}}$ | $C_{\tilde{G}}$ | $\underset{11}{C_{e\gamma}}$ | $\underset{33}{C_{dG}}$ | $\underset{33}{C_{d\gamma}}$ | $\underset{22}{C_{uG}}$ | $\underset{22}{C_{u\gamma}}$

Table 4: Operator mixing of the bottom quark operators in $5+3$ WET (1st
column) onto dipole operators (2nd row) in $3+1$ and $5+3$ flavor WET is
shown. Here the entries represent the quantity $\eta_{i}^{j}(\mu_{\rm
low},\mu_{\rm ew})$, $\mu_{\rm ew}=91.1876$ GeV, $\mu_{\rm low}=2.0$ GeV .
Entries below $10^{-8}$ are dropped. Note that for the charm and bottom
dipoles, we stop running at $\mu=m_{b}=4.18$ GeV.

####  4.4.4 ${C_{\\!\underset{prrp}{ee}}^{{\tiny V,LR}}}$ onto dipole

The leptonic ${C_{\\!\underset{prrp}{ee}}^{{\tiny V,LR}}}$ operator can break
CP is $p\neq r$. In this case, we have two vectors, namely
$\vec{\mathcal{C}}_{er}=\\{{C_{\\!\underset{pp}{e\gamma}}},{C_{\\!\underset{prrp}{ee}}^{{\tiny
V,LR}}}\\}$ and
$\vec{\mathcal{C}}_{ep}=\\{{C_{\\!\underset{rr}{e\gamma}}},{C_{\\!\underset{prrp}{ee}}^{{\tiny
V,LR}}}\\}$ . One example of the 2-loop Feynman diagrams governing this mixing
is given in diagram (e) of Fig. 3. In the 2-loop ADM (104) one has to make
replacements $q_{u}\to q_{e}$ and $q_{d}\to q_{e}$. However, since these are
purely leptonic operators which exhibit only QED mixing we need to replace
gluons in these diagrams with photons. So we need to take out the color
factors in the original result. We found that the 2-loop $P$-type diagrams
(which are the only ones allowed for the pure leptonic operators) shown in
Fig. 4 of Ciuchini:1993ks has a universal color factor
$C_{F}=(T^{A}T^{A})_{ij}=\delta_{ij}(N^{2}-1)/2N$ .

Therefore, we need to divide the elements of ADM by this overall color factor
$C_{F}$. With all these adjustments, we arrived at the RG equation for
${C_{\\!\underset{pp}{e\gamma}}}$

${16\pi^{2}\mu}({d{C_{\\!\underset{pp}{e\gamma}}}/d\mu})={15e^{3}m_{r}Q_{e}\over
12\pi^{2}}{C_{\\!\underset{prrp}{ee}}^{{\tiny V,LR}}}.$ (113)

Trivially, the same RGE holds for ${C_{\\!\underset{rr}{e\gamma}}}$.

All operators with charm-quark in 4+1 WET
---
$\eta_{i}^{j}(\mu_{\rm low},\mu_{\rm ew})$
$i\downarrow$ | $[C_{uG}]_{11}$ | $[C_{u\gamma}]_{11}$ | $[C_{dG}]_{11}$ | $[C_{dG}]_{22}$ | $[C_{d\gamma}]_{11}$ | $[C_{d\gamma}]_{22}$ | $C_{\tilde{G}}$ | $[C_{e\gamma}]_{11}$ | $[C_{uG}]_{22}$ | $[C_{u\gamma}]_{22}$
$[C^{S1,RR}_{uu}]_{2222}$ | - | - | - | - | - | - | $-3.7\cdot 10^{-4}$ | - | $-1.0\cdot 10^{-1}$ | $-7.0\cdot 10^{-3}$
$[C^{S8,RR}_{uu}]_{2222}$ | - | - | - | - | - | - | $8.5\cdot 10^{-5}$ | - | $2.3\cdot 10^{-2}$ | $-8.5\cdot 10^{-3}$
$[C^{S1,RR}_{uu}]_{1122}$ | $1.3\cdot 10^{-2}$ | $-8.8\cdot 10^{-5}$ | - | - | - | - | $9.2\cdot 10^{-8}$ | - | $2.5\cdot 10^{-5}$ | $-1.6\cdot 10^{-7}$
$[C^{S8,RR}_{uu}]_{1122}$ | $4.4\cdot 10^{-3}$ | $9.7\cdot 10^{-4}$ | - | - | - | - | $3.0\cdot 10^{-8}$ | - | $8.2\cdot 10^{-6}$ | $1.8\cdot 10^{-6}$
$[C^{S1,RR}_{ud}]_{2211}$ | - | - | $1.3\cdot 10^{-2}$ | - | $8.4\cdot 10^{-5}$ | - | $2.0\cdot 10^{-7}$ | - | $5.4\cdot 10^{-5}$ | $-9.5\cdot 10^{-7}$
$[C^{S8,RR}_{ud}]_{2211}$ | - | - | $4.0\cdot 10^{-3}$ | - | $1.1\cdot 10^{-3}$ | - | $5.9\cdot 10^{-8}$ | - | $1.6\cdot 10^{-5}$ | $-2.4\cdot 10^{-6}$
$[C^{S1,RR}_{ud}]_{2222}$ | - | - | - | $1.3\cdot 10^{-2}$ | - | $8.4\cdot 10^{-5}$ | $4.0\cdot 10^{-6}$ | - | $1.1\cdot 10^{-3}$ | $-1.9\cdot 10^{-5}$
$[C^{S8,RR}_{ud}]_{2222}$ | - | - | - | $4.0\cdot 10^{-3}$ | - | $1.1\cdot 10^{-3}$ | $1.2\cdot 10^{-6}$ | - | $3.3\cdot 10^{-4}$ | $-4.9\cdot 10^{-5}$
$[C^{S1,RR}_{uddu}]_{2112}$ | - | - | $-6.4\cdot 10^{-2}$ | - | $-6.1\cdot 10^{-3}$ | - | $-9.4\cdot 10^{-7}$ | - | $-2.6\cdot 10^{-4}$ | $1.7\cdot 10^{-5}$
$[C^{S8,RR}_{uddu}]_{2112}$ | - | - | $6.9\cdot 10^{-3}$ | - | $-4.9\cdot 10^{-3}$ | - | $1.0\cdot 10^{-7}$ | - | $2.7\cdot 10^{-5}$ | $9.0\cdot 10^{-6}$
$[C^{S1,RR}_{uddu}]_{2222}$ | - | - | - | $-6.4\cdot 10^{-2}$ | - | $-6.1\cdot 10^{-3}$ | $-1.9\cdot 10^{-5}$ | - | $-5.2\cdot 10^{-3}$ | $3.5\cdot 10^{-4}$
$[C^{S8,RR}_{uddu}]_{2222}$ | - | - | - | $6.9\cdot 10^{-3}$ | - | $-4.9\cdot 10^{-3}$ | $2.0\cdot 10^{-6}$ | - | $5.5\cdot 10^{-4}$ | $1.8\cdot 10^{-4}$
$[C^{S1,RR}_{uu}]_{1221}$ | $-6.4\cdot 10^{-2}$ | $-3.4\cdot 10^{-3}$ | - | - | - | - | $-4.4\cdot 10^{-7}$ | - | $-1.2\cdot 10^{-4}$ | $-6.3\cdot 10^{-6}$
$[C^{S8,RR}_{uu}]_{1221}$ | $7.2\cdot 10^{-3}$ | $-5.2\cdot 10^{-3}$ | - | - | - | - | $4.9\cdot 10^{-8}$ | - | $1.4\cdot 10^{-5}$ | $-9.8\cdot 10^{-6}$
$[C^{V1,LR}_{uddu}]_{2112}$ | - | - | $-1.3\cdot 10^{-2}$ | - | $-2.3\cdot 10^{-3}$ | - | $-1.7\cdot 10^{-8}$ | - | $-4.8\cdot 10^{-6}$ | $2.0\cdot 10^{-6}$
$[C^{V8,LR}_{uddu}]_{2112}$ | - | - | $4.3\cdot 10^{-3}$ | - | $-8.1\cdot 10^{-3}$ | - | $3.4\cdot 10^{-8}$ | - | $9.2\cdot 10^{-6}$ | $9.2\cdot 10^{-6}$
$[C^{V1,LR}_{uddu}]_{2222}$ | - | - | - | $-1.3\cdot 10^{-2}$ | - | $-2.3\cdot 10^{-3}$ | $-3.4\cdot 10^{-7}$ | - | $-9.4\cdot 10^{-5}$ | $4.0\cdot 10^{-5}$
$[C^{V8,LR}_{uddu}]_{2222}$ | - | - | - | $4.3\cdot 10^{-3}$ | - | $-8.1\cdot 10^{-3}$ | $6.7\cdot 10^{-7}$ | - | $1.8\cdot 10^{-4}$ | $1.8\cdot 10^{-4}$
$[C^{V1,LR}_{uu}]_{1221}$ | $-1.3\cdot 10^{-2}$ | $-3.3\cdot 10^{-3}$ | - | - | - | - | - | - | $-1.9\cdot 10^{-6}$ | $-6.1\cdot 10^{-6}$
$[C^{V8,LR}_{uu}]_{1221}$ | $4.5\cdot 10^{-3}$ | $-8.0\cdot 10^{-3}$ | - | - | - | - | $1.8\cdot 10^{-8}$ | - | $5.0\cdot 10^{-6}$ | $-8.3\cdot 10^{-6}$
$[C_{u\gamma}]_{22}$ | - | - | - | - | - | - | $-4.9\cdot 10^{-4}$ | - | $-1.4\cdot 10^{-1}$ | $8.7\cdot 10^{-1}$
$[C_{uG}]_{22}$ | - | - | - | - | - | - | $5.1\cdot 10^{-3}$ | - | $1.4\cdot 10^{0}$ | $-8.7\cdot 10^{-2}$
$[C^{S,RR}_{eu}]_{1122}$ | - | - | - | - | - | - | - | $-1.5\cdot 10^{-4}$ | - | $5.8\cdot 10^{-8}$
$[C^{T,RR}_{eu}]_{1122}$ | - | - | - | - | - | - | - | $7.4\cdot 10^{-2}$ | $1.5\cdot 10^{-6}$ | $-3.0\cdot 10^{-5}$
| $[C_{uG}]_{11}$ | $[C_{u\gamma}]_{11}$ | $[C_{dG}]_{11}$ | $[C_{dG}]_{22}$ | $[C_{d\gamma}]_{11}$ | $[C_{d\gamma}]_{22}$ | $C_{\tilde{G}}$ | $[C_{e\gamma}]_{11}$ | $[C_{uG}]_{22}$ | $[C_{u\gamma}]_{22}$

Table 5: Operator mixing of the charm quark operators in $4+1$ WET (1st
column) onto dipole operators (2nd row) in $3+1$ and $4+1$ flavor WET is
shown. Here the entries represent the quantity $\eta_{i}^{j}(\mu_{\rm
low},\mu_{\rm ew})$, $\mu_{\rm ew}=91.1876$ GeV, $\mu_{\rm low}=2.0$ GeV.
Entries below $10^{-8}$ are dropped.

###  4.5 Numerical solution for the RGEs

In this section, we provide the numerical solution to the RGEs, which allows
us to quantify the impact of such effects on the low-energy operators that
contribute to EDMs at the tree level. As we have learned in the previous
subsections, 1-loop and 2-loop RG running effects and threshold effects cause
operator mixing leading to the expansion of operator basis for the EDMs.

All semileptonic operators in 5+3 WET
---
| $\eta_{i}^{j}(\mu_{\rm low},\mu_{\rm ew})$ | $\eta_{i}^{j}(\mu_{b},\mu_{\rm ew})$
$i\downarrow$ | $\underset{11}{C_{uG}}$ | $\underset{11}{C_{u\gamma}}$ | $\underset{11}{C_{dG}}$ | $\underset{22}{C_{dG}}$ | $\underset{11}{C_{d\gamma}}$ | $\underset{22}{C_{d\gamma}}$ | $C_{\tilde{G}}$ | $\underset{11}{C_{e\gamma}}$ | $\underset{33}{C_{dG}}$ | $\underset{33}{C_{d\gamma}}$ | $\underset{22}{C_{uG}}$ | $\underset{22}{C_{u\gamma}}$
$[C^{S,RR}_{eu}]_{1111}$ | - | $5.8\cdot 10^{-8}$ | - | - | - | - | - | $-2.9\cdot 10^{-7}$ | - | - | - | -
$[C^{S,RR}_{ed}]_{1111}$ | - | - | - | - | $-2.9\cdot 10^{-8}$ | - | - | $-1.5\cdot 10^{-7}$ | - | - | - | -
$[C^{S,RR}_{ed}]_{1122}$ | - | - | - | - | - | $-2.9\cdot 10^{-8}$ | - | $-3.1\cdot 10^{-6}$ | - | - | - | -
$[C^{S,RL}_{eu}]_{1111}$ | - | - | - | - | - | - | - | - | - | - | - | -
$[C^{S,RL}_{ed}]_{1111}$ | - | - | - | - | - | - | - | - | - | - | - | -
$[C^{S,RL}_{ed}]_{1122}$ | - | - | - | - | - | - | - | - | - | - | - | -
$[C^{T,RR}_{eu}]_{1111}$ | $1.5\cdot 10^{-6}$ | $-3.0\cdot 10^{-5}$ | - | - | - | - | - | $1.4\cdot 10^{-4}$ | - | - | - | -
$[C^{T,RR}_{ed}]_{1111}$ | - | - | $-5.8\cdot 10^{-7}$ | - | $-3.0\cdot 10^{-5}$ | - | - | $-1.5\cdot 10^{-4}$ | - | - | - | -
$[C^{T,RR}_{ed}]_{1122}$ | - | - | - | $-5.8\cdot 10^{-7}$ | - | $-3.0\cdot 10^{-5}$ | - | $-3.0\cdot 10^{-3}$ | - | - | - | -
$[C^{T,RR}_{eu}]_{1122}$ | - | - | - | - | - | - | - | $7.4\cdot 10^{-2}$ | - | - | $6.7\cdot 10^{-7}$ | $-2.6\cdot 10^{-5}$
$[C^{S,RR}_{eu}]_{1122}$ | - | - | - | - | - | - | - | $-1.5\cdot 10^{-4}$ | - | - | - | $4.0\cdot 10^{-8}$
$[C^{T,RR}_{ed}]_{1133}$ | - | - | - | - | - | - | - | $-1.1\cdot 10^{-1}$ | $-3.4\cdot 10^{-7}$ | $-2.6\cdot 10^{-5}$ | - | -
$[C^{T,RR}_{ed}]_{2211}$ | - | - | $-1.2\cdot 10^{-4}$ | - | $-6.3\cdot 10^{-3}$ | - | - | - | - | - | - | -
$[C^{T,RR}_{ed}]_{2222}$ | - | - | - | $-1.2\cdot 10^{-4}$ | - | $-6.3\cdot 10^{-3}$ | - | - | - | - | - | -
$[C^{T,RR}_{ed}]_{2233}$ | - | - | - | - | - | - | $-3.8\cdot 10^{-8}$ | - | $-7.0\cdot 10^{-5}$ | $-5.4\cdot 10^{-3}$ | - | -
$[C^{T,RR}_{ed}]_{3311}$ | - | - | $-2.0\cdot 10^{-3}$ | - | $-1.1\cdot 10^{-1}$ | - | - | - | - | - | - | -
$[C^{T,RR}_{ed}]_{3322}$ | - | - | - | $-2.0\cdot 10^{-3}$ | - | $-1.1\cdot 10^{-1}$ | - | - | - | - | - | -
$[C^{T,RR}_{ed}]_{3333}$ | - | - | - | - | - | - | $-6.4\cdot 10^{-7}$ | - | $-1.2\cdot 10^{-3}$ | $-9.0\cdot 10^{-2}$ | - | -
$[C^{T,RR}_{eu}]_{2211}$ | $3.1\cdot 10^{-4}$ | $-6.3\cdot 10^{-3}$ | - | - | - | - | - | - | - | - | - | -
$[C^{T,RR}_{eu}]_{2222}$ | - | - | - | - | - | - | $1.1\cdot 10^{-6}$ | - | - | - | $1.4\cdot 10^{-4}$ | $-5.3\cdot 10^{-3}$
$[C^{T,RR}_{eu}]_{3311}$ | $5.3\cdot 10^{-3}$ | $-1.1\cdot 10^{-1}$ | - | - | - | - | - | - | - | - | - | -
$[C^{T,RR}_{eu}]_{3322}$ | - | - | - | - | - | - | $1.9\cdot 10^{-5}$ | - | - | - | $2.3\cdot 10^{-3}$ | $-9.0\cdot 10^{-2}$
$[C^{S,RR}_{ed}]_{1133}$ | - | - | - | - | - | - | - | $-8.1\cdot 10^{-5}$ | - | $-2.0\cdot 10^{-8}$ | - | -
$[C^{S,RR}_{ed}]_{2211}$ | - | - | $-3.6\cdot 10^{-8}$ | - | $-6.0\cdot 10^{-6}$ | - | - | - | - | - | - | -
$[C^{S,RR}_{ed}]_{2222}$ | - | - | - | $-3.6\cdot 10^{-8}$ | - | $-6.0\cdot 10^{-6}$ | - | - | - | - | - | -
$[C^{S,RR}_{ed}]_{2233}$ | - | - | - | - | - | - | - | - | - | $-4.1\cdot 10^{-6}$ | - | -
$[C^{S,RR}_{ed}]_{3311}$ | - | - | $-6.0\cdot 10^{-7}$ | - | $-1.0\cdot 10^{-4}$ | - | - | - | - | - | - | -
$[C^{S,RR}_{ed}]_{3322}$ | - | - | - | $-6.0\cdot 10^{-7}$ | - | $-1.0\cdot 10^{-4}$ | - | - | - | - | - | -
$[C^{S,RR}_{ed}]_{3333}$ | - | - | - | - | - | - | - | - | - | $-6.9\cdot 10^{-5}$ | - | -
$[C^{S,RR}_{eu}]_{2211}$ | $-2.6\cdot 10^{-7}$ | $1.2\cdot 10^{-5}$ | - | - | - | - | - | - | - | - | - | -
$[C^{S,RR}_{eu}]_{2222}$ | - | - | - | - | - | - | - | - | - | - | - | $8.2\cdot 10^{-6}$
$[C^{S,RR}_{eu}]_{3311}$ | $-4.4\cdot 10^{-6}$ | $2.0\cdot 10^{-4}$ | - | - | - | - | - | - | - | - | - | -
$[C^{S,RR}_{eu}]_{3322}$ | - | - | - | - | - | - | $-1.6\cdot 10^{-8}$ | - | - | - | - | $1.4\cdot 10^{-4}$
$[C^{V,LR}_{ee}]_{1221}$ | - | - | - | - | - | - | - | $-2.0\cdot 10^{-4}$ | - | - | - | -
$[C^{V,LR}_{ee}]_{1331}$ | - | - | - | - | - | - | - | $-3.2\cdot 10^{-3}$ | - | - | - | -
$[C^{S,RR}_{ee}]_{1221}$ | - | - | - | - | - | - | - | $6.3\cdot 10^{-4}$ | - | - | - | -
$[C^{S,RR}_{ee}]_{1331}$ | - | - | - | - | - | - | - | $1.1\cdot 10^{-2}$ | - | - | - | -
$[C^{S,RR}_{ee}]_{1122}$ | - | - | - | - | - | - | - | $-1.3\cdot 10^{-5}$ | - | - | - | -
$[C^{S,RR}_{ee}]_{1133}$ | - | - | - | - | - | - | - | $-2.2\cdot 10^{-4}$ | - | - | - | -
| $\underset{11}{C_{uG}}$ | $\underset{11}{C_{u\gamma}}$ | $\underset{11}{C_{dG}}$ | $\underset{22}{C_{dG}}$ | $\underset{11}{C_{d\gamma}}$ | $\underset{22}{C_{d\gamma}}$ | $C_{\tilde{G}}$ | $\underset{11}{C_{e\gamma}}$ | $\underset{33}{C_{dG}}$ | $\underset{33}{C_{d\gamma}}$ | $\underset{22}{C_{uG}}$ | $\underset{22}{C_{u\gamma}}$

Table 6: Operator mixing of the semileptonic operators in $5+3$ WET (1st
column) onto dipole operators (2nd row) in $3+1$ and $5+3$ flavor WET is
shown. Here the entries represent the quantity $\eta_{i}^{j}(\mu_{\rm
low},\mu_{\rm ew})$, $\mu_{\rm ew}=91.1876$ GeV, $\mu_{\rm low}=2.0$ GeV .
Entries below $10^{-8}$ are dropped. Note that for the charm and bottom
dipoles, we stop running at $\mu=m_{b}=4.18$ GeV.

In Tabs. 4, 5 and 6, we show the numerical values of the quantity

$\eta_{i}^{j}(\mu_{\rm low},\mu_{\rm
ew})={\text{Im}C_{j}(\mu_{low})\over\text{Im}C_{i}(\mu_{ew})}$ (114)

for $\mu_{low}=2$ GeV and $\mu_{ew}=91$ GeV. In these tables, we focus mainly
on heavy flavor operators, that do not generate EDMs at tree level. In this
case, $C_{j}(\mu_{low})$ stands for the WCs of the dipole and Weinberg
operators that enter at tree-level to the EDM expressions. The
$C_{i}(\mu_{ew})$ represent the all other WET operators which mix with
$C_{j}(\mu_{low})$ via any of the mechanism discussed above. In the 4+1 and
5+3 theories we only show the operators that contain at least one heavy flavor
fermion current. Running effects, in particular QCD running, are also
important for light flavor operators, and they are fully accounted for in the
master formulae that we provide in Section 5.

##  5 EDM Master Formulae in WET

In this section, we derive master formulae for the EDMs for various species
such as electron (via the HfF, ThO and YbF precession frequencies), neutron,
and proton as well as diamagnetic atoms such as Hg, Xe, and Ra in terms of WET
WCs at the EW scale in the JMS basis. By matching the WET to SMEFT and HEFT
these formulae can be used to get the EDMs predictions in a large class of UV
scenarios with linear and non-linear realizations of EW symmetry breaking,
respectively. The main strengths of these formulae are the systematic
inclusion of short-distance effects due to RG running and matching and the
inclusion of the heavy flavor for the quark and leptons which are novel.

Without loss of generality, the EDMs can be expressed as a linear function of
WET WCs as

$d_{X}=\sum_{I\in\rm
dipoles}\alpha_{I}^{X}(\mu_{ew}){C_{I}^{(5)}(\mu_{ew})\left[{\rm
TeV^{-1}}\right]}+\sum_{J\in\rm
4f}\alpha_{J}^{X}(\mu_{ew}){C_{J}^{(6)}(\mu_{ew})\left[{\rm{TeV^{-2}}}\right]}\,,$
(115)

here $X=n,p$ or Hg, Xe, Ra and the indices $I$ and $J$ run over all
contributing dipole and four-fermion operators in the $n_{f}=5+3$ $\Delta F=0$
WET at the EW scale. The complete list of relevant operators can be found in
Sec. 2. The coefficients $\alpha_{I}$ encode the long-distance effects due to
matrix elements and RG running at the 2-loop level as described in the
previous section. These coefficients have units of $e\,\rm cm\,{TeV}^{2}$ or
$e\,\rm cm\,TeV$ for the dim-6 and dim-5 WCs, respectively. The WCs in (115)
have $\rm{TeV}^{-2}$ and TeV-1 units for dim-6 and dim-5 operators. This is
indicated by the units in square brackets in Eq. (115). In this way, the
overall units of $d_{X}$ are $e$ cm. To obtain the EDMs in the $\rm{GeV}^{-1}$
units we need to divide the $\alpha_{I}$ coefficients by a conversion factor
according to ${\rm GeV}^{-1}=6.52\times 10^{-14}e\,\textrm{cm}$.

We can write very similar expressions for the frequencies for the electron
EDM. In this case, the $\alpha_{I}^{X}$ coefficients will have
(mrad/s)$\rm{TeV}^{2}$ or (mrad/s)TeV units. The values for $\alpha_{I}$
coefficients at the EW scale and their theoretical uncertainties for the HfF
precession frequency, the Hg and neutron EDMs are given in Tab. 7, 8, 9 and
10. The master formulae for $d_{p}$, $\omega_{\rm ThO}$ and $\omega_{\rm YbF}$
can be found in appendix A. We do not provide explicit master formulae for
$d_{\rm Xe}$ and $d_{\rm Ra}$, but they have been implemented in flavio.

###  5.1 Theoretical uncertainties

The theoretical uncertainties in the $\alpha_{I}^{X}$ coefficients can stem
from various sources such as the SM parameters and the matrix elements. The
latter arise from both nuclear theory and hadronization uncertainties and, for
most species, they are the dominant source. The uncertainties from atomic
theory are in most cases better understood. As discussed in Section 3, the
quoted hadronic and nuclear uncertainties provide at best a very rough
estimate of the real theoretical uncertainty. In this respect, the field is
less advanced compared to flavor physics, where, for several observables,
lattice QCD matrix elements have been evaluated, or collider physics. Here we
assume theoretical uncertainties to be Gaussian distributed. The estimate of
the theoretical uncertainties is based on the random sampling of the input
parameters and the evaluation of the standard deviation of the corresponding
EDM observables.

###  5.2 $\omega_{\text{HfF}}$

In this subsection, we provide a master formula for $\omega_{\text{HfF}}$
(mrad/s), which is sensitive to the electron EDM, see Tab. 7. Similar formulae
for the $\omega_{\text{ThO}}$ and $\omega_{\text{YbF}}$ are given in App. A.

$\alpha_{I}^{\rm HfF}(\mu_{ew})$ for the $\omega_{\text{HfF}}$
---
$n_{f}=3+1$ | $n_{f}=5+3$
$[C_{e\gamma}]_{11}$ | $(-2.9\pm 0.1)\cdot 10^{12}$ | $[C^{S,RR}_{eu}]_{1122}$ | $(9.2\pm 0.3)\cdot 10^{5}$
$[C^{S,RR}_{ee}]_{1111}$ | $(-2.7\pm 0.1)\cdot 10^{4}$ | $[C^{T,RR}_{eu}]_{1122}$ | $(-3.8\pm 0.2)\cdot 10^{8}$
$[C^{S,RR}_{eu}]_{1111}$ | $(-2.6\pm 0.2)\cdot 10^{8}$ | $[C^{S,RR}_{ed}]_{1133}$ | $(3.7\pm 0.1)\cdot 10^{5}$
$[C^{S,RR}_{ed}]_{1111}$ | $(-2.6\pm 0.3)\cdot 10^{8}$ | $[C^{T,RR}_{ed}]_{1133}$ | $(4.8\pm 0.2)\cdot 10^{8}$
$[C^{S,RR}_{ed}]_{1122}$ | $(-1.9\pm 0.1)\cdot 10^{7}$ | $[C^{S,RR}_{ee}]_{1122}$ | $(5.8\pm 0.2)\cdot 10^{4}$
$[C^{S,RL}_{eu}]_{1111}$ | $(-2.6\pm 0.3)\cdot 10^{8}$ | $[C^{S,RR}_{ee}]_{1133}$ | $(1.3\pm 0.0)\cdot 10^{6}$
$[C^{S,RL}_{ed}]_{1111}$ | $(-2.6\pm 0.2)\cdot 10^{8}$ | $[C^{S,RR}_{ee}]_{1221}$ | $(-2.8\pm 0.1)\cdot 10^{6}$
$[C^{S,RL}_{ed}]_{1122}$ | $(-1.9\pm 0.1)\cdot 10^{7}$ | $[C^{S,RR}_{ee}]_{1331}$ | $(-5.5\pm 0.2)\cdot 10^{7}$
$[C^{T,RR}_{eu}]_{1111}$ | $(2.3\pm 0.1)\cdot 10^{8}$ | $[C^{V,LR}_{ee}]_{1221}$ | $(9.1\pm 0.4)\cdot 10^{5}$
$[C^{T,RR}_{ed}]_{1111}$ | $(-1.2\pm 0.0)\cdot 10^{8}$ | $[C^{V,LR}_{ee}]_{1331}$ | $(1.5\pm 0.1)\cdot 10^{7}$
$[C^{T,RR}_{ed}]_{1122}$ | $(-9.0\pm 0.4)\cdot 10^{7}$ |  | -

Table 7: Master formula for $\omega_{\rm HfF}$ in WET in the JMS basis at the
EW scale. The $\alpha_{I}^{\rm HfF}$ coefficients have units of (mrad/s)$\rm
TeV^{2}$ and (mrad/s)$\rm TeV$ for the four-fermion and dipole operators,
respectively. The $90\%$ CL upper bound on $\omega_{\rm HfF}$ is $0.17$
mrad/s.

Recalling that the $90\%$ CL upper bound on $\omega_{\rm HfF}$ is $0.17$
mrad/s, we can immediately see that the scale of the operators that contribute
at tree level, which comprises all the operators in the $n_{f}=3+1$ column of
Table 7 with the exception of $\big{[}{C}_{ee}^{S,RR}\big{]}_{1111}$, has to
be very high, $\Lambda\sim 10^{4}$ TeV. Even for
$\big{[}{C}_{ee}^{S,RR}\big{]}_{1111}$, which mixes onto $C_{e\gamma}$ at one
loop, $\Lambda$ has to be close to $400$ TeV. The $n_{f}=5+3$ column in Table
7 contains operators with electrons and heavy quarks or electrons and muons or
taus. All these operators are very strongly constrained.

###  5.3  $d_{\rm Hg}$

The master formulae for $d_{\rm Hg}$ are presented in Tab. 8 and Tab. 9 for
the semileptonic and four-quark operators, respectively. From Table 9 we
notice that for all the four-quark operators in Table 1, the central value of
$d_{\rm Hg}$ is larger than the experimental bound, implying that, even in the
worst case scenario, Hg EDM experiments probe four-quark scales of $1$ TeV or
larger. This is true even for four-quark operators with four heavy quarks,
such as $\big{[}{C}_{uddu}^{V1,LR}\big{]}_{2332}$ (with two $b$ and two $c$
quarks) or $\big{[}{C}_{dd}^{S8,RR}\big{]}_{3333}$ (with four $b$ quarks). The
scale of light flavor operators needs in general to be much larger than the
TeV. This picture is complicated by theoretical uncertainties, which are
generally very large.

$\alpha_{I}^{\rm Hg}(\mu_{ew})$ for the Hg EDM: SL operators
---
$n_{f}=3+1$ | $n_{f}=4+1$ | $n_{f}=5+3$
$[C^{S,RR}_{eu}]_{1111}$ | $(8.3\pm 1.5)\cdot 10^{-22}$ | $[C^{T,RR}_{eu}]_{1122}$ | - | $[C^{T,RR}_{ed}]_{1133}$ | -
$[C^{S,RR}_{ed}]_{1111}$ | $(-3.8\pm 1.3)\cdot 10^{-22}$ | $[C^{S,RR}_{eu}]_{1122}$ | - | $[C^{T,RR}_{ed}]_{2211}$ | $(-2.6\pm 0.7)\cdot 10^{-25}$
$[C^{S,RR}_{ed}]_{1122}$ | $(1.7\pm 0.4)\cdot 10^{-23}$ |  | - | $[C^{T,RR}_{ed}]_{2222}$ | $(1.0\pm 0.7)\cdot 10^{-27}$
$[C^{S,RL}_{eu}]_{1111}$ | $(-3.8\pm 1.4)\cdot 10^{-22}$ |  | - | $[C^{T,RR}_{ed}]_{2233}$ | -
$[C^{S,RL}_{ed}]_{1111}$ | $(8.3\pm 1.6)\cdot 10^{-22}$ |  | - | $[C^{T,RR}_{ed}]_{3311}$ | $(-4.8\pm 1.4)\cdot 10^{-24}$
$[C^{S,RL}_{ed}]_{1122}$ | $(1.7\pm 0.4)\cdot 10^{-23}$ |  | - | $[C^{T,RR}_{ed}]_{3322}$ | $(1.9\pm 1.3)\cdot 10^{-26}$
$[C^{T,RR}_{eu}]_{1111}$ | $(-3.7\pm 0.9)\cdot 10^{-23}$ |  | - | $[C^{T,RR}_{ed}]_{3333}$ | $(3.5\pm 2.1)\cdot 10^{-31}$
$[C^{T,RR}_{ed}]_{1111}$ | $(-2.4\pm 0.6)\cdot 10^{-22}$ |  | - | $[C^{T,RR}_{eu}]_{2211}$ | $(5.0\pm 8.0)\cdot 10^{-26}$
$[C^{T,RR}_{ed}]_{1122}$ | $(1.7\pm 0.6)\cdot 10^{-24}$ |  | - | $[C^{T,RR}_{eu}]_{2222}$ | $(-5.9\pm 3.8)\cdot 10^{-31}$
| - |  | - | $[C^{T,RR}_{eu}]_{3311}$ | $(9.1\pm 12.1)\cdot 10^{-25}$
| - |  | - | $[C^{T,RR}_{eu}]_{3322}$ | $(-9.9\pm 5.7)\cdot 10^{-30}$
| - |  | - | $[C^{S,RR}_{ed}]_{1133}$ | -
| - |  | - | $[C^{S,RR}_{ed}]_{2211}$ | $(-2.6\pm 0.6)\cdot 10^{-28}$
| - |  | - | $[C^{S,RR}_{ed}]_{2222}$ | $(1.0\pm 0.6)\cdot 10^{-30}$
| - |  | - | $[C^{S,RR}_{ed}]_{2233}$ | -
| - |  | - | $[C^{S,RR}_{ed}]_{3311}$ | $(-5.2\pm 1.4)\cdot 10^{-27}$
| - |  | - | $[C^{S,RR}_{ed}]_{3322}$ | $(2.0\pm 1.3)\cdot 10^{-29}$
| - |  | - | $[C^{S,RR}_{ed}]_{3333}$ | -
| - |  | - | $[C^{S,RR}_{eu}]_{2211}$ | $(-8.9\pm 5.2)\cdot 10^{-29}$
| - |  | - | $[C^{S,RR}_{eu}]_{2222}$ | -
| - |  | - | $[C^{S,RR}_{eu}]_{3311}$ | $(-1.8\pm 1.0)\cdot 10^{-27}$
| - |  | - | $[C^{S,RR}_{eu}]_{3322}$ | -

Table 8: Master formula for $d_{\rm Hg}$ in terms of semileptonic WET
operators in the JMS basis at the EW scale. The $\alpha_{I}^{\rm Hg}$ have
units of $e\,\rm cm\,\rm TeV^{2}$. The $90\%$ CL upper bound on $d_{\rm Hg}$
is $6.2\times 10^{-30}$ $e$ cm. The entries below $10^{-33}$ have been
dropped.

$\alpha_{I}^{\rm Hg}(\mu_{ew})$ for the Hg EDM
---
$n_{f}=3+1$ | $n_{f}=4+1$ | $n_{f}=5+3$
$C_{\widetilde{G}}$ | $(-2.4\pm 3.9)\cdot 10^{-25}$ | $[C_{uG}]_{22}$ | $(-2.6\pm 1.5)\cdot 10^{-24}$ | $[C_{dG}]_{33}$ | $(-3.8\pm 2.6)\cdot 10^{-25}$
$[C_{dG}]_{11}$ | $(5.0\pm 28.3)\cdot 10^{-20}$ | $[C_{u\gamma}]_{22}$ | $(2.5\pm 1.6)\cdot 10^{-25}$ | $[C_{d\gamma}]_{33}$ | $(-7.3\pm 5.3)\cdot 10^{-27}$
$[C_{dG}]_{22}$ | $(-6.7\pm 4.9)\cdot 10^{-24}$ | $[C^{S1,RR}_{uu}]_{2222}$ | $(1.9\pm 1.0)\cdot 10^{-28}$ | $[C^{V1,LR}_{uddu}]_{1331}$ | $(-1.4\pm 7.4)\cdot 10^{-24}$
$[C_{uG}]_{11}$ | $(4.1\pm 32.9)\cdot 10^{-20}$ | $[C^{S8,RR}_{uu}]_{2222}$ | $(-4.3\pm 2.6)\cdot 10^{-29}$ | $[C^{V8,LR}_{uddu}]_{1331}$ | $(3.2\pm 21.5)\cdot 10^{-25}$
$[C_{d\gamma}]_{11}$ | $(3.5\pm 1.3)\cdot 10^{-20}$ | $[C^{S1,RR}_{uu}]_{1122}$ | $(5.9\pm 46.4)\cdot 10^{-25}$ | $[C^{V1,LR}_{dd}]_{1331}$ | $(-1.6\pm 7.4)\cdot 10^{-24}$
$[C_{d\gamma}]_{22}$ | $(-1.3\pm 0.8)\cdot 10^{-22}$ | $[C^{S8,RR}_{uu}]_{1122}$ | $(1.7\pm 15.9)\cdot 10^{-25}$ | $[C^{V8,LR}_{dd}]_{1331}$ | $(9.5\pm 20.4)\cdot 10^{-25}$
$[C_{u\gamma}]_{11}$ | $(-9.4\pm 21.8)\cdot 10^{-21}$ | $[C^{S1,RR}_{ud}]_{2211}$ | $(7.0\pm 35.1)\cdot 10^{-25}$ | $[C^{V1,LR}_{dd}]_{2332}$ | $(-5.0\pm 3.3)\cdot 10^{-28}$
$[C^{V1,LR}_{uddu}]_{1111}$ | $(8.9\pm 30.3)\cdot 10^{-24}$ | $[C^{S8,RR}_{ud}]_{2211}$ | $(2.6\pm 12.5)\cdot 10^{-25}$ | $[C^{V8,LR}_{dd}]_{2332}$ | $(-1.8\pm 1.1)\cdot 10^{-27}$
$[C^{V8,LR}_{uddu}]_{1111}$ | $(1.4\pm 5.3)\cdot 10^{-23}$ | $[C^{S1,RR}_{ud}]_{2222}$ | $(-1.2\pm 0.6)\cdot 10^{-29}$ | $[C^{S1,RR}_{dd}]_{3333}$ | $(7.8\pm 5.0)\cdot 10^{-29}$
$[C^{V1,LR}_{uddu}]_{1221}$ | $(4.8\pm 14.4)\cdot 10^{-24}$ | $[C^{S8,RR}_{ud}]_{2222}$ | $(-2.3\pm 1.4)\cdot 10^{-28}$ | $[C^{S8,RR}_{dd}]_{3333}$ | $(-1.6\pm 1.2)\cdot 10^{-29}$
$[C^{V8,LR}_{uddu}]_{1221}$ | $(7.4\pm 23.5)\cdot 10^{-24}$ | $[C^{S1,RR}_{uddu}]_{2112}$ | $(-3.1\pm 15.0)\cdot 10^{-24}$ | $[C^{S1,RR}_{dd}]_{1133}$ | $(9.1\pm 52.2)\cdot 10^{-25}$
$[C^{V1,LR}_{dd}]_{1221}$ | $(-4.1\pm 13.9)\cdot 10^{-24}$ | $[C^{S8,RR}_{uddu}]_{2112}$ | $(5.1\pm 201.5)\cdot 10^{-26}$ | $[C^{S8,RR}_{dd}]_{1133}$ | $(2.6\pm 14.4)\cdot 10^{-25}$
$[C^{V8,LR}_{dd}]_{1221}$ | $(-6.2\pm 20.0)\cdot 10^{-24}$ | $[C^{S1,RR}_{uddu}]_{2222}$ | $(1.1\pm 0.6)\cdot 10^{-27}$ | $[C^{S1,RR}_{dd}]_{2233}$ | $(-5.6\pm 3.5)\cdot 10^{-29}$
$[C^{S1,RR}_{uu}]_{1111}$ | $(-5.9\pm 21.7)\cdot 10^{-24}$ | $[C^{S8,RR}_{uddu}]_{2222}$ | $(8.5\pm 6.3)\cdot 10^{-28}$ | $[C^{S8,RR}_{dd}]_{2233}$ | $(1.6\pm 1.1)\cdot 10^{-28}$
$[C^{S8,RR}_{uu}]_{1111}$ | $(1.8\pm 5.6)\cdot 10^{-24}$ | $[C^{S1,RR}_{uu}]_{1221}$ | $(-2.4\pm 15.3)\cdot 10^{-24}$ | $[C^{S1,RR}_{dd}]_{1331}$ | $(-5.7\pm 34.0)\cdot 10^{-24}$
$[C^{S1,RR}_{dd}]_{1111}$ | $(4.9\pm 24.4)\cdot 10^{-24}$ | $[C^{S8,RR}_{uu}]_{1221}$ | $(2.7\pm 14.9)\cdot 10^{-25}$ | $[C^{S8,RR}_{dd}]_{1331}$ | $(1.1\pm 4.2)\cdot 10^{-24}$
$[C^{S8,RR}_{dd}]_{1111}$ | $(-1.5\pm 4.6)\cdot 10^{-24}$ | $[C^{V1,LR}_{uddu}]_{2112}$ | $(-5.5\pm 22.3)\cdot 10^{-25}$ | $[C^{S1,RR}_{dd}]_{2332}$ | $(-6.4\pm 4.7)\cdot 10^{-28}$
$[C^{S1,RR}_{dd}]_{2222}$ | $(-4.4\pm 2.8)\cdot 10^{-29}$ | $[C^{V8,LR}_{uddu}]_{2112}$ | $(-1.7\pm 10.2)\cdot 10^{-25}$ | $[C^{S8,RR}_{dd}]_{2332}$ | $(-1.3\pm 0.8)\cdot 10^{-27}$
$[C^{S8,RR}_{dd}]_{2222}$ | $(-5.3\pm 3.2)\cdot 10^{-29}$ | $[C^{V1,LR}_{uddu}]_{2222}$ | $(3.6\pm 2.4)\cdot 10^{-28}$ | $[C^{S1,RR}_{ud}]_{1133}$ | $(7.7\pm 61.1)\cdot 10^{-25}$
$[C^{S1,RR}_{dd}]_{1122}$ | $(1.3\pm 5.7)\cdot 10^{-24}$ | $[C^{V8,LR}_{uddu}]_{2222}$ | $(1.2\pm 0.8)\cdot 10^{-27}$ | $[C^{S8,RR}_{ud}]_{1133}$ | $(2.6\pm 13.2)\cdot 10^{-25}$
$[C^{S8,RR}_{dd}]_{1122}$ | $(-3.7\pm 14.2)\cdot 10^{-25}$ | $[C^{V1,LR}_{uu}]_{1221}$ | $(-3.6\pm 21.9)\cdot 10^{-25}$ | $[C^{S1,RR}_{uddu}]_{1331}$ | $(-5.1\pm 24.2)\cdot 10^{-24}$
$[C^{S1,RR}_{dd}]_{1221}$ | $(1.1\pm 4.1)\cdot 10^{-24}$ | $[C^{V8,LR}_{uu}]_{1221}$ | $(1.8\pm 6.7)\cdot 10^{-25}$ | $[C^{S8,RR}_{uddu}]_{1331}$ | $(5.6\pm 33.0)\cdot 10^{-25}$
$[C^{S8,RR}_{dd}]_{1221}$ | $(-3.5\pm 13.1)\cdot 10^{-25}$ |  | - | $[C^{S1,RR}_{ud}]_{2233}$ | $(-5.1\pm 2.8)\cdot 10^{-29}$
$[C^{S1,RR}_{ud}]_{1111}$ | $(-2.3\pm 3.6)\cdot 10^{-25}$ |  | - | $[C^{S8,RR}_{ud}]_{2233}$ | $(-1.7\pm 0.9)\cdot 10^{-29}$
$[C^{S8,RR}_{ud}]_{1111}$ | $(7.0\pm 10.8)\cdot 10^{-26}$ |  | - | $[C^{S1,RR}_{uddu}]_{2332}$ | $(3.3\pm 2.0)\cdot 10^{-28}$
$[C^{S1,RR}_{ud}]_{1122}$ | $(-1.4\pm 5.1)\cdot 10^{-24}$ |  | - | $[C^{S8,RR}_{uddu}]_{2332}$ | $(-4.0\pm 1.8)\cdot 10^{-29}$
$[C^{S8,RR}_{ud}]_{1122}$ | $(4.5\pm 18.1)\cdot 10^{-25}$ |  | - | $[C^{V1,LR}_{uddu}]_{2332}$ | $(9.3\pm 4.7)\cdot 10^{-29}$
$[C^{S1,RR}_{uddu}]_{1111}$ | $(-2.4\pm 3.8)\cdot 10^{-25}$ |  | - | $[C^{V8,LR}_{uddu}]_{2332}$ | $(-2.6\pm 1.4)\cdot 10^{-29}$
$[C^{S8,RR}_{uddu}]_{1111}$ | $(7.0\pm 10.6)\cdot 10^{-26}$ |  | - |  | -
$[C^{S1,RR}_{uddu}]_{1221}$ | $(-1.6\pm 6.0)\cdot 10^{-24}$ |  | - |  | -
$[C^{S8,RR}_{uddu}]_{1221}$ | $(4.6\pm 14.6)\cdot 10^{-25}$ |  | - |  | -

Table 9: Master formula for $d_{\rm Hg}$ in WET in the JMS basis at the EW
scale. The $\alpha_{I}^{\rm Hg}$ have units of $e$ cm $\rm TeV^{2}$ and $e$ cm
$\rm TeV$ for the four-fermion and dipole operators, respectively. The $90\%$
CL upper bound on $d_{\rm Hg}$ is $6.2\times 10^{-30}$ $e$ cm.

Table 8 provides a similar message for semileptonic operators, where one can
extract strong constraints on operators such as
$\big{[}{C}_{eu}^{T,RR}\big{]}_{3311}$ (two $\tau$ leptons and two $u$ quarks)
or $\big{[}{C}_{ed}^{T,RR}\big{]}_{3311}$ (two $\tau$ leptons and two $d$
quarks). In this case, however, operators with both heavy leptons and heavy
quarks are less constrained and their scale can drop below the TeV. This is
for example the case of $\big{[}{C}_{eu}^{T,RR}\big{]}_{2222}$ (with a naive
scale of $\Lambda\sim 0.3$ TeV, neglecting theoretical uncertainties) or
$\big{[}{C}_{eu}^{S,RR}\big{]}_{3322}$, with naive scale below the
electroweak.

###  5.4 $d_{n}$

The master formula for the neutron EDM is presented in Tab. 10. As in the case
of $d_{\rm Hg}$, we notice that, if we neglect theoretical uncertainties, all
four-quark operators in Table 1 need to have scales larger than the TeV.

$\alpha_{I}^{n}(\mu_{ew})$ for neutron EDM
---
$n_{f}=3+1$ | $n_{f}=4+1$ | $n_{f}=5+3$
$C_{\widetilde{G}}$ | $(5.5\pm 2.8)\cdot 10^{-22}$ | $[C_{uG}]_{22}$ | $(7.7\pm 3.9)\cdot 10^{-21}$ | $[C_{dG}]_{33}$ | $(1.1\pm 0.5)\cdot 10^{-21}$
$[C_{dG}]_{11}$ | $(-3.7\pm 1.8)\cdot 10^{-17}$ | $[C_{u\gamma}]_{22}$ | $(-7.4\pm 3.4)\cdot 10^{-22}$ | $[C_{d\gamma}]_{33}$ | $(2.0\pm 1.1)\cdot 10^{-23}$
$[C_{dG}]_{22}$ | $(1.5\pm 0.9)\cdot 10^{-20}$ | $[C^{S1,RR}_{uu}]_{2222}$ | $(-5.6\pm 3.0)\cdot 10^{-25}$ | $[C^{V1,LR}_{uddu}]_{1331}$ | $(6.0\pm 3.1)\cdot 10^{-22}$
$[C_{uG}]_{11}$ | $(-1.9\pm 0.9)\cdot 10^{-17}$ | $[C^{S8,RR}_{uu}]_{2222}$ | $(1.3\pm 0.5)\cdot 10^{-25}$ | $[C^{V8,LR}_{uddu}]_{1331}$ | $(1.5\pm 0.9)\cdot 10^{-22}$
$[C_{d\gamma}]_{11}$ | $(-8.9\pm 0.4)\cdot 10^{-17}$ | $[C^{S1,RR}_{uu}]_{1122}$ | $(-2.3\pm 1.1)\cdot 10^{-22}$ | $[C^{V1,LR}_{dd}]_{1331}$ | $(8.1\pm 6.4)\cdot 10^{-22}$
$[C_{d\gamma}]_{22}$ | $(3.0\pm 1.7)\cdot 10^{-19}$ | $[C^{S8,RR}_{uu}]_{1122}$ | $(-3.6\pm 4.1)\cdot 10^{-23}$ | $[C^{V8,LR}_{dd}]_{1331}$ | $(-1.5\pm 0.2)\cdot 10^{-21}$
$[C_{u\gamma}]_{11}$ | $(2.5\pm 0.2)\cdot 10^{-17}$ | $[C^{S1,RR}_{ud}]_{2211}$ | $(-4.8\pm 2.6)\cdot 10^{-22}$ | $[C^{V1,LR}_{dd}]_{2332}$ | $(1.1\pm 0.6)\cdot 10^{-24}$
$[C^{V1,LR}_{uddu}]_{1111}$ | $(-4.3\pm 2.4)\cdot 10^{-22}$ | $[C^{S8,RR}_{ud}]_{2211}$ | $(-2.8\pm 0.7)\cdot 10^{-22}$ | $[C^{V8,LR}_{dd}]_{2332}$ | $(4.1\pm 2.5)\cdot 10^{-24}$
$[C^{V8,LR}_{uddu}]_{1111}$ | $(-6.6\pm 3.7)\cdot 10^{-22}$ | $[C^{S1,RR}_{ud}]_{2222}$ | $(2.9\pm 1.4)\cdot 10^{-26}$ | $[C^{S1,RR}_{dd}]_{3333}$ | $(-2.2\pm 1.1)\cdot 10^{-25}$
$[C^{V1,LR}_{uddu}]_{1221}$ | $(-9.0\pm 4.7)\cdot 10^{-22}$ | $[C^{S8,RR}_{ud}]_{2222}$ | $(5.2\pm 3.3)\cdot 10^{-25}$ | $[C^{S8,RR}_{dd}]_{3333}$ | $(4.6\pm 2.5)\cdot 10^{-26}$
$[C^{V8,LR}_{uddu}]_{1221}$ | $(-1.4\pm 0.7)\cdot 10^{-21}$ | $[C^{S1,RR}_{uddu}]_{2112}$ | $(2.6\pm 1.0)\cdot 10^{-21}$ | $[C^{S1,RR}_{dd}]_{1133}$ | $(-6.4\pm 3.2)\cdot 10^{-22}$
$[C^{V1,LR}_{dd}]_{1221}$ | $(-4.8\pm 2.6)\cdot 10^{-22}$ | $[C^{S8,RR}_{uddu}]_{2112}$ | $(3.8\pm 1.0)\cdot 10^{-22}$ | $[C^{S8,RR}_{dd}]_{1133}$ | $(-9.7\pm 9.5)\cdot 10^{-23}$