A Global View of the EDM Landscape

Skyler Degenkolb1, Nina Elmer2,

Tanmoy Modak2, Margarete Mühlleitner3, and Tilman Plehn2,4

1 Physikalisches Institut, Universität Heidelberg, Germany

2 Institut für Theoretische Physik, Universität Heidelberg, Germany

3 Institute for Theoretical Physics, Karlsruhe Institute of Technology,
Karlsruhe, Germany

4 Interdisciplinary Center for Scientific Computing (IWR), Universität
Heidelberg, Germany

## Abstract

Permanent electric dipole moments (EDMs) are sensitive probes of the symmetry
structure of elementary particles, which in turn is closely tied to the baryon
asymmetry in the universe. A meaningful interpretation framework for EDM
measurements has to be based on effective quantum field theory. We interpret
the measurements performed to date in terms of a hadronic-scale Lagrangian,
using the SFitter global analysis framework. We find that part of this
Lagrangian is constrained very well, while some of the parameters suffer from
too few high-precision measurements. Theory uncertainties lead to weaker model
constraints, but can be controlled within the global analysis.

###### Contents

  1. 1 Introduction
  2. 2 EDM Lagrangian
    1. 2.1 Weak-scale Lagrangian
    2. 2.2 Hadronic-scale Lagrangian
    3. 2.3 Matched Lagrangians for semileptonic interactions
  3. 3 EDM Measurements
    1. 3.1 Nucleons
    2. 3.2 Paramagnetic systems
    3. 3.3 Diamagnetic systems
    4. 3.4 Single parameter ranges
  4. 4 Global analysis
    1. 4.1 SFitter framework
    2. 4.2 Well-constrained model sub-space
    3. 4.3 Hadronic parameters from diamagnetic systems
    4. 4.4 Poorly constrained model parameters
    5. 4.5 Theory uncertainties
  5. 5 Outlook

##  1 Introduction

While the Standard Model (SM) is structurally complete, it fails to explain
key observations and therefore fails to qualify as a complete theory of
elementary particles. The two leading shortcomings are a missing dark matter
agent and a missing explanation of the baryon asymmetry in the Universe. For
the latter, the Sakharov conditions [1] tell us precisely which structures
would be required: (i) C- and CP-violation, (ii) baryon number violation, and
(iii) a deviation from thermal equilibrium. The first condition is especially
interesting, because we can read it off the fundamental Lagrangian and build,
for instance, models for electroweak baryogenesis around it [2, 3, 4, 5]. In
the SM, CP symmetry is violated through the fermion mixing among three
generations and through the adjoint gluon field strength. Measurements of the
neutron EDM, consistent with zero, clearly show that CP violation in QCD is
too small to explain the observed baryon asymmetry [6, 7, 8, 9, 10, 11].
Physics beyond the Standard Model (BSM), explaining the baryon asymmetry,
should then violate CP to a greater extent and with observable consequences.

Over recent years, EDM measurements have been performed on a wide range of
particles, atoms, and molecules [12, 13, 14, 15]. None of them has been able
to confirm a signal for CP violation. To judge their combined impact on BSM
physics, we need to combine them in a consistent framework. Such a global
analysis must start from a Lagrangian and express the experimental limits on
all EDMs in terms of its fundamental parameters. One choice of Lagrangian is
the hadronic-scale Lagrangian, describing the interactions of nucleons, pions,
and electrons at the GeV scale [12, 16, 15, 17]. Alternatively, we can combine
the EDM measurements at the weak scale, using an effective extension of the
renormalizable SM Lagrangian [18, 13, 19, 20, 21, 22, 17]. Going beyond
effective field theories (EFTs), an ultimate link to the cosmological
motivation requires a UV-complete extension of the SM at the weak scale, for
instance leptoquark models [17], extended Higgs sectors [23, 24, 25], left-
right symmetry [26], or supersymmetry [27, 28, 29, 30, 31, 32]. Technically,
all global analyses [33] face similar challenges which we will tackle in this
paper for the hadronic-scale Lagrangian.

We perform a global EDM analysis with the SFitter analysis tool [34, 35, 36,
37, 38], with its focus on the comprehensive treatment of uncertainties. Our
analysis will provide state-of-the-art limits on the multi-dimensional model
parameter space, and it will allow us to judge the impact of new or proposed
measurements and to identify shortcomings in relating measurements to
fundamental parameters111Only a proper fundamental physics interpretation can
make full use of experimental limits., while remaining easy to repeat or adapt
in response to new theoretical or experimental inputs. For this purpose it is
crucial that we interpret all measurements in the same fundamental physics
framework and include all uncertainties, including theory uncertainties, even
though they lack a statistical interpretation [39].

We start by introducing the consistent hadronic-scale Lagrangian with properly
chosen Wilson coefficients for our global analysis in Sec. 2. We then use this
Lagrangian to provide predictions for the measured EDMs, as detailed in Sec.
3. We start our global analysis without theory uncertainties in Sec. 4, to
understand the relations among different EDM measurements in terms of the
hadronic-scale Lagrangian. To extract correlations and limits on single model
parameters we employ a profile likelihood. We find that the current EDM
measurements define a subspace of well-constrained model parameters and an
orthogonal subspace of poorly constrained parameters with narrow correlation
patterns. Adding theory uncertainties on the relations between Lagrangian
parameters and observables in Sec. 4.5 degrades the interpretation in terms of
fundamental physics. We emphasize that this degradation does not cut into the
discovery potential of EDM measurements, as probes of fundamental symmetries
of elementary particles, but it hampers their interpretation as limits in
fundamental physics.

##  2 EDM Lagrangian

Because CP violation is motivated by cosmology and can be related to physics
beyond the SM at and above the weak scale, we start by introducing it into the
weak-scale Lagrangian in Sec. 2.1, relate it to the GeV-scale Lagrangian in
Sec. 2.2, and use simple matching arguments to simplify this hadronic-scale
Lagrangian which we use as the interpretation framework for our global
analysis in Sec. 2.3. A detailed analysis of CP-violating new physics at the
electroweak scale includes a renormalization group evolution from the GeV-
scale to the electroweak scale and will not be part of this first study.
Instead, we will focus here on a global analysis at the hadronic scale and the
role of correlations and theory uncertainties.

###  2.1 Weak-scale Lagrangian

The operators generating CP violation within and beyond the SM-Lagrangian,
neglecting CP violation in the neutrino sector [13], appear in the Lagrangian

$\displaystyle\mathscr{L}_{\text{CPV}}=\mathscr{L}_{\text{CKM}}+\mathscr{L}_{\bar{\theta}}+\mathscr{L}_{\text{dipole}}+\mathscr{L}_{\text{Weinberg}}+\mathscr{L}_{\text{EFT}}\;.$
(1)

The first term represents CP violation at mass dimension four, from the
complex phases in the CKM matrix. The second arises from the gluon field
strength, also at dimension four,

$\displaystyle\mathscr{L}_{\bar{\theta}}=\frac{g_{3}^{2}}{32\pi^{2}}\bar{\theta}\;\text{Tr}(G^{\mu\nu}\tilde{G}_{\mu\nu})\;,$
(2)

where $g_{3}$ is the strong coupling, $G^{\mu\nu}$ is the gluon field
strength,
$\widetilde{G}^{\mu\nu}=\epsilon^{\mu\nu\lambda\sigma}G_{\lambda\sigma}/2$ is
its dual, and $\bar{\theta}$ is the re-scaled CP-violating parameter in QCD.
The bar notation indicates that corrections from the quark mass matrix are
included. In principle $\bar{\theta}$ can also be included as a model
parameter, for instance in testing a specific BSM model [26]. However, we take
the view that the neutron EDM experimentally constrains $\bar{\theta}$ to have
such a small value that this fine-tuning problem requires a proper
explanation. This means there is little to be learned from including
$\bar{\theta}$ as a model parameter in our global analysis.

The other three contributions in Eq.(1) are higher-dimensional and not part of
a renormalizable extension of the SM-Lagrangian. Electric dipole moments of
fermions $d_{f}^{E}$, and chromoelectric dipole moments of quarks $d_{q}^{C}$,
appear at mass dimension five:

$\displaystyle\mathscr{L}_{\text{dipole}}=-\frac{i}{2}F^{\mu\nu}\sum_{f=q,\ell}d_{f}^{E}\;\left(\bar{f}\sigma_{\mu\nu}\gamma_{5}f\right)-\frac{i}{2}g_{3}G^{a}_{\mu\nu}\sum_{f=q}d_{q}^{C}\;\left(\bar{q}\sigma^{\mu\nu}\gamma_{5}T^{a}q\right)\;.$
(3)

where the indices $q$ and $\ell$ denote quarks and leptons of all three
generations. The electromagnetic field strength is $F^{\mu\nu}$. We chose the
metric convention $(1,-\mathbb{1})$ with
$\gamma_{5}=-i\gamma_{0}\gamma_{1}\gamma_{2}\gamma_{3}$. The fermion spins are
$\sigma_{\mu\nu}=i\left[\gamma_{\mu},\gamma_{\nu}\right]/2$, and $T^{a}$ are
the $SU(3)$ generators. The Weinberg operator is again built out of the gluon
field strength and introduces the gluonic chromo-electric dipole moment
$d^{G}$,

$\displaystyle\mathscr{L}_{\text{Weinberg}}=\frac{1}{3}d^{G}\;f_{abc}G^{a}_{\mu\nu}\widetilde{G}^{b\nu\rho}G^{c\;\mu}_{\rho}\;.$
(4)

Additional CP-violation occurs at mass dimension six and higher, generated at
a new physics scale larger than the vacuum expectation value, $\Lambda>v$,

$\displaystyle\mathscr{L}_{\text{EFT}}=\sum_{i}\frac{C_{i}^{(6)}}{\Lambda^{2}}\mathcal{O}_{i}^{(6)}+\mathcal{O}(\Lambda^{-3})\;.$
(5)

Some relevant dimension-6 operators that generate EDMs include the
semileptonic and hadronic 4-fermion operators

$\displaystyle\mathscr{L}_{\text{EFT}}\supset$ $\displaystyle C_{\ell
eqd}\;\left(\bar{L}^{j}e_{R}\right)\;\left(\bar{d}_{R}Q_{j}\right)+C^{(1)}_{\ell
equ}\;\left(\bar{L}^{j}e_{R}\right)\epsilon_{jk}\left(\bar{Q}^{k}u_{R}\right)+C^{(3)}_{\ell
equ}\;\left(\bar{L}^{j}\sigma_{\mu\nu}e_{R}\right)\epsilon_{jk}\left(\bar{Q}^{k}\sigma_{\mu\nu}u_{R}\right)$
$\displaystyle+C^{(1)}_{quqd}\;\left(\bar{Q}^{j}u_{R}\right)\epsilon_{jk}\left(\bar{Q}^{k}d_{R}\right)+C^{(8)}_{quqd}\;\left(\bar{Q}^{j}T^{a}u_{R}\right)\epsilon_{jk}\left(\bar{Q}^{k}T^{a}d_{R}\right)+\text{h.c.}$
(6)

The dipole moments, the Weinberg operator, and the additional 4-fermion
interactions can serve as the basis for an EDM analyses in the SMEFT framework
[12, 28, 29, 30, 18, 13, 23, 19, 20, 21, 22, 17]. As always, the set of
higher-dimensional SMEFT operators that turn out to be most relevant depends
on the high-scale BSM model that the SMEFT represents. For instance, in
supersymmetric models there are no contributions from
$\mathcal{O}^{(8)}_{quqd}$ and $\mathcal{O}^{(3)}_{\ell equ}$ at tree level,
and the relative size of down-type and up-type quark couplings is affected by
a potentially large $\tan\beta$ enhancement.

###  2.2 Hadronic-scale Lagrangian

The challenge with EDMs in view of the Lagrangian of Eq.(1) is that they are
measured far below the electroweak scale, where the propagating degrees of
freedom are leptons, non-relativistic nucleons $N=(p,n)^{T}$ with average mass
$m_{N}$, and pions $\vec{\pi}=(\pi^{+},\pi^{0},\pi^{-})^{T}$ [12, 28, 16, 30,
15, 17].

When we evolve our EFT to the experimentally relevant GeV scale, only the
lepton EDMs $d^{E}_{\ell}\equiv d_{\ell}$ in Eq.(3) of the electroweak
Lagrangian in Eq.(1) remain unchanged. While for the weak-scale Lagrangian the
relation between the three charged leptons, for instance the scaling with the
lepton mass, raises interesting questions, we factorize the muon and tau EDMs
from the hadronic-scale Lagrangian. They can be included in the same
framework, but given the systems for which experimental limits are available
today, the indirect constraints on EDMs of heavy leptons are many orders of
magnitude weaker than direct ones and there is little interplay among the
relevant model parameters.

We split the hadronic-scale Lagrangian describing EDMs at the experimentally
relevant GeV scale into

$\displaystyle\mathscr{L}_{\text{had}}\supset\mathscr{L}_{N,\text{sr}}+\mathscr{L}_{\pi
N}+\mathscr{L}_{eN}-\frac{i}{2}F^{\mu\nu}d_{e}\;\bar{e}\sigma_{\mu\nu}\gamma_{5}e\;.$
(7)

The dipole moments of the nucleons now read

$\displaystyle\mathscr{L}_{N,\text{sr}}=$
$\displaystyle-2\bar{N}\left[d_{p}^{\text{sr}}\frac{1+\tau_{3}}{2}+d_{n}^{\text{sr}}\frac{1-\tau_{3}}{2}\right]S_{\mu}Nv_{\nu}F^{\mu\nu}\;,$
(8)

where $S_{\mu}$ and $v_{\mu}$ are the spin and velocity of the nucleon. The
isoscalar and isovector contributions define $d_{N}^{\text{sr}}$ as the
Lagrangian parameters for the short-range contributions to the proton and
neutron EDMs.

Next come the interactions of pions and nucleons,

$\displaystyle\mathscr{L}_{\pi N}=$
$\displaystyle\bar{N}\Big{[}g^{(0)}_{\pi}\vec{\tau}\cdot\vec{\pi}+g^{(1)}_{\pi}\pi^{0}+g^{(2)}_{\pi}\left(3\tau_{3}\pi^{0}-\vec{\tau}\cdot\vec{\pi}\right)\Big{]}N$
$\displaystyle+C_{1}\;\left(\bar{N}N\right)\;\partial_{\mu}\left(\bar{N}S^{\mu}\bar{N}\right)+C_{2}\;\left(\bar{N}\vec{\tau}N\right)\cdot\partial_{\mu}\left(\bar{N}S^{\mu}\bar{N}\vec{\tau}\right)+\cdots$
(9)

where $\tau$ are the Pauli matrices and we neglect, for example, interactions
with more than one pion. The contribution involving $g^{(2)}_{\pi}$ is
suppressed relative to $g^{(0,1)}_{\pi}$ by one order in the chiral expansion
[40, 41], but can be taken into account in similar fashion. In our
parameterization these interactions contribute, for instance, to the nucleon
EDMs through calculable pion loops [16]. Naive dimensional analysis [42]
suggests that short-range nucleon interactions enter only at NNLO [43] in the
chiral expansion [44] and can be neglected [41]. In Eq.(9) this applies to all
interactions in the second line. One caveat is that a consistent treatment of
long-range effects may require additional short-distance counter terms [43]
that appear as effective short-range nucleon-nucleon forces. These are not
presently taken into account.

Finally, the higher-dimensional operators in Eq.(6) induce effective
interactions that can be organized according to their tensor structure,
isospin character, and their dependence on the electron and nucleon fields and
spins [15, 17]:

$\displaystyle\mathscr{L}_{eN}=$
$\displaystyle-\frac{G_{F}}{\sqrt{2}}\;\left(\bar{e}i\gamma_{5}e\right)\;\bar{N}\left(C_{S}^{(0)}+C_{S}^{(1)}\tau_{3}\right)N$
$\displaystyle+\frac{8G_{F}}{\sqrt{2}}\;v_{\nu}\;\left(\bar{e}\sigma^{\mu\nu}e\right)\;\bar{N}\left(C_{T}^{(0)}+C_{T}^{(1)}\tau_{3}\right)S_{\mu}N$
$\displaystyle-\frac{G_{F}}{\sqrt{2}}\;\left(\bar{e}e\right)\;\frac{\partial^{\mu}}{m_{N}}\left[\bar{N}\left(C_{P}^{(0)}+C_{P}^{(1)}\tau_{3}\right)S_{\mu}N\right]\;.$
(10)

In a heavy baryon expansion, the last line can be dropped at leading order
[13]. However, we retain all three terms since (1) a pion pole enhancement of
the isovector contribution somewhat offsets this hierarchy and (2)
contributions of heavy quarks can also render it relevant for some new physics
models. At this stage, the independent Lagrangian parameters for our global
EDM analysis at the hadronic scale are:

$\displaystyle\Big{\\{}\;d_{e},C_{S}^{(0,1)},C_{T}^{(0,1)},C_{P}^{(0,1)},g_{\pi}^{(0)},g_{\pi}^{(1)},d_{n,p}^{\text{sr}}\;\Big{\\}}\;.$
(11)

###  2.3 Matched Lagrangians for semileptonic interactions

The set of model parameters defined in Eq.(11) can be further simplified by
matching the semileptonic part of the hadronic-scale Lagrangian of Eq.(10) to
the corresponding weak-scale 4-fermion interactions of Eq.(6), both evaluated
for external nucleons. The light quark content in the nucleons is related to
the nucleon Lagrangian as

$\displaystyle g_{S}^{(0)}\;\bar{\psi}_{N}\psi_{N}$
$\displaystyle=\frac{1}{2}\bigl{\langle}N\left|\bar{u}u+\bar{d}d\right|N\bigr{\rangle}$
$\displaystyle g_{S}^{(1)}\;\bar{\psi}_{N}\tau_{3}\psi_{N}$
$\displaystyle=\frac{1}{2}\bigl{\langle}N\left|\bar{u}u-\bar{d}d\right|N\bigr{\rangle}$
$\displaystyle g_{T}^{(0)}\;\bar{\psi}_{N}\sigma_{\mu\nu}\psi_{N}$
$\displaystyle=\frac{1}{2}\bigl{\langle}N\left|\bar{u}\sigma_{\mu\nu}u+\bar{d}\sigma_{\mu\nu}d\right|N\bigr{\rangle}$
$\displaystyle g_{T}^{(1)}\;\bar{\psi}_{N}\sigma_{\mu\nu}\tau_{3}\psi_{N}$
$\displaystyle=\frac{1}{2}\bigl{\langle}N\left|\bar{u}\sigma_{\mu\nu}u-\bar{d}\sigma_{\mu\nu}d\right|N\bigr{\rangle}$
$\displaystyle g_{P}^{(0)}\;\bar{\psi}_{N}\gamma_{5}\psi_{N}$
$\displaystyle=\frac{1}{2}\bigl{\langle}N\left|\bar{u}\gamma_{5}u+\bar{d}\gamma_{5}d\right|N\bigr{\rangle}$
$\displaystyle g_{P}^{(1)}\;\bar{\psi}_{N}\gamma_{5}\tau_{3}\psi_{N}$
$\displaystyle=\frac{1}{2}\bigl{\langle}N\left|\bar{u}\gamma_{5}u-\bar{d}\gamma_{5}d\right|N\bigr{\rangle}\;,$
(12)

which relations define the scalar, tensor, and pseudoscalar nucleon form
factors $g_{S,T,P}^{(0,1)}$. Using them, the hadronic-scale Wilson
coefficients in Eq.(10) can be matched to the SMEFT Wilson coefficients in
Eq.(6) as [13, 15],

$\displaystyle C_{S}^{(0)}$
$\displaystyle=-g_{S}^{(0)}\frac{v^{2}}{\Lambda^{2}}\;\text{Im}\left(C_{\ell
edq}-C^{(1)}_{\ell equ}\right)$ $\displaystyle\qquad\quad C_{S}^{(1)}$
$\displaystyle=\phantom{-}g_{S}^{(1)}\frac{v^{2}}{\Lambda^{2}}\;\text{Im}\left(C_{\ell
edq}+C^{(1)}_{\ell equ}\right)$ $\displaystyle C_{T}^{(0)}$
$\displaystyle=-g_{T}^{(0)}\frac{v^{2}}{\Lambda^{2}}\;\text{Im}\left(C^{(3)}_{\ell
equ}\right)$ $\displaystyle\qquad\quad C_{T}^{(1)}$
$\displaystyle=-g_{T}^{(1)}\frac{v^{2}}{\Lambda^{2}}\;\text{Im}\left(C^{(3)}_{\ell
equ}\right)$ $\displaystyle C_{P}^{(0)}$
$\displaystyle=\phantom{-}g_{P}^{(0)}\frac{v^{2}}{\Lambda^{2}}\;\text{Im}\left(C_{\ell
edq}+C^{(1)}_{\ell equ}\right)$ $\displaystyle\qquad\quad C_{P}^{(1)}$
$\displaystyle=-g_{P}^{(1)}\frac{v^{2}}{\Lambda^{2}}\;\text{Im}\left(C_{\ell
edq}-C^{(1)}_{\ell equ}\right)\;.$ (13)

Here the six couplings $C_{S,T,P}^{(0,1)}$ are expressed in terms of only
three SMEFT Wilson coefficients, implying

$\displaystyle\frac{C_{P}^{(0)}}{g_{P}^{(0)}}=\frac{C_{S}^{(1)}}{g_{S}^{(1)}}\qquad\qquad\frac{C_{T}^{(0)}}{g_{T}^{(0)}}=\frac{C_{T}^{(1)}}{g_{T}^{(1)}}\qquad\qquad\frac{C_{S}^{(0)}}{g_{S}^{(0)}}=\frac{C_{P}^{(1)}}{g_{P}^{(1)}}\;.$
(14)

Only three independent semileptonic parameters actually enter the hadronic-
scale global analysis. We choose them as $C_{S,T,P}^{(0)}$ and combine them
using the known ratios of hadronic matrix elements to construct the full
Lagrangian. In addition to the light quarks described by Eq.(13), the
relations in Eq.(14) also have to include the contributions of heavy quarks.
These contributions are contained in the nucleon form factors, renormalized at
an appropriate mass scale and accounting for the corresponding anomaly
relations.

Note that in this way our global analysis preserves the full dependence on
$C_{S,T,P}^{(0,1)}$, although only $C_{S,T,P}^{(0)}$ appear as model
parameters. This remains the case regardless of isospin-violating effects, and
also when including experimental systems for which the coefficients of
$C_{S}^{(1)}$ differ significantly, as discussed below.

The implementation of Eq.(13) for $C_{S}^{(0,1)}$ is particularly
straightforward, because for each experimental system the effective parameter
that combines the isoscalar and isovector terms is independent of the nuclear
spin,

$\displaystyle C_{S}$ $\displaystyle=C_{S}^{(0)}+\frac{Z-N}{Z+N}\;C_{S}^{(1)}$
$\displaystyle=C_{S}^{(0)}+\frac{Z-N}{Z+N}\frac{g_{S}^{(1)}}{g_{P}^{(0)}}\;C_{P}^{(0)}$
$\displaystyle\text{with}\qquad\frac{g_{S}^{(1)}}{g_{P}^{(0)}}$
$\displaystyle\approx 0.1\,.$ (15)

In the second step we replace $C_{S}^{(1)}$ with $C_{P}^{(0)}$, reflecting
Eq.(13). Since $g_{S}^{(1)}$ is already suppressed relative to $g_{S}^{(0)}$
by the small isospin violation of the nucleon matrix element, one could argue
that $C_{S}^{(1)}\ll C_{S}^{(0)}$. Moreover, in the heavy nuclei of all atomic
and molecular systems for which EDMs have been measured so far, the isoscalar
and isovector contributions occur in approximately the same ratio,
$(Z-N)/(Z+N)\approx-0.2$, so the effective parameter $C_{S}$ is approximately
system-independent. As noted above, we do not rely on these assumptions.

Next, we relate the pseudoscalar and tensor semileptonic interactions in a
similar fashion. We start with the linear combinations for nucleons,
$C^{(n,p)}_{P,T}=C_{P,T}^{(0)}\mp C_{P,T}^{(1)}$, taking the particle physics
convention $\tau_{3}\left|n\right>=-\left|n\right>$). From those, the
coefficients for a given nucleus can be constructed according to the sum over
spins of the constituent nucleons, where $\left<\sigma_{p,n}\right>$
represents the average neutron or proton spin, evaluated for the measured
nuclear state [45, 46].

$\displaystyle C_{P,T}$
$\displaystyle=\frac{C^{(n)}_{P,T}\left<\sigma_{n}\right>+C^{(p)}_{P,T}\left<\sigma_{p}\right>}{\left<\sigma_{n}\right>+\left<\sigma_{p}\right>}\;.$
(16)

For $C_{T}$ we can see from Eq.(13) that the isoscalar and isovector couplings
differ only through the corresponding nucleon form factors. These are
calculated with small theoretical uncertainties in lattice QCD [47], allowing
us to write

$\displaystyle
C_{T}=\left(1-\frac{g_{T}^{(1)}}{g_{T}^{(0)}}\;\frac{\left<\sigma_{n}\right>-\left<\sigma_{p}\right>}{\left<\sigma_{n}\right>+\left<\sigma_{p}\right>}\right)\;C_{T}^{(0)}\qquad\text{with}\qquad\frac{g_{T}^{(1)}}{g_{T}^{(0)}}\approx
1.7\;.$ (17)

Similarly, for $C_{P}$ it can be shown that

$\displaystyle
C_{P}=C_{P}^{(0)}-\frac{g_{P}^{(1)}}{g_{S}^{(0)}}\;\frac{\left<\sigma_{n}\right>-\left<\sigma_{p}\right>}{\left<\sigma_{n}\right>+\left<\sigma_{p}\right>}\;C_{S}^{(0)}\qquad\text{with}\qquad\frac{g_{P}^{(1)}}{g_{S}^{(0)}}\approx
20.2\;.$ (18)

For the derivation of $g_{P}^{(1)}$ in this ratio we follow Ref. [13]. We
consider only the first generation as relevant light quarks, such that
$g_{P}^{(1)}$ is dominated by the $\pi$-pole contribution

$\displaystyle
g_{P}^{(1)}=\frac{g_{A}\bar{m}_{N}}{\bar{m}}\;\frac{m_{\pi}^{2}}{m_{\pi}^{2}-q^{2}}+\text{heavy
quarks}\qquad\qquad\ \bar{m}=\frac{m_{u}+m_{d}}{2}\;.$ (19)

Here $\bar{m}_{N}\approx 940$ MeV is the average nucleon mass, $\bar{m}$ the
average light quark mass, and $g_{A}$ is the (isovector) axial vector coupling
[48]. The coupling $g_{P}^{(0)}$, appearing in Eq.(15), involves the isoscalar
axial coupling $g_{A}^{(0)}$, obtained from the sum rather than the difference
of the light quark axial charges. We extend this by including a light
$s$-quark, where the $\pi$-pole dominance is replaced by an octet $\eta$-pole
with an appropriately modified average light quark mass $m^{*}$

$\displaystyle
g_{P}^{(0)}=\frac{g_{A}^{(0)}\bar{m}_{N}}{m^{*}}\;\frac{m_{\eta}^{2}}{m_{\eta}^{2}-q^{2}}+\text{heavy
quarks}\qquad\qquad m^{*}=\frac{m_{u}+m_{d}+2m_{s}}{3}\;.$ (20)

Note that heavy quark contributions enter differently in $g_{P}^{(0)}$ and
$g_{P}^{(1)}$, and can be derived using the $U(1)_{A}$ axial anomaly together
with the divergence of the anomaly-free axial current $J^{q}_{\mu
5}=\bar{q}\gamma_{\mu}\gamma_{5}q$ for all quarks $q$ [49, 50, 51, 52]. These
represent a relatively minor contribution to $g_{P}^{(1)}$, but due to
suppression of the $\eta$-pole relative to the $\pi$-pole by the factor
$m^{*}/\bar{m}$, contribute to $g_{P}^{(0)}$ at approximately the same level
as the light quarks.

Our final, simplifying assumption is not strictly needed, but is effective in
reducing the number of model parameters by one,

$\displaystyle d_{p}^{\text{sr}}\approx-d_{n}^{\text{sr}}\;.$ (21)

The nuclei of diamagnetic systems, which apart from the neutron itself provide
the strongest constraints on nucleon EDMs, are typically dominated either by a
valence proton or a valence neutron. As for the anomalous magnetic moment, the
short-range nucleon EDMs are assumed to be dominated by the isovector
contribution. This assumption can be relaxed for the purposes of a rigorous
global analysis, provided improved theory uncertainties concerning the
contributions of all nucleons to the overall EDM of each measured system. Of
the present experimental limits, $d_{p}^{\text{sr}}$ is constrained
essentially only by the TlF measurement.

With these simplifications, the set of low-energy parameters given in Eq.(11)
reduces to

$\displaystyle
c_{j}\in\Big{\\{}\;d_{e},C_{S}^{(0)},C_{T}^{(0)},C_{P}^{(0)},g_{\pi}^{(0)},g_{\pi}^{(1)},d_{n}^{\text{sr}}\;\;\Big{\\}}\;.$
(22)

These Lagrangian parameters define the model parameters for our global EDM
analysis.

##  3 EDM Measurements

In terms of the Lagrangian parameters in Eq.(22) we can predict the measured
EDMs $d_{i}$ as linear combinations with system-specific coefficients
$\alpha_{i,c_{j}}$,

$\displaystyle d_{i}=\sum_{c_{j}}\alpha_{i,c_{j}}c_{j}\;.$ (23)

The measurements we analyze are listed in Tab. 1 and discussed below. Unless
otherwise indicated, the isotopes and charge states that we discuss are those
given in Tab. 1. The isotopes that are relevant for these molecular systems
are 180Hf, 232Th, 174Yb, 205Tl, 16O, and 19F. We neglect constraints from the
comparatively weaker experimental bounds from 85Rb [53, 54], Xem [55], PbO
[56], Eu0.5Ba0.5TiO3 [57], and the $\Lambda$ hyperon [58]. The experimental
bounds for the $\mu$ and $\tau$ leptons constrain the corresponding Lagrangian
parameters and factorize from the hadronic-scale Lagrangian. We do not include
them in our first global analysis.

As discussed in the last section, the $\alpha$-values for all
$C_{S,P,T}^{(0,1)}$ can be extracted from the corresponding values for the
basis given in Eq.(22). For this we follow Eqs. (15)-(18) and find for example

$\displaystyle\alpha_{C_{S}^{(0)}}$
$\displaystyle=\alpha_{C_{S}}-\alpha_{C_{P}}\frac{g_{P}^{(1)}}{g_{S}^{(0)}}\frac{\left<\sigma_{n}\right>-\left<\sigma_{p}\right>}{\left<\sigma_{n}\right>+\left<\sigma_{p}\right>}$
$\displaystyle\alpha_{C_{P}^{(0)}}$
$\displaystyle=\alpha_{C_{P}}+\alpha_{C_{S}}\frac{g_{S}^{(1)}}{g_{P}^{(0)}}\frac{Z-N}{Z+N}$
$\displaystyle\alpha_{C_{T}^{(0)}}$
$\displaystyle=\left(1-\frac{g_{T}^{(1)}}{g_{T}^{(0)}}\;\frac{\left<\sigma_{n}\right>-\left<\sigma_{p}\right>}{\left<\sigma_{n}\right>+\left<\sigma_{p}\right>}\right)\;\alpha_{C_{T}}\;,$
(24)

with the $\alpha_{C_{S,P,T}}$ given in Tab. 2.

System $i$ |  Measured $d_{i}$ $\left[\text{$e$\,cm}\right]$ |  Upper limit on $|d_{i}|$ $\left[\text{$e$\,cm}\right]$ | Reference
---|---|---|---
$n$ | $(0.0\pm 1.1_{\text{stat}}\pm 0.2_{\text{syst}})\cdot 10^{-26}$ | $2.2\cdot 10^{-26}$ | [59]
205Tl | $(-4.0\pm 4.3)\cdot 10^{-25}$ | $1.1\cdot 10^{-24}$ | [60]
133Cs | $(-1.8\pm 6.7_{\text{stat}}\pm 1.8_{\text{syst}})\cdot 10^{-24}$ | $1.4\cdot 10^{-23}$ | [61]
HfF+ | $(-1.3\pm 2.0_{\text{stat}}\pm 0.6_{\text{syst}})\cdot 10^{-30}$ | $4.8\cdot 10^{-30}$ | [62]
ThO | $(4.3\pm 3.1_{\text{stat}}\pm 2.6_{\text{syst}})\cdot 10^{-30}$ | $1.1\cdot 10^{-29}$ | [63]
YbF | $(-2.4\pm 5.7_{\text{stat}}\pm 1.5_{\text{syst}})\cdot 10^{-28}$ | $1.2\cdot 10^{-27}$ | [64]
199Hg | $(2.20\pm 2.75_{\text{stat}}\pm 1.48_{\text{syst}})\cdot 10^{-30}$ | $7.4\cdot 10^{-30}$ | [65, 66]
129Xe | $(-1.76\pm 1.82)\cdot 10^{-28}$ | $4.8\cdot 10^{-28}$ | [67, 68]
171Yb | $(-6.8\pm 5.1_{\text{stat}}\pm 1.2_{\text{syst}})\cdot 10^{-27}$ | $1.5\cdot 10^{-26}$ | [69]
225Ra | $(4\pm 6_{\text{stat}}\pm 0.2_{\text{syst}})\cdot 10^{-24}$ | $1.4\cdot 10^{-23}$ | [70]
TlF | $(-1.7\pm 2.9)\cdot 10^{-23}$ | $6.5\cdot 10^{-23}$ | [71]
|  Measured $\omega_{i}$ $\left[\text{$\text{mrad}/s$}\right]$ |  Rescaling factor $x_{i}$ for $d_{i}$ | Reference
HfF+ | $\phantom{-}(-0.0459\pm 0.0716_{\text{stat}}\pm 0.0217_{\text{syst}})^{\dagger}$ | 0.999 | [62]
ThO |  $\phantom{-}(-0.510\pm 0.373_{\text{stat}}\pm 0.310_{\text{syst}}$) | 0.982 | [63]
YbF |  $\phantom{-}(5.30\pm 12.60_{\text{stat}}\pm 3.30_{\text{syst}}$) | 1.12 | [64]
Table 1: Measured EDM values and $95\%$CL ranges. For 129Xe we combine two independent results with similar precision, using inverse-variance weighting. For the paramagnetic molecules, we also provide the measured angular frequencies and the rescaling factor which allows us to use $x_{i}d_{i}$ for each experimentally reported $d_{i}$. For the definition of $x_{i}$, see text. †The frequency for HfF+ is scaled by a factor of 2 relative to Ref.[62], to consistently use Eq.(29) for all systems. System $i$ | $\left<\sigma_{n}\right>$ | $\left<\sigma_{p}\right>$ | $\left<\sigma_{z}\right>^{(0)}$ | $\alpha_{i,C_{S}}\left[\text{$e$ cm}\right]$ | $\alpha_{i,C_{P}}\left[\text{$e$ cm}\right]$ | $\alpha_{i,C_{T}}\left[\text{$e$ cm}\right]$
---|---|---|---|---|---|---
Tl | $0.274$ | $0.726$ | $1$ | $-6.77\cdot 10^{-18}$ | $1.5\cdot 10^{-23}$ | $5\cdot 10^{-21}$
Cs | $-0.206$ | $-0.572$ | $-0.778$ | $7.8\cdot 10^{-19}$ | $2.2\cdot 10^{-23}$ | $9.2\cdot 10^{-21}$
199Hg | $-0.302$ | $-0.032$ | $-0.334$ | $-2.8\cdot 10^{-22}$ | $6\cdot 10^{-23}$ | $1.7\cdot 10^{-20}$
129Xe | $0.73$ | $0.27$ | $1$ | $-6.28\cdot 10^{-23}$ | $1.6\cdot 10^{-23}$ | $5.7\cdot 10^{-21}$
171Yb | $-0.3$ | $-0.034$ | $-0.334$ | $-2.68\cdot 10^{-22}$ | $4.01\cdot 10^{-23}$ | $1.24\cdot 10^{-20}$
225Ra | $0.72$ | $0.28$ | $1$ | $2.9\cdot 10^{-21}$ | $-6.4\cdot 10^{-22}$ | $-1.8\cdot 10^{-19}$
TlF | $0.274$ | $0.726$ | $1$ | $2.9\cdot 10^{-18}$ | $3\cdot 10^{-19}$ | $2.7\cdot 10^{-16}$
Table 2: Effective parameters used as input to our global analysis, as
summarized in Tab. 3. Within a shell model of the nucleus, the quantity
$\left<\sigma_{z}\right>^{(0)}=\left<\sigma_{n}\right>+\left<\sigma_{p}\right>$
is the isoscalar sum of neutron and proton spin projections. (Note that the
shell model is not expected to be reliable for the deformed nuclei 171Yb and
225Ra.) The spin fractions contributing to semileptonic coefficients in TlF
only take into account the 205Tl nucleus, though see Ref. [72] for some
consideration of contributions from the 19F nucleus.

###  3.1 Nucleons

We start with the simplest hadronic systems for which we can measure EDMs, the
nucleons. Their EDMs can be written directly as a Lagrangian term

$\displaystyle-\frac{i}{2}F^{\mu\nu}d_{N}\;\left(\bar{N}\sigma_{\mu\nu}\gamma_{5}N\right)\;,$
(25)

but can also be described by chiral perturbation theory, based on the
hadronic-scale Lagrangian. In that case we consider the nucleon EDMs as
observables that receive contributions from: short-range contributions
$d_{N}^{\text{sr}}$, NNLO pion-loop contributions, and potential direct
contributions to Eq.(25) within and beyond the SM [73],

$\displaystyle d_{n}$
$\displaystyle=d_{n}^{\text{sr}}-\frac{eg_{A}}{8\pi^{2}F_{\pi}}\left[g_{\pi}^{(0)}\left(\ln\frac{m_{\pi}^{2}}{m_{N}^{2}}-\frac{\pi
m_{\pi}}{2m_{N}}\right)-\frac{g_{\pi}^{(1)}}{4}\left(\kappa_{0}-\kappa_{1}\right)\frac{m_{\pi}^{2}}{m_{N}^{2}}\ln\frac{m_{\pi}^{2}}{m_{N}^{2}}\right]$
(26) $\displaystyle d_{p}$
$\displaystyle=d_{p}^{\text{sr}}+\frac{eg_{A}}{8\pi^{2}F_{\pi}}\left[g_{\pi}^{(0)}\left(\ln\frac{m_{\pi}^{2}}{m_{N}^{2}}-\frac{2\pi
m_{\pi}}{m_{N}}\right)-\frac{g_{\pi}^{(1)}}{4}\left(\frac{2\pi
m_{\pi}}{m_{N}}+\left(\frac{5}{2}+\kappa_{0}+\kappa_{1}\right)\frac{m_{\pi}^{2}}{m_{N}^{2}}\ln\frac{m_{\pi}^{2}}{m_{N}^{2}}\right)\right]\;,$

where $F_{\pi}=92$ MeV is the pion decay constant [74], $m_{\pi}=139$ MeV,
$m_{N}=940$ MeV, $g_{A}\approx 1.27$ is the nucleon isovector axial charge,
and the isoscalar and isovector nucleon anomalous magnetic moments are
$\kappa_{0}=-0.12$ and $\kappa_{1}=3.7$. We set the renormalization scale to
the nucleon mass, and the split between proton and neutron mass leads to a
higher-order effect that is presently negligible in relation to other
uncertainties. In terms of weak-scale parameters, finite values of
$g_{\pi}^{(0,1)}$ can be related to a CKM phase or $\bar{\theta}$, but also to
hadronic 4-fermion operators. The input from the presently most sensitive
neutron EDM measurement to our global analysis is given in Tab. 1.

System $i$ | $\alpha_{i,d_{e}}$ | $\alpha_{i,C_{S}^{(0)}}\left[\text{$e$ cm}\right]$ | $\alpha_{i,C_{P}^{(0)}}\left[\text{$e$ cm}\right]$ | $\alpha_{i,C_{T}^{(0)}}\left[\text{$e$ cm}\right]$ | $\alpha_{i,g_{\pi}^{(0)}}\left[\text{$e$ cm}\right]$ | $\alpha_{i,g_{\pi}^{(1)}}\left[\text{$e$ cm}\right]$ | $\alpha_{i,d_{n}^{\text{sr}}}$ | $\alpha_{i,d_{p}^{\text{sr}}}$
---|---|---|---|---|---|---|---|---
$n$ | $-$ | $-$ | $-$ | $-$ | $1.38^{\pm 0.02}\cdot 10^{-14}$ | $2.73^{\pm 0.02}\cdot 10^{-16}$ | $1$ | $-1$
205Tl |  $-558^{\pm 28}$ [75] | $-6.77^{\pm 0.34}\cdot 10^{-18}$ | $1.5^{+2}_{-0.7}\cdot 10^{-19}$ | $8.8^{\pm 0.9}\cdot 10^{-21}$ | n/a | n/a | n/a | n/a
133Cs | $123^{\pm 4}$ | $7.80^{+0.2}_{-0.8}\cdot 10^{-19}$ | $-1.4^{+0.8}_{-2}\cdot 10^{-20}$ | $1.7^{\pm 0.2}\cdot 10^{-20}$ | $-$ | $-$ | $-$ | $-$
199Hg |  $-0.012^{+0.0094}_{-0.002}$ [76, 33] | $-1.26_{-1.2}^{+0.7}\cdot 10^{-21}$ | $6.6_{-0.3}^{+1.2}\cdot 10^{-23}$ | $-6.4_{-4}^{+3}\cdot 10^{-21}$ | $-1.18_{-2.62}^{+0.19}\cdot 10^{-17}$ | $1.6_{-6.5}^{+0}\cdot 10^{-17}$ | $-1.56^{\pm 0.39}\cdot 10^{-4}$ | $-1.56^{\pm 0.39}\cdot 10^{-5}$
129Xe |  $-8^{+0}_{-8}\cdot 10^{-4}$ [77, 33] | $-2.1_{-2.5}^{+1.2}\cdot 10^{-22}$ | $1.7_{-0.4}^{+0.5}\cdot 10^{-23}$ | $1.24_{-0.61}^{+0.78}\cdot 10^{-21}$ | $-0.4_{-23}^{+1.2}\cdot 10^{-19}$ | $-2.2_{-17}^{+1.1}\cdot 10^{-19}$ | $1.7_{-0}^{+0.7}\cdot 10^{-5}$ | $3.51^{\pm 0.88}\cdot 10^{-6}$
171Yb |  ($-0.012^{+0.01145}_{-0.002}$) [78] | $-9.1_{-11}^{+5}\cdot 10^{-22}$ | $4.5_{-1.1}^{+1.8}\cdot 10^{-23}$ | $-4.4_{-2.9}^{+2.2}\cdot 10^{-21}$ | $-9.5^{\pm 2.4}\cdot 10^{-18}$ | $1.3^{\pm 0.33}\cdot 10^{-17}$ | $-1.13^{\pm 0.28}\cdot 10^{-4}$ | $-1.13^{\pm 0.28}\cdot 10^{-5}$
225Ra |  $-0.054^{\pm 0.002}$ [33] | $8.6_{-4.5}^{+9.5}\cdot 10^{-21}$ | $-7.0_{-2.6}^{+1.7}\cdot 10^{-22}$ | $-4.5_{-2.5}^{+2.0}\cdot 10^{-20}$ | $1.7_{-0.8}^{+5.2}\cdot 10^{-15}$ | $-6.9_{-21}^{+3.1}\cdot 10^{-15}$ | $-5.36^{\pm 1.34}\cdot 10^{-4}$ | $-1.11^{\pm 0.28}\cdot 10^{-4}$
TlF |  $81^{\pm 20}$ [71, 51] | $5.6_{-2.5}^{+4.9}\cdot 10^{-18}$ | $2.4_{-1.9}^{+1.0}\cdot 10^{-19}$ | $4.8_{-1.1}^{+1.2}\cdot 10^{-16}$ | $1.9_{-1.4}^{+0.1}\cdot 10^{-14}$ | $-1.6^{\pm 0.4}\cdot 10^{-13}$ | $-9.47^{\pm 2.37}\cdot 10^{-2}$ | $-4.59^{\pm 1.15}\cdot 10^{-1}$
HfF+ | 1 | $9.17^{\pm 0.06}\cdot 10^{-21}$ | $-$ | $-$ | $-$ | $-$ | $-$ | $-$
ThO | 1 | $1.51^{+0}_{-0.2}\cdot 10^{-20}$ | $-$ | $-$ | $-$ | $-$ | $-$ | $-$
YbF | 1 | $8.99^{\pm 0.70}\cdot 10^{-21}$ | $-$ | $-$ | $-$ | $-$ | $-$ | $-$
| $\eta^{(m)}_{i,d_{e}}\left[\dfrac{\text{mrad}}{\text{s $e$ cm}}\right]$ | $k^{(m)}_{i,C_{S}}\left[\dfrac{\text{mrad}}{\text{s}}\right]$ | $\alpha_{i,C_{P}}$ | $\alpha_{i,C_{T}}$ | $\alpha_{i,g_{\pi}^{(0)}}$ | $\alpha_{i,g_{\pi}^{(1)}}$ | $\alpha_{i,d_{n}^{\text{sr}}}$ | $\alpha_{i,d_{p}^{\text{sr}}}$
HfF+ |  $3.49^{\pm 0.14}\cdot 10^{28}$ [76, 79, 80, 81, 82] |  $3.2^{+0.1}_{-0.2}\cdot 10^{8}$ [76, 79, 80] | $-$ | $-$ | $-$ | $-$ | $-$ | $-$
ThO |  $-1.21^{+0.05}_{-0.39}\cdot 10^{29}$ [76, 83, 84, 85]† |  $-1.82^{+0.42}_{-0.27}\cdot 10^{9}$ [83, 76, 86, 87, 85]† | $-$ | $-$ | $-$ | $-$ | $-$ | $-$
YbF |  $-1.96^{\pm 0.15}\cdot 10^{28}$ [76, 86, 87, 88, 89] |  $-1.76^{\pm 0.2}\cdot 10^{8}$ [76, 86, 88, 87] | $-$ | $-$ | $-$ | $-$ | $-$ | $-$

Table 3: Central values for the $\alpha$-parameters defined in Eq.(23). Here
n/a means that we do not know of a reliable prediction, and $-$ means that we
neglect the dependence in our global analysis. †There appears to be an overall
sign issue in the coefficients reported for ThO in Table 4 of Ref. [76]. For
the coefficient of $d_{e}$ in 171Yb, we take the value from the structurally
similar system 199Hg, since no complete dedicated calculation is presently
available (the result of [78] is taken into account via the upper error
bound). The impact of this choice on the global analysis is negligible. The
values for Tl and Cs of $\alpha_{i,C_{T}^{(0)}}$ are estimated by simple
analytical calculations [90], and the uncertainties quoted here are estimated
as approximately twice those arising from the relevant hadronic matrix
elements.

###  3.2 Paramagnetic systems

Paramagnetic atoms and molecules are primarily sensitive to the electron EDM
and the scalar electron-nucleon couplings $C_{S}^{(0,1)}$ of Eq.(10). It is
common to distinguish the case of paramagnetic atoms with the atomic EDMs

$\displaystyle
d_{i}=\alpha_{i,d_{e}}d_{e}+\alpha_{i,C_{S}}C_{S}+\sum_{c_{j}}\alpha_{i,c_{j}}c_{j}\qquad\text{for}\qquad
i\in\\{\text{Tl},\text{Cs}\\}\;$ (27)

from that of paramagnetic molecules, although in both cases the terms
involving the Lagrangian parameters $d_{e}$ and $C_{S}$ dominate. The
experimentally relevant quantity is a phase difference accumulated over some
measurement time, interpreted as a frequency. Measurements with paramagnetic
molecules are often reported as a parity- and time-reversal-violating
frequency shift $\omega_{i}$,

$\displaystyle\omega_{i}=\eta^{(m)}_{i,d_{e}}d_{e}+k^{(m)}_{i,C_{S}}C_{S}+\sum_{c_{j}}\alpha_{i,c_{j}}^{(m)}c_{j}\qquad\text{for}\qquad
i\in\\{\text{HfF${}^{+}$},\text{ThO},\text{YbF}\\}\;.$ (28)

The superscript $(m)$ indicates the molecular systems. We also give these
measured angular frequencies in Tab. 1. This emphasizes that $\omega_{i}$ does
not scale in a simple way with the experimentally applied electric field; it
rather depends on the molecular structure via a so-called effective electric
field $E_{\text{eff}}$ that saturates when the molecule is polarized. Note
that $\alpha_{i,d_{e}}$ is dimensionless, while $\eta^{(m)}_{i,d_{e}}$ is not;
similarly $\alpha_{i,C_{S}}$ has the units of an EDM, while
$k^{(m)}_{i,C_{S}}$ has units of angular frequency.

The remaining contributions to paramagnetic molecule EDMs have weak
dependences on other low-energy constants, and we note that in particular the
EDMs of the $\mu$ and $\tau$ leptons have been indirectly constrained via the
experimental limits from ThO and Hg [91]. However, we are not aware of any
established values for the coefficients of semileptonic or hadronic
parameters. We thus take $\alpha_{i,c_{j}}^{(m)}=0$ in Eq.(28), while for the
paramagnetic atoms we obtain values for semileptonic coefficients either from
the literature or by scaling arguments, as reported in Tabs. 2 and 3.

Now starting with the truncated version of Eq.(28), we adopt the convention
that

$\displaystyle\omega_{i}$
$\displaystyle=\eta^{(m)}_{i,d_{e}}d_{e}+k^{(m)}_{i,C_{S}}C_{S}$
$\displaystyle=-\frac{E_{\text{eff},i}}{\hbar}d_{e}+\frac{W_{i}}{\hbar}C_{S}\;,$
(29)

where all signs, $g$-factors, spin magnitudes, etc. are absorbed into the
coefficients on the right-hand side. It is also common to refer to
$\mathcal{E}_{\text{eff}}$ as the effective electric field, where

$\displaystyle E_{\text{eff}}$
$\displaystyle=\mathcal{E}_{\text{eff}}\operatorname{sgn}(\vec{J}\cdot\hat{n})\left<\hat{n}\cdot\hat{z}\right>,\;$
(30)

and $\vec{J}$ is the total electronic angular momentum, $\hat{n}$ is the
direction of the internuclear axis, and $\hat{z}$ is the direction of the
externally applied electric field. Here $\Omega=\vec{J}\cdot\hat{n}$ is a
quantum number used in molecular term symbols, and
$\left<\hat{n}\cdot\hat{z}\right>$ indicates the degree to which the molecule
is electrically polarized by the applied electric field. Some of the
conventions used for the orientation of the internuclear axis, the internal
electric field, etc. are summarized in Appendix A.3 of Ref. [92].

The truncated expression for the frequency shift thus provides us with
physical expressions for the two constants,

$\displaystyle\eta^{(m)}_{i,d_{e}}=-\frac{E_{\text{eff},i}}{\hbar}\qquad\text{and}\qquad
k^{(m)}_{i,C_{S}}=\frac{W_{i}}{\hbar}\;.$ (31)

The point is that this frequency measurement cannot be directly translated
into a permanent molecular dipole moment, since the molecules are
substantially or entirely electrically polarized during the measurement. The
linear energy shift due to the lab-frame electric field saturates as
$\left<\hat{n}\cdot\hat{z}\right>\rightarrow 1$, and the limiting dependence
on the external fields is that of an induced dipole moment. To express the
frequency difference $\omega_{i}$ in terms of a physically meaningful
permanent electric dipole moment, we define the molecule’s $d_{i}$ in relation
to the effective electric field,

$\displaystyle-E_{\text{eff},i}d_{i}\equiv-
E_{\text{eff},i}d_{e}+W_{i}C_{S}\;.$ (32)

We can use this form to define, in analogy to Eq.(27),

$\displaystyle
d_{i}=d_{e}+\alpha_{i,C_{S}}C_{S}\qquad\Leftrightarrow\qquad\alpha_{i,d_{e}}=1\quad\text{and}\quad\alpha_{i,C_{S}}=-\frac{W_{i}}{E_{\text{eff},i}}=\frac{k^{(m)}_{i,C_{S}}}{\eta^{(m)}_{i,d_{e}}}\;.$
(33)

For our analysis we re-cast the limits from paramagnetic molecules in units of
$e\text{cm}$, such that the coefficients for all systems can be expressed in
the same units, with the side effect that $\alpha_{i,d_{e}}=1$ for all
paramagnetic molecules. This approach also serves two additional purposes: (i)
Many different conventions are in use for these coefficients, sometimes using
the same symbols for different quantities. Since measured quantities are
related in publications to the electron EDM, our choice makes comparisons
across different works relatively straightforward. Of course, all quantities
must be ultimately connected to an experimentally measured phase; (ii) As
first pointed out in Refs. [86, 87], while there is considerable variation in
the literature values for $\eta^{(m)}_{i,d_{e}}$ and $k^{(m)}_{i,C_{S}}$ in a
given system, there is much less variation in their ratio. Dividing Eq.(28) by
$\eta^{(m)}_{i,d_{e}}$ should pass this uncertainty on to all semileptonic and
hadronic coefficients.

Note that $\eta^{(m)}_{i,d_{e}}$ is used both to obtain sole-source limits on
$d_{e}$ from the experimentally measured $\omega_{i}$, and to convert the
experimentally measured frequencies into EDM units for Tab. 1. The
experimental EDM limits for paramagnetic molecules in Tab. 1 are rescaled by
the indicated factor $x_{i}$, which differs from unity when updated values for
$\eta^{(m)}_{i,d_{e}}$ differ from the cited publication. This is typically
the case when improved molecular structure calculations have become available
since the experimental limit was published.

Recommended values for many of the $\alpha_{i,c_{j}}$ are given in Tables
III-V of Ref. [15] and in Table 4 of Ref. [76], including in many cases the
ranges corresponding to theory uncertainties (or at least different reported
values). We give our choices for $\alpha_{i,c_{j}}$ in Tab. 3.

###  3.3 Diamagnetic systems

In contrast to the paramagnetic systems, where sub-leading contributions are
largely neglected due to lack of theory inputs, the small contributions of
$d_{e}$ and $C_{S}^{(0)}$ are taken into account for all diamagnetic systems.
These are typically smaller contributions to the observable EDM in cases where
all electron spins are paired, with the main contributions coming rather from
nucleon EDMs, nuclear forces mediated by pion exchange with strengths
$g_{\pi}^{(0,1,2)}$, and the nuclear-spin-dependent semileptonic interactions
$C_{T}^{(0)}$ (and possibly $C_{P}^{(0)}$). The experimental precision,
especially of Hg, is nevertheless high enough to contribute meaningful
constraining power for $d_{e}$ and $C_{S}^{(0)}$.

Unfortunately it appears that arguments [76] to the effect that diamagnetic
systems add significant complementary constraining power in the $d_{e}-C_{S}$
subspace due to a different sign of the ratio
$\alpha_{i,d_{e}}/\alpha_{i,C_{S}}$ in comparison to paramagnetic systems, do
not in fact hold for Hg or most other systems in Tab. 1. On the other hand,
recent calculations [33] indicate that $\alpha_{i,d_{e}}$ and
$\alpha_{i,C_{S}}$ in some highly correlated systems may indeed have different
sign, thus reviving this possibility. At present, and as can be seen from Tab.
3, Ra is the only measured system where this appears to be the case. It may
thus represent a special case among diamagnetic systems, in that reducing the
error of experimental and theory inputs could have significant impact on the
$d_{e}-C_{S}^{(0)}$ subspace that is complementary to the dominating
paramagnetic constraints.

The contributions of nuclear forces and nucleon EDMs are frequently
interpreted via the Schiff moment of a given nucleus [15], which is more
easily related to nuclear structure parameters [90]. In terms of our model
coefficients, the Schiff moment $S_{i}$ of system $i$ and a system-specific
coefficient $k_{i,S}$ can be expressed related to the corresponding EDM via

$\displaystyle k_{i,S}S_{i}$
$\displaystyle=\sum_{c_{j}\in\\{d_{n,p}^{\text{sr}},g_{\pi}^{(0,1,2)}\\}}\alpha_{i,c_{j}}c_{j}$
$\displaystyle=k_{i,S}\left[s_{i,n}d_{n}+s_{i,p}d_{p}+\frac{m_{N}g_{A}}{F_{\pi}}\left(a_{i,0}g_{\pi}^{(0)}+a_{i,1}g_{\pi}^{(1)}+a_{i,2}g_{\pi}^{(2)}\right)\right]$
$\displaystyle\approx
k_{i,S}\left[s_{i,n}d_{n}^{\text{sr}}+s_{i,p}d_{p}^{\text{sr}}+\frac{m_{N}g_{A}}{F_{\pi}}\left(\tilde{a}_{i,0}g_{\pi}^{(0)}+\tilde{a}_{i,1}g_{\pi}^{(1)}\right)\right],\;$
(34)

where the coefficients $s_{i,N}$ ($N=n,p$) indicate the contributions from
EDMs of unpaired nucleons and the coefficients $a_{i,m}$ ($m=0,1,2$)
parameterize the strength of CP-violating pion exchange, organized by isospin,
for the nucleus of system $i$. Between the second and third lines we drop the
$g_{\pi}^{(2)}$ term as discussed above, and use Eq.(26) to rewrite the
nucleon EDMs themselves and absorb their contributions from $g_{\pi}^{(0)}$
and $g_{\pi}^{(1)}$ into the implicitly re-defined coefficients
$\tilde{a}_{i,m}$.

The coefficient $k_{i,S}$ is calculated for many systems of interest and can,
in principle, be large in heavy and especially deformed nuclei such as 225Ra.
This corresponds to a nuclear-structure induced enhancement of the observable
EDM, which applies to both the nucleon EDMs and the pion-exchange forces. This
implies that among the potentially leading contributions from the six
hadronic-scale model parameters $C_{P,T}^{(0)}$, $d_{n,p}^{\text{sr}}$, and
$g_{\pi}^{(0,1)}$, very different weights can be expected according to the
electronic and nuclear structure of the various diamagnetic systems.
Inspecting Tab. 3 reveals that to a limited extent, this is indeed already the
case. However, this complementarity is not yet fully exploited.

Despite the pion pole enhancement, $C_{P}^{(0)}$ appears suppressed relative
to $C_{T}^{(0)}$ in all measured systems. The coefficients of
$g_{\pi}^{(0,1)}$ are typically of comparable size for a given system,
although possibly different in sign. Unfortunately the coefficients of
$d_{n,p}$ are not well known for most nuclei, although in principle these can
be calculated. For the pion-exchange forces, even in nuclei that have been the
object of many studies, the theory uncertainties are large. This is especially
true for soft nuclei such as our Hg and Xe, in which non-static deformations
can present special challenges.

For nuclei with spin $I>1/2$, a CP-violating nuclear magnetic quadrupole
moment can in principle exist in analogy to a Schiff-moment-induced EDM and be
analyzed within a common global analysis. At present among the experimental
systems which have already been measured, Cs is the one where this effect
would be expected to be most relevant.

As part of a global analysis, a large number of experimental measurements from
complementary diamagnetic systems can disentangle contributions from model
parameters that are relevant at some level for all of them. In this sense the
role of $d_{e}$ and $C_{S}^{(0)}$ takes on a new importance, not only to
further constrain these parameters themselves, but also as an additional
contribution to the EDM that brings along uncertainties which dilute the
constraining power for other model parameters. Diamagnetic molecular systems
such as TlF, or molecular systems containing Schiff-enhanced nuclides,
introduce complementary constraining power to our global analysis.

We finally note that diamagnetic systems unite theory inputs from different
communities. Different conventions for assigning a negative isospin projection
in the nucleon doublet have to be carefully noted when combining calculated
coefficients from different sources, especially when we include semileptonic
interactions together with nuclear forces. Reference [93] makes an effort to
disambiguate a part of this issue for $\alpha_{g_{\pi}^{(0,1,2)},^{129}Xe}$.

It is only possible to establish meaningful constraints in a context where the
notation and conventions are clear, and where the uncertainties associated
with theory inputs have been clearly assessed.

###  3.4 Single parameter ranges

In this section we provide toy ranges of individual model parameters from the
set

$\displaystyle\Big{\\{}\;d_{e},C_{S}^{(0)},C_{T}^{(0)},C_{P}^{(0)},g_{\pi}^{(0)},g_{\pi}^{(1)},d_{n}^{\text{sr}}\;\;\Big{\\}}\;,$
(35)

using the set of EDM measurements one by one. To extract these single
parameter ranges, we utilize Eq.(27) and include all EDM measurements listed
in Tab. 1 and the model dependence from Tab. 3. For the likelihood we assume a
Gaussian form, which means that all limits are quoted as symmetric one-sigma
error bars around the central, best-fit value. While these single-parameter
limits allow us to compare the reach of different measurements for CP
violation as a whole, they do not give the allowed ranges of the individual
model parameters. The reason is that contributions from different model
parameters to the same measurement can cancel. Because an EFT builds on the
assumption that many higher-dimensional operators are induced by a given
physics model, the single-parameter limits are overly optimistic.

System $i$ | $d_{e}\left[\text{$e$ cm}\right]$ | $C_{S}^{(0)}$ | $C_{P}^{(0)}$ | $C_{T}^{(0)}$
---|---|---|---|---
Tl | $\left(7.2\pm 7.7\right)\cdot 10^{-28}$ | $\left(5.9\pm 6.4\right)\cdot 10^{-8}$ | $\left(-2.7\pm 3.0\right)\cdot 10^{-6}$ | $\left(-4.5\pm 4.9\right)\cdot 10^{-5}$
Cs | $\left(-1.4\pm 5.6\right)\cdot 10^{-26}$ | $\left(-2.3\pm 8.9\right)\cdot 10^{-6}$ | $\left(1.3\pm 5\right)\cdot 10^{-4}$ | $\left(-1.1\pm 4.2\right)\cdot 10^{-4}$
199Hg | $\left(-1.8\pm 2.6\right)\cdot 10^{-28}$ | $\left(-1.7\pm 2.5\right)\cdot 10^{-9}$ | $\left(3.4\pm 4.8\right)\cdot 10^{-8}$ | $\left(-3.4\pm 4.9\right)\cdot 10^{-10}$
129Xe | $\left(2.2\pm 2.3\right)\cdot 10^{-25}$ | $\left(8.3\pm 8.7\right)\cdot 10^{-7}$ | $\left(-1.0\pm 1.1\right)\cdot 10^{-5}$ | $\left(-1.4\pm 1.5\right)\cdot 10^{-7}$
171Yb | $\left(5.7\pm 4.4\right)\cdot 10^{-25}$ | $\left(7.5\pm 5.7\right)\cdot 10^{-6}$ | $\left(-1.5\pm 1.2\right)\cdot 10^{-4}$ | $\left(1.6\pm 1.2\right)\cdot 10^{-6}$
225Ra | $\left(-7.4\pm 1.1\right)\cdot 10^{-23}$ | $\left(4.7\pm 7\right)\cdot 10^{-4}$ | $\left(-5.7\pm 8.5\right)\cdot 10^{-3}$ | $\left(-8.9\pm 13\right)\cdot 10^{-5}$
TlF | $\left(-2.1\pm 3.6\right)\cdot 10^{-25}$ | $\left(-3.0\pm 5.1\right)\cdot 10^{-6}$ | $\left(-7.1\pm 12\right)\cdot 10^{-5}$ | $\left(-3.6\pm 6.1\right)\cdot 10^{-8}$
HfF+ | $\left(-1.3\pm 2.1\right)\cdot 10^{-30}$ | $\left(-1.4\pm 2.3\right)\cdot 10^{-10}$ |  |
ThO | $\left(4.3\pm 4.1\right)\cdot 10^{-30}$ | $\left(2.9\pm 2.7\right)\cdot 10^{-10}$ |  |
YbF | $\left(-2.4\pm 5.9\right)\cdot 10^{-28}$ | $\left(-2.7\pm 6.6\right)\cdot 10^{-8}$ |  |
| $g_{\pi}^{(0)}$ | $g_{\pi}^{(1)}$ | $d_{n}^{\text{sr}}$ | $d_{p}^{\text{sr}}$
$n$ | $\left(0\pm 8.1\right)\cdot 10^{-13}$ | $\left(0\pm 4.1\right)\cdot 10^{-11}$ | $\left(0\pm 1.1\right)\cdot 10^{-26}$ | $\left(0\pm 1.1\right)\cdot 10^{-26}$
199Hg | $\left(-1.9\pm 2.7\right)\cdot 10^{-13}$ | $\left(1.4\pm 2.0\right)\cdot 10^{-13}$ | $\left(-1.4\pm 2.0\right)\cdot 10^{-26}$ | $\left(-1.4\pm 2.0\right)\cdot 10^{-25}$
129Xe | $\left(4.4\pm 4.6\right)\cdot 10^{-9}$ | $\left(8\pm 8.3\right)\cdot 10^{-10}$ | $\left(-1.0\pm 1.1\right)\cdot 10^{-23}$ | $\left(-5.0\pm 5.2\right)\cdot 10^{-23}$
171Yb | $\left(7.2\pm 5.5\right)\cdot 10^{-10}$ | $\left(-5.2\pm 4.0\right)\cdot 10^{-10}$ | $\left(6.0\pm 4.6\right)\cdot 10^{-23}$ | $\left(6.0\pm 4.6\right)\cdot 10^{-22}$
225Ra | $\left(2.4\pm 3.5\right)\cdot 10^{-9}$ | $\left(-5.8\pm 8.7\right)\cdot 10^{-10}$ | $\left(-7.5\pm 11\right)\cdot 10^{-21}$ | $\left(-3.6\pm 5.4\right)\cdot 10^{-20}$
TlF | $\left(-9.0\pm 15\right)\cdot 10^{-10}$ | $\left(1.1\pm 1.8\right)\cdot 10^{-10}$ | $\left(1.8\pm 3.1\right)\cdot 10^{-22}$ | $\left(3.7\pm 6.3\right)\cdot 10^{-23}$
Table 4: Single-parameter ranges allowed by each of the EDM measurements given
in Tab. 1.

Using the single-parameter constraints in Tab. 4 we can compare the impact of
different EDM measurements on a given model parameter, with the caveat that
multi-dimensional correlations might change that picture. Starting with the
electron EDM $d_{e}$, the paramagnetic molecules HfF+ and ThO provide the
strongest constraints, while the paramagnetic YbF molecule has a similar
constraining power as the diamagnetic Hg. The same two paramagnetic molecules
lead the constraining power for the scalar electron-nucleon coupling
$C_{S}^{(0)}$, again followed by Hg, YbF, and Tl.

The pseudoscalar and tensor electron-nucleon couplings $C_{P,T}^{(0)}$ can be
probed by the paramagnetic atoms and the diamagnetic systems. This is done at
present by far most efficiently with Hg and then, with much reduced
constraining power, by Tl, Xe, and TlF. It is difficult to estimate the impact
of these measurements on the combination of $C_{P}^{(0)}$ and $C_{T}^{(0)}$,
and we will see in the next section how this more complex dependence affects
the full global analysis.

The three hadronic parameters, $g_{\pi}^{(0,1)}$ and $d_{n}^{\text{sr}}$, are
most strongly constrained by the neutron EDM measurement and again Hg,
suggesting that there will be significant correlations between the leptonic
and hadronic model parameters in the global analysis. The other diamagnetic
systems lead to much weaker limits, but will still be needed to constrain the
3-dimensional hadronic model space. Note that we are fixing the relation
between the short-range neutron and proton parameters in the Lagrangian,
Eq.(21), but from Tab. 3 we know that different measurements probe different
admixtures of these two parameters.

##  4 Global analysis

To combine the available EDM measurements and analyze them in terms of the
hadronic-scale Lagrangian and its parameters, Eq.(22), we use the established
SFitter analysis tool. It constructs a global likelihood with a comprehensive
uncertainty treatment and analyses it in terms of high-dimensional
correlations. Lower-dimensional and one-dimensional likelihoods for the
individual model parameters can be derived by profiling or marginalization,
depending on the preferred statistical framework. If we assume that
experimental uncertainties are Gaussian, profiling and marginalization have to
lead to identical results. For the theory uncertainties, discussed in Sec.
4.5, the difference between the two approaches makes a formal, but not
significant difference.

###  4.1 SFitter framework

SFitter [34, 35, 36] has been developed for global analyses of LHC
measurements in the context of BSM physics [94] and Higgs and top properties
[95, 96, 38, 97], including comprehensive studies of the connection between
EFTs and UV-completions of the SM [98, 99]. It has a focus on its uncertainty
treatment, including theory uncertainties [39], the connection between
Bayesian and frequentist approaches [35, 37], and published experimental
likelihoods [38].

Our first SFitter analysis of EDMs relates the 11 measurements from Tab. 1 to
the seven model parameters in Eq.(22). In general, SFitter includes
statistical, systematic, and theory uncertainties. At the heart of SFitter is
the fully exclusive likelihood as a function of model and nuisance parameters.
All measurements are described as uncorrelated, with the individual
statistical uncertainties described by Poisson or Gaussian likelihood.
Statistical uncertainties are, usually, uncorrelated as well and described by
a Poisson distribution, turning into a Gaussian for high statistics.
Experimental systematics are assumed to have a Gaussian shape, but can be
described by any nuisance parameter. This Gaussian shape is justified for
parameters which are measured elsewhere. We use it for all experimental
uncertainties, assuming that the measurements are at least as much dominated
by statistical uncertainty, with error distribution appropriate for count-rate
limited frequency measurements.

For the EDM analysis the situation is then relatively simple. First, from
Eq.(23) we know that all observables depend on the model parameters linearly.
Second, we can combine the statistical and systematic experimental
uncertainties into the symmetric Gaussian error bars given in Tab. 1. Finally,
we do not have to consider nuisance parameters, if we assume that the
likelihood has a Gaussian form for each independent measurement. This Gaussian
assumption also implies that for uncorrelated uncertainties a profile
likelihood and a Bayesian marginalization will give the same result.

Theory uncertainties have no well-defined likelihood shape, and no maximum,
but they can be thought of as a range [39]. A flat theory uncertainty is not
parametrization-invariant, as one would have expected from a fixed range, but
without a preferred central value we consider it conservative. In SFitter it
is implemented through an allowed shift of the prediction at no cost in the
likelihoods. For the EDM analysis, theory uncertainties affect the
$\alpha$-values with their central values given in Tab. 3.

To construct the exclusive likelihood, SFitter evaluates EDM predictions over
the entire model parameter space. It uses a Markov chain to encode the
likelihood in the distribution of points covering the model space. A helpful
aspect, common to many BSM analyses, is that we can safely assume the global
minimum of the likelihood to be at the SM parameter point. To remove nuisance
parameters or to extract limits on a reduced number of model parameters,
SFitter can employ a profile likelihood or a Bayesian marginalization [35,
37]. These two methods give different results, with the exception of
uncorrelated Gaussians. Profiling over flat theory uncertainties and Gaussian
experimental uncertainties leads to the RFit[100] prescription, profiling over
two parameters with flat likelihood leads to linearly added uncertainties even
for uncorrelated parameters.

While Eq.(23) suggest a homogeneous set of model parameters, the typical sizes
of the model parameters in Eq.(22) and the $\alpha$-values in Tab. 3 can be
extremely different. For numerical reasons, we internally re-scale each model
parameter and each $\alpha$-value such that all model parameters are evaluated
with similar size. Concretely, this means rescaling $d_{e}$ by a factor
$10^{29}$, $C_{S}^{(0)}$ by $10^{9}$, $g_{\pi}^{(1)}$ and $g_{\pi}^{(0)}$ by
$10^{10}$, $C_{T}^{(0)}$ by $10^{8}$, $C_{P}^{(0)}$ by $10^{6}$, and
$d_{n/p}^{\text{sr}}$ by $10^{23}$. These rescalings are also reflected in the
way we present our results.

###  4.2 Well-constrained model sub-space

Figure 1: Correlations from the 4-dimensional analysis of
$\\{d_{e},C_{S}^{(0)},g_{\pi}^{(0)},d_{n}^{\text{sr}}\\}$, based on all EDM
measurements but neglecting theory uncertainties. The ellipses indicate $68\%$
and $95\%$ CL.

As a starting point of the global analysis and to understand the main
features, we consider a sub-space of well-constrained parameters. Following
our discussion in Sec. 3.4 we expect the semileptonic parameters $d_{e}$ and
$C_{S}^{(0)}$ to be constrained well by the paramagnetic molecules HfF+ and
ThO. Similarly, the hadronic parameters $g_{\pi}^{(0)}$ and
$d_{n}^{\text{sr}}$ are strongly constrained by the neutron and Hg EDMs. This
means the model sub-space

$\displaystyle\Big{\\{}\;d_{e},C_{S},g_{\pi}^{(0)},d_{n}^{\text{sr}}\;\Big{\\}}$
(36)

should be constrained well by the full set of measurements given in Tab. 1.

The interesting question is how the constraints on these four model parameters
are correlated. In Fig. 1 we show these correlations extracted as
2-dimensional profile likelihoods from the fully exclusive, 4-dimensional
likelihood. Three structural aspects stick out: (i) a strong anti-correlation
between $d_{e}$ and $C_{S}^{(0)}$; (ii) a strong anti-correlation between
$g_{\pi}^{(0)}$ and $d_{n}^{\text{sr}}$; and (iii) essentially no correlations
between the $\\{d_{e},C_{S}^{(0)}\\}$ and
$\\{g_{\pi}^{(0)},d_{n}^{\text{sr}}\\}$ parameter subsets.

Within the leptonic sector, the strong correlation between $d_{e}$ and
$C_{S}^{(0)}$ and its independence from the remaining parameter space is
expected to remain for the full global analysis. The reason is that it is
induced by the strongest measurements of HfF+ and ThO, and according to our
parameterization as shown in Tabs. 3 and 4 those two measurements are not
affected by any other model parameter. This means the upper-left panel of Fig.
1 factorizes from our global EDM analysis, and we can consider the remaining
model parameters separately and without the HfF+ and ThO measurements.

For the hadronic parameters the situation is different. Again the neutron and
Hg measurements are three orders more constraining than the other
measurements. However, as the two model parameters we could as well have
chosen $g_{\pi}^{(0)}$ vs $g_{\pi}^{(1)}$, without any change in the
conclusion. This means that we have to expand the hadronic parameter space
next, to see what patterns emerge. The parameters $d_{e}$ and $C_{S}^{(0)}$
will be kept factorized for the rest of our global analysis.

###  4.3 Hadronic parameters from diamagnetic systems

Figure 2: Correlations from three 2-dimensional analyses in the
$\\{g_{\pi}^{(0)},g_{\pi}^{(1)},d_{n}^{\text{sr}}\\}$ parameter space, each
based on a different pair of diamagnetic EDM measurements, as indicated by the
color. The ellipses indicate $68\%$ CL, neglecting theory uncertainties.

For the purely hadronic sector, we define a second simplified model parameter
space,

$\displaystyle\Big{\\{}\;g_{\pi}^{(0)},g_{\pi}^{(1)},d_{n}^{\text{sr}}\;\Big{\\}}\;.$
(37)

All three parameters are constrained by the neutron and diamagnetic EDMs,
while we know from the above discussion that the constraints from diamagnetic
systems on $d_{e}$ and $C_{S}^{(0)}$ are weaker than those from their
paramagnetic counterparts.

From Tab. 4 we see that the neutron and and Hg measurements strongly constrain
two of the three hadronic model parameters in Eq. (37). To understand the
correlations structure of the remaining diamagnetic measurements we show the
correlated constraints of the six possible pairs of diamagnetic measurements
on the different 2-dimensional sub-spaces of Eq. (37).

Starting with the left panel of Fig. 2, the different pairs of measurements
constrain the $g_{\pi}^{(0)}$ vs. $g_{\pi}^{(1)}$ plane with different
correlation patterns, implying that a global analysis will constrain the
3-dimensional hadronic subspace much better than any single pair of
measurements. In the center panel, the situation changes when we look at the
correlation with the neutron EDM-parameter $d_{n}^{\text{sr}}$. Three
combinations are aligned to similar negative correlations between
$d_{n}^{\text{sr}}$ and $g_{\pi}^{(0)}$. In the right panel, the situation is
similar for $g_{\pi}^{(1)}$, with a positive correlation and less striking. In
both cases the exception are the combinations of Xe with Ra, TlF, and Yb,
which constrain $d_{n}^{\text{sr}}$ extremely well and without any correlation
with $g_{\pi}^{(0,1)}$. While this sounds like an advantage, we remind
ourselves that from Tab. 4 and Fig. 1 we know that those limits on
$d_{n}^{\text{sr}}$ are three orders of magnitude weaker than what can be
expected from including the neutron and and Hg measurements.

Altogether, Fig. 2 confirms that the constraints of the sub-leading four
diamagnetic measurements on the 3-dimensional hadronic parameter space of Eq.
(37) are correlated in a non-trivial manner. Evaluating these correlations
requires a global analysis of the, formally, over-constraining set of
diamagnetic measurements.

###  4.4 Poorly constrained model parameters

Figure 3: Correlations from the 5-dimensional analysis of
$\\{C_{T}^{(0)},C_{P}^{(0)},g_{\pi}^{(0)},g_{\pi}^{(1)},d_{n}^{\text{sr}}\\}$
and the factorized $d_{e}-C_{S}^{(0)}$ plane from Fig. 1. We ignore the
neutron and Hg measurements, which induce narrow correlation patterns in the
5-dimensional parameters space and do not affect the profiled 2-dimensional
correlations. The ellipses indicate $68\%$ and $95\%$ CL, neglecting theory
uncertainties.

Finally, we combine the effects of all remaining parameters,

$\displaystyle\Big{\\{}\;C_{T}^{(0)},C_{P}^{(0)},g_{\pi}^{(0)},g_{\pi}^{(1)},d_{n}^{\text{sr}}\;\Big{\\}}\;,$
(38)

ignoring the ThO and HfF+ measurements, which constrain the factorized
$d_{e}-C_{S}^{(0)}$ subspace, and also ignoring the neutron and Hg
measurements. The latter constrain the above parameters, but because they are
much stronger than all other measurements, they will induce narrow
correlations in the allowed 5-dimensional parameter space.

Narrow correlations in a higher-dimensional parameter space vanish when we
profile the likelihood onto 2-dimensional correlations or even single model
parameters. As an example, consider a 2-dimensional parameter space
constrained by one strong and one weak measurement. The strong measurement
leads to a narrow correlation between the two parameters. When we extract the
profile likelihood for one model parameter we can adjust the other model
parameter such that the two parameters trace this narrow correlation. This
way, the entire length of the correlation pattern gets projected onto the
1-dimensional profile likelihoods. The weak measurement dominates the
individual profile likelihoods, while the strong measurement allows us to link
the second model parameter from a first model parameter precisely.

In our case this implies that as long as the correlations from the neutron and
Hg measurements cross the entire parameter space, we can ignore these two
measurements and their correlation patterns in the following discussion of
2-dimensional correlations and single-parameter profile likelihoods. Losing
the best few measurements contributing to the global analysis is an
unfortunate effect of the conservative profile likelihood approach, but for
standard error ellipses it should also exist for Bayesian marginalization. The
main difference is that this kind of effect is numerically extremely
challenging to compute using marginalization, while it is, essentially,
trivial for the profile likelihood.

Given these considerations, we are left with five model parameters,
constrained by seven measurements of similar constraining power. We expect
correlation patterns in the allowed parameter space, but no flat directions.
Indeed, in Fig. 3 we see that all Lagrangian parameters are nicely
constrained. The allowed range, for instance for $g_{\pi}^{(0)}$ is of the
order $10^{-9}$. This can be compared to the constraints from Fig. 1, of the
order $10^{-12}$. The same hierarchy of measurements can be observed for
$g_{\pi}^{(1)}$ and for $d_{n}^{\text{sr}}$, as confirmed by Tab. 4. This
means that the 5-dimensional allowed parameter space illustrated by Fig. 3 is
crossed by two correlation patterns, roughly three orders of magnitude more
narrow than the full parameter space. We emphasize that explaining this
extremely narrow correlation poses a fine-tuning problem in model parameter
space, which the profile likelihood does not address.

###  4.5 Theory uncertainties

Figure 4: Correlations from the 4-dimensional analysis of
$\\{d_{e},C_{S}^{(0)},g_{\pi}^{(0)},d_{n}^{\text{sr}}\\}$. The orange curves
show the effect of theory uncertainties on the results of Fig. 1. The ellipses
indicate $68\%$ and $95\%$ CL.

Theory uncertainties always appear when we use quantum field theory to predict
observables, like EDMs, from Lagrangian parameters: no calculation method is
arbitrarily precise, and a variety of systematic errors can effect accuracy.
While there is some hope that we can estimate and control uncertainties for
small expansion parameters, the uncertainties associated with QCD observables
at low energies (whether from lattice calculations or sum-rule estimates) are
much more tricky to estimate. The precision of nuclear physics calculations,
and their links to effective quantum field theory, are also challenging to
quantify. On the other hand we have to estimate all of these uncertainties,
and we can only ignore them after having shown that they are significantly
smaller than the experimental uncertainties of the associated measurements.

Figure 5: Correlations from the 5-dimensional analysis of
$\\{C_{T}^{(0)},C_{P}^{(0)},g_{\pi}^{(0)},g_{\pi}^{(1)},d_{n}^{\text{sr}}\\}$,
and the factorized $d_{e}-C_{S}^{(0)}$ plane from Fig. 4. The orange curves
show the effect of theory uncertainties on the results of Fig 3. The ellipses
indicate $68\%$ and $95\%$ CL.

For our global EDM analysis, theory uncertainties affect the
$\alpha_{i,c_{j}}$ in Eq.(23). The estimated allowable ranges of each
coefficient are given in Tab. 3. We include them in SFitter as uncorrelated
theory uncertainties, an assumption that can be modified if necessary. Because
of the flat likelihood as a function of the theory nuisance parameter, the
profile likelihood approach leads to the theory uncertainties adding linearly,
weighted by the respective model parameter. Profiling over independent
$\alpha$-ranges simplifies the numerical evaluation in two ways: first, any
parameter–observable pair for which $\alpha$ is compatible with zero will
effectively be removed from the global analysis, because the optimal choice of
$\alpha$ will remove all contributions from the corresponding model parameter;
second, even if we cannot choose $\alpha$ such that measurement and prediction
agree, we can choose it to maximize the likelihood and to minimize the impact
of the measurement, which means we choose the smallest allowed absolute value
of $\alpha$.

As in Sec. 4.2 we start with the well-constrained 4-dimensional subspace
$\\{d_{e},C_{S}^{(0)},g_{\pi}^{(0)},d_{n}^{\text{sr}}\\}$. Again, $d_{e}$ and
$C_{S}^{(0)}$ are constrained by the paramagnetic molecules HfF+ and ThO, just
as without theory uncertainties. From Tab. 3 we see that we can ignore the
theory uncertainty in relating the electron EDM parameter $d_{e}$ to these
systems. In the hadronic sector, the theory uncertainties affecting the
relation of $g_{\pi}^{(0)}$ and $d_{n}^{\text{sr}}$ to the neutron and Hg EDMs
is reasonably small, although for the neutron this situation should not be
over-interpreted. The quoted uncertainty arises from propagating ranges for
the constants within Eq.(26), and not from careful evaluations of the chiral
expansion itself. In Fig. 4 we show the numerical impact of the theory
uncertainties on the 2-dimensional correlations. The slightly stronger HfF+
measurement, which determines the width of the correlation pattern, is only
minimally affected by the theory uncertainties, while the larger theory
uncertainties on the ThO measurement extend the length of the ellipse visibly.

Moving on, we can now look at the effect of the theory uncertainties on the
less-constrained hadronic sector for diamagnetic systems, discussed in Sec.
4.3. Here, Tab. 3 shows sizeable, order-one theory uncertainties. In addition,
some of the $\alpha$-values include an allowed zero value when we include
theory uncertainties. Specifically, $g_{\pi}^{(0)}$ will no longer be
constrained by the Xe measurement and $g_{\pi}^{(1)}$ will lose the Hg
constraint. In addition, some of the theory uncertainties in the diamagnetic
sector are large, so we expect sizeable impact.

As for the case without theory uncertainties, the neutron and Hg limits are
much more constraining than the other measurements of the hadronic sector.
Following the argument given in Sec. 4.4, this means that 2-dimensional
correlations and single-parameter limits extracted by profiling the
likelihoods will not be impacted by these strong measurements. The shift in
the constrained 2-dimensional correlations are shown in Fig. 5. In comparison
their effect on the factorized parameters $d_{e}$ and $C_{S}^{(0)}$ (copied
from Fig. 4) is mild, and the constraints on the hadronic model parameters are
consistently weaker. The only exception is $d_{n}^{\text{sr}}$, since as we
know from Sec. 4.3 the measurements are hardly correlated and the theory
uncertainties are smaller than order-one.

##  5 Outlook

EDMs are extremely sensitive, targeted probes of one of the most important
symmetries of elementary particles, directly related to the observed baryon
asymmetry in the Universe. The number of EDM measurements in very different
systems has grown rapidly in recent years, leading to the question of how the
different measurements can contribute to constraining and understanding CP
violation in terms of a fundamental Lagrangian.

We can choose different Lagrangians to answer this question, starting with UV-
complete models versus EFTs. In the absence of a specific hint for BSM physics
we choose an EFT description. Next, we have a choice of different energy
scales with different degrees of freedom. For our first SFitter analysis we
rely on the hadronic-scale Lagrangian, valid at the GeV scale and describing
the interactions of electrons and nucleons. After relating the hadronic-scale
Lagrangian to its weak-scale SMEFT counterpart we want to constrain the seven
Lagrangian parameters given in Fig. 6 through 11 independent measurements
given in Tab. 1.

As a toy analysis we look at single-parameter constraints from individual EDM
measurements. These limits all driven by the same small set of highly
constraining measurements, like the paramagnetic molecular ion HfF+, the
neutron EDM, or the diamagnetic atom Hg. The extremely strong constraints
indicated in Fig. 6 do not allow for a cancellation of contributions from two
model parameters to a given measurement, at the price of creating a fine
tuning problem. While the single-parameter estimates indicate the strength of
an experiment looking for a sign of CP violation, they should not be confused
with a measurement of a given parameter.

Figure 6: Constraints from the global EDM analysis on the parameters of the
hadronic-scale Lagrangian. We show (i) hugely over-constrained single-
parameter ranges allowed by the best available measurement; (ii) over-
optimistic allowed ranges for profiled single parameters, ignoring theory
uncertainties; (iii) allowed ranges for profiled single parameters including
experimental and theory uncertainties.

For our global analysis we use SFitter, with its focus on the statistical
interpretation and a comprehensive uncertainty treatment. First, we ignore all
theory uncertainties and only consider experimental uncertainties as
uncorrelated and Gaussian (combining the statistical and systematic
uncertainties reported in the respective papers). In this case Bayesian
marginalization and profile likelihood give the same result.

In Sec. 4.2 we find that a small set of powerful measurements constrains the
electron-hadron interactions as well as a subset of the hadronic sector. The
correlated limits on the electron EDM parameter $d_{e}$ and the scalar
coupling $C_{S}^{(0)}$ are especially strong and factorize from the hadronic
sector. Next, we find in Sec. 4.3 that the constraints on the hadronic sector
from the diamagnetic systems show rich correlations, motivating our global
analysis described in Sec. 4.4. For the hadronic parameters, the narrow
correlations from the strong neutron and Hg constraints do not appear in
profiled 2-dimensional correlations or single-parameter limits. As a result,
the hadronic model parameters are constrained much worse than the single-
parameter results suggest, as can be in Fig. 6.

Finally, we show the same limits including theory uncertainties. Such theory
uncertainties always appear when we relate measurements to fundamental
Lagrangian parameters. We assume a flat likelihood within allowed ranges of
the factors relating the Lagrangian parameters to the EDM predictions. While
the impact of the theory uncertainties on the factorized semileptonic sector
is relatively mild, the correlated analysis of the hadronic parameters leads
to a further weakening of the constraints on the individual model parameters.

#### Acknowledgments

We gratefully acknowledge extremely helpful discussions and correspondence
with Tim Chupp, Vincenzo Cirigliano, Jordy de Vries, Victor Flambaum, Timo
Fleig, Maxim Pospelov, and Adam Ritz. We also thank Duarte Azevedo for his
contributions at an early phase of this project. We would like to thank the
ECT* Trento for the hospitality while submitting this work and presenting the
results for the time.

NE is funded by the Heidelberg IMPRS Precision Tests of Fundamental
Symmetries. This research is supported by the Deutsche Forschungsgemeinschaft
(DFG, German Research Foundation) under grant 396021762 – TRR 257: Particle
Physics Phenomenology after the Higgs Discovery and through Germany’s
Excellence Strategy EXC 2181/1 – 390900948 (the Heidelberg STRUCTURES
Excellence Cluster). Furthermore, we acknowledge support by the state of
Baden-Württemberg through bwHPC and the German Research Foundation (DFG)
through grant no INST 39/963-1 FUGG (bwForCluster NEMO).

## References

  * [1] A. D. Sakharov, Violation of CP Invariance, C asymmetry, and baryon asymmetry of the universe, Pisma Zh. Eksp. Teor. Fiz. 5 (1967) 32.
  * [2] V. A. Kuzmin, V. A. Rubakov, and M. E. Shaposhnikov, On the Anomalous Electroweak Baryon Number Nonconservation in the Early Universe, Phys. Lett. B 155 (1985) 36.
  * [3] M. E. Shaposhnikov, Baryon Asymmetry of the Universe in Standard Electroweak Theory, Nucl. Phys. B 287 (1987) 757.
  * [4] A. E. Nelson, D. B. Kaplan, and A. G. Cohen, Why there is something rather than nothing: Matter from weak interactions, Nucl. Phys. B 373 (1992) 453.
  * [5] D. E. Morrissey and M. J. Ramsey-Musolf, Electroweak baryogenesis, New J. Phys. 14 (2012) 125003, arXiv:1206.2942 [hep-ph].
  * [6] A. G. Cohen, D. B. Kaplan, and A. E. Nelson, Progress in electroweak baryogenesis, Ann. Rev. Nucl. Part. Sci. 43 (1993) 27, arXiv:hep-ph/9302210.
  * [7] M. B. Gavela, P. Hernandez, J. Orloff, and O. Pene, Standard model CP violation and baryon asymmetry, Mod. Phys. Lett. A 9 (1994) 795, arXiv:hep-ph/9312215.
  * [8] P. Huet and E. Sather, Electroweak baryogenesis and standard model CP violation, Phys. Rev. D 51 (1995) 379, arXiv:hep-ph/9404302.
  * [9] M. B. Gavela, M. Lozano, J. Orloff, and O. Pene, Standard model CP violation and baryon asymmetry. Part 1: Zero temperature, Nucl. Phys. B 430 (1994) 345, arXiv:hep-ph/9406288.
  * [10] M. B. Gavela, P. Hernandez, J. Orloff, O. Pene, and C. Quimbay, Standard model CP violation and baryon asymmetry. Part 2: Finite temperature, Nucl. Phys. B 430 (1994) 382, arXiv:hep-ph/9406289.
  * [11] A. Riotto and M. Trodden, Recent progress in baryogenesis, Ann. Rev. Nucl. Part. Sci. 49 (1999) 35, arXiv:hep-ph/9901362.
  * [12] M. Pospelov and A. Ritz, Electric dipole moments as probes of new physics, Annals Phys. 318 (2005) 119, arXiv:hep-ph/0504231.
  * [13] J. Engel, M. J. Ramsey-Musolf, and U. van Kolck, Electric Dipole Moments of Nucleons, Nuclei, and Atoms: The Standard Model and Beyond, Prog. Part. Nucl. Phys. 71 (2013) 21, arXiv:1303.2371 [nucl-th].
  * [14] N. Yamanaka, B. K. Sahoo, N. Yoshinaga, T. Sato, K. Asahi, and B. P. Das, Probing exotic phenomena at the interface of nuclear and particle physics with the electric dipole moments of diamagnetic atoms: A unique window to hadronic and semi-leptonic CP violation, Eur. Phys. J. A 53 (2017) 3, 54, arXiv:1703.01570 [hep-ph].
  * [15] T. Chupp, P. Fierlinger, M. Ramsey-Musolf, and J. Singh, Electric dipole moments of atoms, molecules, nuclei, and particles, Rev. Mod. Phys. 91 (2019) 1, 015001, arXiv:1710.02504 [physics.atom-ph].
  * [16] E. Mereghetti, J. de Vries, W. H. Hockings, C. M. Maekawa, and U. van Kolck, The Electric Dipole Form Factor of the Nucleon in Chiral Perturbation Theory to Sub-leading Order, Phys. Lett. B 696 (2011) 97, arXiv:1010.4078 [hep-ph].
  * [17] W. Dekens, J. de Vries, M. Jung, and K. K. Vos, The phenomenology of electric dipole moments in models of scalar leptoquarks, JHEP 01 (2019) 069, arXiv:1809.09114 [hep-ph].
  * [18] D. McKeen, M. Pospelov, and A. Ritz, Modified Higgs branching ratios versus CP and lepton flavor violation, Phys. Rev. D 86 (2012) 113004, arXiv:1208.4597 [hep-ph].
  * [19] S. Dwivedi, D. K. Ghosh, B. Mukhopadhyaya, and A. Shivaji, Constraints on CP-violating gauge-Higgs operators, Phys. Rev. D 92 (2015) 9, 095015, arXiv:1505.05844 [hep-ph].
  * [20] Y. T. Chien, V. Cirigliano, W. Dekens, J. de Vries, and E. Mereghetti, Direct and indirect constraints on CP-violating Higgs-quark and Higgs-gluon interactions, JHEP 02 (2016) 011, arXiv:1510.00725 [hep-ph].
  * [21] V. Cirigliano, W. Dekens, J. de Vries, and E. Mereghetti, Is there room for CP violation in the top-Higgs sector?, Phys. Rev. D 94 (2016) 1, 016002, arXiv:1603.03049 [hep-ph].
  * [22] V. Cirigliano, W. Dekens, J. de Vries, and E. Mereghetti, Constraining the top-Higgs sector of the Standard Model Effective Field Theory, Phys. Rev. D 94 (2016) 3, 034031, arXiv:1605.04311 [hep-ph].
  * [23] S. Inoue, M. J. Ramsey-Musolf, and Y. Zhang, CP-violating phenomenology of flavor conserving two Higgs doublet models, Phys. Rev. D 89 (2014) 11, 115023, arXiv:1403.4257 [hep-ph].
  * [24] D. Fontes, M. Mühlleitner, J. C. Romão, R. Santos, J. a. P. Silva, and J. Wittbrodt, The C2HDM revisited, JHEP 02 (2018) 073, arXiv:1711.09419 [hep-ph].
  * [25] W. Altmannshofer, S. Gori, N. Hamer, and H. H. Patel, Electron EDM in the complex two-Higgs doublet model, Phys. Rev. D 102 (2020) 11, 115042, arXiv:2009.01258 [hep-ph].
  * [26] M. J. Ramsey-Musolf and J. C. Vasquez, Left-right symmetry and electric dipole moments. a global analysis, Physics Letters B 815 (2021) 136136.
  * [27] V. D. Barger, T. Falk, T. Han, J. Jiang, T. Li, and T. Plehn, CP violating phases in SUSY, electric dipole moments, and linear colliders, Phys. Rev. D 64 (2001) 056007, arXiv:hep-ph/0101106.
  * [28] J. R. Ellis, J. S. Lee, and A. Pilaftsis, Electric Dipole Moments in the MSSM Reloaded, JHEP 10 (2008) 049, arXiv:0808.1819 [hep-ph].
  * [29] Y. Li, S. Profumo, and M. Ramsey-Musolf, A Comprehensive Analysis of Electric Dipole Moment Constraints on CP-violating Phases in the MSSM, JHEP 08 (2010) 062, arXiv:1006.1440 [hep-ph].
  * [30] K. Cheung, T.-J. Hou, J. S. Lee, and E. Senaha, Higgs Mediated EDMs in the Next-to-MSSM: An Application to Electroweak Baryogenesis, Phys. Rev. D 84 (2011) 015002, arXiv:1102.5679 [hep-ph].
  * [31] S. F. King, M. Muhlleitner, R. Nevzorov, and K. Walz, Exploring the CP-violating NMSSM: EDM Constraints and Phenomenology, Nucl. Phys. B 901 (2015) 526, arXiv:1508.03255 [hep-ph].
  * [32] T. N. Dao, D. N. Le, and M. Mühlleitner, Leptonic anomalous magnetic and electric dipole moments in the CP-violating NMSSM with and without inverse seesaw mechanism, Eur. Phys. J. C 82 (2022) 10, 954, arXiv:2207.12618 [hep-ph].
  * [33] K. Gaul and R. Berger, Global analysis of CP-violation in atoms, molecules and role of medium-heavy systems, arXiv:2312.08858 [hep-ph].
  * [34] R. Lafaye, T. Plehn, and D. Zerwas, SFITTER: SUSY parameter analysis at LHC and LC, arXiv:hep-ph/0404282.
  * [35] R. Lafaye, T. Plehn, M. Rauch, and D. Zerwas, Measuring Supersymmetry, Eur. Phys. J. C 54 (2008) 617, arXiv:0709.3985 [hep-ph].
  * [36] R. Lafaye, T. Plehn, M. Rauch, D. Zerwas, and M. Duhrssen, Measuring the Higgs Sector, JHEP 08 (2009) 009, arXiv:0904.3866 [hep-ph].
  * [37] I. Brivio, S. Bruggisser, N. Elmer, E. Geoffray, M. Luchmann, and T. Plehn, To Profile or To Marginalize – A SMEFT Case Study, SciPost Phys. 16 (2024) 035, arXiv:2208.08454 [hep-ph].
  * [38] N. Elmer, M. Madigan, T. Plehn, and N. Schmal, Staying on Top of SMEFT-Likelihood Analyses, arXiv:2312.12502 [hep-ph].
  * [39] A. Ghosh, B. Nachman, T. Plehn, L. Shire, T. M. P. Tait, and D. Whiteson, Statistical patterns of theory uncertainties, SciPost Phys. Core 6 (2023) 045, arXiv:2210.15167 [hep-ph].
  * [40] J. de Vries, E. Mereghetti, R. G. E. Timmermans, and U. van Kolck, The Effective Chiral Lagrangian From Dimension-Six Parity and Time-Reversal Violation, Annals Phys. 338 (2013) 50, arXiv:1212.0990 [hep-ph].
  * [41] C. M. Maekawa, E. Mereghetti, J. de Vries, and U. van Kolck, The Time-Reversal- and Parity-Violating Nuclear Potential in Chiral Effective Theory, Nucl. Phys. A 872 (2011) 117, arXiv:1106.6119 [nucl-th].
  * [42] A. Manohar and H. Georgi, Chiral Quarks and the Nonrelativistic Quark Model, Nucl. Phys. B 234 (1984) 189.
  * [43] J. de Vries, A. Gnech, and S. Shain, Renormalization of $CP$-violating nuclear forces, Phys. Rev. C 103 (2021) 1, L012501, arXiv:2007.04927 [hep-ph].
  * [44] S. Weinberg, Nuclear forces from chiral Lagrangians, Phys. Lett. B 251 (1990) 288.
  * [45] Y. V. Stadnik and V. V. Flambaum, Nuclear spin-dependent interactions: Searches for WIMP, Axion and Topological Defect Dark Matter, and Tests of Fundamental Symmetries, Eur. Phys. J. C 75 (2015) 3, 110, arXiv:1408.2184 [hep-ph].
  * [46] J. C. Berengut, V. V. Flambaum, and E. M. Kava, Search for variation of fundamental constants and violations of fundamental symmetries using isotope comparisons, Phys. Rev. A 84 (2011) 042510, arXiv:1109.1893 [physics.atom-ph].
  * [47] Flavour Lattice Averaging Group (FLAG), Y. Aoki et al., FLAG Review 2021, Eur. Phys. J. C 82 (2022) 10, 869, arXiv:2111.09849 [hep-lat].
  * [48] HERMES, A. Airapetian et al., Precise determination of the spin structure function g(1) of the proton, deuteron and neutron, Phys. Rev. D 75 (2007) 012007, arXiv:hep-ex/0609039.
  * [49] M. Shifman, A. Vainshtein, and V. Zakharov, Remarks on higgs-boson interactions with nucleons, Physics Letters B 78 (1978) 4, 443.
  * [50] A. Anselm, V. Bunakov, V. Gudkov, and N. Uraltsev, On the neutron electric dipole moment in the weinberg cp-violation model, Physics Letters B 152 (1985) 1, 116.
  * [51] S. M. Barr, T and P odd electron - nucleon interactions and the electric dipole moments of large atoms, Phys. Rev. D 45 (1992) 4148.
  * [52] M. Bauer and T. Plehn, Yet Another Introduction to Dark Matter: The Particle Physics Approach, vol. 959 of Lecture Notes in Physics.  Springer, 2019.  arXiv:1705.01987 [hep-ph].
  * [53] E. S. Ensberg, Experimental upper limit for the permanent electric dipole moment of ${\mathrm{rb}}^{85}$ by optical pumping techniques, Phys. Rev. 164 (Dec, 1967) 270.
  * [54] E. S. Ensberg, Experimental upper limit for the permanent electric dipole moment of ${\mathrm{rb}}^{85}$ by optical-pumping techniques, Phys. Rev. 153 (Jan, 1967) 36.
  * [55] M. A. Player and P. G. H. Sandars, An experiment to search for an electric dipole moment in the 3p2 metastable state of xenon, Journal of Physics B: Atomic and Molecular Physics 3 (dec, 1970) 1620.
  * [56] S. Eckel, P. Hamilton, E. Kirilov, H. W. Smith, and D. DeMille, Search for the electron electric dipole moment using $\Omega$-doublet levels in PbO, Phys. Rev. A 87 (2013) 5, 052130, arXiv:1303.3075 [physics.atom-ph].
  * [57] S. Eckel, A. O. Sushkov, and S. K. Lamoreaux, A limit on the electron electric dipole moment using paramagnetic ferroelectric Eu0.5Ba0.5TiO3, Phys. Rev. Lett. 109 (2012) 193003, arXiv:1208.4420 [physics.atom-ph].
  * [58] L. Pondrom, R. Handler, M. Sheaff, P. T. Cox, J. Dworkin, O. E. Overseth, T. Devlin, L. Schachinger, and K. J. Heller, New Limit on the Electric Dipole Moment of the $\Lambda$ Hyperon, Phys. Rev. D 23 (1981) 814.
  * [59] C. Abel et al., Measurement of the Permanent Electric Dipole Moment of the Neutron, Phys. Rev. Lett. 124 (2020) 8, 081803, arXiv:2001.11966 [hep-ex].
  * [60] B. C. Regan, E. D. Commins, C. J. Schmidt, and D. DeMille, New limit on the electron electric dipole moment, Phys. Rev. Lett. 88 (2002) 071805.
  * [61] S. A. Murthy, D. Krause, Z. L. Li, and L. R. Hunter, New Limits on the Electron Electric Dipole Moment from Cesium, Phys. Rev. Lett. 63 (1989) 965.
  * [62] T. S. Roussy et al., A new bound on the electron’s electric dipole moment, Science 381 (2023) 46, arXiv:2212.11841 [physics.atom-ph].
  * [63] ACME, V. Andreev et al., Improved limit on the electric dipole moment of the electron, Nature 562 (2018) 7727, 355.
  * [64] J. J. Hudson, D. M. Kara, I. J. Smallman, B. E. Sauer, M. R. Tarbutt, and E. A. Hinds, Improved measurement of the shape of the electron, Nature 473 (2011) 493.
  * [65] B. Graner, Y. Chen, E. G. Lindahl, and B. R. Heckel, Reduced Limit on the Permanent Electric Dipole Moment of Hg199, Phys. Rev. Lett. 116 (2016) 16, 161601, arXiv:1601.04339 [physics.atom-ph]. [Erratum: Phys.Rev.Lett. 119, 119901 (2017)].
  * [66] B. Graner, Y. Chen, E. G. Lindahl, and B. R. Heckel, Erratum: Reduced limit on the permanent electric dipole moment of ${}^{199}\mathrm{Hg}$ [phys. rev. lett. 116, 161601 (2016)], Phys. Rev. Lett. 119 (Sep, 2017) 119901.
  * [67] N. Sachdeva et al., New Limit on the Permanent Electric Dipole Moment of 129Xe using 3He Comagnetometry and SQUID Detection, Phys. Rev. Lett. 123 (2019) 14, 143003, arXiv:1902.02864 [physics.atom-ph].
  * [68] F. Allmendinger, I. Engin, W. Heil, S. Karpuk, H.-J. Krause, B. Niederländer, A. Offenhäusser, M. Repetto, U. Schmidt, and S. Zimmer, Measurement of the permanent electric dipole moment of the ${}^{129}\mathrm{Xe}$ atom, Phys. Rev. A 100 (Aug, 2019) 022505.
  * [69] T. A. Zheng, Y. A. Yang, S. Z. Wang, J. T. Singh, Z. X. Xiong, T. Xia, and Z. T. Lu, Measurement of the Electric Dipole Moment of Yb171 Atoms in an Optical Dipole Trap, Phys. Rev. Lett. 129 (2022) 8, 083001, arXiv:2207.08140 [physics.atom-ph].
  * [70] M. Bishof et al., Improved limit on the 225Ra electric dipole moment, Phys. Rev. C 94 (2016) 2, 025501, arXiv:1606.04931 [nucl-ex].
  * [71] D. Cho, K. Sangster, and E. A. Hinds, Search for time reversal symmetry violation in thallium fluoride using a jet source, Phys. Rev. A 44 (1991) 2783.
  * [72] P. V. Coveney and P. G. H. Sandars, PARITY AND TIME VIOLATING INTERACTIONS IN THALLIUM FLUORIDE, J. Phys. B 16 (1983) 3727.
  * [73] J. de Vries, P. Draper, K. Fuyuto, J. Kozaczuk, and B. Lillard, Uncovering an axion mechanism with the EDM portfolio, Phys. Rev. D 104 (2021) 5, 055039, arXiv:2107.04046 [hep-ph].
  * [74] Particle Data Group, R. L. Workman and Others, Review of Particle Physics, PTEP 2022 (2022) 083C01.
  * [75] T. Fleig and L. V. Skripnikov, P , t -violating and magnetic hyperfine interactions in atomic thallium, Symmetry 12 (2020) 4, .
  * [76] T. Fleig and M. Jung, Model-independent determinations of the electron EDM and the role of diamagnetic atoms, JHEP 07 (2018) 012, arXiv:1802.02171 [hep-ph].
  * [77] A.-M. Mårtensson-Pendrill and P. Öster, Calculations of atomic electric dipole moments, Physica Scripta 36 (sep, 1987) 444.
  * [78] V. A. Dzuba, V. V. Flambaum, and S. G. Porsev, Calculation of P,T-odd electric dipole moments for diamagnetic atoms Xe-129, Yb-171, Hg-199, Rn-211, and Ra-225, Phys. Rev. A 80 (2009) 032120, arXiv:0906.5437 [physics.atom-ph].
  * [79] L. V. Skripnikov, Communication: Theoretical study of HfF+ cation to search for the T,P-odd interactions, The Journal of Chemical Physics 147 (07, 2017) 021101, https://pubs.aip.org/aip/jcp/article-pdf/doi/10.1063/1.4993622/13976956/021101_1_online.pdf.
  * [80] T. Fleig, $\mathcal{P},\mathcal{T}$-odd and magnetic hyperfine-interaction constants and excited-state lifetime for $HfF^{+}$, Phys. Rev. A 96 (2017) 4, 040502, arXiv:1706.02893 [physics.atom-ph].
  * [81] A. N. Petrov, N. S. Mosyagin, T. A. Isaev, and A. V. Titov, Theoretical study of Hf F+ in search of the electron electric dipole moment, Phys. Rev. A 76 (2007) 030501, arXiv:physics/0611254.
  * [82] E. R. Meyer, J. L. Bohn, and M. P. Deskevich, Candidate molecular ions for an electron electric dipole moment experiment, Phys. Rev. A 73 (Jun, 2006) 062108.
  * [83] L. V. Skripnikov, Combined 4-component and relativistic pseudopotential study of ThO for the electron electric dipole moment search, The Journal of Chemical Physics 145 (12, 2016) 214301, https://pubs.aip.org/aip/jcp/article-pdf/doi/10.1063/1.4968229/15523128/214301_1_online.pdf.
  * [84] E. R. Meyer and J. L. Bohn, Prospects for an electron electric dipole moment search in metastable ThO and ThF, Phys. Rev. A 78 (2008) 010502, arXiv:0805.0161 [physics.atom-ph].
  * [85] M. Denis and T. Fleig, In search of discrete symmetry violations beyond the standard model: Thorium monoxide reloaded, The Journal of Chemical Physics 145 (12, 2016) 214307, https://pubs.aip.org/aip/jcp/article-pdf/doi/10.1063/1.4968597/15523220/214307_1_online.pdf.
  * [86] V. A. Dzuba, V. V. Flambaum, and C. Harabati, Relations between matrix elements of different weak interactions and interpretation of the parity-nonconserving and electron electric-dipole-moment measurements in atoms and molecules, Phys. Rev. A 84 (Nov, 2011) 052108.
  * [87] V. A. Dzuba, V. V. Flambaum, and C. Harabati, Erratum: Relations between matrix elements of different weak interactions and interpretation of the parity-nonconserving and electron electric-dipole-moment measurements in atoms and molecules [phys. rev. a 84, 052108 (2011)], Phys. Rev. A 85 (Feb, 2012) 029901.
  * [88] A. Sunaga, M. Abe, M. Hada, and B. P. Das, Relativistic coupled-cluster calculation of the electron-nucleus scalar-pseudoscalar interaction constant ${W}_{s}$ in ybf, Phys. Rev. A 93 (Apr, 2016) 042507.
  * [89] M. Abe, G. Gopakumar, M. Hada, B. P. Das, H. Tatewaki, and D. Mukherjee, Application of relativistic coupled-cluster theory to the effective electric field in ybf, Phys. Rev. A 90 (Aug, 2014) 022501.
  * [90] J. Ginges and V. Flambaum, Violations of fundamental symmetries in atoms and tests of unification theories of elementary particles, Physics Reports 397 (2004) 2, 63.
  * [91] Y. Ema, T. Gao, and M. Pospelov, Improved Indirect Limits on Muon Electric Dipole Moment, Phys. Rev. Lett. 128 (2022) 13, 131803, arXiv:2108.05398 [hep-ph].
  * [92] ACME, J. Baron et al., Methods, Analysis, and the Treatment of Systematic Errors for the Electron Electric Dipole Moment Search in Thorium Monoxide, New J. Phys. 19 (2017) 7, 073029, arXiv:1612.09318 [physics.atom-ph].
  * [93] K. Yanase and N. Shimizu, Large-scale shell-model calculations of nuclear Schiff moments of 129Xe and 199Hg, Phys. Rev. C 102 (2020) 6, 065502, arXiv:2006.15142 [nucl-th].
  * [94] A. Butter, S. Murgia, T. Plehn, and T. M. P. Tait, Saving the MSSM from the Galactic Center Excess, Phys. Rev. D 96 (2017) 3, 035036, arXiv:1612.07115 [hep-ph].
  * [95] M. Klute, R. Lafaye, T. Plehn, M. Rauch, and D. Zerwas, Measuring Higgs Couplings from LHC Data, Phys. Rev. Lett. 109 (2012) 101801, arXiv:1205.2699 [hep-ph].
  * [96] A. Biekötter, T. Corbett, and T. Plehn, The Gauge-Higgs Legacy of the LHC Run II, SciPost Phys. 6 (2019) 6, 064, arXiv:1812.07587 [hep-ph].
  * [97] I. Brivio, S. Bruggisser, F. Maltoni, R. Moutafis, T. Plehn, E. Vryonidou, S. Westhoff, and C. Zhang, O new physics, where art thou? A global search in the top sector, JHEP 02 (2020) 131, arXiv:1910.03606 [hep-ph].
  * [98] D. López-Val, T. Plehn, and M. Rauch, Measuring extended Higgs sectors as a consistent free couplings model, JHEP 10 (2013) 134, arXiv:1308.1979 [hep-ph].
  * [99] I. Brivio, S. Bruggisser, E. Geoffray, W. Killian, M. Krämer, M. Luchmann, T. Plehn, and B. Summ, From models to SMEFT and back?, SciPost Phys. 12 (2022) 1, 036, arXiv:2108.01094 [hep-ph].
  * [100] A. Hocker, H. Lacker, S. Laplace, and F. Le Diberder, A New approach to a global fit of the CKM matrix, Eur. Phys. J. C 21 (2001) 225, arXiv:hep-ph/0104062.