$[C^{V8,LR}_{dd}]_{1221}$ | $(-7.4\pm 3.1)\cdot 10^{-22}$ | $[C^{S1,RR}_{uddu}]_{2222}$ | $(-2.5\pm 1.5)\cdot 10^{-24}$ | $[C^{S1,RR}_{dd}]_{2233}$ | $(1.3\pm 0.7)\cdot 10^{-25}$
$[C^{S1,RR}_{uu}]_{1111}$ | $(1.1\pm 0.5)\cdot 10^{-21}$ | $[C^{S8,RR}_{uddu}]_{2222}$ | $(-1.9\pm 1.3)\cdot 10^{-24}$ | $[C^{S8,RR}_{dd}]_{2233}$ | $(-3.7\pm 2.1)\cdot 10^{-25}$
$[C^{S8,RR}_{uu}]_{1111}$ | $(-3.4\pm 1.5)\cdot 10^{-22}$ | $[C^{S1,RR}_{uu}]_{1221}$ | $(8.3\pm 4.2)\cdot 10^{-22}$ | $[C^{S1,RR}_{dd}]_{1331}$ | $(3.5\pm 2.0)\cdot 10^{-21}$
$[C^{S1,RR}_{dd}]_{1111}$ | $(5.7\pm 3.1)\cdot 10^{-22}$ | $[C^{S8,RR}_{uu}]_{1221}$ | $(-2.5\pm 0.4)\cdot 10^{-22}$ | $[C^{S8,RR}_{dd}]_{1331}$ | $(-1.3\pm 0.3)\cdot 10^{-21}$
$[C^{S8,RR}_{dd}]_{1111}$ | $(-1.8\pm 0.9)\cdot 10^{-22}$ | $[C^{V1,LR}_{uddu}]_{2112}$ | $(5.4\pm 1.4)\cdot 10^{-22}$ | $[C^{S1,RR}_{dd}]_{2332}$ | $(1.4\pm 0.8)\cdot 10^{-24}$
$[C^{S1,RR}_{dd}]_{2222}$ | $(1.0\pm 0.6)\cdot 10^{-25}$ | $[C^{V8,LR}_{uddu}]_{2112}$ | $(7.2\pm 0.5)\cdot 10^{-22}$ | $[C^{S8,RR}_{dd}]_{2332}$ | $(2.9\pm 1.6)\cdot 10^{-24}$
$[C^{S8,RR}_{dd}]_{2222}$ | $(1.2\pm 0.6)\cdot 10^{-25}$ | $[C^{V1,LR}_{uddu}]_{2222}$ | $(-8.1\pm 4.5)\cdot 10^{-25}$ | $[C^{S1,RR}_{ud}]_{1133}$ | $(-3.2\pm 1.4)\cdot 10^{-22}$
$[C^{S1,RR}_{dd}]_{1122}$ | $(1.2\pm 0.7)\cdot 10^{-22}$ | $[C^{V8,LR}_{uddu}]_{2222}$ | $(-2.8\pm 1.8)\cdot 10^{-24}$ | $[C^{S8,RR}_{ud}]_{1133}$ | $(-1.4\pm 0.5)\cdot 10^{-22}$
$[C^{S8,RR}_{dd}]_{1122}$ | $(-4.7\pm 2.4)\cdot 10^{-23}$ | $[C^{V1,LR}_{uu}]_{1221}$ | $(6.3\pm 7.9)\cdot 10^{-23}$ | $[C^{S1,RR}_{uddu}]_{1331}$ | $(2.3\pm 1.0)\cdot 10^{-21}$
$[C^{S1,RR}_{dd}]_{1221}$ | $(2.5\pm 1.0)\cdot 10^{-22}$ | $[C^{V8,LR}_{uu}]_{1221}$ | $(-2.6\pm 0.3)\cdot 10^{-22}$ | $[C^{S8,RR}_{uddu}]_{1331}$ | $(-5.8\pm 11.6)\cdot 10^{-23}$
$[C^{S8,RR}_{dd}]_{1221}$ | $(-7.8\pm 2.0)\cdot 10^{-23}$ |  | - | $[C^{S1,RR}_{ud}]_{2233}$ | $(1.5\pm 0.7)\cdot 10^{-25}$
$[C^{S1,RR}_{ud}]_{1111}$ | $(4.2\pm 2.1)\cdot 10^{-22}$ |  | - | $[C^{S8,RR}_{ud}]_{2233}$ | $(4.9\pm 2.0)\cdot 10^{-26}$
$[C^{S8,RR}_{ud}]_{1111}$ | $(-1.3\pm 0.7)\cdot 10^{-22}$ |  | - | $[C^{S1,RR}_{uddu}]_{2332}$ | $(-9.7\pm 4.9)\cdot 10^{-25}$
$[C^{S1,RR}_{ud}]_{1122}$ | $(2.6\pm 1.4)\cdot 10^{-22}$ |  | - | $[C^{S8,RR}_{uddu}]_{2332}$ | $(1.2\pm 0.6)\cdot 10^{-25}$
$[C^{S8,RR}_{ud}]_{1122}$ | $(-8.8\pm 4.4)\cdot 10^{-23}$ |  | - | $[C^{V1,LR}_{uddu}]_{2332}$ | $(-2.7\pm 1.5)\cdot 10^{-25}$
$[C^{S1,RR}_{uddu}]_{1111}$ | $(4.2\pm 2.2)\cdot 10^{-22}$ |  | - | $[C^{V8,LR}_{uddu}]_{2332}$ | $(7.5\pm 3.6)\cdot 10^{-26}$
$[C^{S8,RR}_{uddu}]_{1111}$ | $(-1.3\pm 0.7)\cdot 10^{-22}$ |  | - |  | -
$[C^{S1,RR}_{uddu}]_{1221}$ | $(3.4\pm 1.4)\cdot 10^{-22}$ |  | - |  | -
$[C^{S8,RR}_{uddu}]_{1221}$ | $(-8.4\pm 3.9)\cdot 10^{-23}$ |  | - |  | -

Table 10: Master formula for $d_{n}$ in WET in the JMS basis at the EW scale.
The $\alpha_{I}^{n}$ have units of $e$ cm $\rm TeV^{2}$ and $e$ cm $\rm TeV$
for the four-fermion and dipole operators, respectively. The $90\%$ CL upper
bound on $d_{n}$ is $1.8\times 10^{-26}$ $e$ cm.

##  6 Non-standard Higgs Couplings

In this section, we apply the master formulae derived so far to constrain
possible non-standard Yukawa couplings of the Higgs boson. We consider the
following Yukawa Lagrangian

$\mathcal{L}_{\rm Yuk}^{\rm
eff}=-\left(\sum_{i,j=d,s,b}y_{ij}^{d}~{}\bar{d}_{L,i}d_{R,j}+\sum_{i,j=u,c,t}y_{ij}^{u}~{}\bar{u}_{L,i}u_{R,j}+\sum_{i,j=e,\mu,\tau}y_{ij}^{e}~{}\bar{e}_{L,i}e_{R,j}\right)h+h.c.$
(116)

This Lagrangian can arise in two theoretical frameworks that can be used for
describing non-standard couplings of the Higgs to quarks, lepton, and gauge
bosons. In SMEFT, gauge symmetry is linearly realized, the Higgs boson belongs
to a $SU_{L}(2)$ doublet and operators are organized according to their
canonical dimension. In this framework, the SM Yukawa couplings arise at
dim-4, they can always be chosen to be real and diagonal, and are determined
by the quark and lepton masses. Non-standard Yukawa couplings arise at dim-6
and are given by the Lagrangian Buchmuller:1986zs ; Grzadkowski:2010es ;
Hamoudou:2022tdn

$\displaystyle\mathcal{L}_{h}$
$\displaystyle=(D_{\mu}H)^{\dagger}(D^{\mu}H)-\lambda\left(H^{\dagger}H-{1\over
2}v^{2}\right)^{2}-\left[\bar{Q}\hat{Y}_{u}u_{R}\widetilde{H}+\bar{Q}\hat{Y}_{d}d_{R}H+\bar{L}\hat{Y}_{u}e_{R}H\right.$
(117)
$\displaystyle\left.+(\bar{Q}C_{uH}u_{R}\widetilde{H}+\bar{Q}C_{dH}d_{R}H+\bar{L}C_{eH}e_{R}H)(H^{\dagger}H)+h.c.\right].$

In unitary gauge

$H=\frac{1}{\sqrt{2}}(0,v+h)^{T},$ (118)

where $h$ is the physical Higgs boson field. $C_{fH}$, for $f=u,d,e$, are the
WC of dim-6 SMEFT operators. We can rewrite Eq. (117) in terms of mass
matrices, $M_{f}$, and Yukawa interactions

$\displaystyle M_{u}$
$\displaystyle=\frac{v}{\sqrt{2}}\left(\hat{Y}_{u}+\frac{v^{2}}{2}C_{uH}\right),\qquad
y^{u}=\frac{1}{\sqrt{2}}\left(\hat{Y}_{u}+\frac{3}{2}v^{2}C_{uH}\right)=\frac{M_{u}}{v}+\frac{v^{2}}{\sqrt{2}}C_{uH}\,,$
(119) $\displaystyle M_{d}$
$\displaystyle=\frac{v}{\sqrt{2}}\left(\hat{Y}_{d}+\frac{v^{2}}{2}C_{dH}\right),\qquad
y^{d}=\frac{1}{\sqrt{2}}\left(\hat{Y}_{d}+\frac{3}{2}v^{2}C_{dH}\right)=\frac{M_{d}}{v}+\frac{v^{2}}{\sqrt{2}}C_{dH}\,,$
(120) $\displaystyle M_{e}$
$\displaystyle=\frac{v}{\sqrt{2}}\left(\hat{Y}_{e}+\frac{v^{2}}{2}C_{eH}\right),\qquad
y^{e}=\frac{1}{\sqrt{2}}\left(\hat{Y}_{e}+\frac{3}{2}v^{2}C_{eH}\right)=\frac{M_{e}}{v}+\frac{v^{2}}{\sqrt{2}}C_{eH}.$
(121)

Since at dim-6 the fermion masses and Yukawa couplings are proportional to
different linear combinations of $\hat{Y}_{f}$ and $C_{fH}$, once the mass
matrices are diagonalized, the Yukawas are in general non-diagonal and
complex.

The other possible theoretical framework is the Higgs Effective Field Theory
(HEFT), in which gauge symmetry is realized non-linearly. In this case, the
Higgs field $h$ is a singlet under $SU(2)_{L}$, and the EFT provides a
momentum expansion in $Q/\Lambda$. The LO HEFT Lagrangian is given by
Buchalla:2013rka

$\displaystyle\mathcal{L}_{Uh}$ $\displaystyle=\frac{v^{2}}{4}\langle
D_{\mu}U^{\dagger}D^{\mu}U\rangle(1+F_{U}(h))+\frac{1}{2}\partial_{\mu}h\partial^{\mu}h-V(h)$
(122)
$\displaystyle-v\left[\bar{Q}\left(\hat{Y}_{u}+\sum_{n=1}^{\infty}\hat{Y}_{u}^{(n)}\left({h\over
v}\right)^{n}\right)UP_{+}r+\bar{Q}\left(\hat{Y}_{d}+\sum_{n=1}^{\infty}\hat{Y}_{d}^{(n)}\left({h\over
v}\right)^{n}\right)UP_{-}r\right.$
$\displaystyle\left.+\bar{L}\left(\hat{Y}_{e}+\sum_{n=1}^{\infty}\hat{Y}_{e}^{(n)}\left({h\over
v}\right)^{n}\right)UP_{-}\eta+h.c.\right].$

Here, $F_{U}(h)$ and $V(h)$ are arbitrary functions of the Higgs field

$F_{U}(h)=\sum_{n=1}^{\infty}f_{U,n}\left({h\over v}\right)^{n}\ \,,\quad
V(h)=v^{4}\sum_{n=2}^{\infty}f_{V,n}\left({h\over v}\right)^{n}.$ (123)

Also, the left and right-chiral fields are defined by
$Q=(u_{L},d_{L})^{T},L=(\nu_{L},e_{L})^{T}$, $r=(u_{R},d_{R})^{T}$ and
$\eta=(\nu_{R},e_{R})^{T}$ and $P_{\pm}=(1\pm\tau_{3})/2$, with $\bm{\tau}$
Pauli matrices. $U$ is a matrix containing the Goldstone fields

$U=\textrm{exp}(2i\Phi/v),\qquad\Phi=\phi^{a}\frac{\tau^{a}}{2}=\frac{1}{\sqrt{2}}\begin{pmatrix}\frac{\phi^{0}}{\sqrt{2}}&\phi^{+}\\\
\phi^{-}&{-\frac{\phi^{0}}{\sqrt{2}}}\end{pmatrix},$ (124)

and the action of covariant derivative on $U$ is given by

$D_{\mu}U=\partial_{\mu}+igW_{\mu}U-ig^{\prime}B_{\mu}U\frac{\tau_{3}}{2}.$
(125)

The implications of (122) are that the $hWW$ and $hZZ$ couplings are decoupled
from the $W$ and $Z$ masses. They are parameterized by a free coupling
$\kappa_{V}$, with $\kappa_{V}=1$ in the SM. Similarly, the fermion masses and
Yukawas are characterized by independent matrices, so that, as in the SMEFT,
the Yukawa couplings are in general complex and non-diagonal. In this case, we
have

$M_{f}=v\hat{Y}_{f},\qquad y^{f}=\hat{Y}_{f}^{(1)}.$ (126)

The main difference between the HEFT and SMEFT is that in HEFT the leading
operator (as well as subleading operators) contain an arbitrary number of
Higgs fields, while in SMEFT insertions of $H^{\dagger}H$ are suppressed by
$\Lambda^{2}$ Buchalla:2013rka . Thus, in HEFT, Yukawa couplings and fermion
masses differ already at leading order.

###  6.1 Matching at the EW scale

Integrating out the Higgs boson at tree level leads to several four-fermion
scalar operators with CPV WCs, if the Yukawa couplings are assumed to be
complex Hamoudou:2022tdn .

$\displaystyle{[C^{S1,RR}_{dd}]}_{ijkl}(\mu_{ew})$
$\displaystyle=\frac{1}{2m_{h}^{2}}y_{ij}^{d}y^{d}_{kl}\,,\quad{[C^{S1,RR}_{uu}]}_{ijkl}(\mu_{ew})=\frac{1}{2m_{h}^{2}}y_{ij}^{u}y_{kl}^{u}\,,$
(127) $\displaystyle{[C^{S1,RR}_{ud}]}_{ijkl}(\mu_{ew})$
$\displaystyle=\frac{1}{m_{h}^{2}}y_{ij}^{u}y_{kl}^{d}\,,\quad{[C^{S,RR}_{ee}]}_{ijkl}(\mu_{ew})=\frac{1}{2m_{h}^{2}}y_{ij}^{e}y_{kl}^{e}\,,$
$\displaystyle{[C^{S,RR}_{ed}]}_{ijkl}(\mu_{ew})$
$\displaystyle=\frac{1}{m_{h}^{2}}y_{ij}^{e}y^{d}_{kl}\,,\quad{[C^{S,RR}_{eu}]}_{ijkl}(\mu_{ew})=\frac{1}{m_{h}^{2}}y_{ij}^{e}y^{u}_{kl}\,,$
$\displaystyle{[C^{S,RL}_{ed}]}_{ijkl}(\mu_{ew})$
$\displaystyle=\frac{1}{m_{h}^{2}}y_{ij}^{e}y^{d*}_{lk}\,,\quad{[C^{S,RL}_{eu}]}_{ijkl}(\mu_{ew})=\frac{1}{m_{h}^{2}}y_{ij}^{e}y^{u*}_{lk}\,,$
$\displaystyle{[C^{V1,LR}_{dd}]}_{ijkl}(\mu_{ew})$
$\displaystyle=-\frac{1}{6m_{h}^{2}}y_{il}^{d}y^{d*}_{jk}\,,\quad{[C^{V8,LR}_{dd}]}_{ijkl}(\mu_{ew})=-\frac{1}{m_{h}^{2}}y_{il}^{d}y^{d*}_{jk}\,,$
$\displaystyle{[C^{V1,LR}_{uu}]}_{ijkl}(\mu_{ew})$
$\displaystyle=-\frac{1}{6m_{h}^{2}}y_{il}^{u}y^{u*}_{jk}\,,\quad{[C^{V8,LR}_{uu}]}_{ijkl}(\mu_{ew})=-\frac{1}{m_{h}^{2}}y_{il}^{u}y^{u*}_{jk}\,,$
$\displaystyle{[C^{V1,LR}_{uddu}]}_{ijkl}(\mu_{ew})$
$\displaystyle=-\frac{1}{6m_{h}^{2}}y_{il}^{u}y^{d*}_{jk}\,,\quad{[C^{V8,LR}_{uddu}]}_{ijkl}(\mu_{ew})=-\frac{1}{m_{h}^{2}}y_{il}^{u}y^{d*}_{jk}\,,$
$\displaystyle{[C^{V,LR}_{ee}]}_{ijkl}(\mu_{ew})$
$\displaystyle=-\frac{1}{2m_{h}^{2}}y_{il}^{e}y^{e*}_{jk}.$

Here the matching scale $\mu_{ew}$ can be set to $m_{h}$. Since EDMs can only
constrain the imaginary parts of the WCs, they probe combinations of couplings
such as

$\quad{\rm
Im}(y^{f}_{ij}y^{f^{\prime}}_{kl})=\textrm{Re}(y^{f}_{ij})\textrm{Im}(y^{f^{\prime}}_{kl})+\textrm{Im}(y^{f}_{ij})\textrm{Re}(y^{f^{\prime}}_{kl})\,,\quad
f,f^{\prime}=u,d,e\,.$ (128)

In the SMEFT, the imaginary part of the Yukawa coupling can be non-zero only
at dim-6, so that four-fermion operators become proportional to

$\left.\quad{\rm Im}(y^{f}_{ij}y^{f^{\prime}}_{kl})\right|_{\rm
SMEFT}=\frac{m_{f}}{v}\delta_{ij}\textrm{Im}(y^{f^{\prime}}_{kl})+\frac{m_{f^{\prime}}}{v}\delta_{kl}\textrm{Im}(y^{f}_{ij})+\mathcal{O}\left(\frac{v^{4}}{\Lambda^{4}}\right).$
(129)

In this case, flavor-violating couplings only contribute to EDMs at dim-8 and
can be neglected. Both flavor-conserving and flavor-violating couplings
contribute at leading order in the HEFT. In the following, we analyze the
flavor-conserving and flavor-violating couplings separately. Eq. (127) is the
initial condition for the renormalization group evolution. As discussed in
Section 4, the renormalization group running of these operators below the EW
scale results in contributions to four-fermion operators, and, in particular
in the case of operators with heavy quarks and leptons, to quark and gluon
dipole operators. All these contributions are included in the master formulae.

Beyond tree level, non-standard Yukawa couplings generate dipole contributions
at one loop Hisano:2012cc ; Chien:2015xha . These are proportional to the same
combination of couplings in Eq. (127) and provide a next-to-leading
logarithmic correction, which we can neglect.

Figure 4: Two loop Barr-Zee diagrams contributing to $\Delta F=0$ dipole
operators. Here plain lines denote quarks and leptons lighter than the
electroweak scale, double lines the top quark and double wiggly lines the $W$
boson. Insertions of the fermion Yukawa and of the $hWW$ couplings are denoted
by squares, while gauge couplings are denoted by dots.

At two loops, Barr-Zee diagrams Barr:1990vd provide sizable contributions, and
are sensitive to different combinations of Yukawa couplings. The diagrams are
shown in Fig. 4. Integrating out the top-quark, Higgs boson, and $W$-boson at
$\mu_{ew}$ leads to contributions to the quark and lepton dipole operators
$C_{f\gamma}$ and to the quark chromo-dipole operator $C_{qG}$

$\displaystyle\textrm{Im}\left[C_{f\gamma}\right]_{ii}(\mu_{ew})$
$\displaystyle=-12e\frac{\alpha}{(4\pi)^{3}}q_{f}q^{2}_{t}\frac{1}{m_{t}}\left[f(x_{t})\textrm{Re}(y^{f}_{ii})\textrm{Im}(y^{u}_{33})+g(x_{t})\textrm{Im}(y^{f}_{ii})\textrm{Re}(y^{u}_{33})\right]$
$\displaystyle+2eq_{f}\frac{\alpha}{(4\pi)^{3}}\frac{1}{v}\kappa_{V}(3f(x_{W})+5g(x_{W}))\,\textrm{Im}(y^{f}_{ii})\,,$
(130) $\displaystyle\textrm{Im}\left[C_{qG}\right]_{ii}(\mu_{ew})$
$\displaystyle=2g_{s}\frac{\alpha_{s}}{(4\pi)^{3}}\frac{1}{m_{t}}\left[f(x_{t})\textrm{Re}(y^{q}_{ii})\textrm{Im}(y^{u}_{33})+g(x_{t})\textrm{Im}(y^{q}_{ii})\textrm{Re}(y^{u}_{33})\right],$
(131)

where $x_{i}=m_{i}^{2}/m_{h}^{2}$, and $q_{i}$ is the electric charge. The
loop functions are given by

$\displaystyle
f(z)=\frac{z}{2}\int_{0}^{1}dx\frac{1-2x(1-x)}{x(1-x)-z}\ln\frac{x(1-x)}{z},\qquad
g(z)=\frac{z}{2}\int_{0}^{1}dx\frac{1}{x(1-x)-z}\ln\frac{x(1-x)}{z}.$ (132)

In the SMEFT, $\kappa_{V}=1$, and we can set the real part of the Yukawa
couplings to its dim-4 value. In the HEFT, $\kappa_{V}$ is a free, real
parameter, that needs to be extracted from data. In the HEFT, the loop and
momentum expansions are tied by $\Lambda=4\pi v$ so that the contributions in
Eq. (6.1) arise at NNLO. At this order, one would also expect contributions
from dipole operators in the HEFT Buchalla:2013rka . Since the Barr-Zee
diagrams are finite, we can however consider them in isolation.

###  6.2 Constraints on flavor-violating Higgs couplings

First, we discuss the constraints on flavor-violating Higgs couplings. In the
SMEFT interpretation, flavor-violating Higgs couplings do not contribute to
EDM at dim-6. The bounds in this subsection are thus valid in the HEFT
picture. It is instructive to have linearized formulae of the EDMs directly in
terms of Higgs couplings. Since $d_{\rm Hg}$ and electron EDM impose strongest
limits, we employ our EFT master formulae to obtain expressions for $d_{\rm
Hg}$ and $\omega_{\rm HfF}$. For $d_{\rm Hg}$, we find

$\displaystyle\hfil\displaystyle d_{\rm Hg}\over[{\rm ecm}]$
$\displaystyle\supset{(1.1\pm 4.1)\cdot 10^{-24}\over(2\times 0.125^{2})}{\rm
Im}(y^{d}_{12}y^{d}_{21})+{(-5.7\pm 34)\cdot 10^{-24}\over(2\times
0.125^{2})}{\rm Im}(y^{d}_{13}y^{d}_{31})$ (133) $\displaystyle+{(-6.4\pm
4.7)\cdot 10^{-28}\over(2\times 0.125^{2})}{\rm
Im}(y^{d}_{23}y^{d}_{32})+{(-2.4\pm 15.3)\cdot 10^{-24}\over(2\times
0.125^{2})}{\rm Im}(y^{u}_{12}y^{u}_{21})\,,$

while, for $\omega_{\textrm{HfF}}$, we have

${\omega_{\textrm{HfF}}\over[\textrm{mrad/s}]}\supset{(9.1\pm 0.4)\cdot
10^{5}\over(2\times 0.125^{2})}{\rm Im}(y^{e}_{12}y^{e}_{21})+{(1.5\pm
0.1)\cdot 10^{7}\over(2\times 0.125^{2})}{\rm Im}(y^{e}_{13}y^{e}_{31}).$
(134)

Similarly, one can find expressions for the other EDM observables. Only a
specific combination of the Higgs coupling can be probed with EDMs. To get an
estimate of the upper bounds on these couplings one can compare the above
expressions with the experimental limits, e.g. for $\left|d_{\rm
Hg}\right|<6.2\times 10^{-30}$ $e$ cm and $|\omega_{\textrm{HfF}}|<0.17$
mrad/s.

Figure 5: EDM and flavor constraints on the CP and flavor violating Higgs
couplings. Note that the black line, brown and golden lines are truncated to
increase the statistics.

We also perform a combined fit of FV Higgs couplings to all EDM observables
($d_{n}$, $d_{\rm Hg}$, $d_{\rm Xe}$, $d_{\rm Ra}$, $\omega_{\rm
HfF},\omega_{\rm YbF},\omega_{\rm ThO}$) using the current and future upper
limits given in Tab. 2. For this purpose, we assume only one combination of
the couplings to be non-zero at a time. The results are presented in Tab. 11
(column 2).

In column 3, we indicate the origin of the EDMs in terms of low-energy WET
operators at $\mu=2$ GeV. At the EW scale, the scalar and vector operators are
generated by integrating out the Higgs as shown in the matching relations
(127). Some of these give a leading contribution to the EDMs through direct
four-fermion matrix elements (see (3.4.1)- (60)) for the light flavors
$f,f^{\prime}=u,d,s$. 1-loop QCD and QED effects through the operator mixing
onto dipoles are also present for the light flavors. However, for the heavier
generations, $c$ and $b$, the contribution to the EDMs arises solely due to
operator mixing at 1-loop level (see (80)-(82)). Apart from that, as shown in
column 3, the Weinberg operator also contributes for the case of heavier
generations due to threshold corrections of the $b$ and $c$ dipoles onto it at
the $m_{b}$ and $m_{c}$ scales (see Eqs. (89) and (90)).

In Fig. 5 we show a comparison of the constraints from the EDMs and $\Delta
M_{d}$ on the CP + flavor violating down quark Higgs couplings in the $bd$
sector. We see that current $d_{n}$ and $d_{\rm Hg}$ bounds are weaker than
the constraints from $\Delta M_{d}$. However, a neutron EDM bound at the level
of $10^{-27}$ $e$ cm will be highly competitive with $B$-$\bar{B}$
oscillations. For other sectors ($bs,sd$, and $uc$) the flavor bounds
Blankenburg:2012ex are found to be stronger.

Generic FV Higgs couplings
---
couplings | current limit | future limit | operators at hadronic scale
Im($y^{d}_{12}y_{21}^{d}$) | $1.2\cdot 10^{-7}$ | $7.0\cdot 10^{-9}$ | $\big{[}{C}_{d\gamma(G)}\big{]}_{11,22},\big{[}{C}_{dd}^{S1(8),RR}\big{]}_{1221},\big{[}{C}_{dd}^{S1(8),RR}\big{]}_{1122}$ |
Im($y_{13}^{d}y_{31}^{d}$) | $2.1\cdot 10^{-8}$ | $6.0\cdot 10^{-10}$ | $\big{[}{C}_{\widetilde{G}}\big{]},\big{[}{C}_{d\gamma(G)}\big{]}_{11}$ |
Im($y_{23}^{d}y_{32}^{d}$) | $1.5\cdot 10^{-4}$ | $1.4\cdot 10^{-6}$ | $\big{[}{C}_{\widetilde{G}}\big{]},\big{[}{C}_{d\gamma(G)}\big{]}_{22}$ |
Im($y_{12}^{u}y_{21}^{u}$) | $5.0\cdot 10^{-8}$ | $2.3\cdot 10^{-9}$ | $\big{[}{C}_{\widetilde{G}}\big{]},\big{[}{C}_{u\gamma(G)}\big{]}_{11}$ |
Im($y_{12}^{e}y_{21}^{e}$) | $1.1\cdot 10^{-9}$ | $1.1\cdot 10^{-10}$ | $\big{[}{C}_{e\gamma}\big{]}_{11}$ |
Im($y_{13}^{e}y_{31}^{e}$) | $6.0\cdot 10^{-11}$ | $6.0\cdot 10^{-12}$ | $\big{[}{C}_{e\gamma}\big{]}_{11}$ |

Table 11: Current and future EDM constraints on the combinations of the
generic flavor violating Higgs couplings (column 2) at $1\sigma$ level . In
the column 3, we also indicate the origin of EDM contributions at $\mu=2$ GeV.
The matching scale is set to the Higgs mass.

###  6.3 EDM constraints on flavor-conserving Higgs couplings

For the flavor-conserving Yukawas, we get contributions from the tree-level
matching onto four-fermion, given in Eq. (127), and from the 2-loop Barr-Zee
diagrams, in Eq. (6.1). Using these one can directly use the master formulae
to obtain the EDMs in terms of the FC Yukawas, similar to the expressions in
(133) and (134).

We note some unique features of BZ contributions, which are in contrast to the
tree-level matching, associated with the particular combination of the Higgs
couplings appearing in the matching relations. First, the BZ diagrams involve
a top-quark Yukawa coupling which is correlated to the other quarks as well as
leptonic Yukawas. Secondly, the BZ effects can also correlate fermion Yukawas
to the $W$-Higgs couplings. These features are absent in the tree-level
matching and therefore allow additional constraints on the FC Higgs couplings.
Recall that, the EDMs through tree-level Higgs exchange (127) probe following
combination of FC couplings

$\quad{\rm
Im}(y^{f}_{ii}y^{f^{\prime}}_{jj})=\textrm{Re}(y^{f}_{ii})\textrm{Im}(y^{f^{\prime}}_{jj})+\textrm{Im}(y^{f}_{jj})\textrm{Re}(y^{f^{\prime}}_{ii})\,,\quad
f,f^{\prime}=u,d,e\,,$ (135)

here $f$ can be equal to $f^{\prime}$. Therefore, for $f=f^{\prime}$ and
$i=j$, one can also probe individual couplings (more precisely a combination
of its Re and Im parts: $\textrm{Re}(y^{f}_{ii})\textrm{Im}(y^{f}_{ii})$) in
an uncorrelated manner to other couplings, under the assumption that only a
single coupling is turned on. Overall, the tree-level and 2-loop BZ diagram
effects can test very different parts of the low-energy Higgs effective
coupling parameter space. In Tab. 12 we show upper bounds at $1\sigma$ level
on the various combinations of the generic FC Higgs couplings.

Generic FC Higgs couplings
---
couplings | current limit | future limit | operators at hadronic scale
Im$(y_{11}^{d}y_{11}^{d})$ | $2.4\cdot 10^{-8}$ | $2.6\cdot 10^{-9}$ |  | $\big{[}{C}_{dd}^{S1(8),RR}\big{]}_{1111},\big{[}{C}_{d\gamma(G)}\big{]}_{11}$
Im$(y_{22}^{d}y_{22}^{d})$ | $2.0\cdot 10^{-3}$ | $1.9\cdot 10^{-5}$ |  | $\big{[}{C}_{dd}^{S1(8),RR}\big{]}_{2222},\big{[}{C}_{d\gamma(G)}\big{]}_{22}$
Im$(y_{33}^{d}y_{33}^{d})$ | $1.1\cdot 10^{-3}$ | $9.0\cdot 10^{-6}$ |  | $\big{[}{C}_{\widetilde{G}}\big{]}$
Im$(y_{11}^{u}y_{11}^{u})$ | $2.6\cdot 10^{-8}$ | $1.6\cdot 10^{-9}$ |  | $\big{[}{C}_{uu}^{S1(8),RR}\big{]}_{1111},\big{[}{C}_{u\gamma(G)}\big{]}_{11}$
Im$(y_{22}^{u}y_{22}^{u})$ | $4.0\cdot 10^{-4}$ | $3.5\cdot 10^{-6}$ |  | $\big{[}{C}_{\widetilde{G}}\big{]}$
Im$(y_{11}^{e}y_{11}^{e})$ | $1.2\cdot 10^{-7}$ | $1.2\cdot 10^{-8}$ |  | $\big{[}{C}_{ee}^{S,RR}\big{]}_{1111},\big{[}{C}_{e\gamma}\big{]}_{11}$

Table 12: Current and future EDM constraints on the combinations of the
generic flavor conserving Higgs couplings at $1\sigma$ level (to obtain
$90\%$CL these numbers have to be multiplied by factor 1.6 ). In column 3, we
also indicate the origin of leading contributions to the EDMs at $\mu=2$GeV.
It is worth mentioning that the EDMs can also probe other combinations of the
Yukawa couplings which are not shown here. In that case, the scalar or vector
operators with left-right chiralities are generated at the EW scale (see e.g.
(127)). The electron EDM dominates the constraints on the leptonic and
semileptonic operators. The matching scale is set to the Higgs mass.

###  6.4 LHC constraints on flavor-conserving Higgs couplings

Finally, we want to compare the bounds in Table 12 with complementary
constraints from the LHC. Non-standard Yukawa couplings affect Higgs
production and decay processes and are thus constrained by measurements of the
Higgs properties at the LHC ATLAS:2022vkf ; CMS:2022dwd . For a given Higgs
production mechanism, $i\rightarrow H$, followed by the decay of the Higgs to
the final state $f$, the signal strength in the presence of non-standard
interactions is defined as

$\mu_{i\rightarrow H\rightarrow
f}=\mu_{i}\,\mu_{f}=\left(1+\frac{\sigma_{i\rightarrow H}}{\sigma^{\rm
SM}_{i\rightarrow H}}\right)\frac{1+\frac{\Gamma_{H\rightarrow f}}{\Gamma^{\rm
SM}_{H\rightarrow f}}}{1+\frac{\Gamma_{\textrm{tot}}}{\Gamma^{\rm
SM}_{\textrm{tot}}}},$ (136)

where $\sigma^{\rm SM}$ and $\sigma$ are, respectively, the production cross
sections in the SM and the correction induced by non-standard interactions.
$\Gamma^{\rm SM}_{H\rightarrow f}$ are the decay widths in the channel $f$ and
$\Gamma^{\rm SM}_{\textrm{tot}}$ the Higgs total width. In the SM, the Yukawas
of the light $u$, $d$, and $s$ quarks are too small to significantly affect
Higgs production and decay. Direct constraints on $H\rightarrow c\bar{c}$ are
sensitive to charm Yukawas about ten times larger than the SM expectation
ATLAS:2022ers ; CMS:2022fxs , while the $b$ and $t$ Yukawas are compatible
with the SM ATLAS:2022vkf ; CMS:2022dwd . In the HEFT, the quark masses and
Yukawas are decoupled from each other. Light quark Yukawas can thus affect the
Higgs production via the partonic processes $\bar{q}q\rightarrow H$, which
falls in the same category at gluon fusion, and
$q\bar{q}^{(\prime)}\rightarrow HV$, with $V=W,Z$. They also affect the decay
signal strength by modifying the Higgs total width. The top Yukawa can be
probed by the loop processes $gg\rightarrow H$, $H\rightarrow\gamma\gamma$,
$H\rightarrow\gamma Z$, and, at tree level, via $t\bar{t}H$ production.

At $\sqrt{S}=13$ TeV, the gluon fusion SM cross section is
LHCHiggsCrossSectionWorkingGroup:2016ypw

$\sigma^{\rm SM}_{ggH}=48.58^{+2.22}_{-3.27}\pm 1.56\,{\rm pb}.$

where the cross-section is evaluated at the factorization and renormalization
scales $\mu_{F}=\mu_{R}=m_{h}/2$. The first error includes the contribution of
missing orders and EW corrections, while the second error denotes the PDF
uncertainties.

Non-standard Yukawas affect the production cross-section in several ways. A
nonstandard Yukawa coupling of the top quark affects gluon fusion production.
The real part of the coupling simply rescales the SM contribution. In the
large quark mass limit, the imaginary part of the top Yukawa coupling
contributes to gluon fusion via the operator $hG\tilde{G}$. Neglecting
$\mathcal{O}(m_{h}^{2}/m_{t}^{2})$ corrections and at leading order in QCD,
the contribution of the imaginary part of the top Yukawa to the gluon fusion
cross section is a factor of $9/4$ larger than that of the real part. Higher
order QCD corrections are different for scalar and pseudoscalar couplings, and
the expression of the difference at NLO and NNLO can be found in Ref.
Anastasiou:2002wq ; Harlander:2002vv .

The contributions of light quarks with scalar and pseudoscalar Yukawas at NNLO
in QCD have been computed in Ref. Harlander:2003ai . In this case, the cross-
section is proportional to the absolute value of the Yukawa coupling, up to
corrections proportional to $m_{q}/m_{h}$, which we neglect. We write the
production cross-section as

$\displaystyle\sigma_{ggH}$ $\displaystyle=\left(({\rm
Re}\,y^{u}_{33})^{2}+\frac{9}{4}({\rm
Im}\,y^{u}_{33})^{2}\right)\frac{1}{\left|(y^{u}_{33})_{\rm
SM}\right|^{2}}\sigma^{\rm SM}_{ggH}+({\rm
Im}\,y^{u}_{33})^{2}\Delta\sigma_{ggH}$
$\displaystyle+\sum_{i=1}^{3}\sigma^{d}_{ii}\left|y^{d}_{ii}\right|^{2}+\sum_{i=1}^{2}\sigma^{u}_{ii}\left|y^{u}_{ii}\right|^{2},$
(137)

with the Yukawa couplings evaluated at the scale $\mu=m_{h}$. The results for
$\sigma_{q\bar{q}}$ at $\sqrt{S}=13$ TeV are summarized in Tab. 13. The
central value of the cross-section is given at the scale
$\mu_{F}=\mu_{R}=m_{h}/2$. The scale uncertainties are obtained by varying
$\mu=\mu_{F}=\mu_{R}$ between $m_{h}/4$ and $m_{h}$, while the pdf and
$\alpha_{s}$ error was evaluated following the PDF4LHC21 recommendation
PDF4LHCWorkingGroup:2022cjn , using 100 replicas in the
PDF4LHC21$\\_$mc$\\_$pdfas PDF set.

$\sigma$ ($10^{3}$ pb) | $\sigma^{u}_{11}$ | $\sigma^{d}_{11}$ | $\sigma^{d}_{22}$ | $\sigma^{u}_{22}$ | $\sigma^{d}_{33}$
---|---|---|---|---|---
central | $120.0$ | $83.3$ | $31.7$ | $12.6$ | $3.9$
scale | ${}^{+2.7}_{-4.8}$ | ${}^{+1.9}_{-3.3}$ | ${}^{+0.8}_{-1.1}$ | ${}^{+0.5}_{-0.5}$ | ${}^{+0.2}_{-0.1}$
pdf | $3.6$ | $3.1$ | $5.3$ | $0.5$ | $0.1$
$\mu_{ggH}$ | $[2.2,2.8]\cdot 10^{3}$ | $[1.5,1.9]\cdot 10^{3}$ | $[5.1,8.2]\cdot 10^{2}$ | $[2.3,3.0]\cdot 10^{2}$ | $[72,91]$
$\sigma_{HV}$ (pb) | $\sigma^{u}_{11}$ | $\sigma^{d}_{11}$ | $\sigma^{d}_{22}$ | $\sigma^{u}_{22}$ | $\sigma^{d}_{33}$
$\sigma_{HW+}$ | 16.0 | 15.3 | 1.99 | 1.44 | 0.09
$\sigma_{HW-}$ | 8.35 | 8.50 | 1.53 | 1.69 | 0.06
$\sigma_{HZ}$ | 3.18 | 2.24 | 0.50 | 0.19 | 0.09
$\mu_{HW}$ | 158 | 154 | 22.8 | 20.3 | 0.97
$\mu_{HZ}$ | 106 | 75 | 17 | 6.37 | 3.02
Table 13: Contribution of non-standard light quark Yukawa couplings to the
Higgs production cross section in the gluon fusion and HV channels.

The corrections to associated $HW$ and $HZ$ production induced by non-standard
Yukawas were computed at NLO in QCD in Ref. Alioli:2018ljm

$\sigma_{HV}=\kappa_{V}\sigma^{\rm
SM}_{HV}+\sum_{i=1}^{3}\sigma^{d}_{ii,\,HV}\left|y^{d}_{ii}\right|^{2}+\sum_{i=1}^{2}\sigma^{u}_{ii,\,HV}\left|y^{u}_{ii}\right|^{2},$
(138)

with LHCHiggsCrossSectionWorkingGroup:2016ypw

$\displaystyle\sigma^{SM}_{HW^{+}(\ell^{+}\nu)}$
$\displaystyle=94.26^{+0.5\%}_{-0.7\%}\pm 1.8\%\,{\rm
fb},\qquad\sigma^{SM}_{HW^{-}(\ell^{-}\bar{\nu})}=59.83^{+0.4\%}_{-0.7\%}\pm
2.0\%\,{\rm fb},$ $\displaystyle\sigma^{SM}_{HZ^{-}(\ell^{+}\ell^{-})}$
$\displaystyle=29.82^{+3.8\%}_{-3.1\%}\pm 1.6\%\,{\rm fb}.$ (139)

The theoretical uncertainties on the non-standard contributions to
$\sigma_{HW+}$, $\sigma_{HW-}$ and $\sigma_{HZ}$ were shown to be at the 10%
level Alioli:2018ljm . Since the HV channel gives weaker bounds and the
theoretical uncertainties are small, we did not re-evaluate them in this work.
For the corrections to the $t\bar{t}H$ cross-section, we use

$\mu_{t\bar{t}h}=\frac{\left(\textrm{Re}\,y^{u}_{33}\right)^{2}+0.4\left(\textrm{Im}\,y^{u}_{33}\right)^{2}}{\left|(y^{u}_{33})_{\rm
SM}\right|^{2}}.$ (140)

The Yukawa also affect the Higgs decay into fermions. At NLO in QCD, the decay
of a Higgs boson to two quarks is given by Spira:2016ztx

$\Gamma(H\rightarrow
q_{j}\bar{q}_{j})=\frac{3m_{h}}{8\pi}|y^{q}_{jj}|^{2}\left(1+5.67\frac{\alpha_{s}}{\pi}\right)=(18.0\;{\rm
GeV})|y^{q}_{jj}|^{2}.$ (141)

The decay into two leptons has a similar expression, without the
$\mathcal{O}(\alpha_{s})$ correction. For illustration, we use the signal
strength measurements of the ATLAS experiment ATLAS:2022vkf , set
$\kappa_{V}=1$ and $\textrm{Re}\,y^{u}_{33}$ to its SM value. The latter
assumption is justified by analyses of the CP properties of the couplings of
the Higgs boson to the top, which prefer a dominantly real coupling
CMS:2020cga ; ATLAS:2020ior . With these assumptions, a combined fit to the
Yukawas of the electron and of the $u$, $d$, $s$, $c$ and $b$ quarks to ATLAS
data yields

$\displaystyle|y^{e}_{11}|$ $\displaystyle<1.2\cdot
10^{-2},\qquad|y^{u}_{11}|<6.4\cdot 10^{-3},\qquad|y^{d}_{11}|<6.9\cdot
10^{-3},$ $\displaystyle\quad|y^{d}_{22}|$ $\displaystyle<7.1\cdot
10^{-3},\qquad|y^{u}_{22}|<7.1\cdot 10^{-3},$ (142)

at the 90% confidence level, while, for the $b$ quark, we have

$\displaystyle 0.90\cdot 10^{-2}<|y^{d}_{33}|<1.2\cdot 10^{-2}.$ (143)

The bounds are given on couplings at the renormalization scale $\mu=m_{h}$.
For the $u$ and $d$ quarks, corrections to the production cross section and
Higgs total width provide similar effects, while for $s$, $c$ and $b$ quarks
the dominant effect are the corrections to the Higgs partial and total widths.
The data we use are only sensitive to the absolute values of the couplings.

We next perform a joint fit to the Higgs and EDM data in the HEFT framework,
in which fermion masses and Yukawa are independent. In our fit, we fix
$\kappa_{V}$ and the real part of the top Yukawa coupling,
$\textrm{Re}y^{u}_{33}$, to their SM model values. We then perform a
simultaneous fit to the real parts of the $e$, $u$, $d$, $s$, $c$ and $b$
Yukawas and to one imaginary part at a time. We neglect theoretical errors on
EDMs, so that the bounds give an indication of the full potential of EDMs
experiments, but should be taken with some caution. If we include both the
Barr-Zee and four-fermion contributions, at the 90% CL we find

$\displaystyle|\textrm{Im}y^{e}_{11}|<1.2\cdot
10^{-6},\qquad|\textrm{Im}y^{u}_{11}|$ $\displaystyle<1.2\cdot
10^{-6},\qquad|\textrm{Im}y^{d}_{11}|<1.7\cdot 10^{-7},$
$\displaystyle|\textrm{Im}y^{d}_{22}|<7.1\cdot
10^{-3},\qquad|\textrm{Im}y^{u}_{22}|$ $\displaystyle<7.1\cdot
10^{-3},\qquad|\textrm{Im}y^{d}_{33}|<1.2\cdot 10^{-2}.$ (144)

The 90% CL for the real part of the couplings corresponds to the ranges given
in Eqs. (6.4) and (143). We see that for the $e$, $u$ and $d$ Yukawas, the EDM
bound on the imaginary part is much stronger than the bounds from Higgs
observables in Eq. (6.4). For the $s$, $c$ and $b$ quark, however, the bounds
go back to what we obtained from the signal strengths. This is partially due
to cancellations between the Barr-Zee (Eq. (6.1)) and four-fermion (Eq. (127))
contributions to EDMs, which can happen once we let the real part of the
Yukawas deviate from its SM value.

###  6.5 SMEFT Interpretation

The Yukawa couplings in SMEFT are more constrained as compared to the HEFT.
Since the leading non-standard effects to the Yukawa couplings are dim-6
(117), we can not have the double insertion of the Yukawas at this level.
Hence, the contributions to the EDMs through flavor-violating Higgs couplings
in Tab. 11 is essentially a dim-8 effect in the SMEFT power counting, because
for the matching of $\big{[}{C}_{ff}^{S1,RR}\big{]}_{iijj}$ a double insertion
of flavor-violating couplings is required. On the other hand, flavor-
conserving CPV Yukawa couplings generate EDMs at dim-6 in SMEFT. In this case,
one of the Yukawa has to be SM-like. Moreover, since the SM Yukawas are
restricted to be real in the mass basis, one can directly constrain the
individual CPV couplings from EDM observables unlike the generic case
presented in Tab. 12.

Limiting to dim-6 effects in the SMEFT, the matching onto 4f WET operators can
be obtained by setting one of the Yukawas in (127) to its SM value

$y^{d}_{ii,\textrm{SM}}={m_{d_{i}}\over v}\,,\quad
y^{u}_{ii,\textrm{SM}}={m_{u_{i}}\over v}\,,\quad
y^{e}_{ii,\textrm{SM}}={m_{e_{i}}\over v}.$ (145)

Moreover, in the BZ contributions (6.1) we also set $\kappa_{V}=1$. The
current limits on the CPV FC Higgs couplings are given in Tab. 14. For each
case, we present two sets of results using the tree-level (4f) and 2-loop (BZ)
matching. Also, in this case, we neglect theoretical errors. Once they are
under better control, theoretical errors can straightforwardly be added to the
analysis.

CPV FC Higgs couplings in SMEFT at $1\sigma$ level
---
$d_{n}$, Hg, Xe, Ra
matching | Im($y^{d}_{11}$) | Im($y^{d}_{22}$) | Im($y^{d}_{33}$) | Im($y^{u}_{11}$) | Im($y^{u}_{22}$) | Im$(y^{e}_{11})$ | Im$(y^{e}_{22})$ | Im($y^{e}_{33})$
4f | $1.4\cdot 10^{-5}$ | $2.6\cdot 10^{-3}$ | $2.6\cdot 10^{-3}$ | $6.0\cdot 10^{-6}$ | $4.0\cdot 10^{-3}$ | – | – | –
BZ | $9.0\cdot 10^{-8}$ | $7.0\cdot 10^{-4}$ | $9.0\cdot 10^{-3}$ | $1.1\cdot 10^{-7}$ | $1.3\cdot 10^{-3}$ | – | – | –
$\omega_{\rm HfF}$, $\omega_{\rm YbF}$, $\omega_{\rm ThO}$
matching | Im($y^{d}_{11}$) | Im($y^{d}_{22}$) | Im($y^{d}_{33}$) | Im($y^{u}_{11}$) | Im($y^{u}_{22}$) | Im$(y^{e}_{11})$ | Im$(y^{e}_{22})$ | Im($y^{e}_{33})$
4f | $4.0\cdot 10^{-3}$ | $4.0\cdot 10^{-3}$ | $2.0\cdot 10^{-3}$ | $9.0\cdot 10^{-4}$ | $8.0\cdot 10^{-4}$ | $2.6\cdot 10^{-8}$ | $1.6\cdot 10^{-3}$ | $1.0\cdot 10^{-4}$
BZ | – | – | – | – | – | $1.5\cdot 10^{-9}$ | – | –
combined
matching | Im($y^{d}_{11}$) | Im($y^{d}_{22}$) | Im($y^{d}_{33}$) | Im($y^{u}_{11}$) | Im($y^{u}_{22}$) | Im$(y^{e}_{11})$ | Im$(y^{e}_{22})$ | Im($y^{e}_{33})$
4f +BZ | $9.0\cdot 10^{-8}$ | $7.0\cdot 10^{-4}$ | $1.8\cdot 10^{-3}$ | $1.0\cdot 10^{-7}$ | $7.0\cdot 10^{-4}$ | $1.5\cdot 10^{-9}$ | $1.6\cdot 10^{-3}$ | $1.0\cdot 10^{-4}$

Table 14: The current limits on the CPV FC Yukawa couplings resulting due to
tree-level and 2-loop Barr-Zee diagrams up to dim-6 level in SMEFT. The
neutron, Hg, Xe, and Ra EDMs and the frequencies $\omega_{\rm HfF}$,
$\omega_{\rm YbF}$, $\omega_{\rm ThO}$ are included in the fit. The real parts
of the Yukawas are set to their SM values. The matching scale is set to the
Higgs mass.

The imaginary part of the Yukawa couplings of the $u$ and $d$ quarks is
predominantly constrained by the neutron and Hg EDMs, via the contributions
arising from Barr-Zee diagrams. The constraints are very strong, and out of
the reach of existing colliders. The $s$ quark is also mostly constrained by
hadronic EDMs. In this case, the bound is a factor of ten stronger than the
constraints from Higgs observables discussed in Sec. 6.4, so that, once
theoretical errors are considered, the LHC and EDM experiments can probe
similar scales. The $b$-quark Yukawa is mostly constrained by the electron
EDM. These contributions arise from first integrating out the Higgs and
matching onto the scalar four-fermion operator
$\big{[}{C}_{ed}^{S,RR}\big{]}$, which consequently runs into
$\big{[}{C}_{ed}^{T,RR}\big{]}$ and $C_{e\gamma}$. Finally, the $c$-quark
Yukawa receive similar constraints from $d_{n}$, $d_{\rm Hg}$ and $\omega_{\rm
HfF}$. For the $c$ and $b$ Yukawas, the indirect bounds on the Higgs couplings
are complementary to the constraints resulting from the direct Higgs searches
as presented in the previous section.

The bounds in Table 14 are of the same order of magnitude as those presented
in Ref. Chien:2015xha . The main difference is in the treatment of the RGE
between the EW and $\sim$ 2 GeV scale. Ref. Chien:2015xha ignored the role of
four-fermion operators, which re-appeared as unresummed large logarithms of
$m_{q}/m_{h}$ in the matching coefficients. Our analysis is more similar to
Ref. Brod:2023wsh , which, for the $b$ and $c$ quarks, calculated the matching
coefficients and the mixing between $C_{e\gamma}$,
$\big{[}{C}_{eq}^{S,RR}\big{]}$ and $\big{[}{C}_{eq}^{T,RR}\big{]}$ at one
order higher than in our study. The bounds on $y^{d}_{33}$ and $y^{u}_{22}$
obtained in Ref. Brod:2023wsh are very close to those presented in Table 14.

##  7 Summary and Outlook

The inability of the SM to explain the origin of the baryon asymmetry in the
Universe provides a strong motivation to search for new sources of CP
violation beyond the phase of the CKM quark mixing matrix. Electric dipole
moments of leptons, nucleons, atoms, and molecules are extremely sensitive
probes of flavor-diagonal CP violation. Upcoming experiments have the
potential to improve the bounds on the neutron EDM and on EDMs of diamagnetic
atoms by at least one order of magnitude, while molecular experiments will
further constrain the electron EDM and start to probe hadronic EDMs. As they
could offer the first hints of BSM physics, it is important to develop robust
tools for the model-independent interpretation of EDM experiments. In
addition, identifying the detailed features of BSM physics from one or
multiple observations in EDM experiments requires understanding their
correlations with observables in other systems and at other energy scales.
Effective Field Theories provide a natural framework to achieve these goals in
a controlled and systematic way, with minimal reliance on specific BSM models.
In this paper, we discuss EDMs in the Weak EFT and leverage EFT tools such as
renormalization group evolution and matching at the heavy fermion thresholds
to derive master formulae for EDMs in terms of the complete set of $\Delta
F=0$ WET operators at the electroweak scale. This set includes both operators
with light $u$, $d$, and $s$ quarks, which have nonvanishing nucleon matrix
elements and thus contribute to EDM at “tree level”, but also operators with
heavy $b$ and $c$ quarks and leptons of the second and third generations.
Indeed we find that the great majority of CP-violating $\Delta F=0$ operators
with heavy fermions mix into tree-level operators at leading or next to
leading-log. EDM constraints are so strong that even in the case of operators
that contribute at next-to-leading log (meaning two or even more loops) the
operator scale has to be larger than 1 TeV. We now briefly summarize the main
results of this paper.

  * •

The main result of this paper is the master formulae for the HfF, ThO, and YbF
precession frequencies, given in Tables 7 and 15, and for the neutron, proton,
and Hg EDMs, in Tables 8, 9, 10 and 16. These formulae can be readily applied
to EFTs at the electroweak scale (SMEFT or HEFT) or to BSM models by
calculating the WET matching coefficients at the electroweak scale.

The coefficients $\alpha_{I}^{X}(\mu_{ew})$ provided in these tables encode
both short-distance effects, from the running between the electroweak scale
$\mu_{ew}$ and the low energy scales $\mu_{low}=2$ GeV (or the heavy fermion
thresholds), and long-distance effects, from hadronic and nuclear matrix
elements. Concerning the short-distance effects:

  * •

We consider the full WET 1-loop anomalous dimension, which is sufficient to
capture the LL contributions from scalar four-quark operators with two heavy
and two light quarks ( $\big{[}{C}_{uu}^{S1(8),RR}\big{]}$,
$\big{[}{C}_{ud}^{S1(8),RR}\big{]}$, and
$\big{[}{C}_{uddu}^{S1(8),RR}\big{]}$), scalar leptonic operators with two
heavy and two light leptons ($\big{[}{C}_{ee}^{S,RR}\big{]}$) and scalar and
tensor semileptonic operators with light quarks and heavy leptons or light
leptons and heavy quarks ($\big{[}{C}_{eu(d)}^{T,RR}\big{]}$ and
$\big{[}{C}_{eu(d)}^{S,RR}\big{]}$). The 1-loop RGE and the 1-loop matching of
heavy quark QCD dipoles onto the Weinberg three-gluon operator provide NLL
contributions from four-quark scalar operators with four heavy quarks, and
semileptonic scalar and tensor operators with heavy quarks and heavy leptons.

  * •

The vector operators $\big{[}{C}_{uu}^{V1(8),LR}\big{]}$,
$\big{[}{C}_{dd}^{V1(8),LR}\big{]}$, $\big{[}{C}_{uddu}^{V1(8),LR}\big{]}$ and
$\big{[}{C}_{ee}^{V,LR}\big{]}$, with two or four heavy fermions, do not
generate LL contributions. For these operators, we identify NLL contributions
by generalizing 2-loop anomalous dimensions originally derived for
$B\rightarrow X_{s}\gamma$ Cho:1993zb , and by considering the 1-loop matching
at the heavy fermion threshold. After 2-loop running and 1-loop matching onto
the $b$ and $c$ QCD dipole, $\big{[}{C}_{uddu}^{V1(8),LR}\big{]}_{2332}$
generates a NNLL contribution to EDMs.

For the neutron, proton and diamagnetic atoms, the long distance piece of
$\alpha_{I}^{X}(\mu_{ew})$ is affected by large hadronic and nuclear
uncertainties. Our master formulae use up-to-date available informations from
Lattice QCD, QCD sum rules and chiral perturbation theory, and the
implementation in the package flavio allows for the prompt updating of long-
distance matrix elements, as soon as they become available.

As an example, we have applied the master formulae to study the constraints
from EDMs on Higgs couplings. For flavor-violating Higgs couplings, we showed
that, at the moment, $\Delta F=2$ observables provide stronger bounds. Future
EDM experiments, however, will provide competitive constraints, especially in
the $bd$ sector. In the case of flavor conserving Yukawa couplings, we updated
bounds on the imaginary part in two scenarios. In the HEFT scenario, in which
the real part of the Yukawa is also a free parameter, we found that EDMs give
the strongest bounds on the imaginary part of the $e$, $u$, and $d$ Yukawas,
while Higgs observables dominate the bounds on the $s$, $c$ and $b$ Yukawas.
In the SMEFT scenario, in which, up to dimension-eight corrections, the real
part of the Yukawa is fixed to its SM value, EDMs give by far the strongest
bounds. In particular, the electron EDM gives the strongest bounds on the $b$
and $s$ Yukawas, neutron and mercury dominate the $u$ and $d$ bounds, and for
the charm quark both contributions are important.

This work can be extended in several directions. First of all, we have in this
paper relied on a logarithmic counting to organize the contributions to EDMs
arising from the solution of the WET RGEs at one and two loops. This counting
does not take into account the presence of small couplings, such as the QED
coupling, for which the suppression from additional loops is not fully offset
by the large logarithms. It is thus possible that higher loop corrections to
matching and running, while formally subleading, will be numerically more
important that the LL and NLL terms identified here, especially in those cases
in which these effectively arise at three loops (e.g.
$\big{[}{C}_{uddu}^{V1(8),LR}\big{]}_{2332}$ or
$\big{[}{C}_{ud}^{S1(8),RR}\big{]}_{2233}$). Especially for applications to
very sensitive observables as EDMs, it is thus important to extend the
calculations of WET anomalous dimensions and threshold corrections to higher
order.

Secondly, higher order corrections are necessary to constrain the few $\Delta
F=0$ CP-violating operators that are not listed in Table 1. These include
$\big{[}{C}_{eu}^{S,RL}\big{]}_{1122}$, $\big{[}{C}_{eu}^{S,RL}\big{]}_{1133}$
and all the entries with heavy leptons, for which it should be possible to
generalize the 2-loop anomalous dimension for vector operators, and
$\big{[}{C}_{ee}^{S,RR}\big{]}_{iijj}$ with $i,j>1$. In addition, we find very
weak constraints on $\mu$-charm and $\tau$-charm semileptonic operators
($\big{[}{C}_{eu}^{T(S),RR}\big{]}_{2222}$,
$\big{[}{C}_{eu}^{T(S),RR}\big{]}_{3322}$) and $\mu$-bottom and $\tau$-bottom
semileptonic operators ($\big{[}{C}_{ed}^{T(S),RR}\big{]}_{2233}$ and
$\big{[}{C}_{ed}^{T(S),RR}\big{]}_{3333}$). For these classes, it will be
important to find additional paths.

Finally, reducing the theoretical uncertainties is crucial to the success of
the EDM program. This requires, in particular, to address the large hadronic
and nuclear uncertainties that affect the master formulae provided in this
paper. In future the theoretical uncertaintities can be included in the fits
using smelli program Aebischer:2018iyb .

## Acknowledgements

We acknowledge several useful discussions with W. Dekens and M. Misiak. JK and
EM are supported by the U.S. Department of Energy through the Los Alamos
National Laboratory and by the Laboratory Directed Research and Development
program of Los Alamos National Laboratory under project numbers 20220706PRD1
and 20240078DR. Los Alamos National Laboratory is operated by Triad National
Security, LLC, for the National Nuclear Security Administration of U.S.
Department of Energy (Contract No. 89233218CNA000001). We acknowledge support
from the DOE Topical Collaboration “Nuclear Theory for New Physics,” award No.
DE-SC0023663.

##  Appendix A Additional Formulae

In this section, we provide some additional master formulae for the
$\omega_{\rm ThO}$, $\omega_{\rm YbF}$ (15) and proton EDM (16).

$\alpha_{I}^{\rm ThO}(\mu_{ew})$ for the $\omega_{\text{ThO}}$
---
$n_{f}=3+1$ | $n_{f}=5+3$
$[C_{e\gamma}]_{11}$ | $(9.9\pm 2.1)\cdot 10^{12}$ | $[C^{S,RR}_{eu}]_{1122}$ | $(-3.2\pm 0.6)\cdot 10^{6}$
$[C^{S,RR}_{ee}]_{1111}$ | $(9.4\pm 2.1)\cdot 10^{4}$ | $[C^{T,RR}_{eu}]_{1122}$ | $(1.3\pm 0.3)\cdot 10^{9}$
$[C^{S,RR}_{eu}]_{1111}$ | $(1.5\pm 0.3)\cdot 10^{9}$ | $[C^{S,RR}_{ed}]_{1133}$ | $(-1.3\pm 0.2)\cdot 10^{6}$
$[C^{S,RR}_{ed}]_{1111}$ | $(1.5\pm 0.3)\cdot 10^{9}$ | $[C^{T,RR}_{ed}]_{1133}$ | $(-1.7\pm 0.3)\cdot 10^{9}$
$[C^{S,RR}_{ed}]_{1122}$ | $(1.1\pm 0.2)\cdot 10^{8}$ | $[C^{S,RR}_{ee}]_{1122}$ | $(-2.0\pm 0.4)\cdot 10^{5}$
$[C^{S,RL}_{eu}]_{1111}$ | $(1.5\pm 0.3)\cdot 10^{9}$ | $[C^{S,RR}_{ee}]_{1133}$ | $(-4.4\pm 0.7)\cdot 10^{6}$
$[C^{S,RL}_{ed}]_{1111}$ | $(1.5\pm 0.3)\cdot 10^{9}$ | $[C^{S,RR}_{ee}]_{1221}$ | $(9.9\pm 2.0)\cdot 10^{6}$
$[C^{S,RL}_{ed}]_{1122}$ | $(1.1\pm 0.3)\cdot 10^{8}$ | $[C^{S,RR}_{ee}]_{1331}$ | $(1.9\pm 0.4)\cdot 10^{8}$
$[C^{T,RR}_{eu}]_{1111}$ | $(-8.7\pm 1.6)\cdot 10^{8}$ | $[C^{V,LR}_{ee}]_{1221}$ | $(-3.2\pm 0.6)\cdot 10^{6}$
$[C^{T,RR}_{ed}]_{1111}$ | $(4.4\pm 0.8)\cdot 10^{8}$ | $[C^{V,LR}_{ee}]_{1331}$ | $(-5.1\pm 1.0)\cdot 10^{7}$
$[C^{T,RR}_{ed}]_{1122}$ | $(3.1\pm 0.5)\cdot 10^{8}$ |  | -
$\alpha_{I}^{\rm YbF}(\mu_{ew})$ for the $\omega_{\text{YbF}}$
$n_{f}=3+1$ | $n_{f}=5+3$
$[C_{e\gamma}]_{11}$ | $(1.6\pm 0.1)\cdot 10^{12}$ | $[C^{S,RR}_{eu}]_{1122}$ | $(-5.2\pm 0.4)\cdot 10^{5}$
$[C^{S,RR}_{ee}]_{1111}$ | $(1.5\pm 0.1)\cdot 10^{4}$ | $[C^{T,RR}_{eu}]_{1122}$ | $(2.1\pm 0.2)\cdot 10^{8}$
$[C^{S,RR}_{eu}]_{1111}$ | $(1.4\pm 0.2)\cdot 10^{8}$ | $[C^{S,RR}_{ed}]_{1133}$ | $(-2.1\pm 0.2)\cdot 10^{5}$
$[C^{S,RR}_{ed}]_{1111}$ | $(1.4\pm 0.2)\cdot 10^{8}$ | $[C^{T,RR}_{ed}]_{1133}$ | $(-2.7\pm 0.2)\cdot 10^{8}$
$[C^{S,RR}_{ed}]_{1122}$ | $(1.1\pm 0.1)\cdot 10^{7}$ | $[C^{S,RR}_{ee}]_{1122}$ | $(-3.3\pm 0.2)\cdot 10^{4}$
$[C^{S,RL}_{eu}]_{1111}$ | $(1.4\pm 0.2)\cdot 10^{8}$ | $[C^{S,RR}_{ee}]_{1133}$ | $(-7.1\pm 0.5)\cdot 10^{5}$
$[C^{S,RL}_{ed}]_{1111}$ | $(1.4\pm 0.2)\cdot 10^{8}$ | $[C^{S,RR}_{ee}]_{1221}$ | $(1.6\pm 0.1)\cdot 10^{6}$
$[C^{S,RL}_{ed}]_{1122}$ | $(1.1\pm 0.1)\cdot 10^{7}$ | $[C^{S,RR}_{ee}]_{1331}$ | $(3.1\pm 0.2)\cdot 10^{7}$
$[C^{T,RR}_{eu}]_{1111}$ | $(-1.3\pm 0.1)\cdot 10^{8}$ | $[C^{V,LR}_{ee}]_{1221}$ | $(-5.1\pm 0.4)\cdot 10^{5}$
$[C^{T,RR}_{ed}]_{1111}$ | $(6.5\pm 0.5)\cdot 10^{7}$ | $[C^{V,LR}_{ee}]_{1331}$ | $(-8.3\pm 0.7)\cdot 10^{6}$
$[C^{T,RR}_{ed}]_{1122}$ | $(5.1\pm 0.4)\cdot 10^{7}$ |  | -

Table 15: Master Formulae for $\omega_{\rm ThO}$ and $\omega_{\rm YbF}$ in WET
in the JMS basis at the EW scale. The $\alpha_{I}^{\rm ThO}$ and
$\alpha_{I}^{\rm YbF}$ have units of (mrad/s)$\rm TeV^{2}$ and (mrad/s)$\rm
TeV$ for the 4f and dipole operators, respectively. The $90\%$ CL upper bound
on $\omega_{\rm ThO}$ and $\omega_{\rm YbF}$ are $1.3$mrad/s and 23.5mrad/s,
respectively.

$\alpha_{I}^{p}(\mu_{ew})$ for proton EDM
---
$n_{f}=3+1$ | $n_{f}=4+1$ | $n_{f}=5+3$
$C_{\widetilde{G}}$ | $(-7.6\pm 3.8)\cdot 10^{-22}$ | $[C_{uG}]_{22}$ | $(-1.1\pm 0.5)\cdot 10^{-20}$ | $[C_{dG}]_{33}$ | $(-1.5\pm 0.8)\cdot 10^{-21}$
$[C_{dG}]_{11}$ | $(2.1\pm 1.0)\cdot 10^{-17}$ | $[C_{u\gamma}]_{22}$ | $(1.0\pm 0.5)\cdot 10^{-21}$ | $[C_{d\gamma}]_{33}$ | $(-2.9\pm 1.4)\cdot 10^{-23}$
$[C_{dG}]_{22}$ | $(1.5\pm 0.9)\cdot 10^{-20}$ | $[C^{S1,RR}_{uu}]_{2222}$ | $(7.8\pm 3.9)\cdot 10^{-25}$ | $[C^{V1,LR}_{uddu}]_{1331}$ | $(-1.5\pm 0.7)\cdot 10^{-21}$
$[C_{uG}]_{11}$ | $(4.8\pm 1.8)\cdot 10^{-17}$ | $[C^{S8,RR}_{uu}]_{2222}$ | $(-1.8\pm 0.9)\cdot 10^{-25}$ | $[C^{V8,LR}_{uddu}]_{1331}$ | $(-8.1\pm 2.0)\cdot 10^{-22}$
$[C_{d\gamma}]_{11}$ | $(2.3\pm 0.2)\cdot 10^{-17}$ | $[C^{S1,RR}_{uu}]_{1122}$ | $(5.7\pm 2.9)\cdot 10^{-22}$ | $[C^{V1,LR}_{dd}]_{1331}$ | $(-6.1\pm 3.7)\cdot 10^{-22}$
$[C_{d\gamma}]_{22}$ | $(3.0\pm 1.8)\cdot 10^{-19}$ | $[C^{S8,RR}_{uu}]_{1122}$ | $(3.8\pm 8.2)\cdot 10^{-23}$ | $[C^{V8,LR}_{dd}]_{1331}$ | $(5.1\pm 1.0)\cdot 10^{-22}$
$[C_{u\gamma}]_{11}$ | $(-9.3\pm 0.4)\cdot 10^{-17}$ | $[C^{S1,RR}_{ud}]_{2211}$ | $(2.9\pm 1.3)\cdot 10^{-22}$ | $[C^{V1,LR}_{dd}]_{2332}$ | $(1.1\pm 0.6)\cdot 10^{-24}$
$[C^{V1,LR}_{uddu}]_{1111}$ | $(5.9\pm 2.9)\cdot 10^{-22}$ | $[C^{S8,RR}_{ud}]_{2211}$ | $(1.2\pm 0.4)\cdot 10^{-22}$ | $[C^{V8,LR}_{dd}]_{2332}$ | $(4.1\pm 2.2)\cdot 10^{-24}$
$[C^{V8,LR}_{uddu}]_{1111}$ | $(8.9\pm 4.7)\cdot 10^{-22}$ | $[C^{S1,RR}_{ud}]_{2222}$ | $(1.5\pm 1.4)\cdot 10^{-26}$ | $[C^{S1,RR}_{dd}]_{3333}$ | $(3.0\pm 1.5)\cdot 10^{-25}$
$[C^{V1,LR}_{uddu}]_{1221}$ | $(1.1\pm 0.5)\cdot 10^{-21}$ | $[C^{S8,RR}_{ud}]_{2222}$ | $(5.1\pm 2.8)\cdot 10^{-25}$ | $[C^{S8,RR}_{dd}]_{3333}$ | $(-6.4\pm 3.1)\cdot 10^{-26}$
$[C^{V8,LR}_{uddu}]_{1221}$ | $(1.7\pm 0.8)\cdot 10^{-21}$ | $[C^{S1,RR}_{uddu}]_{2112}$ | $(-1.3\pm 0.5)\cdot 10^{-21}$ | $[C^{S1,RR}_{dd}]_{1133}$ | $(3.8\pm 1.9)\cdot 10^{-22}$
$[C^{V1,LR}_{dd}]_{1221}$ | $(5.1\pm 2.7)\cdot 10^{-22}$ | $[C^{S8,RR}_{uddu}]_{2112}$ | $(-3.6\pm 5.8)\cdot 10^{-23}$ | $[C^{S8,RR}_{dd}]_{1133}$ | $(9.6\pm 5.3)\cdot 10^{-23}$
$[C^{V8,LR}_{dd}]_{1221}$ | $(7.8\pm 4.0)\cdot 10^{-22}$ | $[C^{S1,RR}_{uddu}]_{2222}$ | $(-2.4\pm 1.5)\cdot 10^{-24}$ | $[C^{S1,RR}_{dd}]_{2233}$ | $(1.3\pm 0.8)\cdot 10^{-25}$
$[C^{S1,RR}_{uu}]_{1111}$ | $(-1.3\pm 0.7)\cdot 10^{-21}$ | $[C^{S8,RR}_{uddu}]_{2222}$ | $(-1.9\pm 1.2)\cdot 10^{-24}$ | $[C^{S8,RR}_{dd}]_{2233}$ | $(-3.7\pm 2.4)\cdot 10^{-25}$
$[C^{S8,RR}_{uu}]_{1111}$ | $(4.1\pm 2.6)\cdot 10^{-22}$ | $[C^{S1,RR}_{uu}]_{1221}$ | $(-1.8\pm 1.2)\cdot 10^{-21}$ | $[C^{S1,RR}_{dd}]_{1331}$ | $(-2.3\pm 1.1)\cdot 10^{-21}$
$[C^{S1,RR}_{dd}]_{1111}$ | $(-6.2\pm 3.6)\cdot 10^{-22}$ | $[C^{S8,RR}_{uu}]_{1221}$ | $(8.3\pm 1.3)\cdot 10^{-22}$ | $[C^{S8,RR}_{dd}]_{1331}$ | $(5.2\pm 1.6)\cdot 10^{-22}$
$[C^{S8,RR}_{dd}]_{1111}$ | $(1.9\pm 0.9)\cdot 10^{-22}$ | $[C^{V1,LR}_{uddu}]_{2112}$ | $(-2.5\pm 0.9)\cdot 10^{-22}$ | $[C^{S1,RR}_{dd}]_{2332}$ | $(1.5\pm 0.9)\cdot 10^{-24}$
$[C^{S1,RR}_{dd}]_{2222}$ | $(1.0\pm 0.6)\cdot 10^{-25}$ | $[C^{V8,LR}_{uddu}]_{2112}$ | $(-1.5\pm 0.3)\cdot 10^{-22}$ | $[C^{S8,RR}_{dd}]_{2332}$ | $(2.9\pm 1.7)\cdot 10^{-24}$
$[C^{S8,RR}_{dd}]_{2222}$ | $(1.2\pm 0.6)\cdot 10^{-25}$ | $[C^{V1,LR}_{uddu}]_{2222}$ | $(-8.1\pm 4.9)\cdot 10^{-25}$ | $[C^{S1,RR}_{ud}]_{1133}$ | $(8.2\pm 3.7)\cdot 10^{-22}$
$[C^{S1,RR}_{dd}]_{1122}$ | $(-1.4\pm 0.9)\cdot 10^{-22}$ | $[C^{V8,LR}_{uddu}]_{2222}$ | $(-2.8\pm 1.6)\cdot 10^{-24}$ | $[C^{S8,RR}_{ud}]_{1133}$ | $(3.9\pm 1.1)\cdot 10^{-22}$
$[C^{S8,RR}_{dd}]_{1122}$ | $(5.1\pm 2.3)\cdot 10^{-23}$ | $[C^{V1,LR}_{uu}]_{1221}$ | $(-2.9\pm 18.4)\cdot 10^{-23}$ | $[C^{S1,RR}_{uddu}]_{1331}$ | $(-6.2\pm 2.5)\cdot 10^{-21}$
$[C^{S1,RR}_{dd}]_{1221}$ | $(-2.2\pm 0.9)\cdot 10^{-22}$ | $[C^{V8,LR}_{uu}]_{1221}$ | $(9.4\pm 0.6)\cdot 10^{-22}$ | $[C^{S8,RR}_{uddu}]_{1331}$ | $(-1.1\pm 2.8)\cdot 10^{-22}$
$[C^{S8,RR}_{dd}]_{1221}$ | $(6.0\pm 2.6)\cdot 10^{-23}$ |  | - | $[C^{S1,RR}_{ud}]_{2233}$ | $(-2.1\pm 1.2)\cdot 10^{-25}$
$[C^{S1,RR}_{ud}]_{1111}$ | $(-4.8\pm 2.5)\cdot 10^{-22}$ |  | - | $[C^{S8,RR}_{ud}]_{2233}$ | $(-6.8\pm 3.3)\cdot 10^{-26}$
$[C^{S8,RR}_{ud}]_{1111}$ | $(1.5\pm 0.8)\cdot 10^{-22}$ |  | - | $[C^{S1,RR}_{uddu}]_{2332}$ | $(1.4\pm 0.6)\cdot 10^{-24}$
$[C^{S1,RR}_{ud}]_{1122}$ | $(-3.0\pm 1.7)\cdot 10^{-22}$ |  | - | $[C^{S8,RR}_{uddu}]_{2332}$ | $(-1.7\pm 0.9)\cdot 10^{-25}$
$[C^{S8,RR}_{ud}]_{1122}$ | $(1.1\pm 0.5)\cdot 10^{-22}$ |  | - | $[C^{V1,LR}_{uddu}]_{2332}$ | $(3.8\pm 1.8)\cdot 10^{-25}$
$[C^{S1,RR}_{uddu}]_{1111}$ | $(-4.9\pm 2.5)\cdot 10^{-22}$ |  | - | $[C^{V8,LR}_{uddu}]_{2332}$ | $(-1.0\pm 0.5)\cdot 10^{-25}$
$[C^{S8,RR}_{uddu}]_{1111}$ | $(1.5\pm 0.8)\cdot 10^{-22}$ |  | - |  | -
$[C^{S1,RR}_{uddu}]_{1221}$ | $(-5.1\pm 1.8)\cdot 10^{-22}$ |  | - |  | -
$[C^{S8,RR}_{uddu}]_{1221}$ | $(9.7\pm 5.3)\cdot 10^{-23}$ |  | - |  | -

Table 16: Master Formula for $d_{p}$ in WET in the JMS basis at the EW scale.
The $\alpha_{I}^{p}$ have units of e-cm.$\rm TeV^{2}$ and e-cm.$\rm TeV$ for
the four-fermion and dipole operators, respectively.

## References

  * (1) A. D. Sakharov, Violation of CP Invariance, C asymmetry, and baryon asymmetry of the universe, Pisma Zh. Eksp. Teor. Fiz. 5 (1967) 32–35.
  * (2) 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) 345–381, [hep-ph/9406288].
  * (3) M. B. Gavela, P. Hernandez, J. Orloff, and O. Pene, Standard model CP violation and baryon asymmetry, Mod. Phys. Lett. A 9 (1994) 795–810, [hep-ph/9312215].
  * (4) 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) 382–426, [hep-ph/9406289].
  * (5) P. Huet and E. Sather, Electroweak baryogenesis and standard model CP violation, Phys. Rev. D 51 (1995) 379–394, [hep-ph/9404302].
  * (6) I. B. Khriplovich, Quark Electric Dipole Moment and Induced $\theta$ Term in the Kobayashi-Maskawa Model, Phys. Lett. B 173 (1986) 193–196.
  * (7) A. Czarnecki and B. Krause, Neutron electric dipole moment in the standard model: Valence quark contributions, Phys. Rev. Lett. 78 (1997) 4339–4342, [hep-ph/9704355].
  * (8) T. Mannel and N. Uraltsev, Loop-Less Electric Dipole Moment of the Nucleon in the Standard Model, Phys. Rev. D 85 (2012) 096002, [arXiv:1202.6270].
  * (9) C.-Y. Seng, Reexamination of The Standard Model Nucleon Electric Dipole Moment, Phys. Rev. C 91 (2015), no. 2 025502, [arXiv:1411.1476].
  * (10) M. Pospelov and A. Ritz, Electric dipole moments as probes of new physics, Annals Phys. 318 (2005) 119–169, [hep-ph/0504231].
  * (11) T. Chupp, P. Fierlinger, M. Ramsey-Musolf, and J. Singh, Electric Dipole Moments of the Atoms, Molecules, Nuclei and Particles, arXiv:1710.02504.
  * (12) R. Alarcon et al., Electric dipole moments and the search for new physics, in Snowmass 2021, 3, 2022.  arXiv:2203.08103.
  * (13) A. Buras, Gauge Theory of Weak Decays.  Cambridge University Press, 6, 2020.
  * (14) E. E. Jenkins, A. V. Manohar, and P. Stoffer, Low-Energy Effective Field Theory below the Electroweak Scale: Operators and Matching, JHEP 03 (2018) 016, [arXiv:1709.04486].
  * (15) E. E. Jenkins, A. V. Manohar, and P. Stoffer, Low-Energy Effective Field Theory below the Electroweak Scale: Anomalous Dimensions, JHEP 01 (2018) 084, [arXiv:1711.05270].
  * (16) W. Buchmuller and D. Wyler, Effective Lagrangian Analysis of New Interactions and Flavor Conservation, Nucl. Phys. B268 (1986) 621–653.
  * (17) B. Grzadkowski, M. Iskrzynski, M. Misiak, and J. Rosiek, Dimension-Six Terms in the Standard Model Lagrangian, JHEP 1010 (2010) 085, [arXiv:1008.4884].
  * (18) G. Buchalla, O. Catà, and C. Krause, Complete Electroweak Chiral Lagrangian with a Light Higgs at NLO, Nucl. Phys. B 880 (2014) 552–573, [arXiv:1307.5017]. [Erratum: Nucl.Phys.B 913, 475–478 (2016)].
  * (19) G. Buchalla, O. Cata, A. Celis, M. Knecht, and C. Krause, Complete One-Loop Renormalization of the Higgs-Electroweak Chiral Lagrangian, Nucl. Phys. B 928 (2018) 93–106, [arXiv:1710.06412].
  * (20) I. B. Khriplovich and S. K. Lamoreaux, CP violation without strangeness: Electric dipole moments of particles, atoms, and molecules.  1997\.
  * (21) 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) 21–74, [arXiv:1303.2371].
  * (22) 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) 50–96, [arXiv:1212.0990].
  * (23) J. Brod, U. Haisch, and J. Zupan, Constraints on CP-violating Higgs couplings to the third generation, JHEP 11 (2013) 180, [arXiv:1310.1385].
  * (24) 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].
  * (25) J. Brod, J. M. Cornell, D. Skodras, and E. Stamou, Global constraints on Yukawa operators in the standard model effective theory, JHEP 08 (2022) 294, [arXiv:2203.03736].
  * (26) H. Bahl, E. Fuchs, S. Heinemeyer, J. Katzy, M. Menen, K. Peters, M. Saimpert, and G. Weiglein, Constraining the ${\mathcal{C}}{\mathcal{P}}$ structure of Higgs-fermion couplings with a global LHC fit, the electron EDM and baryogenesis, Eur. Phys. J. C 82 (2022), no. 7 604, [arXiv:2202.11753].
  * (27) E. Fuchs, M. Losada, Y. Nir, and Y. Viernik, $CP$ violation from $\tau$, $t$ and $b$ dimension-6 Yukawa couplings - interplay of baryogenesis, EDM and Higgs physics, JHEP 05 (2020) 056, [arXiv:2003.00099].
  * (28) J. Brod, Z. Polonsky, and E. Stamou, A Precise Electron EDM Constraint on CP-odd Heavy-Quark Yukawas, arXiv:2306.12478.
  * (29) V. Cirigliano, W. Dekens, J. de Vries, and E. Mereghetti, Constraining the top-Higgs sector of the Standard Model Effective Field Theory, Phys. Rev. D94 (2016), no. 3 034031, [arXiv:1605.04311].
  * (30) K. Fuyuto and M. Ramsey-Musolf, Top Down Electroweak Dipole Operators, arXiv:1706.08548.
  * (31) Y. Ema, T. Gao, and M. Pospelov, Improved indirect limits on charm and bottom quark EDMs, JHEP 07 (2022) 106, [arXiv:2205.11532].
  * (32) U. Haisch and G. Koole, Beautiful and charming chromodipole moments, JHEP 09 (2021) 133, [arXiv:2106.01289].
  * (33) W. Dekens and J. de Vries, Renormalization Group Running of Dimension-Six Sources of Parity and Time-Reversal Violation, JHEP 05 (2013) 149, [arXiv:1303.3156].
  * (34) V. Cirigliano, A. Crivellin, W. Dekens, J. de Vries, M. Hoferichter, and E. Mereghetti, CP Violation in Higgs-Gauge Interactions: From Tabletop Experiments to the LHC, Phys. Rev. Lett. 123 (2019), no. 5 051801, [arXiv:1903.03625].
  * (35) U. Haisch and A. Hala, Bounds on CP-violating Higgs-gluon interactions: the case of vanishing light-quark Yukawa couplings, JHEP 11 (2019) 117, [arXiv:1909.09373].
  * (36) S. Alioli, V. Cirigliano, W. Dekens, J. de Vries, and E. Mereghetti, Right-handed charged currents in the era of the Large Hadron Collider, JHEP 05 (2017) 086, [arXiv:1703.04751].
  * (37) M. Endo and D. Ueda, Nuclear EDM from SMEFT flavor-changing operator, JHEP 04 (2020) 053, [arXiv:1911.10805].
  * (38) S. Fajfer, J. F. Kamenik, N. Košnik, A. Smolkovič, and M. Tammaro, New Physics in CP violating and flavour changing quark dipole transitions, JHEP 10 (2023) 133, [arXiv:2306.16471].
  * (39) J. Kley, T. Theil, E. Venturini, and A. Weiler, Electric dipole moments at one-loop in the dimension-6 SMEFT, Eur. Phys. J. C 82 (2022), no. 10 926, [arXiv:2109.15085].
  * (40) D. M. Straub, flavio: a Python package for flavour and precision phenomenology in the Standard Model and beyond, arXiv:1810.08132.
  * (41) G. Buchalla, A. J. Buras, and M. E. Lautenbacher, Weak decays beyond leading logarithms, Rev. Mod. Phys. 68 (1996) 1125–1144, [hep-ph/9512380].
  * (42) J. Aebischer, A. Crivellin, M. Fael, and C. Greub, Matching of gauge invariant dimension-six operators for $b\to s$ and $b\to c$ transitions, JHEP 05 (2016) 037, [arXiv:1512.02830].
  * (43) W. Dekens and P. Stoffer, Low-energy effective field theory below the electroweak scale: matching at one loop, JHEP 10 (2019) 197, [arXiv:1908.05295].
  * (44) J. Aebischer, J. Kumar, and D. M. Straub, Wilson: a Python package for the running and matching of Wilson coefficients above and below the electroweak scale, Eur. Phys. J. C78 (2018), no. 12 1026, [arXiv:1804.05033].
  * (45) J. Aebischer et al., WCxf: an exchange format for Wilson coefficients beyond the Standard Model, arXiv:1712.05298.
  * (46) R. D. Peccei and H. R. Quinn, Constraints Imposed by CP Conservation in the Presence of Instantons, Phys. Rev. D16 (1977) 1791–1797.
  * (47) R. D. Peccei and H. R. Quinn, CP Conservation in the Presence of Instantons, Phys. Rev. Lett. 38 (1977) 1440–1443.
  * (48) J. Bühler and P. Stoffer, One-loop matching of CP-odd four-quark operators to the gradient-flow scheme, JHEP 08 (2023) 194, [arXiv:2304.00985].
  * (49) ACME Collaboration, V. Andreev et al., Improved limit on the electric dipole moment of the electron, Nature 562 (2018), no. 7727 355–360.
  * (50) T. S. Roussy et al., An improved bound on the electron’s electric dipole moment, Science 381 (2023), no. 6653 adg4084, [arXiv:2212.11841].
  * (51) D. M. Kara, I. J. Smallman, J. J. Hudson, B. E. Sauer, M. R. Tarbutt, and E. A. Hinds, Measurement of the electron’s electric dipole moment using YbF molecules: methods and data analysis, New J. Phys. 14 (2012) 103051, [arXiv:1208.4507].
  * (52) C. Abel et al., Measurement of the Permanent Electric Dipole Moment of the Neutron, Phys. Rev. Lett. 124 (2020), no. 8 081803, [arXiv:2001.11966].
  * (53) W. C. Griffith, M. D. Swallows, T. H. Loftus, M. V. Romalis, B. R. Heckel, and E. N. Fortson, Improved Limit on the Permanent Electric Dipole Moment of Hg-199, Phys. Rev. Lett. 102 (2009) 101601, [arXiv:0901.2328].
  * (54) 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), no. 14 143003, [arXiv:1902.02864].
  * (55) R. H. Parker et al., First Measurement of the Atomic Electric Dipole Moment of 225Ra, Phys. Rev. Lett. 114 (2015), no. 23 233002, [arXiv:1504.07477].
  * (56) T. Chupp and M. Ramsey-Musolf, Electric Dipole Moments: A Global Analysis, Phys. Rev. C91 (2015), no. 3 035502, [arXiv:1407.1064].
  * (57) G. Arrowsmith-Kron et al., Opportunities for Fundamental Physics Research with Radioactive Molecules, arXiv:2302.02165.
  * (58) n2EDM Collaboration, N. J. Ayres et al., The design of the n2EDM experiment: nEDM Collaboration, Eur. Phys. J. C 81 (2021), no. 6 512, [arXiv:2101.08730].
  * (59) D. Wurm et al., The PanEDM Neutron Electric Dipole Moment Experiment at the ILL, EPJ Web Conf. 219 (2019) 02006, [arXiv:1911.09161].
  * (60) J. W. Martin, Current status of neutron electric dipole moment experiments, J. Phys. Conf. Ser. 1643 (2020), no. 1 012002.
  * (61) T. M. Ito et al., Performance of the upgraded ultracold neutron source at Los Alamos National Laboratory and its implication for a possible neutron electric dipole moment experiment, Phys. Rev. C 97 (2018), no. 1 012501, [arXiv:1710.05182].
  * (62) J. Alexander et al., The storage ring proton EDM experiment, arXiv:2205.00830.
  * (63) 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), no. 16 161601, [arXiv:1601.04339]. [Erratum: Phys.Rev.Lett. 119, 119901 (2017)].
  * (64) K. Kumar, Z.-T. Lu, and M. J. Ramsey-Musolf, Working Group Report: Nucleons, Nuclei, and Atoms, in Workshop on Fundamental Physics at the Intensity Frontier, pp. 159–214, 12, 2013.  arXiv:1312.5416.
  * (65) C. Alexandrou, S. Bacchio, M. Constantinou, J. Finkenrath, K. Hadjiyiannakou, K. Jansen, G. Koutsou, and A. Vaquero Aviles-Casco, Nucleon axial, tensor, and scalar charges and $\sigma$-terms in lattice QCD, Phys. Rev. D 102 (2020), no. 5 054517, [arXiv:1909.00485].
  * (66) W. Dekens, E. E. Jenkins, A. V. Manohar, and P. Stoffer, Non-perturbative effects in $\mu\to e\gamma$, JHEP 01 (2019) 088, [arXiv:1810.05675].
  * (67) J. Aebischer, W. Dekens, E. E. Jenkins, A. V. Manohar, D. Sengupta, and P. Stoffer, Effective field theory interpretation of lepton magnetic and electric dipole moments, JHEP 07 (2021) 107, [arXiv:2102.08954].
  * (68) V. Cirigliano, K. Fuyuto, C. Lee, E. Mereghetti, and B. Yan, Charged Lepton Flavor Violation at the EIC, JHEP 03 (2021) 256, [arXiv:2102.06176].
  * (69) M. Knecht and A. Nyffeler, Resonance estimates of O(p**6) low-energy constants and QCD short distance constraints, Eur. Phys. J. C 21 (2001) 659–678, [hep-ph/0106034].
  * (70) V. Mateu and J. Portoles, Form-factors in radiative pion decay, Eur. Phys. J. C 52 (2007) 325–338, [arXiv:0706.1039].
  * (71) O. Cata and V. Mateu, Novel patterns for vector mesons from the large-N(c) limit, Phys. Rev. D 77 (2008) 116009, [arXiv:0801.4374].
  * (72) Muon (g-2) Collaboration, G. W. Bennett et al., An Improved Limit on the Muon Electric Dipole Moment, Phys. Rev. D 80 (2009) 052008, [arXiv:0811.1207].
  * (73) Belle Collaboration, K. Inami et al., An improved search for the electric dipole moment of the $\tau$ lepton, JHEP 04 (2022) 110, [arXiv:2108.11543].
  * (74) Particle Data Group Collaboration, R. L. Workman and Others, Review of Particle Physics, PTEP 2022 (2022) 083C01.
  * (75) M. Hoferichter, J. Ruiz de Elvira, B. Kubis, and U.-G. Meißner, Roy–Steiner-equation analysis of pion–nucleon scattering, Phys. Rept. 625 (2016) 1–88, [arXiv:1510.06039].
  * (76) Flavour Lattice Averaging Group (FLAG) Collaboration, Y. Aoki et al., FLAG Review 2021, Eur. Phys. J. C 82 (2022), no. 10 869, [arXiv:2111.09849].
  * (77) S. Degenkolb, N. Elmer, T. Modak, M. Mühlleitner, and T. Plehn, A Global View of the EDM Landscape, arXiv:2403.02052.
  * (78) 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].
  * (79) R. Gupta, B. Yoon, T. Bhattacharya, V. Cirigliano, Y.-C. Jang, and H.-W. Lin, Flavor diagonal tensor charges of the nucleon from (2+1+1)-flavor lattice QCD, Phys. Rev. D 98 (2018), no. 9 091501, [arXiv:1808.07597].
  * (80) T. Bhattacharya, V. Cirigliano, R. Gupta, E. Mereghetti, J.-S. Yoo, and B. Yoon, Quark chromoelectric dipole moment operator on the lattice, Phys. Rev. D 108 (2023), no. 7 074507, [arXiv:2304.09929].
  * (81) M. Abramczyk, S. Aoki, T. Blum, T. Izubuchi, H. Ohki, and S. Syritsyn, Lattice calculation of electric dipole moments and form factors of the nucleon, Phys. Rev. D 96 (2017), no. 1 014501, [arXiv:1701.07792].
  * (82) SymLat Collaboration, J. Kim, T. Luu, M. D. Rizik, and A. Shindler, Nonperturbative renormalization of the quark chromoelectric dipole moment with the gradient flow: Power divergences, Phys. Rev. D 104 (2021), no. 7 074516, [arXiv:2106.07633].
  * (83) M. Pospelov and A. Ritz, Neutron EDM from electric and chromoelectric dipole moments of quarks, Phys. Rev. D63 (2001) 073015, [hep-ph/0010037].
  * (84) O. Lebedev, K. A. Olive, M. Pospelov, and A. Ritz, Probing CP violation with the deuteron electric dipole moment, Phys. Rev. D 70 (2004) 016003, [hep-ph/0402023].
  * (85) K. Fuyuto, J. Hisano, and N. Nagata, Neutron electric dipole moment induced by strangeness revisited, Phys. Rev. D 87 (2013), no. 5 054018, [arXiv:1211.5228].
  * (86) C.-Y. Seng, J. de Vries, E. Mereghetti, H. H. Patel, and M. Ramsey-Musolf, Nucleon electric dipole moments and the isovector parity- and time-reversal-odd pion–nucleon coupling, Phys. Lett. B 736 (2014) 147–153, [arXiv:1401.5366].
  * (87) J. de Vries, E. Mereghetti, C.-Y. Seng, and A. Walker-Loud, Lattice QCD spectroscopy for hadronic CP violation, Phys. Lett. B 766 (2017) 254–262, [arXiv:1612.01567].
  * (88) T. Bhattacharya, V. Cirigliano, R. Gupta, E. Mereghetti, and B. Yoon, Dimension-5 CP-odd operators: QCD mixing and renormalization, Phys. Rev. D92 (2015), no. 11 114026, [arXiv:1502.07325].
  * (89) E. Mereghetti, C. J. Monahan, M. D. Rizik, A. Shindler, and P. Stoffer, One-loop matching for quark dipole operators in a gradient-flow scheme, JHEP 04 (2022) 050, [arXiv:2111.11449].
  * (90) D. A. Demir, M. Pospelov, and A. Ritz, Hadronic EDMs, the Weinberg operator, and light gluinos, Phys. Rev. D67 (2003) 015007, [hep-ph/0208257].
  * (91) U. Haisch and A. Hala, Sum rules for CP-violating operators of Weinberg type, JHEP 11 (2019) 154, [arXiv:1909.08955].
  * (92) T. Bhattacharya, V. Cirigliano, R. Gupta, E. Mereghetti, and B. Yoon, Calculation of neutron electric dipole moment due to the QCD topological term, Weinberg three-gluon operator and the quark chromoelectric moment, PoS LATTICE2021 (2022) 567, [arXiv:2203.03746].
  * (93) V. Cirigliano, E. Mereghetti, and P. Stoffer, Non-perturbative renormalization scheme for the $CP$ -odd three-gluon operator, JHEP 09 (2020) 094, [arXiv:2004.03576].
  * (94) O. L. Crosas, C. J. Monahan, M. D. Rizik, A. Shindler, and P. Stoffer, One-loop matching of the CP-odd three-gluon operator to the gradient flow, Phys. Lett. B 847 (2023) 138301, [arXiv:2308.16221].
  * (95) R. J. Crewther, P. Di Vecchia, G. Veneziano, and E. Witten, Chiral Estimate of the Electric Dipole Moment of the Neutron in Quantum Chromodynamics, Phys. Lett. B 88 (1979) 123. [Erratum: Phys.Lett.B 91, 487 (1980)].
  * (96) J. Dragos, T. Luu, A. Shindler, J. de Vries, and A. Yousif, Confirming the Existence of the strong CP Problem in Lattice QCD with the Gradient Flow, Phys. Rev. C 103 (2021), no. 1 015202, [arXiv:1902.03254].
  * (97) T. Bhattacharya, V. Cirigliano, R. Gupta, E. Mereghetti, and B. Yoon, Contribution of the QCD $\Theta$-term to the nucleon electric dipole moment, Phys. Rev. D 103 (2021), no. 11 114507, [arXiv:2101.07230].
  * (98) 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) 117–160, [arXiv:1106.6119].
  * (99) J. de Vries, E. Epelbaum, L. Girlanda, A. Gnech, E. Mereghetti, and M. Viviani, Parity- and Time-Reversal-Violating Nuclear Forces, Front. in Phys. 8 (2020) 218, [arXiv:2001.09050].
  * (100) W. C. Haxton and E. M. Henley, ENHANCED T VIOLATING NUCLEAR MOMENTS, Phys. Rev. Lett. 51 (1983) 1937.
  * (101) J. Engel, J. L. Friar, and A. C. Hayes, Nuclear octupole correlations and the enhancement of atomic time reversal violation, Phys. Rev. C 61 (2000) 035502, [nucl-th/9910008].
  * (102) J. de Vries, E. Mereghetti, and A. Walker-Loud, Baryon mass splittings and strong CP violation in SU(3) Chiral Perturbation Theory, Phys. Rev. C 92 (2015), no. 4 045201, [arXiv:1506.06247].
  * (103) V. Baluni, CP Violating Effects in QCD, Phys. Rev. D 19 (1979) 2227–2230.
  * (104) M. Pospelov, Best values for the CP odd meson nucleon couplings from supersymmetry, Phys. Lett. B 530 (2002) 123–128, [hep-ph/0109044].
  * (105) V. Cirigliano, W. Dekens, J. de Vries, and E. Mereghetti, An $\varepsilon^{\prime}$ improvement from right-handed currents, Phys. Lett. B767 (2017) 1–9, [arXiv:1612.03914].
  * (106) W. Dekens, J. de Vries, and S. Shain, CP-violating axion interactions in effective field theory, JHEP 07 (2022) 014, [arXiv:2203.11230].
  * (107) V. Cirigliano, W. Dekens, M. Graesser, and E. Mereghetti, Neutrinoless double beta decay and chiral $SU(3)$, Phys. Lett. B 769 (2017) 460–464, [arXiv:1701.01443].
  * (108) A. Nicholson et al., Heavy physics contributions to neutrinoless double beta decay from QCD, Phys. Rev. Lett. 121 (2018), no. 17 172501, [arXiv:1805.02634].
  * (109) V. Spevak, N. Auerbach, and V. V. Flambaum, Enhanced T odd P odd electromagnetic moments in reflection asymmetric nuclei, Phys. Rev. C 56 (1997) 1357–1369, [nucl-th/9612044].
  * (110) G. Barton, Notes on the static parity nonconserving internucleon potential, Nuovo Cim. 19 (1961) 512–527.
  * (111) I. S. Towner and A. C. Hayes, P, T violating nuclear matrix elements in the one meson exchange approximation, Phys. Rev. C 49 (1994) 2391–2397, [nucl-th/9402026].
  * (112) Y.-H. Song, R. Lazauskas, and V. Gudkov, Nuclear electric dipole moment of three-body systems, Phys. Rev. C 87 (2013), no. 1 015501, [arXiv:1211.3762].
  * (113) Z. Yang, E. Mereghetti, L. Platter, M. R. Schindler, and J. Vanasse, Electric dipole moments of three-nucleon systems in the pionless effective field theory, Phys. Rev. C 104 (2021), no. 2 024002, [arXiv:2011.01885].
  * (114) J. de Vries, A. Gnech, and S. Shain, Renormalization of $CP$-violating nuclear forces, Phys. Rev. C 103 (2021), no. 1 L012501, [arXiv:2007.04927].
  * (115) M. Hubert and T. Fleig, Electric dipole moments generated by nuclear Schiff moment interactions: A reassessment of the atoms Xe129 and Hg199 and the molecule TlF205, Phys. Rev. A 106 (2022), no. 2 022817, [arXiv:2203.04618].
  * (116) V. S. Prasannaa, R. Mitra, and B. K. Sahoo, Reappraisal of P, T-odd parameters from the improved calculation of electric dipole moment of 225Ra atom, J. Phys. B 53 (2020), no. 19 195004.
  * (117) V. F. Dmitriev and R. A. Sen’kov, Schiff moment of the mercury nucleus and the proton dipole moment, Phys. Rev. Lett. 91 (2003) 212303, [nucl-th/0306050].
  * (118) K. Yanase, N. Shimizu, K. Higashiyama, and N. Yoshinaga, Constraints on the nuclear Schiff moment from its correlation with electromagnetic properties, arXiv:2210.08498.
  * (119) 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].
  * (120) T. Fleig and M. Jung, $\mathcal{P,T}$ -odd interactions in atomic 129Xe and phenomenological applications, Phys. Rev. A 103 (2021), no. 1 012807, [arXiv:2009.07730].
  * (121) K. Gaul and R. Berger, Global analysis of CP-violation in atoms, molecules and role of medium-heavy systems, arXiv:2312.08858.
  * (122) K. Yanase, N. Yoshinaga, K. Higashiyama, and N. Yamanaka, Electric dipole moment of 199Hg atom from $P$, $CP$-odd electron-nucleon interaction, Phys. Rev. D 99 (2019), no. 7 075021, [arXiv:1805.00419].
  * (123) J. Menendez, D. Gazit, and A. Schwenk, Spin-dependent WIMP scattering off nuclei, Phys. Rev. D 86 (2012) 103511, [arXiv:1208.1094].
  * (124) J. S. M. Ginges and V. V. Flambaum, Violations of fundamental symmetries in atoms and tests of unification theories of elementary particles, Phys. Rept. 397 (2004) 63–154, [physics/0309054].
  * (125) E. Braaten, C.-S. Li, and T.-C. Yuan, The Evolution of Weinberg’s Gluonic CP Violation Operator, Phys. Rev. Lett. 64 (1990) 1709.
  * (126) M. Ciuchini, E. Franco, G. Martinelli, L. Reina, and L. Silvestrini, Scheme independence of the effective Hamiltonian for $b\to s\gamma$ and $b\to sg$ decays, Phys. Lett. B316 (1993) 127–136, [hep-ph/9307364].
  * (127) P. L. Cho and M. Misiak, b —$>$ s gamma decay in SU(2)-L x SU(2)-r x U(1) extensions of the Standard Model, Phys. Rev. D 49 (1994) 5894–5903, [hep-ph/9310332].
  * (128) W. Buchmüller, R. Rückl, and D. Wyler, Leptoquarks in Lepton - Quark Collisions, Phys. Lett. B191 (1987) 442–448. [Erratum: Phys. Lett.B448,320(1999)].
  * (129) S. Hamoudou, J. Kumar, and D. London, Dimension-8 SMEFT matching conditions for the low-energy effective field theory, JHEP 03 (2023) 157, [arXiv:2207.08856].
  * (130) J. Hisano, K. Tsumura, and M. J. S. Yang, QCD Corrections to Neutron Electric Dipole Moment from Dimension-six Four-Quark Operators, Phys. Lett. B713 (2012) 473–480, [arXiv:1205.2212].
  * (131) S. M. Barr and A. Zee, Electric Dipole Moment of the Electron and of the Neutron, Phys. Rev. Lett. 65 (1990) 21–24. [Erratum: Phys. Rev. Lett.65,2920(1990)].
  * (132) G. Blankenburg, J. Ellis, and G. Isidori, Flavour-Changing Decays of a 125 GeV Higgs-like Particle, Phys. Lett. B 712 (2012) 386–390, [arXiv:1202.5704].
  * (133) ATLAS Collaboration, G. Aad et al., A detailed map of Higgs boson interactions by the ATLAS experiment ten years after the discovery, Nature 607 (2022), no. 7917 52–59, [arXiv:2207.00092]. [Erratum: Nature 612, E24 (2022)].
  * (134) CMS Collaboration, A. Tumasyan et al., A portrait of the Higgs boson by the CMS experiment ten years after the discovery., Nature 607 (2022), no. 7917 60–68, [arXiv:2207.00043].
  * (135) ATLAS Collaboration, G. Aad et al., Direct constraint on the Higgs-charm coupling from a search for Higgs boson decays into charm quarks with the ATLAS detector, Eur. Phys. J. C 82 (2022) 717, [arXiv:2201.11428].
  * (136) CMS Collaboration, A. Tumasyan et al., Search for Higgs Boson and Observation of Z Boson through their Decay into a Charm Quark-Antiquark Pair in Boosted Topologies in Proton-Proton Collisions at s=13 TeV, Phys. Rev. Lett. 131 (2023), no. 4 041801, [arXiv:2211.14181].
  * (137) LHC Higgs Cross Section Working Group Collaboration, D. de Florian et al., Handbook of LHC Higgs Cross Sections: 4. Deciphering the Nature of the Higgs Sector, arXiv:1610.07922.
  * (138) C. Anastasiou and K. Melnikov, Pseudoscalar Higgs boson production at hadron colliders in NNLO QCD, Phys. Rev. D 67 (2003) 037501, [hep-ph/0208115].
  * (139) R. V. Harlander and W. B. Kilgore, Production of a pseudoscalar Higgs boson at hadron colliders at next-to-next-to leading order, JHEP 10 (2002) 017, [hep-ph/0208096].
  * (140) R. V. Harlander and W. B. Kilgore, Higgs boson production in bottom quark fusion at next-to-next-to leading order, Phys. Rev. D 68 (2003) 013001, [hep-ph/0304035].
  * (141) PDF4LHC Working Group Collaboration, R. D. Ball et al., The PDF4LHC21 combination of global PDF fits for the LHC Run III, J. Phys. G 49 (2022), no. 8 080501, [arXiv:2203.05506].
  * (142) S. Alioli, W. Dekens, M. Girard, and E. Mereghetti, NLO QCD corrections to SM-EFT dilepton and electroweak Higgs boson production, matched to parton shower in POWHEG, JHEP 08 (2018) 205, [arXiv:1804.07407].
  * (143) M. Spira, Higgs Boson Production and Decay at Hadron Colliders, Prog. Part. Nucl. Phys. 95 (2017) 98–159, [arXiv:1612.07651].
  * (144) CMS Collaboration, A. M. Sirunyan et al., Measurements of $\mathrm{t\bar{t}}H$ Production and the CP Structure of the Yukawa Interaction between the Higgs Boson and Top Quark in the Diphoton Decay Channel, Phys. Rev. Lett. 125 (2020), no. 6 061801, [arXiv:2003.10866].
  * (145) ATLAS Collaboration, G. Aad et al., $CP$ Properties of Higgs Boson Interactions with Top Quarks in the $t\bar{t}H$ and $tH$ Processes Using $H\rightarrow\gamma\gamma$ with the ATLAS Detector, Phys. Rev. Lett. 125 (2020), no. 6 061802, [arXiv:2004.04545].
  * (146) J. Aebischer, J. Kumar, P. Stangl, and D. M. Straub, A Global Likelihood for Precision Constraints and Flavour Anomalies, arXiv:1810.07698.