1em captionlabel caption DO-TH 23/07

# A Precise Electron EDM Constraint on CP-odd Heavy-Quark Yukawas

Joachim Brod<EMAIL_ADDRESS>Department of Physics, University of
Cincinnati, Cincinnati, OH 45221, USA Zachary Polonsky
<EMAIL_ADDRESS>Physik-Institut, Universität Zürich, CH-8057
Zürich, Switzerland Emmanuel Stamou<EMAIL_ADDRESS>Fakultät
Physik, TU Dortmund, D-44221 Dortmund, Germany

###### Abstract

CP-odd Higgs couplings to bottom and charm quarks arise in many extensions of
the standard model and are of potential interest for electroweak baryogenesis.
These couplings induce a contribution to the electron EDM. The experimental
limit on the latter then leads to a strong bound on the CP-odd Higgs
couplings. We point out that this bound receives large QCD corrections, even
though it arises from a leptonic observable. We calculate the contribution of
CP-odd Higgs couplings to the bottom and charm quarks in renormalisation-group
improved perturbation theory at next-to-leading order in the strong
interaction, thereby reducing the uncertainty to a few percent.

##  1 Introduction

The precise determination of the Yukawa couplings of the Higgs boson to all
fermions has been a focus of particle physics since the discovery of the Higgs
boson in 2012 [1, 2]. In the Standard Model (SM) all Yukawa couplings are
aligned with the fermion masses and thus real, but multiple extensions of the
SM induce non-trivial phases. This is of particular interest as these phases
(mainly in the Yukawa couplings to the third fermion generation) play a
leading role in models of electroweak baryogenesis [3]. In this article, we
focus on the bottom- and charm-quark Yukawa couplings.

It is well-known that the present bounds on the electric dipole moment (EDM)
of the electron place strong constraints on CP-violating phases in the quark
Yukawa couplings [4, 5, 6, 7, 8]. However, it is less well-appreciated (and
maybe somewhat surprising) that the heavy-quark contributions to the electron
EDM receive large QCD corrections, leading to a large implicit uncertainty in
the current constraints. In this section, we briefly review the current
situation. In the remainder of this article, we calculate the leading
logarithmic (LL) and next-to-leading logarithmic (NLL) QCD corrections,
thereby reducing the presently ${\cal O}(1)$ uncertainty to a few percent.

For the purpose of this work, we assume a modification of the SM heavy-quark
Yukawa couplings of the form111 This parameterisation of pseudoscalar Higgs
couplings should be thought of as either the dimension-four part of the so-
called Higgs Effective Field Theory [9], the electroweak chiral Lagrangian[10]
in unitarity gauge for the electroweak sector, or as the leading term arising
from the dimension-six SMEFT operators of the form
$H^{\dagger}H\overline{Q}_{L,i}{H}d_{R,i}$ and
$H^{\dagger}H\overline{Q}_{L,i}{\tilde{H}}u_{R,i}$, where $H$ denotes the
Higgs doublet in the unbroken phase of electroweak gauge symmetry, while
$Q_{L,i}$ and $d_{R,i}$/$u_{R,i}$ with $i=2,3$ represent the left-handed quark
doublet and the right-handed quark fields of the second or third generation,
respectively. For further details see also the discussion in Ref. [8].

$\mathscr{L}_{h{q_{h}}{q_{h}}}=-\frac{y^{\text{SM}}_{q_{h}}}{\sqrt{2}}\kappa_{q_{h}}{\bar{q}_{h}}\left(\cos\phi_{q_{h}}+i\gamma_{5}\sin\phi_{q_{h}}\right){q_{h}}\,h\,.$
(1)

Here, ${q_{h}}=b,c$ denotes the bottom- or charm-quark field and $h$ the
physical Higgs field. Moreover, $y^{\text{SM}}_{q_{h}}\equiv
m_{q_{h}}e/\sqrt{2}s_{w}M_{W}$ is the SM Yukawa, with $e$ the positron charge,
$s_{w}$ the sine of the weak mixing angle, and $m_{q}$ and $M_{W}$ the heavy-
quark and $W$-boson masses, respectively. The real parameter
$\kappa_{q_{h}}\geq 0$ parameterises modifications to the absolute value of
the Yukawa coupling, while the phase $\phi_{q_{h}}\in[0,2\pi)$ parameterises
CP violation. The SM corresponds to the values $\kappa_{q_{h}}=1$ and
$\phi_{q_{h}}=0$.

Figure 1: Barr–Zee diagram that contributes to the electron EDM from a CP-odd
heavy-quark Yukawa coupling with the Higgs (red square).

Virtual heavy quarks with CP-odd Higgs couplings induce an electron EDM
$d_{e}$, defined by

$\mathscr{L}_{\mathrm{e}ff}=-d_{e}\,\frac{i}{2}\,\bar{e}\sigma^{\mu\nu}\gamma_{5}e\,F_{\mu\nu}\,,$
(2)

with $\sigma^{\mu\nu}\equiv\tfrac{i}{2}[\gamma^{\mu},\gamma^{\nu}]$, via the
well-known Barr–Zee diagrams [11]. Calculating the two-loop Barr–Zee diagrams
(see Figure 1) keeping the non-zero heavy-quark mass and expanding the loop
functions for small quark mass and small external momenta leads to [4]

$\begin{split}d_{e}&\simeq-12eQ_{e}Q_{q_{h}}^{2}\,\frac{\alpha_{e}}{(4\pi)^{3}}\sqrt{2}G_{F}m_{e}\,\kappa_{q_{h}}\sin\phi_{q_{h}}\,x_{q_{h}}\left(\log^{2}x_{q_{h}}+\frac{\pi^{2}}{3}\right)\,,\end{split}$
(3)

up to higher orders in the ratio $x_{q_{h}}\equiv m_{q_{h}}^{2}/M_{h}^{2}$
(with $M_{h}\sim 125\,$GeV the Higgs-boson mass). Here, $\alpha_{e}$ is the
fine-structure constant, $Q_{q_{h}}$ is the charge of the heavy quark with
$Q_{b}=-1/3$ and $Q_{c}=+2/3$, and $Q_{e}=-1$ is the electron charge. Barr–Zee
diagrams with an internal $Z$ boson also affect the electron EDM, however,
they lead to a contribution that is suppressed with respect to those with an
internal photon by the small coupling of the $Z$ boson to electrons [4].

Using the ACME result, $|d_{e}|<1.1\times 10^{-29}\,e\,\text{cm}$ (at 90% CL)
[12], Eq. (3) implies the bounds $\kappa_{b}|\sin\phi_{b}|\leq 0.41$ and
$\kappa_{c}|\sin\phi_{c}|\leq 1.1$ at the 90% CL for the bottom- and charm-
quark case, respectively. However, to obtain these bounds, numerical values
for the heavy-quark masses must be chosen. It is not clear a priori at which
scale the quark masses should be evaluated; $\mu=M_{h}$ and $\mu=m_{q_{h}}$
would be obvious choices. For the bounds above, we used the values
$m_{b}(M_{h})$ and $m_{c}(M_{h})$; choosing $m_{b}(m_{b})$ and $m_{c}(m_{c})$
instead leads to the significantly stronger bounds
$\kappa_{b}|\sin\phi_{b}|\leq 0.23$ and $\kappa_{c}|\sin\phi_{c}|\leq 0.35$ at
the 90% CL. The differences arise from the large QCD running of the quark
masses between the two scales $\mu=M_{h}$ and $\mu=m_{q_{h}}$. This indicates
that the QCD corrections to Eq. (3) are large, even though the electron EDM is
a leptonic observable. By our explicit calculation, we will show that the
predicted value for $d_{e}$ after resolving the ambiguity lies somewhere in
between the values obtained by using the two scales $M_{h}$ and $m_{q_{h}}$.

We give here a brief outline of the main ideas of the calculation. First,
notice that the result in Eq. (3), which was obtained by a two-loop, fixed-
order calculation, is numerically dominated by the large quadratic logarithms
$\log^{2}x_{q_{h}}$. This logarithmic contribution can be reproduced by the
one-loop QED renormalisation-group (RG) evolution in an appropriate effective
theory (see Section 2), truncated at order $\alpha_{e}^{2}$. The second term
in Eq. (3), $\pi^{2}/3$, has no logarithmic dependence on the mass ratio and
is formally of next-to-next-to-leading-logarithmic (NNLL) order in RG-improved
perturbation theory. Numerically, this term gives a $\sim 6\%$ correction to
the logarithmic contribution to $d_{e}/e$ in the bottom case and a $\sim
3.5\%$ correction in the charm case. On the other hand, we have seen above
that the choice of different renormalisation scales for the quark masses
represents a much larger uncertainty, of the order of $100\%$.

We, thus, conclude that the QCD corrections to the Barr–Zee diagrams are
large, and we expect that these corrections are dominated by leading QCD
logarithms, since the product $\alpha_{s}\log x_{q_{h}}$ is large
($\alpha_{s}$ denotes the strong coupling constant). These logarithms can be
reliably calculated using RG-improved perturbation theory. The result with
resummed leading QCD logarithms has the schematic form
$\alpha_{e}^{2}\log^{2}x_{q_{h}}(1+\alpha_{s}\log
x_{q_{h}}+\alpha_{s}^{2}\log^{2}x_{q_{h}}+\ldots)$. Here, the first term
reflects the quadratic logarithm in Eq. (3), while all other terms correspond
to the leading logarithms of the diagrams obtained by dressing the Barr-Zee
diagrams, Figure 1, with an arbitrary number of gluons.

It is, however, well-known that one can only consistently fix the scale and
scheme dependence of the input parameters by going beyond the LL
approximation. Hence, we will also perform the NLL calculation, reducing the
the uncertainty down to the percent level. This corresponds roughly to the
size of the correction that we have estimated above using the $\pi^{2}/3$ term
in the fixed-order result, which can be viewed as part of the NNLL order
result in RG-improved perturbation theory. This term is thus not included in
our NLL calculation.

Our calculation will show that the QCD perturbation series converges well, as
might be expected for a leptonic observable. In addition, we emphasize that
this is a complete calculation – no further hadronic input (such as the
lattice matrix elements required for hadronic EDMs) is needed. The final
result for $d_{e}/e$ lies between the two values obtained from the naive
computation, determining the heavy quark masses at the different scales. All
these results are illustrated in Figure 4 of Section 4.

This paper is organised as follows: in Section 2, we introduce the effective
theories used in the computation. In Section 3, we present the detailed
results of our calculation, namely, the initial conditions at the electroweak
scale, the calculation of the anomalous dimensions, and the threshold
corrections at the heavy-quark thresholds. We also show the final analytical
results in a compact form. In Section 4, we present the numerical results of
the calculation including updated bounds on the CP-odd heavy-quark Yukawa
couplings. We conclude in Section 5. Additional information is presented in
two appendices: In App. A, we collect the unphysical operators used in the
computation of the anomalous-dimension matrix and in App. B, we show the
results for renormalisation constants entering the calculation.

##  2 Effective Theories Below the Weak Scale

A precise determination of the electron EDM in the presence of CP-odd Higgs
couplings to the bottom and charm quarks requires an effective field theory
that allows to sum large QCD logarithms to all orders in the strong coupling
constant and includes the effect of the combined $\alpha_{s}$ and $\alpha_{e}$
RG evolution. Based on the “full” Lagrangian in Eq. (1), we construct the
effective Lagrangian below the electroweak scale, $\mu_{\text{ew}}$, by
integrating out the heavy degrees of freedom of the SM (the top quark, the
weak gauge bosons, and the Higgs). EDMs are then induced by non-renormalisable
operators that are CP odd. In the current work, we focus on the part of the
effective Lagrangian that is relevant for predicting the electron EDM in the
presence of CP-odd Yukawas couplings to the bottom and charm quarks. In this
case, the relevant effective Lagrangian reads

$\mathscr{L}_{\text{eff},{q_{h}}}=-\sqrt{2}G_{\mathrm{F}}\,\Big{(}C_{1}^{e{q_{h}}}O_{1}^{e{q_{h}}}+C_{1}^{{q_{h}}e}O_{1}^{{q_{h}}e}+C_{2}^{e{q_{h}}}O_{2}^{e{q_{h}}}+C_{3}^{e}O_{3}^{e}\Big{)}+\ldots\,,$
(4)

where the four linearly independent operators are

$\displaystyle O_{1}^{e{q_{h}}}$
$\displaystyle=(\bar{e}e)\,({\bar{q}_{h}}\,i\gamma_{5}{q_{h}})\,,$
$\displaystyle O_{1}^{{q_{h}}e}$
$\displaystyle=({\bar{q}_{h}}{q_{h}})\,(\bar{e}\,i\gamma_{5}e)\,,$
$\displaystyle O_{2}^{e{q_{h}}}$
$\displaystyle=\frac{1}{2}\epsilon^{\mu\nu\rho\sigma}(\bar{e}\sigma_{\mu\nu}e)\,({\bar{q}_{h}}\,\sigma_{\rho\sigma}{q_{h}})\,,$
$\displaystyle O_{3}^{e}$
$\displaystyle=\frac{Q_{e}}{2}\frac{m_{q_{h}}}{e}\,(\bar{e}\sigma^{\mu\nu}e)\,\tilde{F}_{\mu\nu}\,,$
(5)

and $C_{1}^{e{q_{h}}}$, $C_{1}^{{q_{h}}e}$, $C_{2}^{e{q_{h}}}$, and
$C_{3}^{e}$ are the corresponding Wilson coefficients. Additional CP-odd
operators that are suppressed by additional factors of
$m_{\text{light}}/m_{q_{h}}$ (where $m_{\text{light}}$ corresponds to a light
quark mass) are denoted by the ellipsis. We defined the electron dipole
operator with a factor of the running quark mass $m_{q_{h}}\equiv
m_{q_{h}}(\mu)$, to avoid awkward ratios of quark and lepton masses in the
anomalous dimensions. Throughout this work the $\gamma_{5}$ matrix is defined
by

$\gamma_{5}\equiv\frac{i}{4!}\epsilon_{\mu\nu\rho\sigma}\gamma^{\mu}\gamma^{\nu}\gamma^{\rho}\gamma^{\sigma}\,,$
(6)

where $\epsilon^{\mu\nu\rho\sigma}$ is the totally antisymmetric Levi-Civita
tensor in four space-time dimensions with
$\epsilon_{0123}=-\epsilon^{0123}=1$, and we use the notation
$\widetilde{F}^{\mu\nu}=\tfrac{1}{2}\epsilon^{\mu\nu\rho\sigma}F_{\rho\sigma}$.
We treat $\gamma_{5}$ within the “Larin” scheme; for the details and
subtleties we refer to Ref. [6]. The non-standard sign convention for
$O_{3}^{q}$ is related to our definition of the covariant derivative acting on
fermion fields $f$,

$D_{\mu}\equiv\partial_{\mu}-ig_{s}T^{a}G_{\mu}^{a}+ieQ_{f}A_{\mu}\,.$ (7)

The basis of all flavour-diagonal, CP-odd operators is closed under the QED
and QCD RG flow, as both interactions conserve CP and flavour. Below the
electroweak scale, there is a tower of effective theories relevant for
predicting the electron EDM. For the bottom case, ${q_{h}}=b$, we employ the
effective Lagrangian in Eq. (4) for the five-flavour theory, while for the
charm case, ${q_{h}}=c$, we employ Eq. (4) for both the five-flavour and the
four-flavour theory (see discussion in Section 3.3 for the threshold
corrections on couplings and Wilson coefficients at the bottom-quark scale).

The effective theory below the heavy-quark scale, $\mu_{{q_{h}}}$, does not
contain four-fermion operators with heavy quarks, and we use the modified
effective Lagrangian

$\widetilde{\mathscr{L}}_{\text{eff},{q_{h}}}=-\sqrt{2}G_{\mathrm{F}}\widetilde{C}_{3}^{e}\widetilde{O}_{3}^{e}+\ldots\,,$
(8)

in which – as opposed to Eq. (4) – the dipole operator is defined with the
conventional factor $m_{e}$:

$\widetilde{O}_{3}^{e}=\frac{Q_{e}}{2}\frac{m_{e}}{e}\,(\bar{e}\sigma^{\mu\nu}e)\,\tilde{F}_{\mu\nu}\,.$
(9)

Note that for ${q_{h}}=b$ the Lagrangian in Eq. (8) refers to the four-flavour
Lagrangian, while for ${q_{h}}=c$ to the three-flavour one. This definition of
$\widetilde{\mathscr{L}}_{\text{eff},{q_{h}}}$ implies that (cf. Eq. (2))

$\frac{d_{e}}{e}=-\sqrt{2}G_{\mathrm{F}}\frac{m_{e}}{4\pi\alpha_{e}}\widetilde{C}_{3}^{e}\,.$
(10)

In the next section, we describe how the RG evolution within the effective
field theories relates $d_{e}$ to the parameters $\kappa_{q_{h}}$ and
$\phi_{q_{h}}$ of the “full” theory in Eq. (1).

##  3 Renormalisation Group Evolution

Our goal is the summation of all leading and next-to-leading logarithms via
RG-improved perturbation theory. The calculation proceeds in the following
steps.

First, we integrate out the Higgs and weak gauge bosons together with the top
quark at the electroweak scale, $\mu_{\text{ew}}\sim M_{h}$. This matching
calculation at $\mu_{\text{ew}}$ induces the initial conditions for the five-
flavour Wilson coefficients appearing in Eq. (4). We collect them in Section
3.1. Subsequently, we perform the RG evolution from $\mu_{\text{ew}}$ down to
the bottom-quark threshold, $\mu_{b}\sim m_{b}(m_{b})$. The anomalous
dimensions relevant for the mixed QCD–QED RG evolution at NLL accuracy are
computed here for the first time, see Section 3.2. The next step depends on
whether ${q_{h}}=b$ or ${q_{h}}=c$. For the bottom case, ${q_{h}}=b$, we match
directly to the four-flavour version of the Lagrangian in Eq. (8) to obtain
the prediction of the electron EDM. The relevant threshold corrections at
$\mu_{b}$ are discussed in Section 3.3. For the charm case, ${q_{h}}=c$, we
must instead match at $\mu_{b}$ to the four-flavour version of the Lagrangian
with four-fermion operators in Eq. (4) and additionally perform the RG
evolution in the four-flavour theory from $\mu_{b}$ down to $\mu_{c}\sim
m_{c}(m_{c})$. The corresponding anomalous dimensions are also given in
section 3.2. Finally, we match to the three-flavour version of the Lagrangian
in Eq. (8) to obtain the prediction of the electron EDM.

The calculations of the amplitudes relevant for computing the initial
conditions of Wilson coefficients and the hitherto unknown anomalous
dimensions have been performed with self-written FORM [13] routines, i.e.,
MaRTIn [14], implementing the two-loop recursion algorithms presented in Refs.
[15, 16]. The amplitudes were generated using QGRAF [17].

The RG evolution between the different matching scales is significantly more
involved than in applications in which the electromagnetic coupling,
$\alpha_{e}$, can be neglected. The reason is that the leading contribution to
the electron EDM contains a term with two powers of the large logarithm, i.e,
$\log^{2}x_{q_{h}}$, see Eq. (3). Within the effective theory, this term is
obtained by a LL QED calculation, truncated at order $\alpha_{e}^{2}$.
However, the numerically relevant corrections to this result are not further
electromagnetic $\alpha_{e}^{n}\log^{n}$ corrections, but large,
logarithmically enhanced QCD corrections which must be summed to all orders to
obtain an accurate prediction. Therefore, to properly account for the
numerically relevant corrections we must consistently solve the mixed QCD–QED
RG equations.

This can be achieved consistently using the general formalism developed in
Ref. [18]. The main idea is that, since $\alpha_{s}\log x_{q_{h}}$ is ${\cal
O}(1)$, such products of QCD coupling times large logarithms must be resummed
for an accurate prediction. By contrast, the product $\alpha_{e}\log
x_{q_{h}}$ is small and does not require resummation. The large logarithms
appearing in $\alpha_{e}\log x_{q_{h}}$ are expressed in terms of the resummed
$\alpha_{s}\log x_{q_{h}}$, i.e. $\alpha_{e}\log
x_{q_{h}}=(\alpha_{e}/\alpha_{s})\times\alpha_{s}\log x_{q_{h}}$. As a
consequence, the conventional expansion in terms of $\alpha_{s}$ and
$\alpha_{e}$ is replaced by an expansion in $\alpha_{s}$ and
$\kappa\equiv\alpha_{e}/\alpha_{s}$. Therefore, in this setup the Wilson
coefficients at some scale $\mu$ have the expansion

$C_{i}(\mu)=\sum_{n,m=0}^{\infty}\tilde{\alpha}_{s}(\mu)^{n}\kappa(\mu)^{m}\,C^{(nm)}_{i}(\mu)\,,$
(11)

with $\tilde{\alpha}_{s}\equiv\alpha_{s}/4\pi$. By implementing the mixed RG
evolution we will compute the $C^{(nm)}_{i}(\mu)$ coefficients relevant for
the electron EDM; for details we refer to Ref. [18]. The analytical results
are presented in Section 3.4.

###  3.1 Initial Conditions at the Weak Scale

Figure 2: Examples of leading order (left) and next-to-leading order (right)
Feynman diagrams that contribute to the initial conditions of the four-fermion
sector in the EFT. CP-odd heavy-quark Yukawa couplings are indicated by red
squares.

We augment the SM by flavour-conserving, CP-odd Higgs Yukawa couplings to the
heavy quarks ${q_{h}}=b,c$, as parameterised in Eq. (1). At a scale
$\mu_{\text{ew}}\approx M_{h}$ we integrate out the heavy degrees of freedom
of the SM and match to the effective five-flavour theory relevant for the
electron EDM in Eq. (4). The initial conditions for the Wilson coefficients
relevant for our calculation are obtained by evaluating the tree-level and
one-loop Feynman diagrams such as those shown in Figure 2; we find

$\displaystyle C_{1}^{e{q_{h}}}(\mu_{\text{ew}})$
$\displaystyle=-\kappa_{q_{h}}\sin\phi_{q_{h}}\frac{m_{e}m_{{q_{h}}}}{M_{h}^{2}}+{\cal
O}(\alpha_{s}^{2},\alpha_{s}\alpha_{e},\alpha_{e}^{2})\,,$ (12) $\displaystyle
C_{1}^{{q_{h}}e}(\mu_{\text{ew}})$ $\displaystyle=0+{\cal
O}(\alpha_{s}^{2},\alpha_{s}\alpha_{e},\alpha_{e}^{2})\,,$ (13)
$\displaystyle\begin{split}C_{2}^{e{q_{h}}}(\mu_{\text{ew}})&=\frac{\alpha_{e}}{4\pi}\left(\frac{3}{2}+\log\frac{\mu_{\text{ew}}^{2}}{M_{h}^{2}}\right)\frac{m_{e}m_{{q_{h}}}}{M_{h}^{2}}Q_{e}Q_{q_{h}}\kappa_{q_{h}}\sin\phi_{q_{h}}+{\cal
O}(\alpha_{s}^{2},\alpha_{s}\alpha_{e},\alpha_{e}^{2})\,,\end{split}$ (14)
$\displaystyle C_{3}^{e}(\mu_{\text{ew}})$ $\displaystyle=0+{\cal
O}(\alpha_{e}^{2})\,.$ (15)

All parameters appearing above correspond to parameters in the five-flavour
theory evaluated at the scale $\mu_{\text{ew}}$, i.e.,
$m_{q_{h}}=m_{{q_{h}},\text{5fl}}(\mu_{\text{ew}})$ and
$\alpha_{e}=\alpha_{e,\text{5fl}}(\mu_{\text{ew}})$. We treat the heavy quarks
and the electron as massless at the electroweak matching scale; therefore, no
powers of $m_{e}/M_{h}$ or $m_{q_{h}}/M_{h}$ appear in the results. The
explicit factors of the electron and heavy-quark masses arise by expressing
the Yukawa couplings in Eq. (1) in terms of $m_{e}$ and $m_{q_{h}}$. We have
included all terms up to corrections of quadratic order in the strong and
electromagnetic coupling constants, as only these are required for our
analysis. The ${\mathcal{O}}(\alpha_{e})$ coefficient, Eq. (14), is well-
defined only after specifying the basis of evanescent operators which can be
found in App. A.

In Section 3.4, we will use the framework of Ref. [18] to solve the mixed
QCD–QED RG as an expansion in $\tilde{\alpha}_{s}$ and $\kappa$. Based on Eqs.
(12)-(14) we find the contributing expansion coefficients in Eq. (11)

$\begin{split}C_{1}^{e{q_{h}},(00)}(\mu_{\text{ew}})&=-\kappa_{q_{h}}\sin\phi_{q_{h}}\frac{m_{e}m_{{q_{h}}}}{M_{h}^{2}}\,,\\\
C_{2}^{e{q_{h}},(11)}(\mu_{\text{ew}})&=\left(\frac{3}{2}+\log\frac{\mu_{\text{ew}}^{2}}{M_{h}^{2}}\right)\frac{m_{e}m_{{q_{h}}}}{M_{h}^{2}}Q_{e}Q_{q_{h}}\kappa_{q_{h}}\sin\phi_{q_{h}}\,.\end{split}$
(16)

###  3.2 Anomalous Dimensions

To solve the RG in the effective theories below the electroweak scale we need
to include the running of coupling constants, mass anomalous dimensions, and
the mixing of operators. In this section, we collect the results that enter
the analysis at NLL accuracy.

The running of the Wilson coefficients from the electroweak matching scale
down to the relevant quark scale is governed by the RG equation

$\frac{dC_{i}}{d\log\mu}=C_{j}\gamma_{ji}\,,$ (17)

where $\gamma_{ji}$ are the components of the anomalous dimension matrix
(ADM). We choose the ordering of Wilson coefficients as

$\vec{C}=(C_{1}^{e{q_{h}}},~{}C_{1}^{{q_{h}}e},~{}C_{2}^{e{q_{h}}},~{}C_{3}^{e})^{T}\,.$
(18)

The ADM admits an expansion in powers of222 Note that for the ADM we do not
expand in terms of $\tilde{\alpha}_{s}$ and $\kappa$ as for Wilson
coefficients. $\tilde{\alpha}_{s}\equiv\alpha_{s}/4\pi$ and
$\tilde{\alpha}_{e}\equiv\alpha_{e}/4\pi$,

$\gamma=\sum_{\begin{subarray}{c}n,m=0\\\ n+m\geq
1\end{subarray}}\gamma^{(nm)}\tilde{\alpha}_{s}(\mu)^{n}\tilde{\alpha}_{e}(\mu)^{m}\,,$
(19)

with $\tilde{\alpha}_{s}$, $\tilde{\alpha}_{e}$, and $\gamma^{(nm)}$ depending
on the number of active fermion flavours in the effective theory. Using this
expansion, the ADM can be organised by loop order. In general, it depends on
the number of active fermion flavours and on the flavour of the heavy quark
${q_{h}}$. Below we present our results entering the five- and four-flavour RG
evolution required for the case ${q_{h}}=b$ and ${q_{h}}=c$.

By explicit calculation, we find at one loop

$\displaystyle\gamma^{(01)}$
$\displaystyle=\begin{pmatrix}-6(Q_{e}^{2}+Q_{q_{h}}^{2})&0&-2Q_{e}Q_{q_{h}}&0\\\\[5.0pt]
0&-6(Q_{e}^{2}+Q_{q_{h}}^{2})&-2Q_{e}Q_{q_{h}}&0\\\\[5.0pt]
-48Q_{q_{h}}Q_{e}&-48Q_{q_{h}}Q_{e}&2(Q_{e}^{2}+Q_{q_{h}}^{2})&-48\frac{Q_{q_{h}}}{Q_{e}}\\\\[5.0pt]
0&0&0&10Q_{e}^{2}+6Q_{q_{h}}^{2}-2\beta_{e}^{(0)}\end{pmatrix}\,,$ (20)
$\displaystyle\gamma^{(10)}$
$\displaystyle=\begin{pmatrix}-6C_{F}&0&0&0\\\\[5.0pt]
0&-6C_{F}&0&0\\\\[5.0pt] 0&0&2C_{F}&0\\\\[5.0pt]
0&0&0&\gamma_{{q_{h}},s}^{(0)}\end{pmatrix}\,,$ (21)

with
$\beta_{e}^{(0)}=-\frac{4}{3}\Big{(}N_{c}n_{d}Q_{b}^{2}+N_{c}n_{u}Q_{c}^{2}+n_{\ell}Q_{e}^{2}\Big{)}$
and $\gamma_{{q_{h}},s}^{(0)}=6C_{F}$. Moreover, $N_{c}=3$ is the number of
quark colors, $n_{u}$ is the number of active up-type quarks, $n_{d}$ is the
number of active down-type quarks, $n_{\ell}$ is the number of active charged
leptons and $C_{F}\equiv(N_{c}^{2}-1)/2N_{c}=4/3$. In both the bottom and
charm cases, we have $n_{u}=2$ and $n_{\ell}=3$.

Figure 3: Examples of two-loop QCD (left), mixed QED–QCD (middle), and two-
loop QED (right) Feynman diagrams entering the computation of the anomalous
dimension matrix. Effective operator insertions are represented by the
unshaded boxes.

At two-loop, the pure QCD ADM is the same for the bottom- and charm-quark case
and depends on the number of active quark flavours,

$\gamma^{(20)}=\begin{pmatrix}\frac{724}{9}-\frac{88n_{d}}{9}&0&0&0\\\\[5.0pt]
0&-\frac{1132}{9}+\frac{40n_{d}}{9}&0&0\\\\[5.0pt]
0&0&\frac{1964}{27}-\frac{104n_{d}}{27}&0\\\\[5.0pt]
0&0&0&\frac{1132}{9}-\frac{40n_{d}}{9}\end{pmatrix}\,.$ (22)

For the bottom case, we only need the five-flavour theory, i.e., we always fix
$n_{d}=3$. For the charm case, we must solve the RG in both five- and four-
flavour theory in which case $n_{d}=3$ and $n_{d}=2$, respectively.

For the two-loop mixed QCD–QED and the pure two-loop QED ADMs, explicit
factors of the quark charges result in different results for the bottom and
charm cases. For notational simplicity, we quote the ADMs for the two cases
separately after having substituted the electric charges. For the bottom, we
find, after fixing $n_{d}=3$

$\gamma^{(11)}_{[{q_{h}}=b]}=\begin{pmatrix}-\frac{8}{9}&0&-\frac{16}{9}&0\\\\[5.0pt]
0&-\frac{8}{9}&\frac{16}{3}&0\\\\[5.0pt]
128&-\frac{128}{3}&-\frac{152}{27}&-\frac{448}{3}\\\\[5.0pt]
0&0&0&40\end{pmatrix}\,,\quad\gamma^{(02)}_{[{q_{h}}=b]}=\begin{pmatrix}\frac{2420}{81}&-\frac{14}{3}&\frac{392}{81}&-8\\\\[5.0pt]
-\frac{14}{3}&-\frac{7820}{81}&\frac{8}{81}&-\frac{8}{3}\\\\[5.0pt]
\frac{64}{27}&\frac{3136}{27}&-\frac{17234}{243}&\frac{464}{9}\\\\[5.0pt]
0&0&0&-\frac{8704}{81}\end{pmatrix}\,.$ (23)

For the charm, we find, keeping the $n_{d}$-dependence explicit

$\displaystyle\gamma^{(11)}_{[{q_{h}}=c]}$
$\displaystyle=\begin{pmatrix}-\frac{32}{9}&0&\frac{32}{9}&0\\\\[5.0pt]
0&-\frac{32}{9}&-\frac{32}{3}&0\\\\[5.0pt]
-256&\frac{256}{3}&-\frac{608}{27}&\frac{896}{3}\\\\[5.0pt]
0&0&0&32+\frac{32n_{d}}{9}\end{pmatrix}\,,$ (24)
$\displaystyle\gamma^{(02)}_{[{q_{h}}=c]}$
$\displaystyle=\begin{pmatrix}-\frac{439}{81}+\frac{4n_{d}}{81}&-\frac{56}{3}&-\frac{688}{81}-\frac{8n_{d}}{81}&-32\\\\[5.0pt]
-\frac{56}{3}&-\frac{5879}{81}-\frac{316n_{d}}{81}&-\frac{208}{81}-\frac{8n_{d}}{81}&-\frac{32}{3}\\\\[5.0pt]
-\frac{1664}{27}-\frac{64n_{d}}{27}&-\frac{5504}{27}-\frac{64n_{d}}{27}&-\frac{25661}{243}-\frac{676n_{d}}{243}&-\frac{256}{9}\\\\[5.0pt]
0&0&0&-\frac{8647}{81}-\frac{404n_{d}}{81}\end{pmatrix}\,.$ (25)

Some representative Feynman diagrams used to compute the two-loop ADMs are
shown in Figure 3. The two-loop ADM are well-defined only after specifying the
basis of evanescent operators; this is done in App. A.

All two-loop results in this section are new, to the best of our knowledge.
The one-loop RG evolution has also been calculated in Ref. [19].

###  3.3 Threshold Corrections

When performing the running of the Wilson coefficients, we must also integrate
out the heavy quarks at the respective scales. This leads to threshold
corrections to the gauge couplings and quark masses, as well as to the Wilson
coefficients. Below we describe the origin of the threshold corrections for
each case, and collect the corresponding results. Due to the mixed RG
evolution, the matching of the effective theories must also be performed as an
expansion in $\tilde{\alpha}_{s}$ and $\kappa$, cf. Eq. (11). We stress that
the results presented below are only applicable for our specific case in which
the only non-vanishing initial conditions are the ones in section 3.1. Other
UV completions, with more contributions to the initial conditions, can receive
additional threshold corrections (for instance, if
$C^{e}_{3}(\mu_{\text{ew}})\neq 0$ at one-loop).

#### The Bottom-Quark Case

In the case of an anomalous bottom-quark Yukawa coupling, ${q_{h}}=b$, the
only relevant threshold below the weak scale is at $\mu_{b}$ in which we match
the five-flavour version of the Lagrangian in Eq. (4) to the four-flavour
version of the Lagrangian in Eq. (8). There are three effects that induce a
non-trivial correction to the matching onto $\widetilde{C}_{3}^{e}$: the
threshold correction for $\alpha_{s}$ when matching from the five-flavour onto
the four-flavour theories (the corresponding one for $\alpha_{e}$ does not
contribute because in our case $C_{3,\text{5fl}}^{e,(01)}(\mu_{b})=0$); the
different normalization of the dipole operators $O^{e}_{3}$ and
$\widetilde{O}^{e}_{3}$ in the two theories, which leads to a factor of
$m_{b}/m_{e}$; and a threshold correction from one-loop insertions of
$O^{be}_{2}$ in the five-flavour theory. Details can be found in the analogous
discussion in Ref. [6].

Taking all of these effects into account, we find for the electron dipole
operator in the four-flavour theory at $\mu_{b}$

$\displaystyle\widetilde{C}_{3}^{e}(\mu_{b})=$
$\displaystyle\kappa_{\text{4fl}}^{2}\frac{m_{b}}{m_{e}}C_{3,\text{5fl}}^{e,(02)}(\mu_{b})$
$\displaystyle+$
$\displaystyle{\tilde{\alpha}_{s,\text{4fl}}}\kappa_{\text{4fl}}^{2}\frac{m_{b}}{m_{e}}\biggl{(}C_{3,\text{5fl}}^{e,(12)}(\mu_{b})+24\frac{Q_{b}}{Q_{e}}\log\frac{\mu_{b}^{2}}{m_{b}^{2}}C_{2,\text{5fl}}^{be,(01)}(\mu_{b})$
$\displaystyle\qquad\qquad\qquad\qquad-\frac{1}{2}\gamma_{b,s}^{(0)}\log\frac{\mu_{b}^{2}}{m_{b}^{2}}C_{3,\text{5fl}}^{e,(02)}(\mu_{b})-2\delta\alpha_{s}\log\frac{\mu_{b}^{2}}{m_{b}^{2}}C_{3,\text{5fl}}^{e,(02)}(\mu_{b})\biggr{)}$
(26) $\displaystyle=$
$\displaystyle\kappa_{\text{4fl}}^{2}\frac{m_{b}}{m_{e}}C_{3,\text{5fl}}^{e,(02)}(\mu_{b})$
$\displaystyle+$
$\displaystyle{\tilde{\alpha}_{s,\text{4fl}}}\kappa_{\text{4fl}}^{2}\frac{m_{b}}{m_{e}}\biggl{(}C_{3,\text{5fl}}^{e,(12)}(\mu_{b})+8\log\frac{\mu_{b}^{2}}{m_{b}^{2}}C_{2,\text{5fl}}^{be,(01)}(\mu_{b})-\frac{16}{3}\log\frac{\mu_{b}^{2}}{m_{b}^{2}}C_{3,\text{5fl}}^{e,(02)}(\mu_{b})\biggr{)}\,,$
(27)

with $\gamma_{b,s}^{0}=6C_{F}$ the leading-order QCD mass anomalous dimension,
the $\alpha_{s}$ threshold correction $\delta\alpha_{s}=2/3$, and
$m_{b}=m_{b}(m_{b})$ in the five-flavour theory in the $\overline{\text{MS}}$
scheme. We indicate by explicit subscripts “5fl” and “4fl” in which effective
theory the various quantities are defined. The couplings in Eq. (27) are
evaluated at $\mu_{b}$, i.e., $\tilde{\alpha}_{s,\text{4fl}}(\mu_{b})$ and
$\kappa_{\text{4fl}}(\mu_{b})$.

#### The Charm-Quark Case

In the case of an anomalous charm-quark Yukawa coupling, ${q_{h}}=b$, there
are threshold corrections both at the bottom- and the charm-quark thresholds,
$\mu_{b}$ and $\mu_{c}$, respectively.

At $\mu_{b}$, the effective Lagrangian in the five- and four-flavour theories
are the same (see Eq. (4)), and at NLL accuracy the only relevant effect is
the decoupling of $\alpha_{s}$. The only non-trivial matching condition at the
bottom-quark scale then reads for the charm-quark case, ${q_{h}}=c$,

$C_{3,\text{4fl}}^{e}(\mu_{b})=\kappa_{\text{4fl}}^{2}C_{3,\text{5fl}}^{e,(02)}(\mu_{b})+\tilde{\alpha}_{s,\text{4fl}}\kappa_{\text{4fl}}^{2}\biggl{(}C_{3,\text{5fl}}^{e,(12)}(\mu_{b})-2\delta\alpha_{s}\log\frac{\mu_{b}^{2}}{m_{b}^{2}}C_{3,\text{5fl}}^{e,(02)}(\mu_{b})\biggr{)}\,.$
(28)

Other operators beyond $O_{3}^{e}$ also receive threshold corrections.
However, none of these terms enter the final three-flavour value of
$\widetilde{C}_{3}^{e}(\mu_{c})$ at NLL order, so we do not list them here.

At the charm-quark threshold, $\mu_{c}$, the threshold correction from
matching the four-flavour onto the three-flavour theory with the single
operator in Eq. (8) is analogous to the bottom case. Accordingly, we find for
${q_{h}}=c$

$\displaystyle\widetilde{C}_{3}^{e}(\mu_{c})=$
$\displaystyle\kappa_{\text{3fl}}^{2}\frac{m_{c}}{m_{e}}C_{3,\text{4fl}}^{e,(02)}(\mu_{c})$
$\displaystyle+$
$\displaystyle{\tilde{\alpha}_{s,\text{3fl}}}\kappa_{\text{3fl}}^{2}\frac{m_{c}}{m_{e}}\biggl{(}C_{3,\text{4fl}}^{e,(12)}(\mu_{c})+24\frac{Q_{c}}{Q_{e}}\log\frac{\mu_{c}^{2}}{m_{c}^{2}}C_{2,\text{4fl}}^{ce,(01)}(\mu_{c})$
$\displaystyle\qquad\qquad\qquad\qquad-\frac{1}{2}\gamma_{c,s}^{(0)}\log\frac{\mu_{c}^{2}}{m_{c}^{2}}C_{3,\text{4fl}}^{e,(02)}(\mu_{c})-2\delta\alpha_{s}\log\frac{\mu_{c}^{2}}{m_{c}^{2}}C_{3,\text{4fl}}^{e,(02)}(\mu_{c})\biggr{)}$
(29) $\displaystyle=$
$\displaystyle\kappa_{\text{3fl}}^{2}\frac{m_{c}}{m_{e}}C_{3,\text{4fl}}^{e,(02)}(\mu_{c})$
$\displaystyle+$
$\displaystyle{\tilde{\alpha}_{s,\text{3fl}}}\kappa_{\text{3fl}}^{2}\frac{m_{c}}{m_{e}}\biggl{(}C_{3,\text{4fl}}^{e,(12)}(\mu_{c})-16\log\frac{\mu_{c}^{2}}{m_{c}^{2}}C_{2,\text{4fl}}^{ce,(01)}(\mu_{c})-\frac{16}{3}\log\frac{\mu_{c}^{2}}{m_{c}^{2}}C_{3,\text{4fl}}^{e,(02)}(\mu_{c})\biggr{)}\,,$
(30)

where $\gamma_{c,s}^{0}=6C_{F}$, $\delta\alpha_{s}=2/3$, and
$m_{c}=m_{c}(m_{c})$ in the four-flavour theory in the $\overline{\text{MS}}$
scheme.

###  3.4 Analytic Solution of the RG

In this section, we show the final result for the electron EDM Wilson
coefficients after implementing the mixed RG evolution (see discussion above
Eq. (11)) using the ADMs from section 3.2, and including the threshold
corrections from section 3.3. We obtain obtain the exact analytical result for
the contribution to the electron dipole moment up to
$\mathcal{O}(\alpha_{s}\kappa^{2})$ including the resummation of QCD
logarithms:

$\frac{d_{e}}{e}=-\sqrt{2}G_{\mathrm{F}}\frac{m_{e}}{4\pi\alpha_{e}}\left[\kappa^{2}\widetilde{C}_{3}^{e,(02)}(\mu_{q_{h}})+\tilde{\alpha}_{s}\kappa^{2}\widetilde{C}_{3}^{e,(12)}(\mu_{q_{h}})+{\cal
O}\big{(}\tilde{\alpha}_{s}^{2}\kappa^{2},\kappa^{3}\big{)}\right]\,,$ (31)

with $\alpha_{e}$, $\tilde{\alpha}_{s}$, and
$\kappa\equiv\alpha_{e}/\alpha_{s}$ evaluated at the scale $\mu_{q_{h}}$ in
the four- and three-flavour theory for the bottom- and charm-quark case,
respectively. The coefficients $\widetilde{C}_{3}^{e,(mn)}(\mu_{q_{h}})$ are
functions of the initial conditions at $\mu_{\text{ew}}$ and of ratios of the
values of $\alpha_{s}$ at different scales. For brevity we introduce
corresponding compact notations

$\displaystyle\bar{C}_{iX}^{(mn)}\equiv{C}_{i,\text{5fl}}^{X,(mn)}(\mu_{\text{ew}})\,,\qquad\eta_{5}\equiv\frac{\alpha_{s,\text{5fl}}(\mu_{\text{ew}})}{\alpha_{s,\text{5fl}}(\mu_{b})}\,,\qquad\eta_{4}\equiv\frac{\alpha_{s,\text{4fl}}(\mu_{b})}{\alpha_{s,\text{4fl}}(\mu_{c})}\,.$

In both the bottom and charm-quark cases, the operator $\widetilde{O}_{e}^{3}$
in Eq. (8) below the respective heavy-quark scale is modified only by QED
running effects which are negligibly small.

#### Bottom-Quark Result

In the bottom-quark case, ${q_{h}}=b$, we find for the coefficients in Eq.
(31)

$\displaystyle\widetilde{C}_{3}^{e,(02)}(\mu_{b})$
$\displaystyle=\frac{m_{b}}{m_{e}}\bar{C}_{1eb}^{(00)}\biggl{(}\frac{12}{77}\eta_{5}^{-\frac{12}{23}}-\frac{8}{35}\eta_{5}^{-\frac{19}{23}}+\frac{4}{55}\eta_{5}^{-\frac{34}{23}}\biggr{)}\,,$
(32) $\displaystyle\widetilde{C}_{3}^{e,(12)}(\mu_{b})$
$\displaystyle=\frac{m_{b}}{m_{e}}\bar{C}_{1eb}^{(00)}\left(\frac{248}{105}\eta_{5}^{-\frac{19}{23}}-\frac{152}{77}\eta_{5}^{-\frac{12}{23}}-\frac{64}{165}\eta_{5}^{-\frac{34}{23}}\right)\log\frac{\mu_{b}^{2}}{m_{b}^{2}}$
$\displaystyle+\frac{m_{b}}{m_{e}}\bar{C}_{1eb}^{(00)}\biggl{(}\frac{37864}{40733}\eta_{5}^{\frac{11}{23}}-\frac{314306}{166635}\eta_{5}^{\frac{4}{23}}+\frac{3779848}{261855}\eta_{5}^{-\frac{11}{23}}$
$\displaystyle\qquad\qquad~{}~{}\quad-\frac{2044442}{122199}\eta_{5}^{-\frac{12}{23}}+\frac{199636}{55545}\eta_{5}^{-\frac{19}{23}}-\frac{29848}{87285}\eta_{5}^{-\frac{34}{23}}\biggr{)}$
$\displaystyle+\frac{m_{b}}{m_{e}}\bar{C}_{2eb}^{(11)}\biggl{(}\frac{8}{5}\eta_{5}^{-\frac{11}{23}}-\frac{8}{5}\eta_{5}^{\frac{4}{23}}\biggr{)}\,,$
(33)

where $m_{b}=m_{b}(m_{b})$ in the five-flavour theory and in the
$\overline{\textrm{MS}}$ scheme.

#### Charm-Quark Result

In the charm-quark case, ${q_{h}}=c$, we find for the coefficients in Eq. (31)

$\displaystyle\widetilde{C}_{3}^{e,(02)}(\mu_{c})$
$\displaystyle=\frac{m_{c}}{m_{e}}\bar{C}_{1ec}^{(00)}\biggl{(}\frac{16}{39}\eta_{4}^{-\frac{12}{25}}\eta_{5}^{-\frac{12}{23}}+\frac{64}{357}\eta_{4}^{-\frac{21}{25}}\eta_{5}^{-\frac{12}{23}}+\frac{576}{17017}\eta_{4}^{-\frac{38}{25}}\eta_{5}^{-\frac{12}{23}}$
$\displaystyle\qquad\qquad~{}~{}-\frac{96}{119}\eta_{4}^{-\frac{21}{25}}\eta_{5}^{-\frac{19}{23}}-\frac{64}{595}\eta_{4}^{-\frac{38}{25}}\eta_{5}^{-\frac{19}{23}}+\frac{16}{55}\eta_{4}^{-\frac{38}{25}}\eta_{5}^{-\frac{34}{23}}\biggr{)}\,,$
(34) $\displaystyle\widetilde{C}_{3}^{e,(12)}(\mu_{c})$
$\displaystyle=\frac{m_{c}}{m_{e}}\bar{C}_{1ec}^{(00)}\log\frac{\mu_{b}^{2}}{m_{b}^{2}}\biggl{(}\frac{64}{119}\eta_{4}^{\frac{4}{25}}\eta_{5}^{-\frac{19}{23}}-\frac{64}{119}\eta_{4}^{\frac{4}{25}}\eta_{5}^{-\frac{12}{23}}-\frac{384}{1309}\eta_{4}^{-\frac{13}{25}}\eta_{5}^{-\frac{12}{23}}$
$\displaystyle\qquad\qquad\qquad\quad~{}~{}~{}+\frac{1216}{1785}\eta_{4}^{-\frac{13}{25}}\eta_{5}^{-\frac{19}{23}}-\frac{64}{165}\eta_{4}^{-\frac{13}{25}}\eta_{5}^{-\frac{34}{23}}\biggr{)}$
$\displaystyle+\frac{m_{c}}{m_{e}}\bar{C}_{1ec}^{(00)}\log\frac{\mu_{c}^{2}}{m_{c}^{2}}\biggl{(}\frac{1056}{119}\eta_{4}^{-\frac{21}{25}}\eta_{5}^{-\frac{19}{23}}-\frac{224}{39}\eta_{4}^{-\frac{12}{25}}\eta_{5}^{-\frac{12}{23}}-\frac{704}{357}\eta_{4}^{-\frac{21}{25}}\eta_{5}^{-\frac{12}{23}}$
$\displaystyle\qquad\qquad\qquad\quad~{}~{}~{}-\frac{3072}{17017}\eta_{4}^{-\frac{38}{25}}\eta_{5}^{-\frac{12}{23}}+\frac{1024}{1785}\eta_{4}^{-\frac{38}{25}}\eta_{5}^{-\frac{19}{23}}-\frac{256}{165}\eta_{4}^{-\frac{38}{25}}\eta_{5}^{-\frac{34}{23}}\biggr{)}$
$\displaystyle+\frac{m_{c}}{m_{e}}\bar{C}_{1ec}^{(00)}\biggl{(}\frac{605824}{566559}\eta_{4}^{\frac{4}{25}}\eta_{5}^{\frac{11}{23}}-\frac{1257224}{188853}\eta_{4}^{\frac{4}{25}}\eta_{5}^{\frac{4}{23}}+\frac{151456}{61893}\eta_{4}^{\frac{13}{25}}\eta_{5}^{\frac{11}{23}}$
$\displaystyle\qquad\qquad~{}~{}-\frac{21158428}{16861875}\eta_{4}^{\frac{4}{25}}\eta_{5}^{-\frac{12}{23}}+\frac{3414272}{12894375}\eta_{4}^{\frac{13}{25}}\eta_{5}^{-\frac{12}{23}}+\frac{891392}{5620625}\eta_{4}^{\frac{4}{25}}\eta_{5}^{-\frac{19}{23}}$
$\displaystyle\qquad\qquad~{}~{}-\frac{2514448}{2832795}\eta_{5}^{\frac{4}{23}}\eta_{4}^{-\frac{13}{25}}+\frac{1817472}{9001993}\eta_{5}^{\frac{11}{23}}\eta_{4}^{-\frac{13}{25}}+\frac{15119392}{261855}\eta_{4}^{-\frac{13}{25}}\eta_{5}^{-\frac{11}{23}}$
$\displaystyle\qquad\qquad~{}~{}+\frac{8657588}{219375}\eta_{4}^{-\frac{12}{25}}\eta_{5}^{-\frac{12}{23}}-\frac{578877793792}{5626245625}\eta_{4}^{-\frac{13}{25}}\eta_{5}^{-\frac{12}{23}}-\frac{4759984}{2008125}\eta_{4}^{-\frac{21}{25}}\eta_{5}^{-\frac{12}{23}}$
$\displaystyle\qquad\qquad~{}~{}-\frac{1460352}{10635625}\eta_{4}^{-\frac{38}{25}}\eta_{5}^{-\frac{12}{23}}+\frac{612782088}{196721875}\eta_{4}^{-\frac{13}{25}}\eta_{5}^{-\frac{19}{23}}+\frac{2379992}{223125}\eta_{4}^{-\frac{21}{25}}\eta_{5}^{-\frac{19}{23}}$
$\displaystyle\qquad\qquad~{}~{}+\frac{486784}{1115625}\eta_{4}^{-\frac{38}{25}}\eta_{5}^{-\frac{19}{23}}-\frac{3414272}{18184375}\eta_{4}^{-\frac{13}{25}}\eta_{5}^{-\frac{34}{23}}-\frac{121696}{103125}\eta_{4}^{-\frac{38}{25}}\eta_{5}^{-\frac{34}{23}}\biggr{)}$
$\displaystyle+\frac{m_{c}}{m_{e}}\bar{C}_{2ec}^{(11)}\biggl{(}\frac{48}{17}\eta_{4}^{\frac{4}{25}}\eta_{5}^{\frac{4}{23}}+\frac{32}{85}\eta_{5}^{\frac{4}{23}}\eta_{4}^{-\frac{13}{25}}-\frac{16}{5}\eta_{4}^{-\frac{13}{25}}\eta_{5}^{-\frac{11}{23}}\biggr{)}$
(35)

where $m_{c}=m_{c}(m_{c})$ in the four-flavour theory and in the
$\overline{\textrm{MS}}$ scheme.

##  4 Numerical Results

In this section, we present the numerical results based on the analytic
expressions of the previous section and obtain constraints on CP-odd Higgs
Yukawas to the bottom and charm quarks from the electron EDM. Combining the
initial conditions in section 3.1 with the RG solution in section 3.4 leads to
the electron EDM prediction, cf., Eq. (10),

$\frac{d_{e}}{e}=\frac{\sqrt{2}G_{\mathrm{F}}}{4\pi\alpha_{e}}m_{e}\times\left\\{\begin{array}[]{l}\dfrac{m_{b}(m_{b})m_{b}(\mu_{\text{ew}})}{M_{h}^{2}}\kappa_{b}\sin\phi_{b}\Big{[}\kappa^{2}F^{\text{LL}}_{b}(\eta_{5})\\\
\hskip
80.00012pt\quad\,+\tilde{\alpha}_{s}\kappa^{2}F^{\text{NLL}}_{b}(\eta_{5};\log\frac{\mu_{b}}{m_{b}},\log\frac{\mu_{\text{ew}}}{M_{h}})\Big{]}\\\\[10.00002pt]
\dfrac{m_{c}(m_{c})m_{c}(\mu_{\text{ew}})}{M_{h}^{2}}\kappa_{c}\sin\phi_{c}\Big{[}\kappa^{2}F^{\text{LL}}_{c}(\eta_{5},\eta_{4})\\\
\hskip
80.00012pt\quad\,+\tilde{\alpha}_{s}\kappa^{2}F^{\text{NLL}}_{c}(\eta_{4},\eta_{5};\log\frac{\mu_{c}}{m_{c}},\log\frac{\mu_{b}}{m_{b}},\log\frac{\mu_{\text{ew}}}{M_{h}})\Big{]}\end{array}\right\\}\\\
+{\cal O}\big{(}\tilde{\alpha}_{s}^{2}\kappa^{2},\kappa^{3}\big{)}\,.$ (36)

where the heavy quark masses at the electroweak scale are given by their NLL
running relations

$\begin{split}&\frac{m_{b}(\mu_{\text{ew}})}{m_{b}(m_{b})}=\eta_{5}^{\frac{12}{23}}\Bigg{[}1+\frac{\alpha_{s,5\text{fl}}(\mu_{b})}{4\pi}\Bigg{(}\frac{7462}{1587}\big{(}\eta_{5}-1\big{)}-4\log\frac{\mu_{b}^{2}}{m_{b}^{2}}\Bigg{)}\Bigg{]}\,,\\\\[5.0pt]
&\frac{m_{c}(\mu_{\text{ew}})}{m_{c}(m_{c})}=\eta_{4}^{\frac{12}{25}}\eta_{5}^{\frac{12}{23}}\Bigg{[}1+\frac{\alpha_{s,4\text{fl}}(\mu_{c})}{4\pi}\Bigg{(}\frac{7462}{1587}\eta_{4}\eta_{5}-\frac{213392}{330625}\eta_{4}-\frac{7606}{1875}-4\log\frac{\mu_{c}^{2}}{m_{c}^{2}}\Bigg{)}\Bigg{]}\,.\end{split}$
(37)

This result demonstrates explicitly how our RG-improved calculation removes
the ambiguity of the fixed-order computation by resumming the large
$\alpha_{s}$ logarithms and thus defining the scale at which the masses are
evaluated: in contrast to the fixed-order result in Eq. (3), $d_{e}$ is here
proportional to $m_{q_{h}}(m_{q_{h}})m_{q_{h}}(\mu_{\text{ew}})/M_{h}^{2}$ and
the functions $F_{{q_{h}}}^{\text{(N)LL}}$, which contain the $\alpha_{s}$
resummation and do not depend on the large logarithms $\log m_{q_{h}}/M_{h}$.
Expanding the LL result of Eq. (31) in $\alpha_{s}(\mu_{\text{ew}})$ we
recover exactly the $\log^{2}x_{q}$ term in Eq. (3). On the other hand, we
cannot reproduce the $\pi^{2}/3$ term in Eq. (3) since it must arise from the
higher-order terms in Eq. (31) of order $\kappa^{2}\tilde{\alpha}_{s}^{2}$,
which are counted as NNLL in the QCD resummation.

Table 1: Input parameters used in evaluating the low-scale Wilson coefficient
$\widetilde{C}_{3}^{e}(\mu_{q_{h}})$ and equivalently $d_{e}/e$. All values
are taken from Ref. [20]; running parameters are evaluated in the
$\overline{\text{MS}}$ scheme.
Parameter | Value | Parameter | Value
---|---|---|---
$\alpha_{s}(M_{Z})$ | $0.1179$ | $\alpha_{e}(M_{Z})$ | $1/127.952$
$G_{\mathrm{F}}$ |  $1.1663787\times 10^{-5}$ GeV-2 | $M_{Z}$ |  $91.1876$ GeV
$M_{h}$ |  $125.25$ GeV | $m_{e}$ |  $5.1099895$$\times 10^{-4}$ GeV
$m_{b}(m_{b})$ |  $4.18$ GeV | $m_{c}(m_{c})$ |  $1.27$ GeV

The full expressions for $F^{\text{(N)LL}}_{q_{h}}$ are readily extracted from
the results in section 3. For convenience, we give their numerical values for
the special case of $\mu_{\text{ew}}=M_{h}$, $\mu_{b}=m_{b}(m_{b})$, and
$\mu_{c}=m_{c}(m_{c})$. In this case, the logarithms in
$F^{\text{NLL}}_{q_{h}}$ vanish and the only dependence is on $\eta_{5/4}$. We
find

$\displaystyle F_{b}^{\text{LL}}$ $\displaystyle=-0.0202\,,$ $\displaystyle
F_{b}^{\text{NLL}}$ $\displaystyle=-0.0952\,,$ (38) $\displaystyle
F_{c}^{\text{LL}}$ $\displaystyle=-0.383\,,$ $\displaystyle
F_{c}^{\text{NLL}}$ $\displaystyle=-0.685\,,$ (39)

where we used the values $\eta_{5}=0.506$ and $\eta_{4}=0.605$ obtained by
solving the mixed QCD–QED RG equations at two-loop accuracy using the
numerical input in Table 1.

Next, we use the full analytic expressions to estimate the uncertainty in
predicting $d_{e}$, and provide the corresponding bounds on the anomalous CP-
odd ${q_{h}}$-Yukawa couplings. We estimate the uncertainty due to the
truncation of the perturbation series in two ways:

  * •

The dependence on the matching scales cancels in our result to the order we
calculated ($\tilde{\alpha}_{s}\kappa^{2}$). The residual scale dependence is
sensitive to higher-order terms in the perturbation series. Therefore, we
evaluate the Wilson coefficient $\widetilde{C}^{e}_{3}$ at the fixed low scale
$m_{q_{h}}(m_{q_{h}})$, and separately vary all matching scales
($\mu_{\text{ew}}$ and $\mu_{b}$ for the bottom-quark case; $\mu_{\text{ew}}$,
$\mu_{b}$, and $\mu_{c}$ for the charm-quark case). We fix all scales that are
not varied to their “central” values ($\mu_{\text{ew}}=M_{h}$ and
$\mu_{q_{h}}=m_{q_{h}}(m_{q_{h}})$). The maximal residual scale dependence
then provides our first uncertainty estimate on $\widetilde{C}^{e}_{3}$ or,
equivalently, on $d_{e}$. The ranges for the scale variations are chosen as
$\mu_{\text{ew}}\in[60~{}\text{GeV},~{}250~{}\text{GeV}]$,
$\mu_{b}=[2~{}\text{GeV},~{}8~{}\text{GeV}]$, and
$\mu_{c}=[1~{}\text{GeV},~{}2~{}\text{GeV}]$. The scale variations are shown
in Figure 4 and further discussed below. (The $\mu_{b}$ variation is not
explicitly shown for the charm-quark case, as it looks very similar to the
$\mu_{c}$ variation.)

  * •

There is a further ambiguity in our result that would only be resolved by a
NNLL calculation: we can evaluate the NLL correction to $d_{e}$, i.e., the
whole term proportional to $F^{\text{NLL}}_{q_{h}}$ in Eq. (31), using either
two- or one-loop values for all masses and couplings. We use this numerical
difference as a further way to estimate the uncertainty; this difference
effectively smears the lines obtained for the NLL scale variation, as
described above, into the red bands shown in Figure 4.

As central values for $\widetilde{C}^{e}_{3}$ or equivalently $d_{e}$ at NLL
accuracy we take the average of the maximal and minimal values obtained by all
scale variations and differences as described above. Half of that difference
is then assigned as the theoretical uncertainty associated with missing
higher-order terms. This leads to

$\dfrac{d_{e}}{e}={\kappa_{q_{h}}\sin\phi_{q_{h}}}\times\begin{cases}(3.03\pm
0.13)\cdot 10^{-29}~{}\text{cm}\qquad\text{for}\quad{q_{h}}=b\,,\\\ (1.39\pm
0.03)\cdot 10^{-29}~{}\text{cm}\qquad\text{for}\quad{q_{h}}=c\,.\end{cases}$
(40)

Figure 4: Residual scale dependence of the electric dipole moment induced by
CP-odd bottom-quark couplings (upper two panels) and CP-odd charm-quark
couplings (lower two panels). In the left two panels, the variation of the
matching scale at the bottom and charm thresholds is shown, respectively,
while the right two panels show the dependence on the electroweak matching
scale. In all plots, the dashed lines show the scale variation of the LL
result. The scale variation of the NLL results is indicated by the red bands.
The boundaries of the red bands are obtained by the evaluating the NLL scale
variation in two ways, as discussed in the main text. Finally, the two gray
dotted lines show the fixed-order results that have been used in literature so
far; they correspond to evaluating the quark masses in Eq. (3) at the
electroweak scale (the line marked “high”) and at the heavy-quark threshold
(the line marked “low”), and neglecting all other QCD corrections.

The corresponding results are further illustrated in the two upper and two
lower plots of Figure 4 for the bottom- and charm-quark cases, respectively.
The plots on the left show the scale dependence of $d_{e}/e$ on $\mu_{q_{h}}$
at LL (dashed, black line) and NLL (red band) accuracy. The plots on the right
show the corresponding dependence on $\mu_{\text{ew}}$. For the LL result we
use one-loop values for all masses and couplings, and both one- and two-loop
values for the NLL result, as discussed above. The comparison of LL and NLL
transparently shows how the NLL computation presented here drastically reduces
the large uncertainties associated to QCD corrections.

Furthermore, the gray dotted lines in Figure 4 show the naive result of the
fixed-order calculation, Eq. (3), that has been used so far in the literature.
The two lines correspond to using different values for the heavy-quark mass
$m_{q_{h}}$: the line marked “high” corresponds to using the value evaluated
at the electroweak scale ($m_{q_{h}}(M_{h})$), while line marked “low”
corresponds to using the mass evaluated at the low scale
($m_{q_{h}}(m_{q_{h}})$). The spread of these three lines illustrates the
level of ambiguity in the fixed-order result. The NLL computation of the
current work removes this ambiguity almost entirely.

The ACME collaboration has constrained the electron EDM to $|d_{e}|<1.1\times
10^{-29}$ at 90% confidence level [12]. To perform a rough combination of this
experimental constraint with our derived theory uncertainties we interpret the
measurement as a Gaussian centered at zero and the above bound as the
correspodning 90% confidence level (CL) interval, i.e., including negative
values for $d_{e}$. Adding the corresponding experimental “$1\sigma$”
uncertainty in quadrature with the theory uncertainty we find based on an one-
parameter $\chi^{2}$ function in terms of $\kappa_{q_{h}}\sin\phi_{q_{h}}$

$\displaystyle\kappa_{b}|\sin\phi_{b}|$ $\displaystyle<0.22\quad\text{[at
68.27\% CL]}\,,$ $\displaystyle\kappa_{b}|\sin\phi_{b}|<0.36\quad\text{[at
90\% CL]}\,,$ (41) $\displaystyle\kappa_{c}|\sin\phi_{c}|$
$\displaystyle<0.48\quad\text{[at 68.27\% CL]}\,,$
$\displaystyle\kappa_{c}|\sin\phi_{c}|<0.79\quad\text{[at 90\% CL]}\,.$ (42)

##  5 Conclusions

The experimental bound on the electron EDM [12] translates into strong
constraints on new CP-violating phases in various extensions of the SM (the SM
contribution to the electron EDM is estimated to lie nine orders of magnitude
below the current experimental sensitivity [21, 22]). CP-violating phases such
as those in Eq. (1) appear in several well-motivated beyond standard model
theories (see e.g. Refs. [9, 10]), and the electron EDM is capable of placing
stringent bounds on these phases [4, 7, 8, 6].

However, implicit in these electron EDM bounds is a large ${\cal O}(1)$ QCD
uncertainty that has so far been neglected, even though it leads to sizeable
ambiguities in the resulting constraints. Fortunately, since the electron EDM
is a leptonic observable, this ambiguity can be removed systematically by a
perturbative calculation, without the need for additional non-perturbative
information. In this work, we have calculated the contribution to the electron
EDM of CP-odd Higgs couplings to the bottom and charm quarks in RG-improved
perturbation theory, summing the leading and next-to-leading large logarithms
proportional to the strong coupling constant. This calculation has reduced the
residual ambiguity in the bound to the level of a few percent, as discussed in
detail in Section 4. The perturbation series shows good convergence, as
expected for a leptonic observable. If, in the future, a non-zero electron EDM
were observed, the error could be further reduced by summing the next-to-next-
to-leading QCD logarithms, as well as taking into account the QED evolution of
the electric dipole moment below the heavy-quark thresholds.

## Acknowledgments

We thank Luca Merlo for a discussion that triggered this project. Z.P.
acknowledges financial support from the European Research Council (ERC) under
the European Union’s Horizon 2020 research and innovation program under grant
agreement 833280 (FLAY), and from the Swiss National Science Foundation (SNF)
under contract 200020-204428. J.B. acknowledges support in part by DoE grant
DE-SC0011784.

##  Appendix A Unphysical Operators

For completeness, we list here the “unphysical” operators entering our
calculation in intermediate steps. They are called “unphysical” because they
vanish either via the equations of motion (e.o.m.) of the quark fields for
onshell external states, or via algebraic relations that are valid in $d=4$,
but not in $d\neq 4$.

###  A.1 E.o.m.-vanishing Operators

These operators have matrix-elements that vanish via the e.o.m. of the quark
field. The following two gauge-invariant operators enter our computation at
the two-loop level:

$\begin{split}N_{1}^{e}&=\frac{m_{q_{h}}}{2e^{2}}\bar{e}\bigl{[}\stackrel{{\scriptstyle\leftarrow}}{{\not{D}}}\stackrel{{\scriptstyle\leftarrow}}{{\not{D}}}i\gamma_{5}+i\gamma_{5}\not{D}\not{D}\bigr{]}e\,,\\\
N_{2}^{e}&=\frac{m_{q_{h}}}{2e^{2}}\bar{e}\bigl{[}\stackrel{{\scriptstyle\leftarrow}}{{\not{D}}}\stackrel{{\scriptstyle\leftarrow}}{{D^{\sigma}}}\gamma^{\mu}\gamma^{\nu}\gamma^{\rho}-\gamma^{\mu}\gamma^{\nu}\gamma^{\rho}D^{\sigma}\not{D}\bigr{]}e\,\,\epsilon_{\mu\nu\rho\sigma}\,.\end{split}$
(43)

The covariant derivative acting on electron fields is defined as

$D_{\mu}\equiv\partial_{\mu}+ieQ_{e}A_{\mu}\,,$ (44)

with $Q_{e}=-1$ the electron electrical charge.

###  A.2 Evanescent Operators

Next we list the evanescent operators that enter our computation at one- and
two-loop order. The leading-order ADM does not depend on their definition, but
the next-to-leading-order ADM does. In the $q$–$q$–$A$ sector we need the
operator

$\begin{split}{E}_{\gamma}^{e}&=\frac{Q_{e}}{4}\frac{m_{q_{h}}}{e}\bar{e}\\{\sigma^{\mu\nu},i\gamma_{5}\\}e\,F_{\mu\nu}-Q_{3}^{e}\,.\end{split}$
(45)

The evanescent operators required in the $e$–${q_{h}}$ sector read

$\begin{split}{E}_{1}^{e{q_{h}}}&=\frac{1}{2}(\bar{e}\gamma_{[\mu}\gamma_{\nu]}e)\,({\bar{q}_{h}}\\{\gamma^{[\mu}\gamma^{\nu]},i\gamma_{5}\\}{q_{h}})+O_{2}^{e{q_{h}}}\,,\\\
{E}_{1}^{{q_{h}}e}&=\frac{1}{2}({\bar{q}_{h}}\gamma_{[\mu}\gamma_{\nu]}{q_{h}})\,(\bar{e}\\{\gamma^{[\mu}\gamma^{\nu]},i\gamma_{5}\\}e)+O_{2}^{e{q_{h}}}\,,\\\
{E}_{2}^{e{q_{h}}}&=\frac{}{}\big{[}(\bar{e}\gamma_{[\mu}\gamma_{\nu]}\gamma^{[\rho}\gamma^{\sigma]}e)\,({\bar{q}_{h}}\gamma^{[\mu}\gamma^{\nu]}\gamma^{[\tau}\gamma^{\upsilon]}{q_{h}})+(\bar{e}\gamma^{[\rho}\gamma^{\sigma]}\gamma_{[\mu}\gamma_{\nu]}e)\,({\bar{q}_{h}}\gamma^{[\tau}\gamma^{\upsilon]}\gamma^{[\mu}\gamma^{\nu]}{q_{h}})\big{]}\epsilon_{\rho\sigma\tau\upsilon}\\\
&\quad-48(O_{1}^{e{q_{h}}}+O_{1}^{{q_{h}}e})+16O_{2}^{e{q_{h}}}\,,\\\
{E}_{3}^{e{q_{h}}}&=\frac{1}{2}(\bar{e}\gamma_{[\mu}\gamma_{\nu}\gamma_{\rho}\gamma_{\sigma]}e)\,({\bar{q}_{h}}\\{\gamma^{[\mu}\gamma^{\nu}\gamma^{\rho}\gamma^{\sigma]},i\gamma_{5}\\}{q_{h}})-24O_{1}^{{q_{h}}e}\,,\\\
{E}_{3}^{{q_{h}}e}&=\frac{1}{2}({\bar{q}_{h}}\gamma_{[\mu}\gamma_{\nu}\gamma_{\rho}\gamma_{\sigma]}{q_{h}})\,(\bar{e}\\{\gamma^{[\mu}\gamma^{\nu}\gamma^{\rho}\gamma^{\sigma]},i\gamma_{5}\\}e)-24O_{1}^{e{q_{h}}}\,,\\\
{E}_{4}^{e{q_{h}}}&=\frac{1}{2}(\bar{e}\gamma_{[\mu}\gamma_{\nu}\gamma_{\rho}\gamma_{\sigma}\gamma_{\tau}\gamma_{\upsilon]}e)\,({\bar{q}_{h}}\\{\gamma^{[\mu}\gamma^{\nu}\gamma^{\rho}\gamma^{\sigma}\gamma^{\tau}\gamma^{\upsilon]},i\gamma_{5}\\}{q_{h}})\,,\\\
{E}_{4}^{{q_{h}}e}&=\frac{1}{2}({\bar{q}_{h}}\gamma_{[\mu}\gamma_{\nu}\gamma_{\rho}\gamma_{\sigma}\gamma_{\tau}\gamma_{\upsilon]}{q_{h}})\,(\bar{e}\\{\gamma^{[\mu}\gamma^{\nu}\gamma^{\rho}\gamma^{\sigma}\gamma^{\tau}\gamma^{\upsilon]},i\gamma_{5}\\}e)\,,\\\
{E}_{5}^{e{q_{h}}}&=\frac{1}{2}\big{[}(\bar{e}\gamma_{[\mu}\gamma_{\nu}\gamma_{\rho}\gamma_{\sigma]}\gamma^{[\tau}\gamma^{\upsilon]}e)\,({\bar{q}_{h}}\gamma^{[\mu}\gamma^{\nu}\gamma^{\rho}\gamma^{\sigma]}\gamma^{[\zeta}\gamma^{\xi]}{q_{h}})\\\
&\qquad+(\bar{e}\gamma^{[\tau}\gamma^{\upsilon]}\gamma_{[\mu}\gamma_{\nu}\gamma_{\rho}\gamma_{\sigma]}e)\,({\bar{q}_{h}}\gamma^{[\zeta}\gamma^{\xi]}\gamma^{[\mu}\gamma^{\nu}\gamma^{\rho}\gamma^{\sigma]}{q_{h}})\big{]}\epsilon_{\tau\upsilon\zeta\xi}+48O_{2}^{e{q_{h}}}\,.\end{split}$
(46)

The square brackets denote antisymmetrisation normalised as

$\gamma_{[\mu_{1},...,\mu_{n}]}\equiv\frac{1}{n!}\sum_{\sigma}(-1)^{\sigma}\gamma_{\mu_{\sigma(1)}}\ldots\gamma_{\mu_{\sigma(n)}}\,.$

###  A.3 Operators Related to the Infrared Rearrangement

The last class of unphysical operators arises because our use of infrared
rearrangement breaks gauge invariance in intermediate steps of the
calculation. At the renormalisable level this method generates two gauge non-
invariant operators corresponding to a gluon-mass and a photon-mass term, i.e.

$\mathscr{L}\supset\frac{1}{2}Z_{\text{IRA},g}Z_{G}\,m_{\text{IRA}}^{2}G_{\mu}^{a}G^{\mu,\,a}+\frac{1}{2}Z_{\text{IRA},\gamma}Z_{A}\,m_{\text{IRA}}^{2}A_{\mu}A^{\mu}\,.$
(47)

The mass, $m_{\text{IRA}}$, is completely artificial and drops out of all
physical results, and $Z_{\text{IRA},g}$, $Z_{\text{IRA},\gamma}$ are two
additional renormalisation constants [23]. One-loop insertions of the
dimension-five and dimension-six operators can induce further gauge-invariant,
higher dimension operators that are relics of the infrared rearrangement. For
our calculation, the only relevant one is

$\begin{split}P^{e}&=m_{q_{h}}\frac{m_{\text{IRA}}^{2}}{e^{2}}\bar{e}i\gamma_{5}e\,.\end{split}$
(48)

##  Appendix B Renormalization Constants

The following are the SM counterterms necessary for the calculation:

$\displaystyle Z_{e}^{(0,1)}$
$\displaystyle=\frac{2}{3\epsilon}\Big{[}N_{c}\big{(}n_{d}Q_{d}^{2}+n_{u}Q_{u}^{2}\big{)}+n_{\ell}Q_{e}^{2}\Big{]}\,,$
(49) $\displaystyle Z_{e}^{(0,2)}$
$\displaystyle=\frac{1}{\epsilon^{2}}\bigg{\\{}\frac{2}{3}\Big{[}N_{c}\big{(}n_{d}Q_{d}^{2}+n_{u}Q_{u}^{2}\big{)}+n_{\ell}Q_{e}^{2}\Big{]}^{2}+\epsilon\Big{[}n_{\ell}Q_{e}^{4}+N_{c}\big{(}n_{d}Q_{b}^{4}+n_{u}Q_{c}^{4}\big{)}\Big{]}\bigg{\\}}\,,$
(50) $\displaystyle Z_{A}^{(0,1)}$
$\displaystyle=-\frac{4}{3\epsilon}\Big{[}N_{c}\big{(}n_{d}Q_{d}^{2}+n_{u}Q_{u}^{2}\big{)}+n_{\ell}Q_{e}^{2}\Big{]}\,,$
(51) $\displaystyle Z_{A}^{(1,1)}$
$\displaystyle=\frac{1}{\epsilon}\big{(}1-N_{c}^{2}\big{)}\big{(}n_{d}Q_{d}^{2}+n_{u}Q_{u}^{2}\big{)}\,,$
(52) $\displaystyle Z_{A}^{(0,2)}$
$\displaystyle=-\frac{2}{\epsilon}\bigg{[}n_{\ell}Q_{e}^{4}+N_{c}\big{(}n_{d}Q_{d}^{4}+n_{u}Q_{u}^{4}\big{)}\bigg{]}\,,$
(53) $\displaystyle Z_{m_{A}}^{(0,1)}$
$\displaystyle=-\frac{8}{3\epsilon}\Big{[}N_{c}\big{(}n_{d}Q_{d}^{2}+n_{u}Q_{u}^{2}\big{)}+n_{\ell}Q_{e}^{2}\Big{]}\,,$
(54) $\displaystyle Z_{{q_{h}}}^{(1,0)}$
$\displaystyle=-\frac{1}{\epsilon}C_{F}\xi_{g}\,,$ (55) $\displaystyle
Z_{f}^{(0,1)}$ $\displaystyle=-\frac{1}{\epsilon}Q_{f}^{2}\xi_{a}\,,$ (56)
$\displaystyle Z_{{q_{h}}}^{(1,1)}$
$\displaystyle=\frac{1}{\epsilon^{2}}C_{F}Q_{q_{h}}^{2}\bigg{(}\xi_{g}\xi_{a}+\frac{3\epsilon}{2}\bigg{)}\,,$
(57) $\displaystyle Z_{f}^{(0,2)}$
$\displaystyle=\frac{1}{\epsilon^{2}}Q_{f}^{2}\bigg{\\{}\frac{\xi_{a}^{2}}{2}Q_{f}^{2}+\epsilon\bigg{(}\frac{3}{4}Q_{f}^{2}+n_{\ell}Q_{e}^{2}+N_{c}\big{(}n_{d}Q_{d}^{2}+n_{u}Q_{u}^{2}\big{)}\bigg{)}\bigg{\\}}\,,$
(58) $\displaystyle Z_{m_{q_{h}}}^{(1,0)}$
$\displaystyle=-\frac{3}{\epsilon}C_{F}\,,$ (59) $\displaystyle
Z_{m_{f}}^{(0,1)}$ $\displaystyle=-\frac{3}{\epsilon}Q_{f}^{2}\,,$ (60)
$\displaystyle Z_{m_{q_{h}}}^{(1,1)}$
$\displaystyle=\frac{9}{\epsilon^{2}}C_{F}Q_{q_{h}}^{2}\bigg{(}1-\frac{\epsilon}{6}\bigg{)}\,,$
(61) $\displaystyle Z_{m_{q_{h}}}^{(2,0)}$
$\displaystyle=-\frac{1}{\epsilon^{2}}\frac{C_{F}}{2N_{c}}\bigg{[}\frac{1}{2}\bigg{(}9-31N_{c}^{2}+4N_{c}(n_{u}+n_{d})\bigg{)}-\frac{\epsilon}{12}\bigg{(}9-203N_{c}^{2}+20N_{c}(n_{u}+n_{d})\bigg{)}\bigg{]}\,,$
(62)
$\displaystyle\begin{split}Z_{m_{f}}^{(0,2)}&=\frac{1}{\epsilon^{2}}Q_{f}^{2}\bigg{\\{}\frac{9}{2}Q_{f}^{2}-2\big{(}n_{\ell}Q_{e}^{2}+N_{c}n_{d}Q_{d}^{2}+N_{c}n_{u}Q_{u}^{2}\big{)}\\\
&\hskip
60.00009pt-\epsilon\bigg{(}\frac{3}{4}Q_{f}^{2}-\frac{5}{3}\big{(}n_{\ell}Q_{e}^{2}+N_{c}n_{d}Q_{d}^{2}+N_{c}n_{u}Q_{u}^{2}\big{)}\bigg{)}\bigg{\\}}\,,\end{split}$
(63)

where we have organised the contributions according to

$Z_{i}=\sum_{m,n}\tilde{\alpha}_{s}^{m}\tilde{\alpha}_{e}^{n}Z_{i}^{(m,n)}\,.$
(64)

and $f$ denotes either a charged-lepton or a quark field.

To obtain the two-loop anomalous dimension of the physical sector we need
certain one-loop renormalisation constants involving unphysical operators. We
collect them in this appendix. We write $Z_{x\to y}$ where the subscripts $x$
and $y$ symbolize sets of Wilson coefficients, for which we use the following
notation and standard ordering:

$\begin{split}P=\\{&C_{1}^{e{q_{h}}},\,C_{1}^{{q_{h}}e},\,C_{2}^{e{q_{h}}},\,C_{3}^{e}\\}\,,\\\
E=\\{&C_{E_{1}^{e{q_{h}}}},\,C_{E_{1}^{{q_{h}}e}},\,C_{E_{2}^{e{q_{h}}}},\,C_{E_{\gamma}^{e}}\\}\,,\\\
M=\\{&C_{P^{e}}\\}\,,\\\ N=\\{&C_{N_{1}^{e}},\,C_{N_{2}^{e}}\\}\,.\end{split}$
(65)

The first necessary input is the mixing of the physical operators into all the
evanescent operators that are generated at one-loop. Using the same subscript
notation as above, the renormalisation constants read

$Z^{(0,1)}_{P\to E}=\begin{pmatrix}Q_{e}Q_{q_{h}}&0&0&0\\\
0&Q_{e}Q_{q_{h}}&0&0\\\ 0&0&-\frac{1}{2}Q_{e}Q_{q_{h}}&0\\\
0&0&0&-4Q_{e}^{2}\end{pmatrix}\,,$ (66)

and $Z^{(1,0)}_{P\to E}=0$. The remaining mixing of physical operators into
evanescent operators is zero at one-loop. Furthermore, the finite part of the
mixing of evanescent into physical operators is subtracted by finite
counterterms [24]. They read

$Z^{(0,1)}_{E\to P}=\begin{pmatrix}4Q_{e}Q_{q_{h}}&0&0&0\\\
0&4Q_{e}Q_{q_{h}}&0&0\\\ 0&0&-8Q_{e}Q_{q_{h}}&0\\\
0&0&0&-\frac{2}{3}Q_{e}^{2}\end{pmatrix}\,.$ (67)

The remaining finite mixing of evanescent into physical operators is zero at
one-loop.

Furthermore, we need the mixing constants of the physical operators into the
operators arising from infrared rearrangement; they are found to be

$Z^{(0,1)}_{P\to M}=\begin{pmatrix}0\\\ -12\\\ 0\\\
-12Q_{e}^{2}\end{pmatrix}\,.$ (68)

All other mixing constants of physical into the IRA operators are zero.

Finally, we need the mixing constants of the physical operators into the
e.o.m.-vanishing operators. They are uniquely fixed by the ${q_{h}}\to{q_{h}}$
Greens function. We find

$Z^{(0,1)}_{P\to N}=\begin{pmatrix}0&0\\\ 0&0\\\ 0&0\\\
-2Q_{e}^{2}&-\frac{1}{6}Q_{e}^{2}\end{pmatrix}\,.$ (69)

The two-loop anomalous-dimension matrix is given in terms of the one- and two-
loop renormalisation constants by

$\displaystyle\gamma^{(0,2)}$
$\displaystyle=4Z^{(0,2;1)}-2Z^{(0,1;1)}Z^{(0,1;0)}\,,$ (70)
$\displaystyle\gamma^{(1,1)}$
$\displaystyle=4Z^{(1,1;1)}-2Z^{(0,1;1)}Z^{(1,0;0)}-2Z^{(1,0;1)}Z^{(0,1;0)}\,,$
(71) $\displaystyle\gamma^{(2,0)}$
$\displaystyle=4Z^{(2,0;1)}-2Z^{(1,0;1)}Z^{(1,0;0)}\,,$ (72)

where we have further separated the renormalisation constants according to
their poles, i.e.

$Z_{i}^{(m,n)}=\sum_{r}\frac{1}{\epsilon^{r}}Z_{i}^{(m,n;r)}$ (73)

The quadratic poles of the two-loop diagrams are fixed by the poles of the
one-loop diagrams via

$\begin{split}Z^{(0,2;2)}&=\frac{1}{2}Z^{(0,1;1)}Z^{(0,1;1)}-\frac{1}{2}\beta_{e}Z^{(0,1;1)}\,,\\\
Z^{(1,1;2)}&=\frac{1}{2}Z^{(0,1;1)}Z^{(1,0;1)}+\frac{1}{2}Z^{(1,0;1)}Z^{(0,1;1)}\,,\\\
Z^{(2,0;2)}&=\frac{1}{2}Z^{(1,0;1)}Z^{(1,0;1)}-\frac{1}{2}\beta_{0}Z^{(1,0;1)}\,,\end{split}$
(74)

where $\beta_{0}=\frac{11}{3}N_{c}-\frac{2}{3}N_{\\!f}$. As a check of our
calculation, we computed these poles directly and verified that they satisfy
Eq. (74).

## References

  * [1] ATLAS collaboration, G. Aad et al., _Observation of a new particle in the search for the Standard Model Higgs boson with the ATLAS detector at the LHC_ , _Phys. Lett. B_ 716 (2012) 1–29, [1207.7214].
  * [2] CMS collaboration, S. Chatrchyan et al., _Observation of a New Boson at a Mass of 125 GeV with the CMS Experiment at the LHC_ , _Phys. Lett. B_ 716 (2012) 30–61, [1207.7235].
  * [3] 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, [2003.00099].
  * [4] J. Brod, U. Haisch and J. Zupan, _Constraints on CP-violating Higgs couplings to the third generation_ , _JHEP_ 11 (2013) 180, [1310.1385].
  * [5] D. Egana-Ugrinovic and S. Thomas, _Higgs Boson Contributions to the Electron Electric Dipole Moment_ , 1810.08631.
  * [6] J. Brod and E. Stamou, _Electric dipole moment constraints on CP-violating heavy-quark Yukawas at next-to-leading order_ , _JHEP_ 07 (2021) 080, [1810.12303].
  * [7] H. Bahl, E. Fuchs, S. Heinemeyer, J. Katzy, M. Menen, K. Peters et al., _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) 604, [2202.11753].
  * [8] 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, [2203.03736].
  * [9] F. Feruglio, _The Chiral approach to the electroweak interactions_ , _Int. J. Mod. Phys. A_ 8 (1993) 4937–4972, [hep-ph/9301281].
  * [10] G. Buchalla, O. Catà and C. Krause, _Complete Electroweak Chiral Lagrangian with a Light Higgs at NLO_ , _Nucl. Phys. B_ 880 (2014) 552–573, [1307.5017].
  * [11] S. M. Barr and A. Zee, _Electric Dipole Moment of the Electron and of the Neutron_ , _Phys. Rev. Lett._ 65 (1990) 21–24.
  * [12] ACME collaboration, V. Andreev et al., _Improved limit on the electric dipole moment of the electron_ , _Nature_ 562 (2018) 355–360.
  * [13] J. A. M. Vermaseren, _New features of FORM_ , math-ph/0010025.
  * [14] J. Brod, E. Stamou and T. Steudtner, _Massive Recursive Tensor Integrator (MaRTIn)_ , _to be published_ (2022) .
  * [15] A. I. Davydychev and J. B. Tausk, _Two loop selfenergy diagrams with different masses and the momentum expansion_ , _Nucl. Phys._ B397 (1993) 123–142.
  * [16] C. Bobeth, M. Misiak and J. Urban, _Photonic penguins at two loops and $m_{t}$ dependence of $BR[B\to X_{s}l^{+}l^{-}]$_, _Nucl. Phys._ B574 (2000) 291–330, [hep-ph/9910220].
  * [17] P. Nogueira, _Automatic Feynman graph generation_ , _J. Comput. Phys._ 105 (1993) 279–289.
  * [18] T. Huber, E. Lunghi, M. Misiak and D. Wyler, _Electromagnetic logarithms in $\bar{B}\to X_{s}l^{+}l^{-}$_, _Nucl. Phys. B_ 740 (2006) 105–137, [hep-ph/0512066].
  * [19] E. E. Jenkins, A. V. Manohar and P. Stoffer, _Low-Energy Effective Field Theory below the Electroweak Scale: Anomalous Dimensions_ , _JHEP_ 01 (2018) 084, [1711.05270].
  * [20] Particle Data Group collaboration, P. A. Zyla et al., _Review of Particle Physics_ , _PTEP_ 2020 (2020) 083C01.
  * [21] M. E. Pospelov and I. B. Khriplovich, _Electric dipole moment of the W boson and the electron in the Kobayashi-Maskawa model_ , _Sov. J. Nucl. Phys._ 53 (1991) 638–640.
  * [22] M. Pospelov and A. Ritz, _CKM benchmarks for electron electric dipole moment experiments_ , _Phys. Rev. D_ 89 (2014) 056006, [1311.5537].
  * [23] K. G. Chetyrkin, M. Misiak and M. Munz, _Beta functions and anomalous dimensions up to three loops_ , _Nucl. Phys._ B518 (1998) 473–494, [hep-ph/9711266].
  * [24] M. J. Dugan and B. Grinstein, _On the vanishing of evanescent operators_ , _Phys. Lett._ B256 (1991) 239–244.