The man who doesn't read good books has no advantage over the man who can't
read them.

Mark Twain


compat=1.0.0

UMN-TH-4123/22

FTPI-MINN-22-14

Improved indirect limits on
charm and bottom quark EDMs

Yohei Emaa, Ting Gaob, Maxim Pospelova,b

a | _William I. Fine Theoretical Physics Institute, School of Physics and Astronomy,_
---|---
| _University of Minnesota, Minneapolis, M 55455, USA_
b | _School of Physics and Astronomy, University of Minnesota, Minneapolis, MN 55455, USA_

We derive indirect limits on the charm and bottom quark electric dipole
moments (EDMs) from paramagnetic AMO and neutron EDM experiments. The charm
and bottom quark EDMs generate $CP$-odd photon-gluon operators and light quark
EDMs at the $c$\- and $b$-quark mass thresholds. These $CP$-odd operators
induce the $CP$-odd semi-leptonic operator $C_{S}$ and the neutron EDM below
the QCD scale that are probed by the paramagnetic and neutron EDM experiments,
respectively. The bound from $C_{S}$ is $|d_{c}|<1.3\times
10^{-20}\,e\,\mathrm{cm}$ for the charm quark and $|d_{b}|<7.6\times
10^{-19}\,e\,\mathrm{cm}$ for the bottom quark, with its uncertainty estimated
as 10 %. The neutron EDM provides a stronger bound, $|d_{c}|<6\times
10^{-22}\,e\,\mathrm{cm}$ and $|d_{b}|<2\times 10^{-20}\,e\,\mathrm{cm}$,
though with a larger hadronic uncertainty.

###### Contents

  1. 1 Introduction
  2. 2 $CP$-odd operators from heavy quark EDMs
    1. 2.1 $CP$-odd photon-gluon operators
    2. 2.2 Light quark EDMs
  3. 3 Paramagnetic EDM
  4. 4 Neutron EDM
    1. 4.1 Light quark EDM contribution
    2. 4.2 $CP$-odd photon-gluon operator contribution
    3. 4.3 Constraint on heavy quark EDM
  5. 5 Conclusion
  6. A Convention
  7. B Technical details
    1. B.1 Derivation of $CP$-odd photon-gluon operators
    2. B.2 Borel transformation and IR divergence

##  1 Introduction

Searches for electric dipole moments (EDMs) play a crucial role in probing
physics beyond the Standard Model (SM). Given the smallness of the SM
contribution from the Cabbibo-Kobayashi-Maskawa phase, EDMs of elementary
particles are ideal observables to look for new physics that violates $CP$
symmetry (see [1] for a review).

EDMs of elementary particles may be classified into three categories: lepton
EDMs, light quark EDMs and heavy quark EDMs, where “light” and “heavy” are in
comparison with the QCD scale. Among the lepton EDMs, the electron EDM is
tested by paramagnetic EDM experiments such as the ACME experiments [2], while
the muon and tau EDMs can be probed indirectly through paramagnetic and
diamagnetic EDM experiments [3, 4, 5]. Motivated by the muon $g-2$ anomaly [6,
7], the muon EDM has been and will be measured by the storage ring experiments
as well [8, 9, 10, 11, 12]. Hadronic EDMs can be induced by the EDMs of light
quarks, as well as by other operators such as chromo-EDMs (CEDMs) and three-
gluon $CP$-odd operators. The light quark EDMs, in particular the up and down
quark EDMs, contribute to the neutron EDM and are constrained by the neutron
EDM experiments [13, 14]. Discussion of the strange quark contribution was
typically focused on the color EDM contributions [15, 16], while no reliable
estimate of the neutron EDM induced by the strange quark exists at this point.
Among the heavy quark EDMs, the top quark EDM is arguably most deeply
investigated. In particular, due to its large Yukawa coupling, the top quark
EDM generates light fermion EDMs through diagrams with an intermediate Higgs,
which allows one to indirectly constrain the top quark EDM [17, 18]. The light
quark CEDMs directly contribute to the neutron EDM, while the heavy quark
CEDMs generate, _e.g._ , the three-gluon Weinberg operator at the threshold,
which induce the neutron EDM after the confinement [19, 20, 21, 22, 23, 24].

Direct measurements of the heavy quark EDMs are difficult due to their short
lifetimes. The current strongest direct bound is from $e^{+}e^{-}\to q\bar{q}$
at LEP and is only of the order of $10^{-17}\,e\,\mathrm{cm}$ [25]. Recently,
an LHC based experiment is proposed that aims at directly measuring charm
baryon dipole moments [26, 27, 28, 29, 30], potentially improving the direct
bounds on the heavy quark EDMs. Given this situation, in this paper, we
address indirect limits on the heavy quark EDMs. The heavy quark EDMs induce
several $CP$-odd operators below the heavy quark mass scale. These operators
generate observable signals in neutron/atomic/molecule EDM experiments, which
allows us to impose indirect limits on the heavy quark EDMs. Thus, our goal of
this paper is to understand the current status of indirect limits on the heavy
quark EDMs.

Indirect limits on the charm and bottom quark EDMs were previously considered
in [31] and the CEDM case was also analyzed in [32]. There the authors derived
limits based on the constraints on the heavy quark CEDMs. Indeed, the charm
and bottom quark EDMs well above the heavy quark mass scales, say 1 TeV,
generate the CEDM operators through the renormalization group (RG) running at
a lower energy scale. These CEDMs in turn source the three-gluon Weinberg
operator after integrating out the charm and bottom quark. The Weinberg
operator then generates the neutron EDM, and this allows one to derive limits
on the charm and bottom quark EDMs. We may phrase it a re-interpretation of
bounds on the heavy quark CEDMs at the heavy quark mass threshold, with an
assumption that there is no cancellation between the EDM and CEDM
contributions (see also discussion at the end of Sec. 3). In contrast, in this
paper, we study $CP$-odd operators generated from the EDM operators at the
heavy quark mass threshold. Therefore our consideration directly applies to
the heavy quark EDMs at the quark mass threshold and is independent of [31].
Even though the $CP$-odd operators induced by the heavy quark EDMs are
formally higher dimensional, suppressions from the charm and bottom quark
masses are not severe as there is only a little hierarchy between the quark
masses and the QCD scale.

In order to make our limits robust, we derive indirect limits on charm and
bottom quark EDMs based on multiple observables, paramagnetic and neutron
EDMs. Indirect limits always have a potential of having a cancellation among
different operators. For instance, paramagnetic EDM experiments are sensitive
to only a particular linear combination of the electron EDM $d_{e}$ and the
$CP$-odd semi-leptonic operator $C_{S}$ (see Sec. 3). Therefore a new physics
contribution to $d_{e}$ and $C_{S}$ can be such that the linear contribution
almost vanishes even though each term is sizable. Deriving constraints from
two completely different observables, paramagnetic and neutron EDMs, makes the
probability of having such a cancellation less likely. We also minimize the
QCD uncertainty as much as possible. For this purpose the paramagnetic EDM
plays an essential role as it is sensitive to the heavy quark EDM through the
semi-leptonic operator $C_{S}$. The estimation of $C_{S}$ is not polluted by
hadronic uncertainties, and its uncertainty is theoretically well under
control.

The structure of this paper is as follows. In Sec. 2, we derive $CP$-odd
operators generated by a heavy quark EDM after integrating out the heavy
quark. In particular, we see that the heavy quark EDM generates $CP$-odd
photon-gluon operators and light quark EDMs. We restrict ourselves above the
QCD scale in this section, and thus the operators are written in terms of
light quarks and gluons. In Sec. 3, we calculate the semi-leptonic $CP$-odd
operator $C_{S}$ induced from the $CP$-odd photon-gluon operator below the QCD
scale. This allows us to derive a clean constraint on the heavy quark EDMs
from the paramagnetic EDM experiments. Sec. 4 is devoted to an estimation of
the neutron EDM induced by the $CP$-odd photon-gluon operators and the light
quark EDMs. There we derive limits on the heavy quark EDMs from the neutron
EDM experiments, which are stronger than the paramagnetic EDM experiments but
have a larger hadronic uncertainties. The conventions of this paper is
summarized in App. A, while some technical details of our computation can be
found in App. B.

##  2 $CP$-odd operators from heavy quark EDMs

In this section, we compute $CP$-odd operators induced by integrating out
charm and bottom quarks with EDMs. The Lagrangian is given by

$\displaystyle\mathcal{L}=\bar{Q}\left[i\not{D}-m_{Q}-\frac{id_{Q}}{2}\sigma_{\mu\nu}\gamma_{5}F^{\mu\nu}\right]Q-\frac{1}{4}F^{\mu\nu}F_{\mu\nu}-\frac{1}{4}G^{a\mu\nu}G_{\mu\nu}^{a},$
(2.1)

where $Q$ is the heavy quark with its mass $m_{Q}$ and its EDM $d_{Q}$. The
covariant derivative is defined by

$\displaystyle iD_{\mu}$
$\displaystyle=i\partial_{\mu}+g_{s}G_{\mu}-eQ_{Q}A_{\mu},\quad
G_{\mu}=T^{a}G_{\mu}^{a},$ (2.2)

where $G_{\mu}^{a}$ is the gluon field with its field strength
$G_{\mu\nu}^{a}$ and coupling $g_{s}$, and $A_{\mu}$ is the photon field with
its field strength $F_{\mu\nu}$ and coupling $e$. The SU(3) generator is
denoted by $T^{a}$ while the U(1) charge by $Q_{Q}$, with $Q_{c}=Q_{t}=+2/3$
and $Q_{b}=-1/3$. See App. A for more details on our conventions. Throughout
this paper, “heavy” means that the mass is larger than the QCD scale. Although
our discussion applies equally to the top quark, it has another contribution
that puts a stronger constraint on the EDM [17, 18]. Therefore our main focus
is on the charm and bottom quarks, _i.e._ , $Q=c,b$.

Once we integrate out the heavy quark, $d_{Q}$ generates several $CP$-odd
operators. For our purpose, there are two types of operators that are
important: $CP$-odd photon-gluon operators and light quark EDMs. Below the QCD
scale, the former generates the neutron EDM and the semi-leptonic $CP$-odd
interaction $C_{S}$ that induces paramagnetic EDMs, while the latter
contributes to the neutron EDM. In this section, we keep ourselves above the
QCD scale. The operators are then written in terms of the quarks and gluons.
We will deal with nonperturbative effects at the QCD scale in the subsequent
sections.

###  2.1 $CP$-odd photon-gluon operators

Figure 1: The effective action to linear order in $d_{Q}$ after integrating
out the heavy quark. The thick line on the left hand side indicates the full
heavy quark propagator with the covariant derivative $D_{\mu}$, and the cross
dot indicates the EDM operator insertion. The full propagator is expanded with
respect to the field strengths, which results in the $CP$-odd photon-gluon
operators as shown on the right hand side.

The heavy quark EDM induces $CP$-odd photon-gluon operators through the
diagrams in Fig. 1. The effective action after integrating out the heavy
quark, to linear order in $d_{Q}$, is given by

$\displaystyle
S_{\mathrm{eff}}=-\frac{id_{Q}}{2}\mathrm{Tr}\left[\frac{1}{i\not{D}-m_{Q}}\sigma_{\mu\nu}\tilde{F}^{\mu\nu}\right],$
(2.3)

where the trace is taken over the spinor, the color and the spacetime. We
further expand this expression with respect to the field strength
$H_{\mu\nu}$, given by

$\displaystyle H_{\mu\nu}=g_{s}G_{\mu\nu}-eQ_{Q}F_{\mu\nu},$ (2.4)

which satisfies $[iD_{\mu},iD_{\nu}]=iH_{\mu\nu}$. To the lowest order, we
obtain (see App. B.1 for derivation)

$\displaystyle\mathcal{L}_{\mathrm{eff}}=\frac{d_{Q}}{48\pi^{2}m_{Q}^{3}}\mathrm{tr}_{c}\left[-H_{\mu\nu}H^{\mu\nu}H_{\rho\sigma}\tilde{F}^{\rho\sigma}+2{H^{\mu}}_{\nu}{H^{\nu}}_{\rho}{H^{\rho}}_{\sigma}{\tilde{F}^{\sigma}}_{\mu}\right],$
(2.5)

where $\mathrm{tr}_{c}$ is the trace only over the color. This contains
$CP$-odd photon-gluon operators as well as a $CP$-odd pure photon operator.
The operator quadratic in the photon field is given by

$\displaystyle\mathcal{L}_{G^{2}F\tilde{F}}$
$\displaystyle=\frac{eQ_{Q}g_{s}^{2}d_{Q}}{24\pi^{2}m_{Q}^{3}}\mathrm{tr}_{c}\left[F_{\mu\nu}G^{\mu\nu}\tilde{F}_{\rho\sigma}G^{\rho\sigma}-{\tilde{F}^{\mu}}_{\nu}{G^{\nu}}_{\rho}{F^{\rho}}_{\sigma}{G^{\sigma}}_{\mu}\right],$
(2.6)

while the one linear in the photon field is given by

$\displaystyle\mathcal{L}_{G^{3}\tilde{F}}$
$\displaystyle=\frac{g_{s}^{3}d_{Q}}{48\pi^{2}m_{Q}^{3}}\mathrm{tr}_{c}\left[-G_{\mu\nu}G^{\mu\nu}G_{\rho\sigma}\tilde{F}^{\rho\sigma}+2{G^{\mu}}_{\nu}{G^{\nu}}_{\rho}{G^{\rho}}_{\sigma}{\tilde{F}^{\sigma}}_{\mu}\right].$
(2.7)

Carrying out color traces explicitly, one finds that (2.6) contains
$\delta^{ab}$, while (2.7) has $d^{abc}$ structure. In that sense, (2.6) would
exist for any choice of the gauge group for $G_{\mu\nu}$ including a
$\mathrm{U}(1)$, while (2.7) requires $N\geq 3$ for $\text{SU}(N)$, which
includes of course the color group of the Standard Model.

As we will see, below the QCD scale, operator (2.6) contributes to
paramagnetic EDMs through the semi-leptonic $CP$-odd operator $C_{S}$ while
(2.7) contributes to the neutron EDM. The implication of the $CP$-odd pure
photon operator is studied in detail in [5] in the context of the muon EDM.
This operator puts only a subdominant constraint on the heavy quark EDM, and
thus we do not discuss it any further in this paper.

###  2.2 Light quark EDMs

Figure 2: An example of the three-loop diagrams that generate the light quark
EDMs. The cross dot indicates the heavy quark EDM $d_{Q}$ insertion, the wavy
line is the external photon, the closed solid line is the heavy quark and the
upper solid line is the light quark, respectively. There are five additional
diagrams that are permutations of the gluon lines attached to the light quark
line.

The heavy quark EDM induces the up and down quark EDMs at the three-loop level
through the diagram in Fig. 2 (and its permutations), where the closed loop is
the heavy quark and the cross dot indicates the heavy quark EDM insertion. In
our case the heavy quark EDM operator already contains a derivative acting on
the external photon, and thus we can ignore the momentum flow of the photon.
The mass and the external momentum of the light quark are small compared to
the heavy quark mass scale, and this allows us to expand the diagram with
respect to the light quark mass and momentum. To linear order, the amplitude
is schematically given by

$\displaystyle i\mathcal{M}_{3\mathchar
45\relax\mathrm{loop}}=i\tilde{F}^{\mu\nu}p^{\rho}\bar{q}S_{\mu\nu\rho}q,$
(2.8)

where $q=u,d$ is the light quark field and $p$ is the light quark momentum.
Here we used the equation of motion to express quantities that explicitly
depend on the light quark mass $m_{q}$ in terms of $p$. After projecting
$S_{\mu\nu\rho}$ onto the EDM part, we obtain the light quark EDM $d_{q}$ as

$\displaystyle
d_{q}=\frac{m_{q}}{96}\mathrm{tr}\left[S_{\mu\nu\rho}\left\\{\sigma^{\mu\nu},\gamma^{\rho}\right\\}\right],$
(2.9)

where the trace is taken only over the spinor index, and $\\{..,..\\}$
represents anticommutator. We evaluate this expression with the dimensional
regularization. Since we expanded the amplitude with respect to the external
momentum and the light quark mass, $S_{\mu\nu\rho}$ depends only on the heavy
quark mass and hence is given by the vacuum integrals. In our case, among six
propagators, three are massive and three are massless. In this case, we can
reduce the vacuum integrals to two irreducible master integrals [33]. We use
FIRE6 [34] to reduce the vacuum integrals by integration by parts, and use
analytical expressions of the master integrals in [35]. We then obtain

$\displaystyle d_{q}$
$\displaystyle=\frac{5\left(8\zeta(3)-7\right)}{72}\times\left(\frac{\alpha_{s}}{\pi}\right)^{3}\frac{m_{q}}{m_{Q}}d_{Q},$
(2.10)

where $\alpha_{s}=g_{s}^{2}/4\pi$ and $\zeta(3)\simeq 1.202$ with $\zeta(z)$
the Riemann zeta function. We have checked that all the gauge-dependent part
of the gluon propagator cancels in our computation. Moreover, although each
master integral is divergent, all the divergences cancel and the final result
is finite. If we drop the color factors, this is equivalent to one
contribution to the electron EDM from the muon/tau EDM, the diagram (a) in
[3]. Our result differs from [3] and we will come back to the origin of this
difference in a separate publication [36]. Notice that $d_{q}$ is suppressed
only by $1/m_{Q}$ since the loop integral is dominated by the loop momentum of
order $m_{Q}$, while the $CP$-odd photon-gluon operators that we computed in
Sec. 2.1 are suppressed by $1/m_{Q}^{3}$. Therefore the light quark EDM is
expected to be more important for heavier quarks, although there is another
(and a dominant) contribution in the case of the top quark as we mentioned at
the beginning of this section.

##  3 Paramagnetic EDM

In this section we study a constraint on the heavy quark EDM from paramagnetic
atomic/molecule EDM experiments, in particular the ACME experiment [2]. The
paramagnetic EDM experiments are sensitive only to a linear combination of two
operators, the electron EDM $d_{e}$ and the semi-leptonic $CP$-odd operator
$C_{S}$ given by

$\displaystyle\mathcal{L}=C_{S}\frac{G_{F}}{\sqrt{2}}\bar{e}i\gamma_{5}e\left(\bar{p}p+\bar{n}n\right),$
(3.1)

where $G_{F}$ is the Fermi constant, $e$ is the electron (one should not
confuse it with the electromagnetic coupling), $p$ is the proton and $n$ is
the neutron, respectively. It is convenient to define an _equivalent_ electron
EDM as [37]

$\displaystyle d_{e}^{(\mathrm{equiv})}=d_{e}+C_{S}\times 1.5\times
10^{-20}\,e\,\mathrm{cm},$ (3.2)

where we focus on ThO molecule. The current bound on the equivalent electron
EDM is given by $|d_{e}^{(\mathrm{equiv})}|<1.1\times
10^{-29}\,e\,\mathrm{cm}$ [2].

Figure 3: The diagram that generates the $CP$-odd semi-leptonic operator
$C_{S}$. The photons are attached to the electron line and generate the
structure $\bar{e}i\gamma_{5}e$, while the gluons feed into the nucleon $N$.

We now compute $C_{S}$ induced by the $CP$-odd photon-gluon operator, in
particular the two-photon two-gluon operator, below the QCD scale. The
effective Lagrangian is given by

$\displaystyle\mathcal{L}_{G^{2}F\tilde{F}}=\frac{eQ_{Q}g_{s}^{2}d_{Q}}{48\pi^{2}m_{Q}^{3}}G_{\mu\nu}^{a}G^{a}_{\alpha\beta}\left(F^{\mu\nu}\tilde{F}^{\alpha\beta}-F^{\nu\alpha}\tilde{F}^{\beta\mu}\right).$
(3.3)

Because our starting point here is explicitly isospin symmetric, it will
result in the same $C_{S}$ coupling for neutrons and protons. The nucleon
matrix element of the gluon part is given by

$\displaystyle\langle
N\rvert\frac{g_{s}^{2}}{32\pi^{2}}G^{a}_{\mu\nu}G^{a}_{\alpha\beta}\lvert
N\rangle=-\frac{m_{N}}{108}\left(\eta_{\alpha\mu}\eta_{\beta\nu}-\eta_{\alpha\nu}\eta_{\beta\mu}\right)\bar{N}N+\cdots,$
(3.4)

where $N$ is the nucleon field (either $p$ or $n$) with $m_{N}$ its mass, and
$\cdots$ denotes the traceless tensor part that is irrelevant for our purpose.
Here we used that, in the chiral limit, the one-loop trace anomaly dominantly
contributes to the nucleon mass,

$\displaystyle\langle
N\rvert\frac{g_{s}^{2}}{32\pi^{2}}G_{\mu\nu}^{a}G^{a\mu\nu}\lvert
N\rangle=-\frac{m_{N}}{9}\bar{N}N,$ (3.5)

where the coefficient in the right-hand-side is related to the beta function
[38]. The photons are attached to the electron and induce the operator
$\bar{e}i\gamma_{5}e$ at one-loop [39] as shown in Fig. 3. This integral is
logarithmically divergent, which is regulated by the heavy quark mass scale.
Therefore, to the leading logarithmic accuracy, we obtain

$\displaystyle
C_{S}\frac{G_{F}}{\sqrt{2}}=-\frac{4Q_{Q}\alpha^{2}}{27}\frac{m_{N}m_{e}}{m_{Q}^{3}}\log\left(\frac{m_{Q}}{m_{e}}\right)\frac{d_{Q}}{e},$
(3.6)

where $\alpha=e^{2}/4\pi$ and $m_{e}$ is the electron mass. By requiring
$|{d_{e}^{(\mathrm{equiv})}}|<1.1\times 10^{-29}\,e\,\mathrm{cm}$, we obtain

$\displaystyle\left|{d_{c}}\right|<1.3\times 10^{-20}\,e\,\mathrm{cm},$ (3.7)

for the charm quark and

$\displaystyle\left|{d_{b}}\right|<7.6\times 10^{-19}\,e\,\mathrm{cm},$ (3.8)

for the bottom quark, respectively. The heavy quark EDM also induces the
electron EDM at three-loop [3, 40], but its contribution to
$d_{e}^{(\mathrm{equiv})}$ is negligible compared to $C_{S}$.

We now estimate the precision of our calculation. There is an uncertainty
associated with Eq. (3.5), which is valid only in the chiral limit. However,
the quark contribution to the nucleon mass is less than 10 % [41], and hence
the uncertainty of Eq. (3.5) is also less than 10 %. Thanks to the lattice
computations, these contributions are rather precisely known and the
uncertainty in the non-perturbative matrix element can be further reduced by
including the light quark contributions into our computation. Another
uncertainty originates from our photon-loop computation, where we include only
the leading logarithmic term. The uncertainty associated with this treatment
may be estimated by changing the cut-off scale from $m_{Q}$ to $2m_{Q}$, which
results in $\lesssim 10\,\%$ change of the result for both the charm and
bottom quarks. This can be further improved by performing the full two-loop
computation, with the heavy quark and photon loops at the same time.
Therefore, the precision of our $C_{S}$ calculation is estimated to be 10 %
and this can be further improved by properly including the quark contribution
to the nucleon mass and performing the full two-loop computation of the
photon-loop.

Our constraint is stronger than the one derived from $e^{+}e^{-}\to q\bar{q}$
at LEP by several orders of magnitude [25]. Although our constraint is weaker
than [31] at its face value, there are two caveats. First, as we mentioned in
the introduciton, the constraint in [31] is a re-interpretation of the bounds
on the CEDMs $\tilde{d}_{Q}$ at the heavy quark mass scale, _i.e._ ,
$\tilde{d}_{Q}(m_{Q})$. Both the quark EDM and CEDM at high energy scale
$\Lambda_{\mathrm{NP}}$ contribute to the CEDM at lower energy due to the RG
running, and hence $\tilde{d}_{Q}(m_{Q})$ is given as a linear combination of
$d_{Q}(\Lambda_{\mathrm{NP}})$ and $\tilde{d}_{Q}(\Lambda_{\mathrm{NP}})$.
With only this information, one can put a constraint on only this particular
linear combination of $d_{Q}(\Lambda_{\mathrm{NP}})$ and
$\tilde{d}_{Q}(\Lambda_{\mathrm{NP}})$. Therefore the authors assume that
there is no huge cancellation between $d_{Q}(\Lambda_{\mathrm{NP}})$ and
$\tilde{d}_{Q}(\Lambda_{\mathrm{NP}})$, which allow them to derive a
constraint solely on $d_{Q}(\Lambda_{\mathrm{NP}})$ and thus on
$d_{Q}(m_{Q})$. Since our constraint directly applies to $d_{Q}(m_{Q})$, it is
an independent constraint, without any assumptions on the relative size of
$d_{Q}(\Lambda_{\mathrm{NP}})$ and $\tilde{d}_{Q}(\Lambda_{\mathrm{NP}})$.
Second, it is a complicated task to estimate the size of the neutron EDM
induced by the Weinberg operator. Since [31] relies on the constraints on
$\tilde{d}_{Q}(m_{Q})$ from the Weinberg operator, it has a large hadronic
uncertainty. In contrast, as we have just emphasized, our computation of
$C_{S}$ is precise and its uncertainty is well under control. Given the rapid
progress of the paramagnetic EDM experiments, $C_{S}$ will continue providing
important and clean constraints on the heavy quark EDMs. In the next section,
we will see that the neutron EDM provides a stronger constraint than $C_{S}$,
comparable to [31], though with a larger hadronic uncertainty.

##  4 Neutron EDM

In Sec. 2 we have seen that the heavy quark EDM generates the $CP$-odd photon-
gluon operator and the light quark EDM. Below the QCD scale, these $CP$-odd
operators in turn generate the neutron EDM $d_{n}$, whose experimental upper
bound is [14]\natural1\natural1$\natural$1 The 199Hg EDM experiment puts a
similar constraint on the neutron EDM [4].

$\displaystyle\left|{d_{n}}\right|<1.8\times 10^{-26}\,e\,\mathrm{cm}.$ (4.1)

In this section we translate this upper bound to the bound on the heavy quark
EDM by estimating the size of the induced neutron EDM, with effects of the
nonperturbative QCD taken into account (to the extent it is possible).

###  4.1 Light quark EDM contribution

We first study the neutron EDM induced by the light quark EDM
$d_{n}^{(d_{q})}$. Both the QCD sum rule and the quark model suggest that [42,
1, 43]

$\displaystyle d_{n}^{(d_{q})}$
$\displaystyle=\frac{1}{3}\left(4d_{d}-d_{u}\right).$ (4.2)

QCD sum rule calculations [42, 43] and more recently lattice calculations [44]
give support to this simple formula within $\sim 30\%$ accuracy. We thus
obtain

$\displaystyle d_{n}^{(d_{q})}$
$\displaystyle=\frac{5(8\zeta(3)-7)}{72}\left(\frac{\alpha_{s}}{\pi}\right)^{3}\frac{4m_{d}-m_{u}}{3m_{Q}}d_{Q}.$
(4.3)

As we will see, there is another contribution to the neutron EDM from the
$CP$-odd photon-gluon operator that leads to the QCD condensate power
corrections in the light quark propagator. The rest of this section is devoted
to estimate the size of this contribution.

###  4.2 $CP$-odd photon-gluon operator contribution

In Sec. 2.1 we saw that the heavy quark EDM generates the $CP$-odd photon-
gluon operator

$\displaystyle\mathcal{L}_{G^{3}\tilde{F}}$
$\displaystyle=\frac{g_{s}^{3}d_{Q}}{48\pi^{2}m_{Q}^{3}}\mathrm{tr}_{c}\left[-G_{\mu\nu}G^{\mu\nu}G_{\rho\sigma}\tilde{F}^{\rho\sigma}+2{G^{\mu}}_{\nu}{G^{\nu}}_{\rho}{G^{\rho}}_{\sigma}{\tilde{F}^{\sigma}}_{\mu}\right].$
(4.4)

Below the QCD scale the gluons confine and condense, and this operator feeds
into the neutron EDM. We use the QCD sum rule technique [45] to estimate the
size of the neutron EDM induced by this operator.

The starting point of the QCD sum rule is to define the two-point correlator

$\displaystyle\Pi(p)$ $\displaystyle\equiv i\int d^{4}xe^{ip\cdot x}\langle
0|\mathcal{T}\left\\{\eta(x),\bar{\eta}(0)\right\\}|0\rangle.$ (4.5)

The interpolating function $\eta$ is typically chosen as

$\displaystyle\eta(x)=j_{1}(x)+\beta j_{2}(x),$ (4.6)

and has an overlap with the neutron one-particle state. The functions $j_{1}$
and $j_{2}$ have the same quantum numbers as the neutron and are given by

$\displaystyle j_{1}$
$\displaystyle=2\epsilon_{ijk}\left(d_{i}^{T}\mathcal{C}\gamma_{5}u_{j}\right)d_{k},\quad
j_{2}=2\epsilon_{ijk}\left(d_{i}^{T}\mathcal{C}u_{j}\right)\gamma_{5}d_{k},$
(4.7)

where $i,j,k$ are the color indices with $\epsilon_{ijk}$ the totally anti-
symmetric tensor, and $\mathcal{C}$ is the charge conjugate matrix (see App. A
for its definition). Finally $\beta$ is a redundant parameter that should
disappear in an exact computation. In practice, one can choose $\beta$, _e.g._
, to optimize the convergence of the operator production expansion (OPE). The
QCD sum rule relies on the quark-hadron duality and evaluate this two point
correlator $\Pi(p)$ in the hadronic (phenomenological) side and the quark
(OPE) side. On the phenomenological side, $\Pi(p)$ is expressed in terms of
hadronic quantities such as the neutron mass and dipole moments. On the OPE
side, $\Pi(p)$ is evaluated in terms of perturbative quarks and gluons with
non-perturbative effects included in the form of the QCD condensates. One then
obtains an estimation of the hadronic quantities by equating these two
expressions, after the Borel transformation to reduce effects of excited
states. In our case, we have external electromagnetic field $F_{\mu\nu}$
multiplying three Lorentz structures that in principle we can use to derive
the sum rules: $\not{p}\sigma_{\mu\nu}\not{p}$,
$\\{\not{p},\sigma_{\mu\nu}\\}$ and $\sigma_{\mu\nu}$. In this paper, we focus
on the sum rule that follows from $\not{p}\sigma_{\mu\nu}\not{p}$ since it
depends most strongly on the momentum and the vacuum susceptibilities are less
relevant [46]. Moreover, it is computationally simple to derive the QCD sum
rule based on this structure. In the following we evaluate $\Pi(p)$ in the OPE
and phenomenological sides, respectively.

#### OPE side.

We denote the light quark propagator as

$\displaystyle\left\langle
0|\mathrm{T}\left\\{q_{i}(x),\bar{q}_{j}(y)\right\\}|0\right\rangle\equiv\delta_{ij}S(x,y),$
(4.8)

where only the color-diagonal contribution is relevant for the QCD sum rule
based on $\not{p}\sigma_{\mu\nu}\not{p}$ in our case. The $CP$-odd photon-
gluon operator does not distinguish the up and down quarks, and thus we
collectively denote $u$ and $d$ as $q$. With this expression, we obtain

$\displaystyle\Pi_{\mathrm{OPE}}(p)=-24i\int d^{4}xe^{ip\cdot x}$
$\displaystyle\left\\{\mathrm{tr}\left[\gamma_{5}S^{c}\gamma_{5}S\right]S+S\gamma_{5}S^{c}\gamma_{5}S+\beta^{2}\left(\gamma_{5}SS^{c}S\gamma_{5}+\mathrm{tr}\left[S^{c}S\right]\gamma_{5}S\gamma_{5}\right)\right.$
$\displaystyle\left.+\beta\left(SS^{c}\gamma_{5}S\gamma_{5}+\gamma_{5}S\gamma_{5}S^{c}S+\mathrm{tr}\left[SS^{c}\gamma_{5}\right]S\gamma_{5}+\mathrm{tr}\left[S^{c}S\gamma_{5}\right]\gamma_{5}S\right)\right\\},$
(4.9)

where
$S^{c}_{\alpha\beta}\equiv\left(\mathcal{C}S^{T}\mathcal{C}\right)_{\alpha\beta}$
and $S^{T}_{\alpha\beta}\equiv S_{\beta\alpha}$ with $\alpha$ and $\beta$
being the spinor indices, and the trace is taken only over the spinor index.
In our case the light quark propagator has two contributions

$\displaystyle S$ $\displaystyle=S^{(0)}+S^{(\cancel{{CP}})},$ (4.10)

where $S^{(0)}$ corresponds to the free quark propagator and is given by

$\displaystyle S^{(0)}(x)$ $\displaystyle=\frac{i\not{x}}{2\pi^{2}x^{4}},$
(4.11)

while $S^{(\cancel{{CP}})}$ is the $CP$-odd part which we compute from now.

The source inserted into the correlator has three gluon fields, and we follow
the general idea of [23] where the Weinberg operator contribution to $d_{n}$
was first evaluated, and where two gluons are treated perturbatively, while
the third gluon field contributes to the quark-gluon condensate. With the QCD
condensate background, we can write down the diagram

$\displaystyle S_{ij}^{(\cancel{{CP}})}(p)=\leavevmode\hbox to74.38pt{\vbox
to20.32pt{\pgfpicture\makeatletter\hbox{\hskip
37.18866pt\lower-10.15863pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{
}\nullfont\pgfsys@beginscope\pgfsys@invoke{ }\hbox to0.0pt{ {}{{}}{}{{{}}
{}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{
}\definecolor[named]{pgfstrokecolor}{rgb}{.5,.5,.5}\pgfsys@color@gray@stroke{.5}\pgfsys@invoke{
}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@moveto{36.98866pt}{0.0pt}\pgfsys@curveto{36.98866pt}{5.50006pt}{20.42853pt}{9.95863pt}{0.0pt}{9.95863pt}\pgfsys@curveto{-20.42853pt}{9.95863pt}{-36.98866pt}{5.50006pt}{-36.98866pt}{0.0pt}\pgfsys@curveto{-36.98866pt}{-5.50006pt}{-20.42853pt}{-9.95863pt}{0.0pt}{-9.95863pt}\pgfsys@curveto{20.42853pt}{-9.95863pt}{36.98866pt}{-5.50006pt}{36.98866pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\diagram*{ (a) --
[fermion, momentum=$\scriptstyle p$] (b), (b) -- [fermion] (d), (b) --
[insertion={1.0}] (d), (b) -- [gluon, momentum=$\scriptstyle p$] (c), (c) --
[gluon, insertion={1.0}] (e), (c) -- [photon, insertion={1.0}] (f), (c) --
[gluon, momentum=$\scriptstyle p$] (h), (i) -- [fermion] (h) -- [fermion,
momentum=$\scriptstyle p$] (j), (i) -- [fermion, insertion={0.0}] (h), };
\hss}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ } \feynman[inline = (base.a),
horizontal=a to c] \vertex[label=270:${\scriptstyle j}$] (a); \vertex[right =
of a] (b); \vertex[right = of b] (c); \vertex[below = of b] (d); \vertex[below
=of c] (e); \vertex[above =of c] (f);
{{}}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{1,1,1}\pgfsys@color@gray@fill{1}\pgfsys@invoke{
}{{}{}{{{}{}}}{{}{}} {{}{{}}}{{}{}}{}{{}{}} {\pgfsys@beginscope\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{1,1,1}\pgfsys@color@gray@fill{1}\pgfsys@invoke{
}\pgfsys@rect{-4.02074pt}{-3.33301pt}{6.66602pt}{6.66602pt}\pgfsys@fill\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-0.68773pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} \vertex[right =of c]
(h); \vertex[below =of h] (i); \vertex[label=270:${\scriptstyle i}$, right =of
h] (j); \pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}},$ (4.12)

where the crosses indicate the background fields, either the external photon
or the QCD condensation of the quarks and gluons, and the cross dot is the
insertion of the $CP$-odd photon-gluon operator (4.4). Since the QCD vacuum
does not violate the Lorentz and color symmetries, we have

$\displaystyle\left\langle q_{i}\bar{q}_{j}G^{a}_{\mu\nu}\right\rangle$
$\displaystyle=-\frac{1}{192}T^{a}_{ij}\sigma_{\mu\nu}\left\langle\bar{q}\sigma\cdot
Gq\right\rangle,$ (4.13)

where $\sigma\cdot G=\sigma_{\mu\nu}G^{\mu\nu}$, and $\langle\cdots\rangle$
corresponds to the vacuum expectation value. We thus obtain

$\displaystyle S^{(\cancel{{CP}})}_{ij}(p)$
$\displaystyle=\delta_{ij}S^{(\cancel{{CP}})}(p)=i\delta_{ij}\frac{5\alpha_{s}^{2}d_{Q}}{864m_{Q}^{3}}\frac{g_{s}\langle\bar{q}\sigma\cdot
Gq\rangle}{p^{2}}\frac{i}{\not{p}}\left[\sigma\cdot\tilde{F}-\frac{2p^{\mu}p^{\nu}}{p^{2}}\tilde{F}_{\mu\alpha}{\sigma_{\nu}}^{\alpha}\right]\frac{i}{\not{p}}.$
(4.14)

Its Fourier transformation has an IR divergence, which in the dimensional
regularization leads

$\displaystyle S^{(\cancel{{CP}})}(x)$
$\displaystyle=\frac{5\alpha_{s}^{2}d_{Q}}{27648\pi^{2}m_{Q}^{3}}g_{s}\langle\bar{q}\sigma\cdot
Gq\rangle\Gamma(\epsilon_{\mathrm{IR}})\left(-\Lambda^{2}_{\mathrm{IR}}x^{2}\right)^{-\epsilon_{\mathrm{IR}}}\tilde{F}\cdot\sigma,$
(4.15)

where we take $d_{\mathrm{IR}}=4+2\epsilon_{\mathrm{IR}}$ and keep only the
logarithmic terms, with $\Lambda_{\mathrm{IR}}$ the IR cut-off scale. Due to
the sensitivity to IR scale, calculations of further terms in the OPE are not
possible. As a consequence, this evaluation should be viewed as an estimate,
which cannot be systematically improved within QCD sum rule method. With the
explicit forms of $S^{(0)}$ and $S^{(\cancel{{CP}})}$, we obtain

$\displaystyle\Pi_{\mathrm{OPE}}(p)=-\frac{5\alpha_{s}^{2}d_{Q}}{2^{12}3^{3}\pi^{4}m_{Q}^{3}}\left\langle\bar{q}\sigma\cdot
g_{s}Gq\right\rangle\left[\left(1-\beta\right)^{2}\not{p}\tilde{F}\cdot\sigma\not{p}-18\left(1-\beta^{2}\right)p^{2}\tilde{F}\cdot\sigma\right]I(p^{2};\epsilon_{\mathrm{IR}},\epsilon_{\mathrm{UV}}),$
(4.16)

where we perform the dimensional regularization to tame the UV divergence with
$d_{\mathrm{UV}}=4-2\epsilon_{\mathrm{UV}}$, and

$\displaystyle I(p^{2};\epsilon_{\mathrm{IR}},\epsilon_{\mathrm{UV}})$
$\displaystyle\equiv\Gamma(\epsilon_{\mathrm{IR}})\left(-\frac{\Lambda_{\mathrm{IR}}^{2}}{p^{2}}\right)^{-\epsilon_{\mathrm{IR}}}\Gamma(-\epsilon_{\mathrm{UV}}-\epsilon_{\mathrm{IR}})\left(-\frac{\mu^{2}}{p^{2}}\right)^{-\epsilon_{\mathrm{UV}}},$
(4.17)

with $\mu$ the renormalization scale. We then perform the Borel
transformation, defined as [47]

$\displaystyle\mathcal{B}\left[\Pi(p^{2}=-P^{2})\right]$
$\displaystyle=\frac{1}{\pi}\int_{0}^{\infty}\frac{dP^{2}}{M^{2}}e^{-P^{2}/M^{2}}\mathrm{Im}\left[\Pi(p)\right]_{p^{2}=-P^{2}}.$
(4.18)

We thus obtain the $\not{p}\sigma_{\mu\nu}\not{p}$ part as

$\displaystyle\mathcal{B}\left[\Pi_{\mathrm{OPE}}(p^{2}=-P^{2})\right]_{\not{p}\tilde{F}\cdot\sigma\not{p}}$
$\displaystyle=-\frac{5\alpha_{s}^{2}d_{Q}}{2^{12}3^{3}\pi^{4}m_{Q}^{3}}\left(1-\beta\right)^{2}\left\langle\bar{q}\sigma\cdot
g_{s}Gq\right\rangle\log\left(\frac{M^{2}}{\Lambda_{\mathrm{IR}}^{2}}\right).$
(4.19)

Taking the imaginary part of
$I(p^{2};\epsilon_{\mathrm{IR}},\epsilon_{\mathrm{UV}})$ is somewhat subtle,
and we provide details in App. B.2. There we also clarify the physical meaning
of the scale $\Lambda_{\mathrm{IR}}$; it should be identified with the mass of
the constituent quarks. This expression is to be compared with the
phenomenological expression.

#### Phenomenological side.

On the phenomenological side, the two-point correlator with the neutron EDM
insertion is expressed as

$\displaystyle\Pi_{\mathrm{pheno}}(p)=-\lambda_{n}^{2}\left[\frac{\not{p}+m_{n}}{p^{2}-m_{n}^{2}}-\frac{d_{n}}{2(p^{2}-m_{n}^{2})^{2}}\not{p}\tilde{F}\cdot\sigma\not{p}+\cdots\right],$
(4.20)

where $m_{n}$ is the neutron mass and $\lambda_{n}$ parametrizes the overlap
between the interpolating function $\eta$ and the neutron one-particle state.
After the Borel transformation, we obtain

$\displaystyle\mathcal{B}\left[\Pi_{\mathrm{pheno}}(p^{2}=-P^{2})\right]_{\not{p}\tilde{F}\cdot\sigma\not{p}}$
$\displaystyle=\frac{\lambda_{n}^{2}d_{n}}{2M^{4}}e^{-m_{n}^{2}/M^{2}}+\cdots,$
(4.21)

where $\cdots$ expresses contributions from excited states which we ignore in
the following.

#### Sum rule.

The QCD sum rule of the neutron EDM is obtained by equating Eqs. (4.19) and
(4.21). We thus obtain

$\displaystyle\frac{\lambda_{n}^{2}d_{n}^{(G^{3}\tilde{F})}}{2M^{4}}e^{-m_{n}^{2}/M^{2}}$
$\displaystyle=-\frac{5\alpha_{s}^{2}d_{Q}}{2^{12}3^{3}\pi^{4}m_{Q}^{3}}\left(1-\beta\right)^{2}\left\langle\bar{\psi}\sigma\cdot
g_{s}G\psi\right\rangle\log\left(\frac{M^{2}}{\Lambda_{\mathrm{IR}}^{2}}\right),$
(4.22)

where $d_{n}^{(\tilde{F}G^{3})}$ is the neutron EDM induced by the $CP$-odd
photon-gluon operator (to distinguish it from the one induced by the light
quark EDM). We may use the Ioffe formula for the nucleon mass [48,
49]\natural2\natural2$\natural$2 The Ioffe formula in [49] is derived based on
$\chi_{\mathrm{SR}}/2$ which is equivalent to our $-\eta/2$ with $\beta=-1$.
Therefore the normalization of $\lambda_{n}$ differs by a factor two.

$\displaystyle\frac{\lambda_{n}^{2}m_{n}}{M^{4}}e^{-m_{n}^{2}/M^{2}}=-\frac{7-2\beta-5\beta^{2}}{16\pi^{2}}\langle\bar{q}q\rangle,$
(4.23)

to eliminate $\lambda_{n}$. Then we obtain the QCD sum rule estimation of the
neutron EDM

$\displaystyle d_{n}^{(G^{3}\tilde{F})}$
$\displaystyle=d_{Q}\times\frac{5\alpha_{s}^{2}}{2^{7}3^{3}\pi^{2}}\frac{m_{n}m_{0}^{2}}{m_{Q}^{3}}\frac{1-\beta}{7+5\beta}\log\left(\frac{M^{2}}{\Lambda_{\mathrm{IR}}^{2}}\right),$
(4.24)

where we used $\langle q\sigma\cdot
g_{s}Gq\rangle=m_{0}^{2}\langle\bar{q}q\rangle$ with
$m_{0}^{2}=0.8\,\mathrm{GeV}^{2}$ [50].

###  4.3 Constraint on heavy quark EDM

The neutron EDM has two contributions, induced by the light quark EDM and the
$CP$-odd photon-gluon operators, and is given by

$\displaystyle d_{n}=d_{n}^{(d_{q})}+d_{n}^{(G^{3}\tilde{F})},$ (4.25)

where

$\displaystyle d_{n}^{(d_{q})}$
$\displaystyle=d_{Q}\times\frac{5(8\zeta(3)-7)}{72}\left(\frac{\alpha_{s}}{\pi}\right)^{3}\frac{4m_{d}-m_{u}}{3m_{Q}},\quad
d_{n}^{(G^{3}\tilde{F})}=d_{Q}\times\frac{5\alpha_{s}^{2}}{2^{7}3^{3}\pi^{2}}\frac{m_{n}m_{0}^{2}}{m_{Q}^{3}}\log\left(\frac{M^{2}}{\Lambda_{\mathrm{IR}}^{2}}\right),$
(4.26)

and we take $\beta=-1$ following [48, 51, 52]. The parameters should be
evaluated at $m_{Q}$ for the former contribution, and at the scale close to
$\Lambda_{\mathrm{QCD}}$ for the latter contribution. We use
$m_{c}=1.27\,\mathrm{GeV}$, $m_{b}=4.18\,\mathrm{GeV}$,
$\alpha_{s}(m_{c})=0.38$ and $\alpha_{s}(m_{b})=0.223$ [53]. The light quark
masses (in the $\overline{\mathrm{MS}}$ scheme) also depend on the energy
scale, and we take $m_{u}(m_{c})=2.5\,\mathrm{MeV}$
$m_{d}(m_{c})=5.4\,\mathrm{MeV}$, $m_{u}(m_{b})=1.8\,\mathrm{MeV}$ and
$m_{d}(m_{b})=4.0\,\mathrm{MeV}$ that we obtain by running the light quark
masses at $2\,\mathrm{GeV}$ following [53]. Finally we take $\alpha_{s}=0.5$,
$\Lambda_{\mathrm{IR}}=300\,\mathrm{MeV}$ and $M=800\,\mathrm{MeV}$ for the
QCD sum rule estimation for definiteness. For these values, the contribution
from the $CP$-odd photon-gluon operator is larger by a factor of $16$ and $10$
for the charm and bottom quarks, respectively. In the bottom quark case, the
explicit quark mass suppression is compensated by the running of the strong
coupling, and the $CP$-odd photon-gluon operator is still larger than the
light quark EDM even with the relative suppression factor $1/m_{b}^{2}$. By
requiring that $|d_{n}|<1.8\times 10^{-26}\,e\,\mathrm{cm}$, we obtain

$\displaystyle\left|{d_{c}}\right|$ $\displaystyle<6\times
10^{-22}\,e\,\mathrm{cm},$ (4.27)

for the charm quark, and

$\displaystyle\left|{d_{b}}\right|$ $\displaystyle<2\times
10^{-20}\,e\,\mathrm{cm},$ (4.28)

for the bottom quark, respectively.

The constraint from $d_{n}$ is stronger than that from $C_{S}$ by more than an
order of magnitude. However, one should note that the estimation of $d_{n}$
has a large hadronic uncertainty. Indeed, the final result is affected by a
factor two if we use the sum rule of the nucleon kinetic term instead of the
mass term for $\lambda_{n}$. There are also uncertainties related to the
choice of $M$ and $\beta$ which can again result in a factor of a few
difference in the final result. Therefore our constraint here should be
understood as an order-of-magnitude estimation and the numerical factor should
be taken with care. In contrast, our calculation of $C_{S}$ is far more
precise. Its uncertainty is estimated to be $\sim 10\,\%$ and can be further
reduced if needed (see the end of Sec. 3). In this sense, the constraints from
$d_{n}$ and $C_{S}$ are complementary to each other; $d_{n}$ puts a stronger
constraint on the heavy quark EDM, while the calculation of $C_{S}$ is cleaner
and its uncertainty is well under control.

Our constraint on $d_{c}$ is stronger than [31] by a factor two, while the one
on $d_{b}$ is weaker by a factor two. However, as we noted in the introduction
and the end of Sec. 3, our constraint directly applies to $d_{c}(m_{c})$ and
$d_{b}(m_{b})$, and is independent of the one in [31].

##  5 Conclusion

In this paper, we have derived indirect limits on the charm and bottom quark
EDMs. The charm and bottom quark EDMs generate the $CP$-odd photon-gluon
operators and the light quark EDMs after integrating out the charm and bottom
quarks. Photon-gluon operators contribute to the semi-leptonic $CP$-odd
operator $C_{S}$ (and ultimately to paramagnetic AMO EDMs) as well as to the
neutron EDM at a non-perturbative level. Quark EDM dominantly contributes to
the neutron and nuclear EDMs. Performing our evaluation and using the current
limits, we obtain

$\displaystyle\left|{d_{c}}\right|$ $\displaystyle<1.3\times
10^{-20}\,e\,\mathrm{cm},\quad\left|{d_{b}}\right|<7.6\times
10^{-19}\,e\,\mathrm{cm},$ (5.1)

from the paramagnetic EDM experiments, and

$\displaystyle\left|{d_{c}}\right|$ $\displaystyle<6\times
10^{-22}\,e\,\mathrm{cm},\quad\left|{d_{b}}\right|<2\times
10^{-20}\,e\,\mathrm{cm},$ (5.2)

from the neutron EDM experiment, respectively. Although the constraint from
the neutron EDM is stronger, it has a larger hadronic uncertainty. The
uncertainty of the constraint from the paramagnetic EDM is estimated as 10 %
and can be improved if needed, while the uncertainty from the neutron EDM can
be a factor of a few. Our constraint is independent of the one given in [31]
in the sense that our constraint directly applies to the EDM operators at the
quark mass threshold. By assuming a simple scaling of
$d_{Q}/e\propto(\alpha/\pi)m_{Q}/\Lambda^{2}_{Q}$, we may translate our
constraint as a lower bound on $CP$-odd new physics scale:
$\Lambda_{c}>70\,\mathrm{GeV}$ and $\Lambda_{b}>20\,\mathrm{GeV}$ from the
paramagnetic EDM experiment, and $\Lambda_{c}>300\,\mathrm{GeV}$ and
$\Lambda_{b}>100\,\mathrm{GeV}$ from the neutron EDM experiment.

Our result provides an important benchmark to overcome for the LHC based
measurements of the charmed baryon EDMs [26, 27, 28, 29, 30]. The idea of
using the bent crystal technique for studying electromagnetic properties of
baryons containing a heavy quark is very appealing. However, given the
strength of the bounds derived in our work, and the necessity to satisfy
independent constraints from $d_{n}$ and $C_{S}$ (hence removing a chance of
accidentally large $d_{c(b)}$ due to cancellations), one may want to re-
evaluate the main goal of the charmed baryon experiment. While it will be
difficult to match the indirect sensitivity to $d_{c(b)}$, the planned
measurement may achieve sufficient accuracy to extract the values of the
magnetic moments $\mu_{c(b)}$ and compare it with the QCD predictions.

#### Acknowledgements

Y.E. and M.P. are supported in part by U.S. Department of Energy Grant No. de-
sc0011842. M.P. would like to thank Dr. K. Melnikov for the advice in
evaluating loop contributions. The Feynman diagrams in this paper are
generated by TikZ-Feynman [54].

##  Appendix A Convention

Here we summarize our conventions used in this paper. The field strengths are
given by

$\displaystyle F_{\mu\nu}$
$\displaystyle=\partial_{\mu}A_{\nu}-\partial_{\nu}A_{\mu},\quad
G_{\mu\nu}^{a}=\partial_{\mu}G_{\nu}^{a}-\partial_{\nu}G^{a}_{\mu}+g_{s}f^{abc}G_{\mu}^{b}G_{\nu}^{c},\quad
G_{\mu\nu}=G_{\mu\nu}^{a}T^{a},$ (A.1)

where $f^{abc}$ is the SU(3) structure constant. The SU(3) generator satisfies

$\displaystyle\left[T^{a},T^{b}\right]=if^{abc}T^{c},\quad\mathrm{tr}_{\mathrm{c}}\left[T^{a}T^{b}\right]=\frac{\delta^{ab}}{2}.$
(A.2)

The dual field strengths are defined as

$\displaystyle\tilde{F}^{\mu\nu}=\frac{1}{2}\epsilon^{\mu\nu\rho\sigma}F_{\rho\sigma},\quad\tilde{G}^{\mu\nu}=\frac{1}{2}\epsilon^{\mu\nu\rho\sigma}G_{\rho\sigma},$
(A.3)

with $\epsilon^{0123}=+1$. We take the gamma matrix as

$\displaystyle\left\\{\gamma^{\mu},\gamma^{\nu}\right\\}=2\eta^{\mu\nu},\quad\eta_{\mu\nu}=\mathrm{diag}(+,-,-,-),\quad\gamma_{5}=i\gamma^{0}\gamma^{1}\gamma^{2}\gamma^{3}=-\frac{i}{4!}\epsilon^{\mu\nu\rho\sigma}\gamma_{\mu}\gamma_{\nu}\gamma_{\rho}\gamma_{\sigma},$
(A.4)

and $\sigma_{\mu\nu}$ as

$\displaystyle\sigma_{\mu\nu}=\frac{i}{2}\left(\gamma_{\mu}\gamma_{\nu}-\gamma_{\nu}\gamma_{\mu}\right)=\frac{i}{2}\left[\gamma_{\mu},\gamma_{\nu}\right].$
(A.5)

The charge conjugation matrix $\mathcal{C}$ satisfies

$\displaystyle\left(\gamma^{\mu}\right)^{T}\mathcal{C}=-\mathcal{C}\gamma^{\mu},$
(A.6)

such that $\psi^{T}\mathcal{C}$ has the same Lorentz transformation property
as $\bar{\psi}$. In the Weyl representation it is given by

$\displaystyle\mathcal{C}=i\gamma^{0}\gamma^{2}.$ (A.7)

##  Appendix B Technical details

In this appendix we provide some technical details that we omit in the main
text.

###  B.1 Derivation of $CP$-odd photon-gluon operators

In this subsection we provide derivation of the $CP$-odd photon-gluon
operators (2.5) after integrating out the heavy quarks (see [55] for an
extensive review on this procedure). Our starting point is Eq. (2.3). We can
simplify it as

$\displaystyle S_{\mathrm{eff}}$
$\displaystyle=-\frac{id_{Q}}{2}\mathrm{Tr}\left[\frac{i\not{D}+m_{Q}}{i\not{D}+m_{Q}}\frac{1}{i\not{D}-m_{Q}}\sigma_{\mu\nu}\tilde{F}^{\mu\nu}\right]=-\frac{id_{Q}m_{Q}}{2}\mathrm{Tr}\left[\frac{1}{\left(iD\right)^{2}-m_{Q}^{2}+\frac{1}{2}\sigma\cdot
H}(\sigma\cdot\tilde{F})\right],$ (B.1)

where $(iD)^{2}=iD_{\mu}iD^{\mu}$ and we used that the traces of odd
$\gamma$’s vanish in the second equality. We further expand the denominator
with respect to $H$, and obtain up to fourth-order in the gauge fields

$\displaystyle S_{\mathrm{eff}}$
$\displaystyle=-\frac{id_{Q}m_{Q}}{2}\left(T_{2}+T_{3}+T_{4}\right),$ (B.2)

where

$\displaystyle T_{2}$
$\displaystyle=-\frac{1}{2}\mathrm{Tr}\left[\frac{1}{\left(iD\right)^{2}-m_{Q}^{2}}(\sigma\cdot
H)\frac{1}{\left(iD\right)^{2}-m_{Q}^{2}}(\sigma\cdot\tilde{F})\right],$ (B.3)
$\displaystyle T_{3}$
$\displaystyle=+\frac{1}{4}\mathrm{Tr}\left[\frac{1}{\left(iD\right)^{2}-m_{q}^{2}}(\sigma\cdot
H)\frac{1}{\left(iD\right)^{2}-m_{Q}^{2}}(\sigma\cdot
H)\frac{1}{\left(iD\right)^{2}-m_{Q}^{2}}(\sigma\cdot\tilde{F})\right],$ (B.4)
$\displaystyle T_{4}$
$\displaystyle=-\frac{1}{8}\mathrm{Tr}\left[\frac{1}{\left(iD\right)^{2}-m_{Q}^{2}}(\sigma\cdot
H)\frac{1}{\left(iD\right)^{2}-m_{Q}^{2}}(\sigma\cdot
H)\frac{1}{\left(iD\right)^{2}-m_{Q}^{2}}(\sigma\cdot
H)\frac{1}{\left(iD\right)^{2}-m_{Q}^{2}}(\sigma\cdot\tilde{F})\right].$ (B.5)

The term linear in $\sigma_{\mu\nu}$ vanishes after taking the trace of the
spinor index. In order to compute $T_{2}$, we use the following identity [55]:

$\displaystyle\mathrm{Tr}\left[\frac{1}{\left((iD)^{2}-m_{q}^{2}\right)^{3}}f(H,F)\right]$
$\displaystyle=\mathrm{Tr}\left[\frac{1}{\left((i\partial)^{2}-m_{q}^{2}\right)^{3}}f(H,F)\right]-\mathrm{Tr}\left[\frac{1}{\left((iD)^{2}-m_{q}^{2}\right)^{5}}H_{\mu\nu}H^{\mu\nu}f(H,F)\right],$
(B.6)

where $f(H,F)$ is an arbitrary function of $H_{\mu\nu}$ and $F_{\mu\nu}$. We
thus obtain

$\displaystyle\frac{d}{dm_{Q}^{2}}T_{2}$
$\displaystyle=-\mathrm{Tr}\left[\frac{1}{((i\partial)^{2}-m_{Q}^{2})^{3}}\left(\sigma\cdot
H\right)\sigma\cdot\tilde{F}\right]+\mathrm{Tr}\left[\frac{1}{((iD)^{2}-m_{Q}^{2})^{5}}H_{\mu\nu}H^{\mu\nu}\left(\sigma\cdot
H\right)\sigma\cdot\tilde{F}\right].$ (B.7)

where we ignore the derivatives acting on $F$. We can now replace $iD$ by
$i\partial$ in the last term to the order of our interest, and obtain

$\displaystyle\frac{d}{dm_{Q}^{2}}T_{2}$ $\displaystyle=-i\int
d^{4}x\left[\frac{3eQ_{Q}}{4\pi^{2}m_{Q}^{2}}F_{\mu\nu}\tilde{F}^{\mu\nu}+\frac{1}{24\pi^{2}m_{Q}^{6}}\mathrm{tr}_{c}\left[H_{\mu\nu}H^{\mu\nu}H_{\rho\sigma}\tilde{F}^{\rho\sigma}\right]\right].$
(B.8)

By integrating this we obtain

$\displaystyle T_{2}$ $\displaystyle=i\int
d^{4}x\left[\frac{3eQ_{Q}}{4\pi^{2}}\ln\left(\frac{M_{R}^{2}}{m_{Q}^{2}}\right)F_{\mu\nu}\tilde{F}^{\mu\nu}+\frac{1}{48\pi^{2}m_{Q}^{4}}\mathrm{tr}_{c}\left[H_{\mu\nu}H^{\mu\nu}H_{\rho\sigma}\tilde{F}^{\rho\sigma}\right]\right],$
(B.9)

where the additional mass scale $M_{R}$ comes from the regularization which we
did not write down explicitly. Next we compute $T_{3}$. By ignoring the
derivatives acting on $\tilde{F}_{\mu\nu}$, we obtain

$\displaystyle T_{3}=\frac{1}{4}\mathrm{Tr}$
$\displaystyle\left[\frac{1}{((iD)^{2}-m_{Q}^{2})^{3}}\left(\sigma\cdot
H\right)^{2}(\sigma\cdot\tilde{F})+\frac{1}{((iD)^{2}-m_{Q}^{2})^{4}}\left[(iD)^{2},\sigma\cdot
H\right](\sigma\cdot H)(\sigma\cdot\tilde{F})\right.$
$\displaystyle\left.-\frac{1}{((iD)^{2}-m_{Q}^{2})^{5}}\left[(iD)^{2},\sigma\cdot
H\right]\left[(iD)^{2},\sigma\cdot
H\right]\sigma\cdot\tilde{F}+\cdots\right],$ (B.10)

where the dots indicate terms that contain more than two commutators and thus
correspond to higher dimensional operators that are out of our interest. The
first term is easily seen to generate only $\mathcal{O}(H^{4}\tilde{F})$
operators with the help of the identity (B.6). The second term is expanded as

$\displaystyle\mathrm{Tr}\left[\frac{1}{((iD)^{2}-m_{Q}^{2})^{4}}\left[(iD)^{2},\sigma\cdot
H\right](\sigma\cdot H)(\sigma\cdot\tilde{F})\right]=\mathrm{Tr}$
$\displaystyle\left[\frac{1}{((iD)^{2}-m_{Q}^{2})^{4}}iD^{\alpha}\left[iD_{\alpha},\sigma\cdot
H\right](\sigma\cdot H)\sigma\cdot\tilde{F}\right.$
$\displaystyle\left.+iD^{\alpha}\frac{1}{((iD)^{2}-m_{Q}^{2})^{4}}\left[iD_{\alpha},\sigma\cdot
H\right](\sigma\cdot H)\sigma\cdot\tilde{F}\right.$
$\displaystyle\left.+\frac{1}{((iD)^{2}-m_{Q}^{2})^{4}}\left[iD^{\alpha},\sigma\cdot
H\right]\left[iD_{\alpha},\sigma\cdot H\right]\sigma\cdot\tilde{F}\right].$
(B.11)

The first two terms induce only higher order terms [55], and thus we ignore
them. The third term in Eq. (B.10) already contains three field strengths and
two commutators, and hence we can ignore any further commutators between $D$
and $H$. By taking the angular average, we obtain

$\displaystyle\mathrm{Tr}\left[\frac{1}{((iD)^{2}-m_{Q}^{2})^{5}}\left[(iD)^{2},\sigma\cdot
H\right]\left[(iD)^{2},\sigma\cdot H\right]\sigma\cdot\tilde{F}\right]$
$\displaystyle=\mathrm{Tr}\left[\frac{(iD)^{2}}{((iD)^{2}-m_{Q}^{2})^{5}}\left[iD^{\alpha},\sigma\cdot
H\right]\left[iD_{\alpha},\sigma\cdot H\right]\sigma\cdot\tilde{F}\right].$
(B.12)

By combining them, we obtain

$\displaystyle T_{3}$
$\displaystyle=-\frac{m_{Q}^{2}}{4}\mathrm{Tr}\left[\frac{1}{((iD)^{2}-m_{Q}^{2})^{5}}\left[iD^{\alpha},\sigma\cdot
H\right]\left[iD_{\alpha},\sigma\cdot H\right]\sigma\cdot\tilde{F}\right]$
$\displaystyle=-\frac{g_{s}^{2}}{24\pi^{2}m_{Q}^{4}}\int
d^{4}x\,\mathrm{tr}_{c}\left[(\mathcal{D}^{\alpha}G_{\mu\rho})(\mathcal{D}_{\alpha}{G_{\nu}}^{\rho})\right]\tilde{F}^{\mu\nu}=0.$
(B.13)

Finally we compute $T_{4}$. To the order of our interest, we can simply
replace $iD$ by $i\partial$ in the denominator. It is then easy to see that

$\displaystyle T_{4}=-\frac{i}{48\pi^{2}m_{Q}^{4}}\int
d^{4}x\,\mathrm{tr}_{c}\left[3H_{\mu\nu}H^{\mu\nu}H_{\rho\sigma}\tilde{F}^{\rho\sigma}-4{H^{\mu}}_{\nu}{H^{\nu}}_{\rho}{H^{\rho}}_{\sigma}{\tilde{F}^{\sigma}}_{\mu}\right].$
(B.14)

Therefore we obtain

$\displaystyle S_{\mathrm{eff}}=\frac{d_{Q}}{48\pi^{2}m_{Q}^{3}}\int
d^{4}x\,\mathrm{tr}_{c}\left[-H_{\mu\nu}H^{\mu\nu}H_{\rho\sigma}\tilde{F}^{\rho\sigma}+2{H^{\mu}}_{\nu}{H^{\nu}}_{\rho}{H^{\rho}}_{\sigma}{\tilde{F}^{\sigma}}_{\mu}\right],$
(B.15)

where we ignored the quadratic term that is irrelevant for our purpose. If we
drop the gluons, it correctly reduces to the result in [5].

###  B.2 Borel transformation and IR divergence

In this subsection we discuss the Borel transformation of Eq. (4.16) that
contains both the UV and IR divergences. As the Borel transformation is
related to the imaginary part, it is equivalent to taking the imaginary part
of

$\displaystyle I(p^{2};\epsilon_{\mathrm{IR}},\epsilon_{\mathrm{UV}})$
$\displaystyle\equiv\Gamma(\epsilon_{\mathrm{IR}})\left(-\frac{\Lambda_{\mathrm{IR}}^{2}}{p^{2}}\right)^{-\epsilon_{\mathrm{IR}}}\Gamma(-\epsilon_{\mathrm{UV}}-\epsilon_{\mathrm{IR}})\left(-\frac{\Lambda_{\mathrm{UV}}^{2}}{p^{2}}\right)^{-\epsilon_{\mathrm{UV}}}.$
(B.16)

The limit of this function at $\epsilon_{\mathrm{UV}}\to
0,\>\epsilon_{\mathrm{IR}}\to 0$ is not well defined. For example, if the
limit is taken along the line
$\epsilon_{\mathrm{UV}}=a\,\epsilon_{\mathrm{IR}}$ with fixed $a$, one would
get an $a$-dependent result:

$\underset{\epsilon_{\mathrm{IR}}\rightarrow
0}{\text{lim}}I(p^{2};\epsilon_{\mathrm{IR}},a\,\epsilon_{\mathrm{IR}})=-\frac{1}{2\left(1+a\right)}\left[\text{log}^{2}\left(-\frac{\Lambda_{\mathrm{IR}}^{2}}{p^{2}}\right)+2a\text{log}\left(-\frac{\Lambda_{\mathrm{IR}}^{2}}{p^{2}}\right)\text{log}\left(-\frac{\Lambda_{\mathrm{UV}}^{2}}{p^{2}}\right)+a^{2}\text{log}^{2}\left(-\frac{\Lambda_{\mathrm{UV}}^{2}}{p^{2}}\right)\right],$
(B.17)

where we only kept the double logarithmic terms. The purpose of this
subsection is to understand the correct prescription of evaluating the
imaginary part of this function.

In order to understand the correct prescription, it is helpful to consider a
simpler example: a scalar three-body decay. Indeed, the two point correlator
$\Pi(p)$ contains three quark propagators, and thus the imaginary part of this
function is related a three-body decay phase space integral. Therefore we
consider the following Lagrangian

$\displaystyle\mathcal{L}=\frac{1}{2}\left(\partial\phi\right)^{2}-\frac{m_{\phi}^{2}}{2}\phi^{2}+\frac{1}{2}\left(\partial\chi\right)^{2}-\frac{m_{\chi}^{2}}{2}\chi^{2}-\frac{\lambda}{6}\phi\chi^{3},$
(B.18)

and study the three-body decay $\phi\to 3\chi$. The two-loop diagram is
evaluated as

$\displaystyle i\mathcal{M}_{2}$ $\displaystyle=\leavevmode\hbox to0pt{\vbox
to0pt{\pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower 0.0pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{
}\nullfont\pgfsys@beginscope\pgfsys@invoke{ }\hbox to0.0pt{ {}{{}}{}{{{}}
{}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{
}\definecolor[named]{pgfstrokecolor}{rgb}{.5,.5,.5}\pgfsys@color@gray@stroke{.5}\pgfsys@invoke{
}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@moveto{36.98866pt}{0.0pt}\pgfsys@curveto{36.98866pt}{5.50006pt}{20.42853pt}{9.95863pt}{0.0pt}{9.95863pt}\pgfsys@curveto{-20.42853pt}{9.95863pt}{-36.98866pt}{5.50006pt}{-36.98866pt}{0.0pt}\pgfsys@curveto{-36.98866pt}{-5.50006pt}{-20.42853pt}{-9.95863pt}{0.0pt}{-9.95863pt}\pgfsys@curveto{20.42853pt}{-9.95863pt}{36.98866pt}{-5.50006pt}{36.98866pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\diagram*{ (a) --
[fermion, momentum=$\scriptstyle p$] (b), (b) -- [fermion] (d), (b) --
[insertion={1.0}] (d), (b) -- [gluon, momentum=$\scriptstyle p$] (c), (c) --
[gluon, insertion={1.0}] (e), (c) -- [photon, insertion={1.0}] (f), (c) --
[gluon, momentum=$\scriptstyle p$] (h), (i) -- [fermion] (h) -- [fermion,
momentum=$\scriptstyle p$] (j), (i) -- [fermion, insertion={0.0}] (h), };
\hss}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ } \feynman[inline] \vertex(p1);
\vertex[right = 0.75 of p1] (v1); \vertex[right = 0.5 of v1] (c);
\vertex[above = 0.5 of c] (v2); \vertex[below = 0.5 of c] (v4); \vertex[right
= 0.5 of c] (v3); \vertex[right = 0.75 of v3] (p2); \diagram*{ (p1) --
[scalar, momentum=$\scriptstyle p$] (v1) -- (v3) -- [scalar,
momentum=$\scriptstyle p$] (p2), (v1) -- [quarter left] (v2) -- [quarter left]
(v3) -- [quarter left] (v4) -- [quarter left] (v1), };
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}=\frac{i\lambda^{2}}{6}\int\frac{d^{4}l_{1}}{(2\pi)^{4}}\int\frac{d^{4}l_{2}}{(2\pi)^{4}}D_{F}(l_{1})D_{F}(l_{2})D_{F}(l_{1}+l_{2}-p),$
(B.19)

where $D_{F}(p)$ is the Feynman propagator,

$\displaystyle iD_{F}(p)=\frac{i}{p^{2}-m_{\chi}^{2}+i0},$ (B.20)

and the solid line is $\chi$ while the dashed line is $\phi$. The cutting rule
tells us that

$\displaystyle\mathrm{Im}\mathcal{M}_{2}$
$\displaystyle=\frac{1}{2}\frac{\lambda^{2}}{6}\int d\Pi_{\mathrm{LIPS}},$
(B.21)

where $d\Pi_{\mathrm{LIPS}}$ is the three-body Lorentz invariant phase space
integral. This is expanded with respect to $m_{\chi}^{2}/m_{\phi}^{2}$ as

$\displaystyle\int d\Pi_{\mathrm{LIPS}}$
$\displaystyle=\frac{1}{32\pi^{3}}\left[\frac{m_{\phi}^{2}}{8}+\frac{3m_{\chi}^{2}}{4}\left(\log\left(\frac{4m_{\chi}^{2}}{m_{\phi}^{2}}\right)-1\right)+\cdots\right],$
(B.22)

where we used $p^{2}=m_{\phi}^{2}$. We now evaluate the left hand side of Eq.
(B.21) in an analogous way as the main text. We may expand the propagator as

$\displaystyle
iD_{F}(p)=\frac{i}{p^{2}-m_{\chi}^{2}}=i\left(\frac{1}{p^{2}}+\frac{1}{p^{2}}m_{\chi}^{2}\frac{1}{p^{2}}+\cdots\right),$
(B.23)

where we omit $i\epsilon$ for notational simplicity. In the coordinate space
this is given by

$\displaystyle iD_{F}(x)$
$\displaystyle=-\frac{1}{4\pi^{2}x^{2}}-\frac{m_{\chi}^{2}}{16\pi^{2}}\Gamma(\epsilon_{\mathrm{IR}})\left(-\Lambda_{\mathrm{IR}}^{2}x^{2}\right)^{-\epsilon_{\mathrm{IR}}}+\cdots.$
(B.24)

The first order term in $m_{\chi}^{2}$ is IR divergent and this is analogous
to our propagator with the $CP$-odd operator insertion in Sec. 4.2. The two-
loop amplitude is given by

$\displaystyle i\mathcal{M}_{2}(p)$
$\displaystyle=\frac{\left(i\lambda\right)^{2}}{6}\mu_{\mathrm{UV}}^{d_{\mathrm{UV}}-4}\int
d^{d_{\mathrm{UV}}}x\,e^{ip\cdot x}iD_{F}(x)iD_{F}(x)iD_{F}(x).$ (B.25)

To the first order in $m_{\chi}^{2}$ we obtain

$\displaystyle\left.\mathcal{M}_{2}\right|_{\mathcal{O}(m_{\chi}^{2})}$
$\displaystyle=-\frac{\lambda^{2}m_{\chi}^{2}}{512\pi^{4}}I(p^{2};\epsilon_{\mathrm{IR}},\epsilon_{\mathrm{UV}}).$
(B.26)

Note that we get exactly the same function
$I(p^{2};\epsilon_{\mathrm{IR}},\epsilon_{\mathrm{UV}})$ here. Eq. (B.21)
tells us that

$\displaystyle\left.\mathrm{Im}\mathcal{M}_{2}\right|_{\mathcal{O}(m_{\chi}^{2})}$
$\displaystyle=\left[\frac{1}{2}\frac{\lambda^{2}}{6}\int
d\Pi_{\mathrm{LIPS}}\right]_{\mathcal{O}(m_{\chi}^{2})}=\frac{\lambda^{2}m_{\chi}^{2}}{512\pi^{3}}\log\left(\frac{m_{\chi}^{2}}{m_{\phi}^{2}}\right),$
(B.27)

where we focus on the leading logarithmic term on the right hand side. Thus
the correct prescription of evaluating the imaginary part of $I$ is

$\displaystyle\mathrm{Im}\left[I(p^{2};\epsilon_{\mathrm{IR}},\epsilon_{\mathrm{UV}})\right]$
$\displaystyle=-\pi\log\left(\frac{\Lambda_{\mathrm{IR}}^{2}}{p^{2}}\right),$
(B.28)

where we identify $\Lambda_{\mathrm{IR}}^{2}=m_{\chi}^{2}$ in the present
case. This agrees with the Borel transformation formula in [52]. For the
example in Eq.(B.17), this prescription is equivalent to neglecting the real
part of the UV logarithm.

Our discussion clarifies the physical meaning of $\Lambda_{\mathrm{IR}}$ that
appears in Eq. (4.19). This IR divergence originates from the phase space
integral and is regulated by the mass of the daughter particles. In the
neutron EDM case, the daughter particles are the constituent quarks. Therefore
$\Lambda_{\mathrm{IR}}$ is identified with the mass of the constituent up and
down quarks that we take $\Lambda_{\mathrm{IR}}=300\,\mathrm{MeV}$ in the main
text.

## References

  * [1] M. Pospelov and A. Ritz, “Electric dipole moments as probes of new physics,” Annals Phys. 318 (2005) 119–169, arXiv:hep-ph/0504231.
  * [2] ACME Collaboration, V. Andreev et al., “Improved limit on the electric dipole moment of the electron,” Nature 562 no. 7727, (2018) 355–360.
  * [3] A. G. Grozin, I. B. Khriplovich, and A. S. Rudenko, “Electric dipole moments, from e to tau,” Phys. Atom. Nucl. 72 (2009) 1203–1205, arXiv:0811.1641 [hep-ph].
  * [4] 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 no. 16, (2016) 161601, arXiv:1601.04339 [physics.atom-ph]. [Erratum: Phys. Rev. Lett.119,no.11,119901(2017)].
  * [5] Y. Ema, T. Gao, and M. Pospelov, “Improved Indirect Limits on Muon Electric Dipole Moment,” Phys. Rev. Lett. 128 no. 13, (2022) 131803, arXiv:2108.05398 [hep-ph].
  * [6] T. Aoyama et al., “The anomalous magnetic moment of the muon in the Standard Model,” Phys. Rept. 887 (2020) 1–166, arXiv:2006.04822 [hep-ph].
  * [7] Muon g-2 Collaboration, B. Abi et al., “Measurement of the Positive Muon Anomalous Magnetic Moment to 0.46 ppm,” Phys. Rev. Lett. 126 no. 14, (2021) 141801, arXiv:2104.03281 [hep-ex].
  * [8] 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 [hep-ex].
  * [9] Y. K. Semertzidis et al., “Sensitive search for a permanent muon electric dipole moment,” in KEK International Workshop on High Intensity Muon Sources (HIMUS 99), pp. 81–96.  12, 1999.  arXiv:hep-ph/0012087.
  * [10] H. Iinuma, H. Nakayama, K. Oide, K.-i. Sasaki, N. Saito, T. Mibe, and M. Abe, “Three-dimensional spiral injection scheme for the g-2/EDM experiment at J-PARC,” Nucl. Instrum. Meth. A 832 (2016) 51–62.
  * [11] M. Abe et al., “A New Approach for Measuring the Muon Anomalous Magnetic Moment and Electric Dipole Moment,” PTEP 2019 no. 5, (2019) 053C02, arXiv:1901.03047 [physics.ins-det].
  * [12] A. Adelmann et al., “Search for a muon EDM using the frozen-spin technique,” arXiv:2102.08838 [hep-ex].
  * [13] E. M. Purcell and N. F. Ramsey, “On the Possibility of Electric Dipole Moments for Elementary Particles and Nuclei,” Phys. Rev. 78 (1950) 807–807.
  * [14] C. Abel et al., “Measurement of the Permanent Electric Dipole Moment of the Neutron,” Phys. Rev. Lett. 124 no. 8, (2020) 081803, arXiv:2001.11966 [hep-ex].
  * [15] V. M. Khatsimovsky, I. B. Khriplovich, and A. R. Zhitnitsky, “Strange Quarks in the Nucleon and Neutron Electric Dipole Moment,” Z. Phys. C 36 (1987) 455.
  * [16] K. Fuyuto, J. Hisano, N. Nagata, and K. Tsumura, “QCD Corrections to Quark (Chromo)Electric Dipole Moments in High-scale Supersymmetry,” JHEP 12 (2013) 010, arXiv:1308.6493 [hep-ph].
  * [17] V. Cirigliano, W. Dekens, J. de Vries, and E. Mereghetti, “Is there room for CP violation in the top-Higgs sector?,” Phys. Rev. D 94 no. 1, (2016) 016002, arXiv:1603.03049 [hep-ph].
  * [18] V. Cirigliano, W. Dekens, J. de Vries, and E. Mereghetti, “Constraining the top-Higgs sector of the Standard Model Effective Field Theory,” Phys. Rev. D 94 no. 3, (2016) 034031, arXiv:1605.04311 [hep-ph].
  * [19] S. Weinberg, “Larger Higgs Exchange Terms in the Neutron Electric Dipole Moment,” Phys. Rev. Lett. 63 (1989) 2333.
  * [20] D. Chang, W.-Y. Keung, C. S. Li, and T. C. Yuan, “QCD Corrections to CP Violation From Color Electric Dipole Moment of $b$ Quark,” Phys. Lett. B 241 (1990) 589–592.
  * [21] G. Boyd, A. K. Gupta, S. P. Trivedi, and M. B. Wise, “Effective Hamiltonian for the Electric Dipole Moment of the Neutron,” Phys. Lett. B 241 (1990) 584–588.
  * [22] M. Dine and W. Fischler, “Constraints on New Physics From Weinberg’s Analysis of the Neutron Electric Dipole Moment,” Phys. Lett. B 242 (1990) 239–244.
  * [23] D. A. Demir, M. Pospelov, and A. Ritz, “Hadronic EDMs, the Weinberg operator, and light gluinos,” Phys. Rev. D 67 (2003) 015007, arXiv:hep-ph/0208257.
  * [24] F. Sala, “A bound on the charm chromo-EDM and its implications,” JHEP 03 (2014) 061, arXiv:1312.2589 [hep-ph].
  * [25] A. E. Blinov and A. S. Rudenko, “Upper Limits on Electric and Weak Dipole Moments of tau-Lepton and Heavy Quarks from e+ e- Annihilation,” Nucl. Phys. B Proc. Suppl. 189 (2009) 257–259, arXiv:0811.2380 [hep-ph].
  * [26] V. G. Baryshevsky, “The possibility to measure the magnetic moments of short-lived particles (charm and beauty baryons) at LHC and FCC energies using the phenomenon of spin rotation in crystals,” Phys. Lett. B 757 (2016) 426–429.
  * [27] F. J. Botella, L. M. Garcia Martin, D. Marangotto, F. M. Vidal, A. Merli, N. Neri, A. Oyanguren, and J. R. Vidal, “On the search for the electric dipole moment of strange and charm baryons at LHC,” Eur. Phys. J. C 77 no. 3, (2017) 181, arXiv:1612.06769 [hep-ex].
  * [28] A. S. Fomin et al., “Feasibility of measuring the magnetic dipole moments of the charm baryons at the LHC using bent crystals,” JHEP 08 (2017) 120, arXiv:1705.03382 [hep-ph].
  * [29] E. Bagli et al., “Electromagnetic dipole moments of charged baryons with bent crystals at the LHC,” Eur. Phys. J. C 77 no. 12, (2017) 828, arXiv:1708.08483 [hep-ex]. [Erratum: Eur.Phys.J.C 80, 680 (2020)].
  * [30] S. Aiola et al., “Progress towards the first measurement of charm baryon dipole moments,” Phys. Rev. D 103 no. 7, (2021) 072003, arXiv:2010.11902 [hep-ex].
  * [31] H. Gisbert and J. Ruiz Vidal, “Improved bounds on heavy quark electric dipole moments,” Phys. Rev. D 101 no. 11, (2020) 115010, arXiv:1905.02513 [hep-ph].
  * [32] U. Haisch and G. Koole, “Beautiful and charming chromodipole moments,” JHEP 09 (2021) 133, arXiv:2106.01289 [hep-ph].
  * [33] D. J. Broadhurst, “Three loop on-shell charge renormalization without integration: Lambda-MS (QED) to four loops,” Z. Phys. C 54 (1992) 599–606.
  * [34] A. V. Smirnov and F. S. Chuharev, “FIRE6: Feynman Integral REduction with Modular Arithmetic,” Comput. Phys. Commun. 247 (2020) 106877, arXiv:1901.07808 [hep-ph].
  * [35] S. P. Martin and D. G. Robertson, “Evaluation of the general 3-loop vacuum Feynman integral,” Phys. Rev. D 95 no. 1, (2017) 016008, arXiv:1610.07720 [hep-ph].
  * [36] Y. Ema, T. Gao and M. Pospelov, _in preparation_.
  * [37] M. Pospelov and A. Ritz, “CKM benchmarks for electron electric dipole moment experiments,” Phys. Rev. D 89 no. 5, (2014) 056006, arXiv:1311.5537 [hep-ph].
  * [38] M. A. Shifman, A. I. Vainshtein, and V. I. Zakharov, “Remarks on Higgs Boson Interactions with Nucleons,” Phys. Lett. B 78 (1978) 443–446.
  * [39] V. V. Flambaum, M. Pospelov, A. Ritz, and Y. V. Stadnik, “Sensitivity of EDM experiments in paramagnetic atoms and molecules to hadronic CP violation,” Phys. Rev. D 102 no. 3, (2020) 035001, arXiv:1912.13129 [hep-ph].
  * [40] A. G. Grozin, I. B. Khriplovich, and A. S. Rudenko, “Upper limits on electric dipole moments of tau-lepton, heavy quarks, and W-boson,” Nucl. Phys. B 821 (2009) 285–290, arXiv:0902.3059 [hep-ph].
  * [41] J. Ellis, N. Nagata, and K. A. Olive, “Uncertainties in WIMP Dark Matter Scattering Revisited,” Eur. Phys. J. C 78 no. 7, (2018) 569, arXiv:1805.09795 [hep-ph].
  * [42] M. Pospelov and A. Ritz, “Neutron EDM from electric and chromoelectric dipole moments of quarks,” Phys. Rev. D 63 (2001) 073015, arXiv:hep-ph/0010037.
  * [43] J. Hisano, J. Y. Lee, N. Nagata, and Y. Shimizu, “Reevaluation of Neutron Electric Dipole Moment with QCD Sum Rules,” Phys. Rev. D 85 (2012) 114044, arXiv:1204.2653 [hep-ph].
  * [44] T. Bhattacharya, V. Cirigliano, S. Cohen, R. Gupta, H.-W. Lin, and B. Yoon, “Axial, Scalar and Tensor Charges of the Nucleon from 2+1+1-flavor Lattice QCD,” Phys. Rev. D 94 no. 5, (2016) 054508, arXiv:1606.07049 [hep-lat].
  * [45] M. A. Shifman, A. I. Vainshtein, and V. I. Zakharov, “QCD and Resonance Physics. Theoretical Foundations,” Nucl. Phys. B 147 (1979) 385–447.
  * [46] H.-x. He and X.-D. Ji, “QCD sum rule calculation for the tensor charge of the nucleon,” Phys. Rev. D 54 (1996) 6897–6902, arXiv:hep-ph/9607408.
  * [47] B. L. Ioffe and A. V. Smilga, “Nucleon Magnetic Moments and Magnetic Properties of Vacuum in QCD,” Nucl. Phys. B 232 (1984) 109–142.
  * [48] B. L. Ioffe, “Calculation of Baryon Masses in Quantum Chromodynamics,” Nucl. Phys. B 188 (1981) 317–341. [Erratum: Nucl.Phys.B 191, 591–592 (1981)].
  * [49] D. B. Leinweber, “QCD sum rules for skeptics,” Annals Phys. 254 (1997) 328–396, arXiv:nucl-th/9510051.
  * [50] V. M. Belyaev and B. L. Ioffe, “Determination of Baryon and Baryonic Resonance Masses from QCD Sum Rules. 1. Nonstrange Baryons,” Sov. Phys. JETP 56 (1982) 493–501.
  * [51] B. L. Ioffe, “ON THE CHOICE OF QUARK CURRENTS IN THE QCD SUM RULES FOR BARYON MASSES,” Z. Phys. C 18 (1983) 67.
  * [52] U. Haisch and A. Hala, “Sum rules for CP-violating operators of Weinberg type,” JHEP 11 (2019) 154, arXiv:1909.08955 [hep-ph].
  * [53] Particle Data Group Collaboration, P. A. Zyla et al., “Review of Particle Physics,” PTEP 2020 no. 8, (2020) 083C01.
  * [54] J. Ellis, “TikZ-Feynman: Feynman diagrams with TikZ,” Comput. Phys. Commun. 210 (2017) 103–123, arXiv:1601.05437 [hep-ph].
  * [55] V. A. Novikov, M. A. Shifman, A. I. Vainshtein, and V. I. Zakharov, “Calculations in External Fields in Quantum Chromodynamics. Technical Review,” Fortsch. Phys. 32 (1984) 585.