Sensitivity of EDM experiments in paramagnetic atoms and molecules
to hadronic CP violation
V. V. Flambaum,1 M. Pospelov,2, 3, 4 A. Ritz,5 and Y. V. Stadnik1, 6, 7
1

School of Physics, University of New South Wales, Sydney 2052, Australia
School of Physics and Astronomy, University of Minnesota, Minneapolis, MN 55455, USA
3
William I. Fine Theoretical Physics Institute, School of Physics and Astronomy,
University of Minnesota, Minneapolis, MN 55455, USA
4
Perimeter Institute for Theoretical Physics, Waterloo, ON N2J 2W9, Canada
5
Department of Physics and Astronomy, University of Victoria, Victoria, BC V8P 5C2, Canada
6Helmholtz Institute Mainz, Johannes Gutenberg University, 55128 Mainz, Germany
7
Kavli Institute for the Physics and Mathematics of the Universe (WPI),The University of Tokyo Institutes for Advanced Study,
The University of Tokyo, Kashiwa, Chiba 277-8583, Japan(Dated: January 1, 2020)

arXiv:1912.13129v1 [hep-ph] 31 Dec 20192

Experiments searching for the electric dipole moment (EDM) of the electron de utilise
atomic/molecular states with one or more uncompensated electron spins, and these paramagnetic
systems have recently achieved remarkable sensitivity to de . If the source of CP violation resides
entirely in the hadronic sector, the two-photon exchange diagrams between electrons and the nucleus
induce CP -odd semileptonic interactions, parametrised by the Wilson coefficient CSP , and provide
the dominant source of EDMs in paramagnetic systems instead of de . We evaluate the CSP coefficients induced by the leading hadronic sources of CP violation, namely nucleon EDMs and CP -odd
pion-nucleon couplings, by calculating the nucleon-number-enhanced CP -odd nuclear polarisability,
employing chiral perturbation theory at the nucleon level and the Fermi-gas model for the nucleus.
This allows us to translate the ACME EDM limits from paramagnetic ThO into novel independent
constraints on the QCD theta term |θ̄| < 3 × 10−8 , proton EDM |dp | < 2 × 10−23 e cm, and colour
EDMs of quarks |d˜u − d˜d | < 2 × 10−24 cm. We note that further experimental progress with EDM
experiments in paramagnetic systems may allow them to rival the sensitivity of EDM experimentswith neutrons and diamagnetic atoms to these quantities.

Introduction. — The origin of the matter-antimatterasymmetry in the universe continues to present us with
one of the primary empirical motivations for new physics.
The observed predominance of matter over anti-matter,quantified by the baryon-to-photon ratio ηB ≈ 6 × 10−10 ,is now precisely determined through cosmological observations. A dynamical explanation for such an asymmetry, under the minimal assumption of the hot Big Bangmodel, requires sources of CP violation as first laid out
by Sakharov more than half a century ago [1]. The known
sources of CP violation in the Standard Model (SM) of
particle physics – the CKM phase and QCD vacuum angle (the latter consistent with zero to high accuracy) –are insufficient for this task, providing empirical motivation for new sources of CP violation that can explain theasymmetry.
Experimental searches for new sources of CP - or P, T violation have a long history, and electric dipole moments(EDMs) of nucleons, atoms and molecules are some ofthe primary observables (see, e.g., [2–5]). Fundamental
sources of T violation can feed into these observables ina variety of ways, depending on the specific features of
the nuclear or atomic system. Searches for a neutron
EDM pre-date even the discovery of P violation in the
weak interactions, but the use of atomic and now molecular systems to probe CP violation through EDM-likeobservables has a more recent history, having been motivated by the possibility of various enhancement mech-anisms in the 1980s [6, 7]. Atomic/molecular EDM experiments are usually classified by whether the relevant
state is either paramagnetic or diamagnetic, with the former principally aimed at probing leptonic sources of CP
violation via the EDMs of unpaired electrons, and thelatter mainly probing hadronic sources of CP violation
via nuclear moments.
The prevailing classification of EDM experiments has
been valuable in assessing their complementarity in probing the full range of potential CP -violating sources. However, in recent years, there has been quite dramatic
progress particularly in the use of paramagnetic molecular states to search for EDMs [8–11]. Indeed, the effective sensitivity to the electron EDM de via atomic andnow molecular EDM experiments has improved by a factor of more than 100 over the past decade. This raises
the question of whether EDM experiments in paramagnetic systems may soon provide significant sensitivity tohadronic sources of CP violation, complementary to thatfrom experiments targeting EDMs in diamagnetic systems and the nuclear Schiff moment [12], and also theneutron EDM [13].
In the present Letter, we address this question quantitatively, by focussing on the sensitivity of paramagnetic EDM experiments, such as ACME [11], to CP -odd

2semileptonic operators of the form,
t GF
s GF
√ ēiγ5 e(p̄p + n̄n) + CSP√ ēiγ5 e(p̄p − n̄n) ,L = CSP2
2(1)
where e, n and p refer to the electron, neutron ands,tproton fields, respectively, and CSP
are the couplings
for the singlet and triplet operators, respectively. The
subscript SP denotes the nucleon-scalar and electronpseudoscalar two-fermion bilinears. The semileptonic operators CSP in (1) arise in the absence of any nuclear
spin and are coherently enhanced by the number of nucleons in the nucleus, singling them out as the primarycontributors to paramagnetic EDMs beyond the electron
EDM, − 2i de ēFµν σ µν γ5 e. Hadronic contributions to de ,e.g. from the QCD θ term, have been considered previously [14, 15], but the semileptonic operators above
provide the leading sensitivity in atomic and molecularexperiments. In particular, the leading source of paramagnetic EDMs due to the CKM phase is the CSP operator[16], mediated by two-photon exchange.
In paramagnetic EDM experiments, the induced shift
of atomic/molecular energy levels under an applied external electric field Eext can be written in the form




Z −N
s
t
∆E = −de Eeff − Wc CSP+
CSP+ · · · , (2)A

where the factors Eeff and Wc are quantities that dependon the small Eext , and Z, N and A denote the proton,
neutron and total nucleon numbers of the nucleus, respectively. They are enhanced by a relativistic violationof the Schiff theorem and (for molecular systems) the polarisability [17], and are now known to good precision fora variety of molecular species, see e.g. [18–24]. The existing null result from the ACME experiment, using ThO,
leads to the following 90% confidence-level constraint on
the effective CSP coupling averaged over the p − n composition of the Th nucleus [11],
p
s
t
n
|CSP
− 0.22CSP| = |0.39CSP+ 0.61CSP| < 7.3 × 10−10 .
(3)
Quite generically, for hadronic sources of CP violation,
the de contribution to atomic/molecular EDMs is subdominant to CSP .
The semileptonic operators in (1) can in turn be induced by the leading sources of CP violation at thehadronic level,ii
Lhadronic = − dn n̄Fµν σ µν γ5 n − dp p̄Fµν σ µν γ5 p
2
2
(0)
(1)
+ ḡπN N N̄ τ a N π a + ḡπN N N̄ N π 0 + ... , (4)where N = (p, n)T is the nucleon doublet, dn,p refers to(0,1)
nucleon EDMs, and ḡπN N are the isovector and isoscalar
CP -odd pion-nucleon couplings, respectively. This formula can also be generalised to include CP -odd inter(0)(1)
actions with the octet η meson, η N̄ (ḡηN N + ḡηN N τ 3 )N .
Thus we aim to determine
(0)

(1)

CSP = CSP (dn , dp , ḡπ/ηN N , ḡπ/ηN N , . . .) ,(5)

FIG. 1. (Color online) CP -violating leading order (LO)semileptonic processes involving the exchange of a π 0 or η
meson. The grey vertex denotes the anomalous coupling (atthe one-loop level) of the π 0 /η meson to the electromagnetic
field, while the magenta vertex denotes the CP -violating coupling with the nucleon.

that can be induced in particular by two-photon exchange diagrams. The hadronic-scale interactions in (4)are in turn induced by more fundamental sources, suchas θQCD , quark EDMs and chromo EDMs [4]. In what
follows, we will examine the leading dependencies in (5),and explore the induced sensitivity to fundamental CP violating hadronic sources.
Semileptonic operators induced by CP -odd nucleon polarisabilities. — When the underlying sources of CP violation are hadronic and the nuclei of interest are spinless,the semileptonic couplings CSP in (1) can be generated
by two-photon exchange processes via CP -odd nucleon
polarisabilities,
1
L = − N̄ (βs + τ 3 βt )N Fµν Feµν
4
= (βp p̄p + βn n̄n)E · B .

(6)
(7)

Application of an external electric field E leads to an induced magnetic dipole moment βE, and the sign in (6,7)
is chosen to coincide with the CP -even polarisability convention, L = αpol E 2 /2.
A complete calculation of the CP -odd nuclear polarisability is a complicated task, but at the nucleon level itcan be performed using chiral perturbation theory. The
leading order (LO) terms arise at O(m−2π ) in the pionmass mπ , as shown in Fig. 1, and are given by
"
#
(0)
2
ḡ
m
F
α
π
ηN
N
(1)
(0)
π
LO
βp(n)
=−
ḡ
+(−)ḡπN N + √,πFπ m2π πN N
3 m2η Fη
(8)
where Fπ ≈ 92 MeV is the pion decay constant, and Fη
is the octet η-meson decay constant, which we take to
be Fη ≈ Fπ . The appearance of the factor α/π in thisformula is due to the one-loop nature of the π 0 γγ vertex.
We have neglected small isospin-breaking effects, η − η 0(1)and π 0 − η mixings, as well as ḡηN N , as only the singlet
contribution of η proves to be important in the concrete
examples below. We next address the first formally subleading correction, which emerges from a charged-pion
loop that interacts with E, while the magnetic moment

3

FIG. 2. (Color online) CP -violating next-to-leading order(NLO) semileptonic processes involving a charged-pion loop.
The magenta vertex again denotes the CP -violating couplingof the pion with the nucleon, while the black vertex denotes
the coupling of the electromagnetic field to the nucleon magnetic dipole moment. The analogous processes with the magenta vertex interchanged with the other pion-nucleon vertex
are implicit.

of the nucleon interacts with B (see Fig. 2). The nextto-leading order (NLO) result arises at O(m−1π ), and isgiven by
(0)

βkN LO =αgA ḡπN N
4Fπ mN mπ−µn /µN for k = p ,µp /µN for k = n ,(9)where gA ≈ 1.3 is the axial triplet coupling, mN is thenucleon mass, µn,p are the nucleon magnetic dipole moments, and µN is the nuclear magneton. We observe that
this answer is numerically rather larger than would have
naively been expected, in part as a result of the large values of µn,p . Also, the CP -odd polarisabilities of neutronsand protons have the same sign, as µn is negative, andso add constructively.
To compute the contributions to CSP , we next perform the integral over the diphoton loop, which is softcompared to the hadronic scales that were integrated out
above, and average the result over the nucleon content ina nucleus. We find, to logarithmic accuracy, a known
result for the semi-leptonic operator in the contact approximation:GF (β)N
3αme
ΛUVZ√ CSP = −βp + βnln. (10)AA
2π
me
2
In the limit of a pointlike and structureless nucleus, theUV cutoff scale ΛUV is different for the LO and NLO
contributions: for the LO terms, it is set by the π/η
form factor (i.e., a hadronic scale related to the ρ meson
mass mρ ), while for the NLO process, ΛUV ≈ mπ due
to the presence of the pion propagators in the chargedpion loop. The nuclear size, which sets the value of theatomic s − p mixing matrix element induced by CSP [25],does not play any role in regularising the integral, which
extends down to ∼ me . We also note that going beyond
the logarithmic approximation in the NLO case would
prevent the factorisation of the photon and pion loops,
and would necessitate a full two-loop calculation.

FIG. 3. (Color online) CP -violating µ − d semileptonic processes with internal nuclear excitations. The black vertex
again denotes the interaction of the electromagnetic field with
the nucleon magnetic dipole moment µ, while the cyan vertex denotes the interaction with the nucleon electric dipole
moment d (generated at the one-loop level). The analogous
processes with the black and cyan vertices interchanged are
implicit.

Thus far, we have neglected the fact that the internal
nuclear dynamics may affect the values of the β coefficients, and also lead to additional contributions to the
CSP coefficients. For example, the pion loop calculationin the NLO process above assumed that the intermediate nucleon propagator is “free”, while in reality it will be
modified by nuclear in-medium effects. Moreover, EDMs
of individual nucleons will lead to semileptonic operatorsthat do not reduce to the simple E · B nuclear polarisability — we now address these types of processes.
Semileptonic operators induced by nucleon EDMs. —Let us consider the semileptonic processes in Fig. 3that correspond to the exchange of two photons between
atomic electrons and nucleons, with internal nuclear excitations. In this case, we assume that the nucleons possess both magnetic (µ) and electric (d) dipole moments,
as defined in (4) for the latter. We consider the simplest
non-interacting Fermi-gas model of the nucleus, and single nucleon excitations. Integrating over the temporalcomponent of the loop momentum leads to the following
result per nucleon,
Z 3
4me αµd
GF (µd)
d k
√ CSP
≈
(11)
4 ,
2
3π
2
|k|
nucleonwhere k is the spatial part of the loop momentum. The
integral in this case is dominated by the residue near
the nucleon pole, with the virtual electron and photons
deeply off-shell. Note also that Schiff’s screening theorem [26] does not apply to the semileptonic processes
under consideration, since the interaction of one of thevirtual photons with the nucleon (or nucleus) is magnetic
in nature.
To generalise to the case of a nucleus with A  1 nucleons, we have to average the product µd over the p − ncontent of the nucleus, and evaluate the integral overthe spatial loop momenta. If the initial nucleon momentum is p, where in the Fermi-gas model |p| ≤ pF , thenthe intermediate nucleon momentum typically lies above
the Fermi surface, |p + q| ≥ pF . This provides the IR

4regularisation of the integral in (11) that can be readily
computed in terms of p = |p| and the Fermi momentum
pF ,



Z 3
4π
1
1
1+x
d k
+
ln
, x = p/pF .
4 = p2(1 − x2 ) 4x
1−x
F
|k|
(12)
Averaging over x with the normalised 3x2 dx distribution
in the interval 0 < x < 1 leaves a logarithmic divergenceas x → 1 from the first term in (12). We regularise
this average by taking into account the finite number ofnucleons in a nucleus, xmax ≈ 1 − 1/(3A):
*Z
+
1
d3 k
3
ln (A) ,
(13)
≈
4
4π
4p
F
|k|
where pF ≈ 250 MeV is the Fermi momentum of thenucleus. The contribution (11), generalised to a heavy
nucleus, then takes the form:√
N µn dn
8 2me α2 ln(A) Z µp dp
(µd)
+
(14)
CSP ≈
GF pF mN
A µN eA µN edp − dn≈ 3.4 × 10−11 ×,(15)
10−24 e cm
where we have averaged the product hµdi over all nucleons in the nucleus. (In the second line above, we havepresented an approximate expression for a heavy nucleus
with A ∼ 200 and Z/A ∼ 0.4, for which the orthogonal
combination dp + dn can be neglected.)
The result (14) does not exhaust all possible nuclear contributions. In particular, CP -violating hadronicsources will also induce contributions to the E · B polarisability from the near-outer-shell nucleons that will depend on the details of the nuclear structure. The calculation of such effects goes beyond the scope of this work.
We believe that the bulk contribution of nucleons, whichis enhanced in the atomic/molecular EDM by the factor
O(A), is adequately captured by our treatment above.
Constraints on CP-violating parameters. — We now
turn the current experimental limit on CSP given in (3)into constraints on the fundamental parameters characterising CP violation in the hadronic sector. Using theµ − d interactions of protons inside the nucleus (14), andneglecting dn which is already constrained directly, wederive a novel bound|dp |ThO < 2 × 10−23 e cm ,(16)

which is only a factor of 100 less stringent than the limit
derived from the constraint on the Hg EDM [12].
Next we address constraints related to the LO π 0 exchange, when the neutron and proton contributions add(1)
constructively, which is the case with the ḡπN N coupling.
We immediately obtain the limit(1)|ḡπN N | < 4 × 10−10 ,

(17)

(1)
that upon use of the QCD sum-rule estimate of ḡπN N (d˜q )[27] can be translated into a limit on the isovector combination of light-quark colour EDMs (CEDMs):|d˜u − d˜d | < 2 × 10−24 cm .

(18)

These limits are less stringent than the ones derived fromHg EDM experiments using constraints on the nuclear
Schiff moment. However, the most recent nuclear calculations of the dependence of the Hg Schiff moment on the(1)ḡπN N coupling indicate significant sensitivity to assumptions about the underlying nuclear structure [28], which
also propagates to substantial uncertainty in the limits
on CEDMs inferred from the Hg EDM bound. Our results in (17,18) provide a completely independent limit(1)
on ḡπN N , with less theoretical uncertainty.
We are now ready to address perhaps the most interesting quantity: the limit on the QCD vacuum angle
provided by EDM experiments in paramagnetic systems.
To this end, we can utilise the chiral-limit results for the(0)(0)
nucleon EDMs, ḡπN N and ḡηN N induced by θ̄:(0)gA ḡπN N eΛUVln,
4π 2 Fπ
mπ
m∗ θ̄
(0)
¯
ḡπN N = −hp|ūu − dd|pi
,
Fπ
m∗ θ̄
(0)
¯ − 2s̄s|pi ,hp|ūu + ddḡηN N = − √3Fη
dp(n) = −(+)

(19)
(20)
(21)

where m∗ = mu md /(mu + md ), and the strange quark
contribution to m∗ has been neglected. The UV cutoff ofthe chiral loop [29] can be taken to be ΛUV ∼ 4πFπ , andthe sub-log corrections have been neglected. [For a morein-depth treatment, one can use QCD sum-rule or latticeestimates of dN (θ̄).] The nucleon matrix elements are
known to some accuracy from hadron spectroscopy and¯lattice calculations. Using (md − mu )hp|ūu − dd|pi≈ 2.5¯
MeV, (md +mu )hp|ūu+ dd|pi/2≈ 38 MeV and hp|s̄s|pi ≈(0)¯0.1hp|ūu + dd|pi[30, 31], one finds ḡπN N ≈ −0.017θ̄, in(0)(0)
good agreement with e.g. [32, 33], and ḡηN N ≈ 5ḡπN N .
With these values, we observe that the LO contributionsof π 0 and η exchange to CSP almost cancel for the p − ncomposition of the Th nucleus, as well as other heavy
nuclei. Given the considerable degree of uncertainty inthe quark bilinear matrix elements, this cancellation can
suppress the naive π 0 exchange contribution by an orderof magnitude or more, rendering the LO result intrinsically very uncertain. However, we can combine the NLO
contribution together with the µ − d contribution to obtain the following prediction for a heavy nucleus withA ∼ 200 and Z/A ∼ 0.4 (which includes nuclei of experimental interest such as Th, Tl, Hg, Hf and Xe):CSP (θ̄) ≈ 0.1LO + 1.0NLO + 1.7(µd) × 10−2 θ̄ ≈ 0.03 θ̄ ,(22)
where the numbers in parentheses show the LO, NLO
and µ − d contributions to CSP , respectively. Each number here can vary by as much as 50% (or more in the case5of the LO contribution) upon varying ΛUV and other parameters. (We also note that the IR scale in the NLO
contribution, mπ , can be renormalised somewhat inside
the nucleus due, e.g., to Pauli blocking, and a shift in the
in-medium value for mπ .) With these caveats, the above
result translates to the following limit on the QCD vacuum angle,|θ̄|ThO . 3 × 10−8 .

(23)This is only a factor of about 100 less stringent than thelimit extracted from neutron EDM experiments.
Discussion. — In this Letter, we have shown that paramagnetic EDM experiments, by virtue of their dramatic
recent gains, are now exhibiting levels of sensitivity tohadronic sources of CP violation that are becoming competitive with experiments focusing directly on the nuclear
Schiff moment and the neutron EDM. When the sourceof CP violation is localised in the hadron sector, it iswell known that the top-quark/Higgs two-loop mechanism can give a large contribution to de [34]. On theother hand, as our paper demonstrates, when the main
mediation mechanism is via light quarks, as is the casewith the theta term and light-quark (C)EDMs, the main
pathway for communicating CP violation to the EDMsof paramagnetic systems is via the CSP operator in (1),while de can be neglected. This sensitivity arises throughthe two-photon generation of CSP that is coherently enhanced by the number of nucleons. We have considered
two distinct two-photon exchange mechanisms for generating such CP -violating semileptonic operators: (i) the
exchange of π 0 and η mesons between atomic electrons
and nucleons, as well as charged-pion loops generating
CP -odd nucleon polarisabilities, and (ii) CP -odd nuclear
excitations due to nucleon EDMs.
The most precise result in our analysis is the constraint(1)
on the triplet ḡπN N coupling, Eq. (17), where the main effect comes from π 0 exchange between unpaired electrons
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model of the nucleus, exploiting the coherent nature of
the effect, as CSP is contributed to by all nucleons inside
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Acknowledgments. — M.P. is grateful to Dr. A. DuttMazumder, his late friend for collaboration over a decadeago on unpublished work considering nucleon polarisabilities and EDMs. The authors also thank Dr. M. Voloshin
for helpful discussions. The work of M.P. and A.R. was
supported in part by NSERC, Canada. Research at the
Perimeter Institute is supported in part by the Government of Canada through NSERC and by the Provinceof Ontario through MEDT. Y.V.S. was supported by a
Humboldt Research Fellowship from the Alexander von
Humboldt Foundation and by the World Premier International Research Center Initiative (WPI), MEXT,
Japan.

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