We propose to search for axion dark matter via the oscillating electric dipole moments that axions induce in atoms and molecules. These moments are produced through the intrinsic oscillating electric dipole moments of nucleons and through the P, T-violating nucleon-nucleon interaction mediated by pion exchange, both of which arise due to the axion-gluon coupling, and also directly through the axion-electron interaction. Axion dark matter may also be sought for through the spin-precession effects that axions produce by directly coupling to fermion spins

Dzuba, Vladimir A.

Flambaum, Victor V.

Roberts, Benjamin M.

Stadnik, Yevgeny V.

English

University of Queensland eSpace

Searching for Axion Dark Matter in Atoms:
Oscillating Electric Dipole Moments and Spin-
Precession Effects
Benjamin M. Roberts1, Yevgeny V. Stadnik1, Victor V. Flambaum1,2, Vladimir A. Dzuba1
1 School of Physics, University of New South Wales, Sydney, Australia
2 Mainz Institute for Theoretical Physics, Johannes Gutenberg University, Mainz, Germany
DOI: http://dx.doi.org/10.3204/DESY-PROC-2015-02/roberts benjamin axions
We propose to search for axion dark matter via the oscillating electric dipole moments
that axions induce in atoms and molecules. These moments are produced through the
intrinsic oscillating electric dipole moments of nucleons and through the P, T -violating
nucleon-nucleon interaction mediated by pion exchange, both of which arise due to the
axion-gluon coupling, and also directly through the axion-electron interaction. Axion dark
matter may also be sought for through the spin-precession effects that axions produce by
directly coupling to fermion spins.
1 Introduction
Astrophysical observations indicate that the matter content of the Universe is overwhelmingly
dominated by dark matter (DM), the energy density of which exceeds that of ordinary matter
by a factor of five. In order to explain the observed abundance of DM, it is reasonable to expect
that DM interacts non-gravitationally with ordinary matter. Searches for weakly interacting
massive particle (WIMP) DM, which look for the scattering of WIMPs off nuclei, have not yet
produced a strong positive result. Further progress with these traditional searches is hindered by
the observation that the sought effects are fourth-power in the underlying interaction strength
between DM and Standard Model (SM) matter, which is known to be extremely small.
We propose to search for other well-motivated DM candidates that include the axion, which
may also resolve the strong CP problem of Quantum Chromodynamics (QCD), by exploiting
effects that are first-power in the interaction strength between the axion and SM matter (by
contrast, haloscope [1] and helioscope [2] methods look for second-power effects, while light-
shining-through-wall methods [3] look for fourth-power effects). We focus on the oscillating
electric dipole moments (EDMs) and spin-precession effects that axions induce in atoms and
molecules. There is strong motivation to search for axions in atomic and related systems via such
signatures — to date, static EDM measurements in atoms, molecules and ultracold neutrons
have served as sensitive probes of new physics beyond the Standard Model (see e.g. the reviews
[4, 5, 6]), while searches for sidereal spin-precession effects with atoms and ultracold neutrons
have placed stringent limits on CPT - and Lorentz-invariance-violating models (see e.g. Ref. [7]
for an overview).
Patras 2015 1120 Axion–WIMP 2015
2 Axion dark matter
Axions produced by the vacuum misalignment mechanism are very cold with no pressure.
Furthermore, if axions are sufficiently light and weakly interacting, then they may survive
until the present day and reside in the observed galactic DM haloes (where they have become
virialised over time with vvirial ∼ 10−3). The number density of ultralight (sub-eV) axions
per de Broglie wavelength readily exceeds unity, na/λ
3
dB  1, meaning that axions behave
as a coherently oscillating classical field, a(t) ' a0 cos(mat − pa · r) on time scales less than
τcoh ∼ 2pi/mav2virial and length scales less than lcoh ∼ 2pi/mavvirial. Thus the couplings of an
oscillating galactic axion field to SM particles produce a number of oscillating signatures which
can be sought for experimentally. As we will see below, the particularly interesting signatures
are those where the observables scale as O ∝ 1/fa with the axion decay constant fa.
The axion couplings to SM particles that are of most interest are the following:
Lint = a
fa
g2
32pi2
GG˜ −
∑
f=e,n,p
Cf
2fa
∂µa f¯γ
µγ5f, (1)
where the first term represents the coupling of the axion field to the gluonic field tensor G and
its dual G˜, and the second term represents the coupling of the derivative of the axion field to the
fermion axial-vector currents. Cf are dimensionless model-dependent coefficients. Typically,
|Cn| ∼ |Cp| ∼ 1 in models of the QCD axion [8]. Within the DFSZ model, where the tree level
coupling of the axion to the electron is non-vanishing, |Ce| ∼ 1 [8]. However, within the KSVZ
model, |Ce| ∼ 10−3, since the tree level coupling vanishes and the dominant effect arises at
the 1-loop level [8]. For more generic axion-like pseudoscalar particles, the coefficients Cf are
essentially free parameters, and the coupling to gluons is generally presumed absent.
Oscillating P,T-violating nuclear electromagnetic moments — The coupling of an
oscillating axion field to the gluon fields, which is described by the first term in Eq. (1), induces
an oscillating EDM of the neutron [9, 10],
dn(t) ' 1.2× 10−16 a0
fa
cos(mat) e · cm, (2)
which in turn induces oscillating P,T -violating nuclear electromagnetic moments. In nuclei,
a second and more efficient mechanism exists for the induction of oscillating electromagnetic
moments by axions — namely, the P,T -violating nucleon-nucleon interaction that is mediated
by pion exchange, with the axion field supplying the oscillating source of P and T violation at
one of the piNN vertices [10] (Fig. 1).
Oscillating atomic and molecular electric dipole moments — Axion-induced oscil-
lating P ,T -odd nuclear electromagnetic moments can in turn induce oscillating EDMs in atoms
and molecules. In diamagnetic species (J = 0), only oscillating nuclear Schiff moments (which
require I ≥ 1/2) produce an oscillating atomic/molecular EDM (oscillating nuclear EDMs are
effectively screened for typical axion masses, as a consequence of Schiff’s theorem [11]). Two
atoms that are of particular experimental interest are 199Hg and 225Ra, for which the axion
induces the following oscillating EDMs [10]:
d(199Hg) = −1.8× 10−19 a0
fa
cos(mat) e · cm, (3)
d(225Ra) = 9.3× 10−17 a0
fa
cos(mat) e · cm, (4)
2 Patras 2015
SEARCHING FOR AXION DARK MATTER IN ATOMS: OSCILLATING ELECTRIC . . .
Axion–WIMP 2015 121
Figure 1: Main process responsible for the induction of oscillating P ,T -odd nuclear electromag-
netic moments by an oscillating axion field. The black vertex on the left is due to the usual
strong P ,T -conserving piNN coupling (gpiNN = 13.5), while the magenta vertex on the right is
due to the axion-induced P ,T -violating piNN coupling (g¯piNN ' 0.027 cos(mat) a0/fa) [10].
with the large enhancement in 225Ra compared with 199Hg due to both collective effects and
small energy separation between members of the relevant parity doublet, which occurs in nu-
clei with octupolar deformation and results in a significant enhancement of the nuclear Schiff
moment [12, 13]. A possible platform to search for the oscillating EDMs of diamagnetic atoms
in ferroelectric solid-state media has been proposed in Ref. [14].
Paramagnetic species (J ≥ 1/2) offer more rich possibilities. Firstly, axion-induced oscil-
lating nuclear magnetic quadrupole moments (which require I ≥ 1) also produce an oscillating
atomic/molecular EDM [15], which is typically larger than that due to an oscillating nuclear
Schiff moment (since magnetic quadrupole moments are not subject to screening of the applied
electric field by atomic/molecular electrons). Secondly, an entirely different mechanism exists
for the induction of oscillating EDMs in paramagnetic species, through the direct interaction
of the axion field with atomic/molecular electrons via the second term in Eq. (1). The µ = 0
component of this second term mixes atomic/molecular states of opposite parity (with both
imaginary and real coefficients of admixture), generating the following oscillating atomic EDM
(due to the real coefficients of admixture) in the non-relativistic approximation for an S1/2 state
[10],
da(t) ∼ −Cea0m
2
aαs
faα
e sin(mat), (5)
where αs is the static scalar polarisability. Fully relativistic Hartree-Fock atomic calculations
are in excellent agreement with the scaling da ∝ αs in Eq. (5) [15, 16]. The imaginary coefficients
of admixture in the perturbed atomic wavefunction produce P -violating, T -conserving effects
in atoms, while the analogous imaginary coefficients of admixture in the perturbed nuclear
wavefunction (due to the axion-nucleon interaction via the µ = 0 component of the second
term in Eq. (1)) produce P -violating, T -conserving nuclear anapole moments [10, 15, 16].
Oscillating spin-precession effects — The coupling of an oscillating axion field to the
fermion axial-vector currents produces the following time-dependent non-relativistic potential
for a spin-polarised source, via the µ = 1, 2, 3 components of the second term in Eq. (1)
Hint(t) =
∑
f=e,n,p
Cfa0
2fa
sin(mat) σf · pa, (6)
which gives rise to spin-precession effects [10, 17, 18]. Deformation of the axion field by the
gravitational field of a massive body also produces a time-dependent potential of the form
Patras 2015 3
BENJAMIN M. ROBERTS, YEVGENY V. STADNIK, VICTOR V. FLAMBAUM, VLADIMIR . . .
122 Axion–WIMP 2015
H ′int(t) ∝ (Cfa0/fa) sin(mat) σf · rˆ, which is directed towards the centre of the gravitating
body [10]. These spin-precession effects can be sought for using a wide range of spin-polarised
systems, including atomic co-magnetometers, ultracold neutrons and torsion pendula. The
nucleon spin contents for nuclei of experimental interest have been performed in Ref. [19] for
the accurate interpretation of laboratory measurements.
Acknowledgments
This work was supported by the Australian Research Council. B. M. R. and V. V. F. are
grateful to the Mainz Institute for Theoretical Physics (MITP) for its hospitality and support.
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4 Patras 2015
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Searching for axion dark matter in atoms: oscillating electric dipole moments and spin-precession effects

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