Electric dipole moments of atoms, molecules, nuclei, and particles

A permanent electric dipole moment (EDM) of a particle or system is a separation of charge along its angular momentum axis and is a direct signal of $T$ violation and, assuming $CPT$ symmetry, $CP$ violation. For over 60 years EDMs have been studied, first as a signal of a parity-symmetry violation and then as a signal of $CP$ violation that would clarify its role in nature and in theory. Contemporary motivations include the role that $CP$ violation plays in explaining the cosmological matter-antimatter asymmetry and the search for new physics. Experiments on a variety of systems have become ever-more sensitive, but provide only upper limits on EDMs, and theory at several scales is crucial to interpret these limits. Nuclear theory provides connections from standard-model and beyond-standard-model physics to the observable EDMs, and atomic and molecular theory reveal how $CP$ violation is manifest in these systems. EDM results in hadronic systems require that the standard-model QCD parameter of $\overline{\theta }$ must be exceptionally small, which could be explained by the existence of axions, also a candidate dark-matter particle. Theoretical results on electroweak baryogenesis show that new physics is needed to explain the dominance of matter in the Universe. Experimental and theoretical efforts continue to expand with new ideas and new questions, and this review provides a broad overview of theoretical motivations and interpretations as well as details about experimental techniques, experiments, and prospects. The intent is to provide specifics and context as this exciting field moves forward.

Figure 1

The connections from a fundamental theory at a high-energy scale to an EDM in a measurable low-energy system. The dashed boxes indicate levels dominated by theory, and the solid boxes identify systems that are the object of current and future experiments. The fundamental $CP$-violating Lagrangian at the top, a combination of SM and BSM physics, is reduced to the set of effective-field-theory Wilson coefficients that characterize interactions at the electroweak energy scale of $\approx 300\mathrm{GeV}$, the vacuum expectation value of the Higgs. The set of low-energy parameters defined in Sec. 2 enters calculations that connect the electroweak-scale Wilson coefficients directly to electrons and nuclei. Finally atomic, molecular, and condensed-matter structure calculations connect the low-energy parameters to the observables in experimentally accessible systems.

Figure 2

Penguin diagram giving the SM CKM $\mathrm{\Delta }S=1$ $CP$-violating effective interaction. Adapted from [323].

Figure 3

Left side: bubble nucleation during first-order electroweak phase transition. Right side: $CP$- and $C$-violating interactions in the $⟨{\varphi }_H⟩=0$ background near the bubble walls that produce baryons. From [281].

Figure 4

Sensitivity of the (left panel) electron EDM and (right panel) neutron EDM to the baryon asymmetry in the MSSM. The horizontal axes give the bino soft mass parameter $M_1$; the vertical axes give the sine of the relative phase of $M_1$, the supersymmetric $\mu$ parameter, and the soft Higgs mass parameter $b$. The green bands indicate the values of these parameters needed to obtain the observed baryon asymmetry. Nearly horizontal lines give contours of constant EDMs. From [247].

Figure 5

Representative chiral loop contribution to the neutron EDM arising from SM CKM $CP$ violation. The $\otimes$ indicates a $CP$-violating $\mathrm{\Delta }S=1$ vertex such as that shown in Fig. 2, while the • corresponds to a $CP$-conserving $\mathrm{\Delta }S=1$ interaction. Adapted from [323].

Figure 6

One-loop fermion EDM generated by coupling to SUSY particles, $f^{\prime }$ and $\chi$. Adapted from [146].

Figure 7

An example of BSM couplings of the CEDM to the gluon field $g$. The crossed-circle indicates interactions that mix the left- and right-handed squarks. From [401].

Figure 8

Example two-loop Barr-Zee diagrams that give rise to a fermion EDM (coupling through $\gamma$) or CEDM (coupling through $g$). Here $\tilde{\tau }$ is the $\tau$ slepton, $\tilde{t}$ and $\tilde{b}$ are squarks, and $\chi$ is the chargino. Adapted from [51].

Figure 9

EDM results for the type II two-Higgs-doublet model. Horizontal axes show the ratio of up- and down-type Higgs vacuum expectation values. Vertical axes show the $CP$-violating phase that mixes of $CP$-even and $CP$-odd scalars. Purple regions are excluded by consistency with electroweak symmetry breaking. Left panel: Constraints from experiments prior to 2014 for $d_e$ (blue), $d_n$ (green), and $d_A{(}^{199}\mathrm{Hg})$ (pink). Middle panel: Impact of improving the experimental sensitivity by 1 order of magnitude, where the blue dashed line indicates the prospective reach of $d_e$. The yellow region indicates the reach of a future $d_A{(}^{225}\mathrm{Ra})$ with a sensitivity of ${10}^{-27}e\mathrm{cm}$. Right panel: The same future constraints but with $d_n$ having 2 orders of magnitude better sensitivity than the present limit. From [208].

Figure 10

Representative long-range, pion-exchange contributions to the neutron EDM. The cross represents the $CP$-violating vertex, while the dot is the $CP$-conserving vertex. Adapted from [322].

Figure 11

Magnetization of a permalloy box with equilibration coils. Left: Original L-shaped configuration with coils at the corners. Right: Distributed L-shaped configuration with coils arranged over the surfaces. The coils are solid lines. The color scale shows the magnetization in the permalloy in arbitrary units. From Z. Sun.

Figure 12

Shielding designed for a proton EDM experiment: (a) Permalloy and compensation coil configurations. The outer layers (1,3) are connected in segments. A correction ring (2) compensates for the magnetic distortions caused by the magnetic equilibration coils (4). The compensation ring equilibration coils are labeled (5). (b) Simulation of magnetic distortions caused by outer-cylinder equilibration coils mitigated by the compensation ring.

Figure 13

Comparison of noise in different shields measured with SQUIDs. “Supercon.” refers to a superconducting cylinder cooled below the transition temperature inside the low magnetic-field environment BMSR-2. The Zuse shield is also at PTB-Berlin ([409]). At very low frequencies, the intrinsic noise of the (different) SQUID systems combined and the integration time to record data for this plot dominate the performance; at high frequencies, the noise level is dominated by the experimental setup to perform the measurement. From M. Burghoff.

Figure 14

Sensitivity vs accuracy of sensors used for low-field magnetometry in EDM measurements in typical state-of-the-art configurations: flux-gate (FG), ${}^3\mathrm{He}$, alkali-metal magnetometers with Cs and K, spin-exchange-relaxation-free (SERF) mangetometers and combinations of ${}^3\mathrm{He}$ and Cs or SQUID readout. The term “sensitivity” is used to describe the response of the sensor to changes in the magnetic field, whereas “accuracy” is used here to describe the absolute accuracy for measuring fields, which is a measure of the stability, which is crucial for EDM experiments.

Figure 15

$B_0$ coil from the panEDM apparatus. The coil consists of a closed box of mumetal with a mumetal cylinder inside the shield. The main coil is wound around the cylinder (see text), which induces azimuthal magnetization in the cylindrical shell; correction coils at the ends of the cylinder compensate for the finite length of the windings and for return current paths. Additional correction coils indicated account for imperfections in the geometry.

Figure 16

UCN source at PF-2 at the Institut Max von Laue—Paul Langevin (ILL) in Grenoble, France. Neutrons from the low-energy tail of the cold-neutron spectrum are guided upward and lose energy in the gravitational potential. The turbine further shifts the spectrum to longer wavelengths to produce UCN that are provided to a number of experiments including the EDM. From communication group ILL.

Figure 17

Single phonon dispersion curve for SF-He with a minimum at $k\approx 2{\mathrm{nm}}^{-1}$ and the free neutron $E_n=(\hslash k{)}^2/2m_n$. The two curves intersect at $E_n=0$ and 1 meV corresponding to $k=0.7{\AA }^{-1}$ or ${\lambda }_0=8.9\AA$.

Figure 18

Schematic diagram of the ILL SUN2 source described in the text. Adapted from [246].

Figure 19

Neutron-EDM apparatus from (a) the ILL-Sussex-Rutherford experiment with one chamber for UCN and a comagnetometer. From [41]. (b) The Gatchina apparatus, as set up at ILL, with two neutron storage chambers so that parallel and antiparallel $E$ and $B$ field orientations are measured simultaneously. From [368]. Both experiments are run with the storage chamber in vacuum at room temperature.

Figure 20

Ramsey’s technique of separated oscillatory fields. The experiment starts out with polarized particles in a stable and uniform magnetic field $B_0$, with a stable external oscillator at frequency ${\omega }_R$ near the Larmor frequency ${\omega }_L$ of the particles in the $B_0$ field. First a $\pi /2$ pulse of oscillating magnetic field ($B_1$) rotates the polarization into the plane normal to $B_0$, creating a superposition of spin-up and spin-down states. The spins and external clock evolve independently until a second $\pi /2$ pulse is applied. The second pulse measures the phase difference between the oscillator and precessing spins that accumulates during the free-precession interval. Time evolves from top to bottom in the figure.

Figure 21

Dependence of the measured $E_z$-odd EDM signal on the magnetic-field gradient, represented by $R^{\prime }$, for two $B_z$ orientations. The $y$ axis labeled “EDM” refers to $d^{\mathrm{mea}s}$ in Eq. (97). The data points with error bars are from a typical data run. The solid red lines are linear fits to Eq. (100) for the data set with $B$ up and $B$ down. From [41].

Figure 22

Contributions to the geometric-phase effect. Top: A nonuniform magnetic field that leads to radial magnetic-field components. Bottom: Particle trajectories and contributions to $B_{\rho }$ from the gradient and motional magnetic fields. The left side shows a positive azimuthal velocity, and the right side shows a negative azimuthal velocity. The average frequency shift for the two counterpropagating trajectories is given by Eq. (96). Adapted from [311].

Figure 23

The effective electric field $E_{\mathrm{eff}}$ in YbF as a function of the applied electric field ($E_{\mathrm{lab}}$ in the text). The fully saturated $E_{\mathrm{eff}}$ corresponds to ${\mathcal{P}}_z=1$. From [200].

Figure 24

Level structure of a molecule in a ${{}^3\mathrm{\Delta }}_1$, $J=1$ state. The laboratory electric field $E_{\mathrm{lab}}$ splits the states by $\pm D_{el}{E}_{\mathrm{lab}}$ into molecular dipoles up and down. This results in the two orientations of the effective internal field $E_{\mathrm{eff}}$, which is directed toward the lighter atom. The specific structure for the ThO experiment is shown; for ${\mathrm{HfF}}^{+}J=3/2$ and the stretched states ($m=\pm 3/2$) move up and down similarly. In a magnetic field $\overrightarrow{B}$ parallel or antiparallel to ${\overrightarrow{E}}_{\mathrm{lab}}$, the $m=\pm 1$ states are further split by $\mu B$ as shown. The EDM $d$ further splits the $m=\pm 1$ states with $E_{\mathrm{eff}}>0$, but reduces the splitting with $E_{\mathrm{eff}}<0$ antiparallel to $B$.

Figure 25

The experimental layout of the ACME ThO experiment. From [48].

Figure 26

The experimental layout of the Seattle ${}^{199}\mathrm{Hg}$ experiment. From [180].

Figure 27

Estimates of intrinsic Schiff moments for octupole deformed nuclei. From [134].

Figure 28

The experimental layout of the ${}^{225}\mathrm{Ra}$ experiment. From [70].

Figure 29

Left: Radium energy-level diagram. Right: Isotope shifts for ${}^{225}\mathrm{Ra}$ ($I=1/2$) and ${}^{226}\mathrm{Ra}$ ($I=0$) relative to the isotopic average; (a). (b), (c) The specific levels of interest. From [70].

Figure 30

Phase-shift data showing the fraction of spin-down ${}^{225}\mathrm{Ra}$ nuclei after the nuclear-spin-precession cycle of $20+\delta$ s. From [70].

Figure 31

Electron EDM $d_e$ as a function of $C_S$ from the experimental results in ThO and ${\mathrm{HfF}}^{+}$ with $1\sigma$ experimental error bars. Also shown are 68% and 95% ${\chi }^2$ contours for all paramagnetic systems including Cs, Tl, and YbF. The top and right axes show the corresponding dimensionless Wilson coefficients ${\delta }_e$ and $\mathrm{Im}{C}_{eq}^{(-)}$ normalized to the squared scale ratio $(v/\mathrm{\Lambda }{)}^2$.

Figure 32

Combinations of hadronic parameters allowed by experimental results for the best values for ${\alpha }_{ij}$ in Table 4 with ${\alpha }_{\mathrm{Hg},{\overline{g}}_{\pi }^{(1)}}=1.6\times {10}^{-17}$ and ${\alpha }_{\mathrm{Ra},{\overline{d}}_n^{\mathrm{s}\mathrm{r}}}=-8\times {10}^{-4}$. The allowed values at 68% C.L. are contained within the ellipses for each pair of parameters.