Nuclear Schiff Moments and CP Violation

A suggestion to improve upper limits on the existence of dipole moments in atoms is discussed. Parity and time reversal violating components in the nuclear force may produce P, T-odd moments in nuclei which, in turn, induce such moments in atoms. In order to improve the sensitivity, one must use atoms with a high Z. Atoms with nuclei that have both quadrupole and octupole deformations in the ground state have enhanced Schiff moments when there are forces that violate the time reversal and reflection symmetries. These Schiff moments produce enhanced dipole moments in the atom.

The current status of electric dipole moments of diamagnetic atoms which involves the synergy between atomic experiments and three different theoretical areas -- particle, nuclear and atomic is reviewed. Various models of particle physics that predict CP violation, which is necessary for the existence of such electric dipole moments, are presented. These include the standard model of particle physics and various extensions of it. Effective hadron level combined charge conjugation (C) and parity (P) symmetry violating interactions are derived taking into consideration different ways in which a nucleon interacts with other nucleons as well as with electrons. Nuclear structure calculations of the CP-odd nuclear Schiff moment are discussed using the shell model and other theoretical approaches. Results of the calculations of atomic electric dipole moments due to the interaction of the nuclear Schiff moment with the electrons and the P and time-reversal (T) symmetry violating tensor-pseudotensor electron-nucleus are elucidated using different relativistic many-body theories. The principles of the measurement of the electric dipole moments of diamagnetic atoms are outlined. Upper limits for the nuclear Schiff moment and tensor-pseudotensor coupling constant are obtained combining the results of atomic experiments and relativistic many-body theories. The coefficients for the different sources of CP violation have been estimated at the elementary particle level for all the diamagnetic atoms of current experimental interest and their implications for physics beyond the standard model is discussed. Possible improvements of the current results of the measurements as well as quantum chromodynamics, nuclear and atomic calculations are suggested.

Electric dipole moments of nucleons, nuclei, and atoms: The Standard Model and beyond

The search for the manifestations of new physics beyond the standard model in atomic and nuclear phenomena is pursued by several experimental groups. The discovery of the atomic electric dipole moment (EDM) would reveal the simultaneous violation of parity and time-reversal invariance. The EDM can be induced by -odd forces between the nucleus and atomic electrons. As the nuclear dipole moment is screened according to the Schiff theorem, the appropriate nuclear operator is the Schiff moment that may exist in nuclei under -violation. We briefly review the current experimental situation and discuss more in detail the ideas concerning possible collective mechanisms for the enhancement of the nuclear Schiff moment. The most promising directions are related to the coexistence of octupole and quadrupole collective modes, either in the form of static deformation or as soft vibrational excitations. The search for enhancement is important for widening the pool of nuclei as candidates for the atomic EDM as well as for development of nuclear many-body theory beyond standard mean-field and random phase approximations.

We have implemented analytical second-moment gradients for Hartree-Fock and multiconfigurational self-consistent-field wave functions. The code is used to calculate atomic dipole moments based on the generalized atomic polar tensor (GAPT) formalism [Phys. Rev. Lett. 62, 1469 (1989)], and the proposal of Dinur and Hagler (DH) for the calculation of atomic multipoles [J. Chem. Phys. 91, 2949 (1989)]. Both approaches display smooth basis-set convergence toward a well-defined basis-set limit and give reasonable electron correlation effects on the calculated atomic properties. However, the atomic charges and atomic dipole moments obtained from the GAPT partitioning scheme are unable to provide even qualitatively meaningful molecular quadrupole moments for some molecules, and thus the atomic multipole moments calculated in this scheme cannot be considered well suited for analyzing the electron density in molecules and for calculating intermolecular interaction energies. In contrast, the DH approach gives atomic charges and dipole moments that by definition exactly reproduce the molecular quadrupole moments. The approach of DH is, however, restricted to planar molecules and thus suffers from not being applicable to molecules of arbitrary shape. Both the GAPT and DH approaches give rather poor results for octupole and hexadecapole moments, indicating that at least atomic quadrupole moments are required for an accurate representation of the molecular charge distribution in terms of atomic electric moments.

New fundamental particles at the mass scale of a few TeV c–2 could account for observed phenomena that cannot be explained by the standard model (SM) of particle physics, including the microscopic origin of dark matter and the macroscopic imbalance of matter over antimatter in the Universe. However, no beyond-the-SM (BSM) particles at the TeV scale have yet been detected at the Large Hadron Collider (LHC). With recent innovations, searches for time-reversal symmetry (T) violation through low-energy precision measurements of electric dipole moments (EDMs) of atoms and molecules have attained the sensitivity to detect indirect signatures of certain particles with masses of more than 10 TeV c–2. In this Perspective, we discuss recent developments in the measurement and interpretation of EDMs, and assess proposed techniques for future experiments that could push experimental limits on T-violating BSM physics to the PeV scale. Electric dipole moments (EDMs) of atoms and molecules are a sensitive probe for sources of time-reversal symmetry violation beyond the standard model. In this Perspective, we review recent progress in EDM searches with atoms and molecules and survey proposed next-generation experiments.

Searches for permanent electric dipole moments (EDMs) of fundamental particles, atoms and molecules are promising experiments to constrain and potentially reveal beyond Standard Model (SM) physics. A non-zero EDM is a direct manifestation of time-reversal (T) violation, and, equivalently, violation of the combined operation of charge-conjugation (C) and parity inversion (P). Identifying new sources of CP violation can help to solve fundamental puzzles of the SM, e.g., the observed baryon-asymmetry in the Universe. Theoretical predictions for magnitudes of EDMs in the SM are many orders of magnitude below current experimental limits. However, many theories beyond the SM require larger EDMs. Experimental results, especially when combined in a global analysis, impose strong constraints on CP violating model parameters. Including an overview of EDM searches, I will focus on the future neutron EDM experiment at TRIUMF (Vancouver). For this effort, the TUCAN (TRIUMF Ultra Cold Advanced Neutron source) collaboration is aiming to build a strong, world leading source of ultra cold neutrons (UCN) based on a unique combination of a spallation target and a superfluid helium UCN converter. Another focus will be the search for an EDM of the diamagnetic atom 129 Xe using a 3 He comagnetometer and SQUID detection. The HeXeEDM collaboration has taken EDM data in 2017 and 2018 in the magnetically shielded room (BMSR-2) at PTB Berlin.

Parity- and time-invariance-violating $(P,T$-odd) nuclear forces produce $P,T$-odd nuclear moments. In turn, these moments can induce electric dipole moments (EDMs) in atoms through the mixing of electron wave functions of opposite parity. The nuclear EDM is screened by atomic electrons. The EDM of an atom with closed electron subshells is induced by the nuclear Schiff moment. Previously the interaction with the Schiff moment has been defined for a pointlike nucleus. No problems arise with the calculation of the electron matrix element of this interaction as long as the electrons are considered to be nonrelativistic. However, a more realistic model obviously involves a nucleus of finite size and relativistic electrons. In this work we have calculated the finite nuclear size and relativistic corrections to the Schiff moment. The relativistic corrections originate from the electron wave functions and are incorporated into a ``nuclear'' moment, which we term the local dipole moment. For ${}^{199}\mathrm{Hg}$ these corrections amount to $\ensuremath{\sim}25%.$ We have found that the natural generalization of the electrostatic potential of the Schiff moment for a finite-size nucleus corresponds to an electric-field distribution which, inside the nucleus, is well approximated as constant and directed along the nuclear spin, and outside the nucleus is zero. Also in this work the ${}^{239}\mathrm{Pu}$ atomic EDM is calculated.

We examine the time-reversal-violating nuclear ``Schiff moment'' that induces electric dipole moments in atoms. After presenting a self-contained derivation of the form of the Schiff operator, we show that the distribution of Schiff strength, an important ingredient in the ground-state Schiff moment, is very different from the electric-dipole-strength distribution, with the Schiff moment receiving no strength from the giant dipole resonance in the Goldhaber-Teller model. We then present shell-model calculations in light nuclei that confirm the negligible role of the dipole resonance and show the Schiff strength to be strongly correlated with low-lying octupole strength. Next, we turn to heavy nuclei, examining recent arguments for the strong enhancement of Schiff moments in octupole-deformed nuclei over that of ${}^{199}\mathrm{Hg},$ for example. We concur that there is a significant enhancement while pointing to effects neglected in previous work (both in the octupole-deformed nuclides and ${}^{199}\mathrm{Hg})$ that may reduce it somewhat, and emphasizing the need for microscopic calculations to resolve the issue. Finally, we show that static octupole deformation is not essential for the development of collective Schiff moments; nuclei with strong octupole vibrations have them as well, and some could be exploited by experiment.

A local atomic electric dipole moment distribution of Si atoms on Si(111)-(7 x 7) surface is clearly resolved by using a new technique called noncontact scanning nonlinear dielectric microscopy. The dc-bias voltage dependence of the atomic dipole moment on the Si(111)-(7 x 7) surface is measured. At the weak applied voltage of -0.5 V, a positive dipole moment is detected on the Si adatom sites, whereas a negative dipole moment is observed at the interstitial sites of inter Si adatoms. Moreover, the quantitative dependence of the surface dipole moment as a function of the applied dc voltage is also revealed at a fixed point above the sample surface. This is the first successful demonstration of direct atomic dipole moment observation achieved in the field of capacitance measurement.

We have calculated the atomic electric dipole moments (EDMs) $d$ of $^{3}\mathrm{He}$ and $^{171}\mathrm{Yb}$ induced by their respective nuclear Schiff moments $S$. Our results are $d(^{3}\mathrm{He})=8.3\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}5}$ and $d(^{171}\mathrm{Yb})=\ensuremath{-}1.9$ in units of ${10}^{\ensuremath{-}17}(S∕e\phantom{\rule{0.2em}{0ex}}{\mathrm{fm}}^{3})\phantom{\rule{0.3em}{0ex}}e\phantom{\rule{0.2em}{0ex}}\mathrm{cm}$. By considering the nuclear Schiff moments induced by the parity- and time-reversal violating nucleon-nucleon interaction, we find $d(^{171}\mathrm{Yb})\ensuremath{\sim}0.6d(^{199}\mathrm{Hg})$. For $^{3}\mathrm{He}$ the nuclear EDM coupled with the hyperfine interaction gives a larger atomic EDM than the Schiff moment. The result for $^{3}\mathrm{He}$ is required for a neutron EDM experiment that is under development, where $^{3}\mathrm{He}$ is used as a comagnetometer. We find that the EDM for $^{3}\mathrm{He}$ is orders of magnitude smaller than the neutron EDM. The result for $^{171}\mathrm{Yb}$ is needed for the planning and interpretation of experiments that have been proposed to measure the EDM of this atom.

We report the atomic electric dipole moment induced by the P, T violating interactions in the nuclear/sub-nuclear level, for 207Pb2+ and 207Pb, owing to the recent interest in the ferroelectric crystal PbTiO3 as one of the candidates for investigating macroscopic P, T-odd effects. In this paper, we calculate the atomic electric dipole moments of 207Pb and Pb2+, parametrized in terms of the P, T-odd coupling parameter, the nuclear Schiff moment (NSM), S, in the frame-work of the coupled-perturbed Hartree–Fock theory. We estimate the Schiff moment of Pb2+ using the experimental result of a system, which is electronically similar to the Pb2+ ion. We present the dominant contributions of the electric dipole moment (EDM) matrix elements and the important correlation effects contributing to the atomic EDM of Pb2+. Our results provide the first ever calculated EDM of the Pb2+ ion, and an estimate of its NSM from which the P, T-odd energy shift in a PbTiO3 crystal can be evaluated.

Parity and time invariance violating (P,T-odd) nuclear forces produce P,T-odd nuclear moments. In turn, these moments can induce electric dipole moments (EDMs) in atoms through the mixing of electron wave functions of opposite parity. The nuclear EDM is screened by atomic electrons. The EDM of an atom with closed electron subshells is induced by the nuclear Schiff moment. Previously the interaction with the Schiff moment has been defined for a point-like nucleus. No problems arise with the calculation of the electron matrix element of this interaction as long as the electrons are considered to be non-relativistic. However, a more realistic model obviously involves a nucleus of finite-size and relativistic electrons. In this work we have calculated the finite nuclear-size and relativistic corrections to the Schiff moment. The relativistic corrections originate from the electron wavefunctions and are incorporated into a new “nuclear” moment, which we term the local dipole moment. For 199Hg these corrections amount to ∼20%. We have found that the natural generalization of the electrostatic potential of the Schiff moment for a finite-size nucleus corresponds to an electric field distribution which, inside the nucleus, is well approximated as constant and directed along the nuclear spin, and outside the nucleus is zero.

Parity and time invariance violating ($P,T$-odd) atomic electric dipole moments (EDM) are induced by interaction between atomic electrons and nuclear $P,T$-odd moments which are produced by $P,T$-odd nuclear forces. The nuclear EDM is screened by atomic electrons. The EDM of a non-relativistic atom with closed electron subshells is induced by the nuclear Schiff moment. For heavy relativistic atoms EDM is induced by the nuclear local dipole moments which differ by 10-50% from the Schiff moments calculated previously. We calculate the local dipole moments for ${^{199}{\rm Hg}}$ and ${^{205}{\rm Tl}}$ where the most accurate atomic and molecular EDM measurements have been performed.

Parity (P) and time reversal (T) violating nuclear forces create P, T -odd moments in expansion of the nuclear electrostatic potential. We derive expression for the nuclear electric octupole field which includes the electron screening correction (similar to the screening term in the Schiff moment). Then we calculate the Z alpha corrections to the Schiff moment which appear due to the finite nuclear size. Such corrections are important in heavy atoms with nuclear charge Z > 50. The Schiff and octupole moments induce atomic electric dipole moments (EDM) and P, T -odd interactions in molecules which are measured in numerous experiments to test CP-violation theories.