Nuclear Electric Dipole Moment Research Articles

A survey of nuclear quadrupole deformation in order to estimate the nuclear MQM and its relative contribution to the atomic EDM

Measurement of a non-zero permanent electric dipole moment (EDM) in fundamental particles, such as in an electron or a neutron, or in nuclei or atoms, can help us gain a handle on the sources of Charge-Parity (CP) violation, both in the Standard Model (SM) and beyond. The nuclear magnetic quadrupole moment (MQM), the central topic of this work, is also CP, P, and T violating. Nucleons and nuclei have a non-zero MQM from sources within the SM, but the nuclear MQM is dramatically enhanced if the nuclei are structurally quadrupole deformed. Multiple sources contribute to an atomic EDM namely: (i) nuclear EDM through its Schiff moment, which is enhanced by nuclear octupole deformation, (ii) CP violating interactions between the electrons and the nuclei, and (iii) the nuclear MQM that contributes to the atomic EDM in atoms with an unpaired valence electron. Our survey of nuclear quadrupole deformation identified 48 isotopes as ideal systems in which to search for a CP violating EDM via their enhanced nuclear MQM. Of these candidates, 223,225\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$^{223,225}$$\\end{document}Fr, 223\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$^{223}$$\\end{document}Ra, 223,225,227\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$^{223,225,227}$$\\end{document}Ac, 229\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$^{229}$$\\end{document}Th, and 229\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$^{229}$$\\end{document}Pa also have maximally enhanced nuclear Schiff moment contribution due to their octupole deformation. Laser cooling of the isotopes of Fr and Ra, among a few others, has already been demonstrated, making 223,225\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$^{223,225}$$\\end{document}Fr and 223\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$^{223}$$\\end{document}Ra some of the best systems in which to measure an EDM.

Exploring CP violation beyond the Standard Model and the PQ quality with electric dipole moments

In some models of physics beyond the Standard Model (SM), one of the leading low energy consequences of the model appears in the form of the chromo-electric dipole moments (CEDMs) of the gluons and light quarks. We examine if these CEDMs can be distinguished from the QCD θ-term through the experimentally measurable nuclear and atomic electric dipole moments (EDMs) in both cases with and without the Peccei-Quinn (PQ) mechanism solving the strong CP problem. We find that the nucleon EDMs show a distinctive pattern when the EDMs are dominantly induced by the light quark CEDMs without the PQ mechanism. In the presence of the PQ mechanism, the QCD θ-parameter corresponds to the vacuum value of the axion field, which might be induced either by CEDMs or by UV-originated PQ breaking other than the QCD anomaly, for instance the PQ breaking by quantum gravity effects. We find that in case with the PQ mechanism the nucleon EDMs have a similar pattern regardless of what is the dominant source of EDMs among the CEDMs and θ-term, unless there is a significant cancellation between the contributions from different sources. In contrast, some nuclei or atomic EDMs can have characteristic patterns significantly depending on the dominant source of EDMs, which may allow identifying the dominant source among the CEDMs and θ-term. Yet, discriminating the gluon CEDM from the QCD θ-parameter necessitates additional knowledge of low energy parameters induced by the gluon CEDM, which is not available at the moment. Our results imply that EDMs can reveal unambiguous sign of CEDMs while identifying the origin of the axion vacuum value, however it requires further knowledge of low energy parameters induced by the gluon CEDM.

Screening of the electric field and nuclear electric dipole moment in nonstationary states of atoms and molecules

Screening of the electric field and nuclear electric dipole moment in nonstationary states of atoms and molecules

Nuclear DFT analysis of electromagnetic moments in odd near doubly magic nuclei

We use the nuclear density functional theory to determine nuclear electric quadrupole and magnetic dipole moments in all one-particle and one-hole neighbours of eight doubly magic nuclei. We align angular momenta along the intrinsic axial-symmetry axis with broken time-reversal symmetry, which allows us to explore fully the self-consistent charge, spin, and current polarisation. Spectroscopic moments are determined for symmetry-restored wave functions and compared with available experimental data. We find that the obtained polarisations do not call for using quadrupole- or dipole-moment operators with effective charges or effective g-factors.

Contribution of the Weinberg-type operator to atomic and nuclear electric dipole moments

The contribution of the CP violating three-gluon Weinberg operator, frac{1}{3!}w{f}^{abc}{epsilon}^{nu rho alpha beta}{G}_{mu nu}^a{G}_{alpha beta}^b{G}_{rho}^{cmu} , to the atomic and nuclear EDMs is estimated using QCD sum rules. After calculating the transition matrix element between the pion and the vacuum through the Weinberg operator, we obtain the long-range CP-odd nuclear force by determining the isovector CP-odd pion-nucleon vertex, using chiral perturbation theory at NLO. The EDMs of 199Hg, 129Xe, and 225Ra atoms, as well as those of 2H and 3He nuclei are finally given including comprehensive uncertainty analysis. While the leading contribution of the 199Hg EDM is given by the intrinsic nucleon EDM, that of 129Xe atom may be dominated by the one-pion exchange CP-odd nuclear force generated by the Weinberg operator. From current experimental data of the 199Hg atomic EDM, we obtain an upper limit on the Weinberg operator magnitude of |w| < 4 × 10−10GeV−2 if we assume that it is the only source of CP violation at the scale μ = 1 TeV.

Electric dipole moment of light nuclei in chiral effective field theory

CP-violating interactions at quark level generate CP-violating nuclear interactions and currents, which could be revealed by looking at the presence of a permanent nuclear electric dipole moment. Within the framework of chiral effective field theory, we discuss the derivation of the CP-violating nuclear potential up to next-to-next-to leading order (N2LO) and the preliminary results for the charge operator up to next-to leading order (NLO). Moreover, we introduce some renormalization argument which indicates that we need to promote the short-distance operator to the leading order (LO) in order to reabsorb the divergences generated by the one pion exchange. Finally, we present some selected numerical results for the electric dipole moments of 2H, 3He and 3H discussing the systematic errors introduced by the truncation of the chiral expansion.

Ab initio calculations of electric dipole moments of light nuclei

In any finite system, the presence of a non-zero permanent electric dipole moment (EDM) would indicate CP violation beyond the small violation predicted in the Standard Model. Here, we use the ab initio no-core shell model (NCSM) framework to theoretically investigate the magnitude of the nuclear EDM. We calculate EDMs of several light nuclei using chiral two- and three-body interactions and a PT-violating Hamiltonian based on a one-meson-exchange model. We present a benchmark calculation for $^3$He, as well as results for the more complex nuclei $^{6,7}$Li, $^9$Be, $^{10,11}$B, $^{13}$C, $^{14,15}$N, and $^{19}$F. Our results suggest that different nuclei can be used to probe different terms of the PT violating interaction. These calculations allow us to suggest which nuclei may be good candidates in the search for a measurable permanent electric dipole moment.

Screening of nucleon electric dipole moments in atomic systems

The electric dipole moments (EDMs) of diamagnetic atoms are expected to be sensitive to charge-parity violation particularly in nuclei through the nuclear Schiff moment. I explicitly demonstrate that the well-known form of the Schiff moment operator originating from the nucleon EDM is obtained by considering the screening of the nucleon EDMs in a neutral atom. Consequently, an additional contribution to the Schiff moment arises from the screening of the nuclear EDM induced by the interaction of the nucleon EDMs with the protons. This correction to the Schiff moment of $^{199}$Hg is evaluated in the independent particle model.

Atomic and molecular transitions induced by axions via oscillating nuclear moments

The interaction of standard model's particles with the axionic Dark Matter field may generate oscillating nuclear electric dipole moments (EDMs), oscillating nuclear Schiff moments and oscillating nuclear magnetic quadrupole moments (MQMs) with a frequency corresponding to the axion's Compton frequency. Within an atom or a molecule an oscillating EDM, Schiff moment or MQM can drive transitions between atomic or molecular states. The excitation events can be detected, for example, via subsequent fluorescence or photoionization. Here we calculate the rates of such transitions. If the nucleus has octupole deformation or quadrupole deformation then the transition rate due to Schiff moment and MQM can be up to $10^{-16}$ transition per molecule per year. In addition, an MQM-induced transition may be of M2-type, which is useful for the elimination of background noise since M2-type transitions are suppressed for photons.

Electric field on nucleus in solids and interaction of CP-violating nuclear electric dipole moment with phonons

In atoms and molecules, the electrons screen the nucleus from the external electric field. However, if the frequency of the electric field reaches the energy of atomic or molecular transition, the electric field at the nucleus may be resonantly enhanced by many orders in magnitude. In this paper, we study the mechanisms of screening or enhancement of electric field acting on the nuclei in solid states. We show that in dielectric crystals the phonon oscillations of the lattice play crucial role for determining the electric field on nuclei in the MHz-THz region. As an application, we propose an experimental scheme for measuring nuclear electric dipole moment in a solid state based on the coherent excitation of acoustic phonons.

Nuclear EDM from SMEFT flavor-changing operator

We study nuclear electric dipole moments induced by ∆F = 1 effective operators in the Standard Model Effective Field Theory. Such contributions arise through renormalization group evolutions and matching conditions at the electroweak symmetry breaking scale. We provide one-loop formulae for the matching conditions. We also discuss correlations of these effects with ∆F = 2 observables such as ϵK and Delta {M}_{B_d} .

Oscillating nuclear electric dipole moment induced by axion dark matter produces atomic and molecular electric dipole moments and nuclear spin rotation

According to the Schiff theorem, a static nuclear electric dipole moment (EDM) does not produce atomic and molecular EDMs. However, interaction with the axion dark matter field generates nuclear EDMs ${\mathbf{d}}_{N}={\mathbf{d}}_{N}^{0}\mathrm{cos}(\ensuremath{\omega}t)$ which oscillate with the frequency $\ensuremath{\omega}={m}_{a}{c}^{2}/\ensuremath{\hbar}$. These oscillating nuclear EDMs generate atomic and molecular EDMs proportional to ${\ensuremath{\omega}}^{2}$. The resulting atomic and molecular EDMs do not, however, lead to nuclear spin rotation in constant external electric fields. On the other hand, an oscillating electric field penetrates atoms and molecules and can interact with nuclear EDMs. Moreover, if the nuclear EDM oscillation frequency is in resonance with the external field, nuclear spin rotation happens, with the rotation angle growing linearly with time.

Screening and enhancement of an oscillating electric field in molecules

According to the Schiff theorem, the atomic electrons completely screen the atomic nucleus from an external static electric field. However, this is not the case if the field is time dependent. Electronic orbitals in atoms either shield the nucleus from an oscillating electric field when the frequency of the field is off the atomic resonances or enhance this field when its frequency approaches an atomic transition energy. In molecules, not only electronic but also rotational and vibrational states are responsible for the screening of oscillating electric fields. As will be shown in this paper, the screening of a low-frequency field inside molecules is much weaker than it appears in atoms owing to the molecular rovibrational states. We systematically study the screening of oscillating electric fields inside diatomic molecules in different frequency regimes, i.e., when the field's frequency is either of the order of rovibrational or electronic transition frequencies. In the resonance case, we demonstrate that the microwave-frequency electric field may be enhanced up to six orders in magnitude due to rovibrational states. We also derive the general formulas for the screening and resonance enhancement of oscillating electric field in polyatomic molecules. Possible applications of these results include nuclear electric dipole moment measurements and stimulation of nuclear reactions by laser light.

Time reversal violating magnetic quadrupole moment in heavy deformed nuclei

The existence of permanent electric dipole moments (EDMs) and magnetic quadrupole moments (MQMs) violate both time reversal invariance ($T$) and parity ($P$). Following the $CPT$ theorem they also violate combined $CP$ symmetry. Nuclear EDM are completely screened in atoms and molecules while interaction between electrons and MQM creates atomic and molecular EDM which can be measured and used to test CP-violation theories. Nuclear MQM are produced by the nucleon-nucleon $T,P$-odd interaction and by nucleon EDM. In this work we study the effect of enhancement of the nuclear MQM due to the nuclear deformation. Using the Nilsson model we calculate the nuclear MQMs for deformed nuclei of experimental interest and the resultant MQM energy shift in diatomic molecules of experimental interest $^{173}$YbF , $^{177,179}$HfF$^+$, $^{181}$TaN, $^{181}$TaO$^+$, $^{229}$ThO and $^{229}$ThF$^+$.

Shielding of an external oscillating electric field inside atoms

According to the Schiff theorem, an external electric field vanishes at the atomic nucleus in a neutral atom, i.e., it is completely shielded by electrons. This makes a nuclear electric dipole moment (EDM) unobservable. In this paper an extension of the Schiff theorem to an oscillating electric field is considered. Such a field can reach the nucleus and interact with the nuclear EDM. An enhancement effect appears if the field is in resonance with an atomic or molecular transition. The shielding by electrons strongly affects low-energy nuclear electric dipole transition amplitudes in different nuclear reactions, including radiative transitions, radiative nucleon capture, photo- or electro-excitation of nuclei, and laser-induced or laser-enhanced nuclear reactions.

Screening of an oscillating external electric field in atoms

We study the screening of a homogeneous oscillating external electric field $E_0$ in noble-gas atoms using atomic many-body calculations. At zero frequency of the oscillations ($\omega=0$) the screened field $E(r)$ vanishes at the nucleus, $E(0)=0$. However, the profile of the field $E(r)$ is complicated, with the magnitude of the field exceeding the external field $E_0$ at certain points. For $\omega >0$ the field $E(r,\omega)$ strongly depends on $\omega$ and at some points may exceed the external field $E_0$ many times. The field at the nucleus is not totally screened and grows with $\omega$ faster than $\omega^2$. It can even be enhanced when $\omega$ comes close to resonance with a frequency of an atomic transition. This field interacts with CP-violating nuclear electric dipole moments creating new opportunities for studying them. The screening of the external field by atomic electrons may strongly suppress (or enhance near an atomic resonance) the low energy nuclear electric dipole transitions.

Lorentz-violating contributions to the nuclear Schiff moment and nuclear EDM

In the context of an atom endowed with nuclear electric dipole moment (EDM), we consider the effects on the Schiff moment of $CPT$-even Lorentz-violating (LV) terms that modify the Coulomb potential. First, we study the modifications on the Schiff moment when the nucleus interacts with the electronic cloud by means of a Coulomb potential altered only by the $P$-even LV components. Next, by supposing the existence of an additional intrinsic LV EDM generated by other LV sources, we assess the corrections to the Schiff moment when the interaction nucleus-electrons runs mediated by a Coulomb potential modified by both the $P$-odd and $P$-even LV components. We then use known estimates and EDM measurements to discuss upper bounds on the new Schiff moment components and the possibility of an intrisic nuclear EDM component ascribed to LV effects.

Lattice QCD spectroscopy for hadronic CP violation

The interpretation of nuclear electric dipole moment (EDM) experiments is clouded by large theoretical uncertainties associated with nonperturbative matrix elements. In various beyond-the-Standard Model scenarios nuclear and diamagnetic atomic EDMs are expected to be dominated by CP-violating pion-nucleon interactions that arise from quark chromo-electric dipole moments. The corresponding CP-violating pion-nucleon coupling strengths are, however, poorly known. In this work we propose a strategy to calculate these couplings by using spectroscopic lattice QCD techniques. Instead of directly calculating the pion-nucleon coupling constants, a challenging task, we use chiral symmetry relations that link the pion-nucleon couplings to nucleon sigma terms and mass splittings that are significantly easier to calculate. In this work, we show that these relations are reliable up to next-to-next-to-leading order in the chiral expansion in both SU(2) and SU(3) chiral perturbation theory. We conclude with a brief discussion about practical details regarding the required lattice QCD calculations and the phenomenological impact of an improved understanding of CP-violating matrix elements.

Permanent electric dipole moments of single-, two- and three-nucleon systems

A nonzero electric dipole moment (EDM) of the neutron, proton, deuteron or helion, in fact, of any finite system necessarily involves the breaking of a symmetry, either by the presence of external fields (i.e., electric fields leading to the case of induced EDMs) or explicitly by the breaking of the discrete parity and time-reflection symmetries in the case of permanent EDMs. We discuss two theorems describing these phenomena and report about the cosmological motivation for an existence of [Formula: see text] breaking beyond what is generated by the Kobayashi–Maskawa mechanism in the Standard Model and what this might imply for the permanent EDMs of the nucleon and light nuclei by estimating a window of opportunity for physics beyond what is currently known. Recent — and in the case of the deuteron even unpublished — results for the relevant matrix elements of nuclear EDM operators are presented and the relevance for disentangling underlying New Physics sources is discussed.

Standard model contribution to the electric dipole moment of the deuteron, 3H, and 3He nuclei

We calculate for the first time the electric dipole moment (EDM) of the deuteron, 3H, and 3He nuclei generated by the one-meson exchange CP-odd nuclear force in the standard model. The effective |ΔS| = 1 four-quark operators are matched to the |ΔS| = 1 standard model processes involving the CP phase of the Cabibbo-Kobayashi-Maskawa matrix at the electroweak scale and run down to the hadronic scale μ = 1 GeV according to the renormalization group evolution in the next-to-leading logarithmic order. At the hadronic scale, the hadron matrix elements are modeled in the factorization approach. We then obtain the one-meson (pion, eta meson, and kaon) exchange CP-odd nuclear force, which is the combination of the |ΔS| = 1 meson-baryon vertices which issue from the penguin operator and the hyperon-nucleon transition. From this CP-odd nuclear force, the nuclear EDM is calculated with the realistic Argonne v18 interaction and the CP-odd nuclear force using the Gaussian expansion method. It is found that the EDMs of light nuclear systems are of order O (10−31)e cm. We also estimate the standard model contribution to other hadronic CP violating observables such as the EDMs of 6Li, 9Be nuclei, and the atomic EDMs of 129Xe, 199Hg, 211Rn, and 225Ra generated through the nuclear Schiff moment. We then analyze the source of theoretical uncertainties and show some possible ways to overcome them.