HfF$^+$ cation is a very promising system to search for the electron electric dipole moment (EDM), and corresponding experiment is carried out by E. Cornell group. Here we theoretically investigate the cation to search for another T,P-odd effect -- the nuclear magnetic quadrpole moment (MQM) interaction with electrons. We report the first accurate ab initio relativistic electronic structure calculations of the molecular parameter $W_M$=0.494 $\frac{10^{33}\mathrm{Hz}}{e~{\rm cm}^2}$ that is required to interpret the experimental data in terms of the MQM of Hf nucleus. For this we have implemented and applied the combined Dirac-Coulomb(-Gaunt) and relativistic effective core potential approaches to treat electron correlation effects from all of the electrons and to take into account high-order correlation effects using the coupled cluster method with single, double, triple and noniterative quadruple cluster amplitudes, CCSDT(Q). We discuss interpretation of the MQM effect in terms of the strength constants of T,P-odd nuclear forces, proton and neutron EDM, QCD parameter $θ$ and quark chromo-EDM.
Nuclei with a quadrupole deformation such as $^{177}$Hf have enhanced weak quadrupole moment which induces the tensor weak electron-nucleus interaction in atoms and molecules. Corresponding parity non-conserving (PNC) effect is strongly enhanced in the $^3Δ_1$ electronic state of the $^{177}$HfF$^+$ cation which has very close opposite parity levels mixed by this tensor interaction. In the present paper we perform relativistic many-body calculations of this PNC effect. It is shown that the tensor weak interaction induced by the weak quadrupole moment gives the dominating contribution to the PN…
Updated constraints on $\mathcal{T}$,$\mathcal{P}$-violating axionlike-particle-mediated electron-electron and electron-nucleus interactions from HfF$^+$ experiment
CP violation effects produced by the nuclear magnetic quadrupole moment (MQM), electron electric dipole moment (EDM) and scalar$-$pseudoscalar nucleus$-$electron neutral current (SP) interaction in $^{177}$Hf$^{19}$F$^+$ and $^{179}$Hf$^{19}$F$^+$ are calculated. The role of the hyperfine interaction is investigated. It is shown that the MQM shift can be distinguished from the electron EDM and SP ones due to the implicit dependence of MQM shift on the hyperfine sublevel. The MQM effect is expressed in terms of the proton (EDM), QCD vacuum angle $θ$ and quark chromo-EDMs.
Parity- and time-reversal-symmetry violating interaction constants required for the interpretation of a recent measurement (arXiv:1704.07928 [physics.atom-ph]) of corresponding symmetry violations in the $Ω=1$ (${^3Δ}_1$) science state of the HfF$^+$ molecular ion are reported. Using a relativistic four-component all-electron multi-reference configuration interaction model the nucleon-electron scalar-pseudoscalar interaction constant is determined as $W_S = 20.0$ [kHz]. An updated result for the electron electric-dipole-moment effective electric field of $|E_{\text{eff}}| = 22.7 \left[\frac{\r…
The existence of permanent electric dipole moments (EDMs) and magnetic quadrupole moments (MQMs) violate both time reversal invariance ($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 nucle…
The CP-violating interaction of the nuclear magnetic quadrupole moment (MQM) of the $^{175}$Lu nucleus with electrons in the molecular cation LuOH$^+$ is studied. The resulting effect is expressed in terms of CP-odd parameters, such as quantum chromodynamics angle $\barθ$, quark electric dipole moment (EDM) and chromo-EDM. For this we have performed a calculation of the nuclear MQM as well as the molecular constant that characterises the interaction of this MQM of $^{175}$Lu with electrons. Additionally, we predict the hyperfine structure constants for the ground electronic state of LuOH$^+$. …
Recent measurements in paramagnetic molecules improved the limit on the electron electric dipole moment (EDM) by an order of magnitude. Time-reversal (T) and parity (P) symmetry violation in molecules may also come from their nuclei. We point out that nuclear T,P-odd effects are amplified in paramagnetic molecules containing deformed nuclei, where the primary effects arise from the T,P-odd nuclear magnetic quadrupole moment (MQM). We perform calculations of T,P-odd effects in the molecules TaN, ThO, ThF$^+$, HfF$^+$, YbF, HgF, and BaF induced by MQMs. We compare our results with those for the …
Nuclear magnetic quadrupole moments (MQMs), like intrinsic electric dipole moments of elementary particles, violate both parity and time-reversal symmetry and therefore probe physics beyond the Standard Model of particle physics. We report on accurate relativistic coupled cluster calculations of the nuclear MQM interaction constants in BaF, YbF, BaOH, and YbOH. We elaborate on estimates of the uncertainty of our results. The implications of experiments searching for nonzero nuclear MQMs are discussed.
Experiments with paramagnetic ground or metastable excited states of molecules (ThO, HfF$^+$, YbF, YbOH, BaF, PbO, etc.) provide strong constraints on electron electric dipole moment (EDM) and coupling constant $C_{SP}$ of contact semileptonic interaction. We compute new contributions to $C_{SP}$ arising from the nucleon EDMs due to combined electric and magnetic electron-nucleon interaction. This allows us to improve limits from the experiments with paramagnetic molecules on the $CP$-violating parameters, such as the proton EDM, $|d_p|< 1.1\times 10^{-23} e\cdot $cm, the QCD vacuum angle, $|\…
CP-violating interaction of the nuclear magnetic quadrupole moment (MQM) with electrons in the ytterbium mono-hydroxide molecule, $^{173}$YbOH, is considered. Both the magnetic quadrupole moment (MQM) of the $^{173}$Yb nucleus and the molecular interaction constant WM are estimated. Electron correlation effects are taken into account within the relativistic Fock-space coupled cluster method. Results are interpreted in terms of the strength constants of CP-violating nuclear forces, neutron EDM, QCD vacuum angle $θ$, quark EDM and chromo-EDM.
Nuclei with an octupole deformation have a non-zero electric octupole moment, electric dipole moment (EDM), Schiff moment and magnetic quadrupole moment (MQM) in the intrinsic frame which rotates with the nucleus. In a state with definite angular momentum in the laboratory frame, these moments are forbidden by parity (P) and time reversal invariance (T) conservation, meaning their expectation values vanish due to nuclear rotation. However, nuclei with an octupole deformation have doublets of close opposite parity rotational states with the same spin, which are mixed by T,P-odd nuclear forces. …
Deformed nuclei possess enhanced moments violating time reversal invariance ($T$) and parity ($P$). Collective magnetic quadrupole moments (MQM) appear in nuclei with a quadrupole deformation (which have ordinary $T$,$P$-conserving collective electric quadrupole moments). Nuclei with an octupole deformation have a collective electric octupole moment, electric dipole moment (EDM), Schiff moment and MQM in the intrinsic frame which rotates with the nucleus. In a state with definite angular momentum in the laboratory frame, these moments are forbidden by $T$ and $P$ conservation, meaning their ex…
Collective effects in deformed atomic nuclei present possible avenues of study on the non-spherical distribution of neutrons and the violation of the local Lorentz invariance. We introduce the weak quadrupole moment of nuclei, related to the quadrupole distribution of the weak charge in the nucleus. The weak quadrupole moment produces tensor weak interaction between the nucleus and electrons and can be observed in atomic and molecular experiments measuring parity nonconservation. The dominating contribution to the weak quadrupole is given by the quadrupole moment of the neutron distribution, t…
Theories that extend the Standard Model of particle physics often introduce new interactions that violate charge-parity (CP) symmetry. CP-violating effects within an atomic nucleus can be probed by measuring its nuclear magnetic quadrupole moment (MQM). The sensitivity of such a measurement is enhanced when using a heavy polar molecule containing a nucleus with quadrupole deformation. We determine how the energy levels of a molecule are shifted by the MQM and how those shifts can be measured. The measurement scheme requires molecules in a superposition of magnetic sub-levels that differ by man…
Nuclear forces violating parity and time reversal invariance P,T-odd produce P,T-odd nuclear moments, for example, the nuclear Schiff moment. In turn, this moment can induce the electric dipole moment (EDM) in the atom. The contribution to the Schiff moment from the soft collective quadrupole and octupole vibrations in spherical nuclei is calculated in the framework of the quasiparticle random phase approximation with separable quadrupole and octupole forces. The values of nuclear Schiff moments predicted for odd 217-221Ra and 217-221Ra isotopes indicate a possibility of enhancement by a facto…
Local Lorentz Invariance violating (LLIV) and Einstein equivalence principle violating (EEPV) effects in atomic experiments are discussed. The EEPV effects are strongly enhanced in the narrow 7.8 eV transition in 229Th nucleus. Nuclear LLIV momentum tensors describing anisotropy in the maximal attainable speed for massive particles (analog of Michelson-Morley experiment for light) are expressed in terms of the experimental values of nuclear quadrupole moments. Calculations for nuclei of experimental interest 133Cs, 85Rb, 87Rb, 201Hg, 131Xe and 21Ne have been performed. The results for 21Ne are…
There are several factors which lead to a huge enhancement of parity and time invariance violating effects in the Ra atom: very close electronic levels of opposite parity, the large nuclear charge Z and the collective nature of T,P-odd nuclear moments. Experiments with Radium may be used to measure it's nuclear anapole, magnetic quadrupole and Schiff moments. Such measurements provide information about parity and time invariance violating nuclear forces and electron-nucleon interactions.
Investigations of CP violation in hadron sector may be done using measurements in the ThO molecule. Recent measurements in this molecule improved the limit on electron EDM by an order of magnitude. Another time reversal (T) and parity (P) violating effect in $^{229}$ThO is induced by the nuclear magnetic quadrupole moment. We have performed nuclear and molecular calculations to express this effect in terms of the strength constants of T,P-odd nuclear forces, neutron EDM, QCD vacuum angle $θ$, quark EDM and chromo-EDM.
The AcOH${}^+$ molecular ion is identified as a prospective system to search for $\mathcal{CP}$-violation effects. According to our study AcOH${}^+$ belongs to the class of laser-coolable polyatomic molecular cations implying the large coherence time in the experiments to study symmetry violating effects of fundamental interactions. We perform both nuclear and high level relativistic coupled cluster electronic structure calculations to express experimentally measurable $\mathcal{T}$,$\mathcal{P}$-violating energy shift in terms of fundamental quantities such as the nuclear magnetic quadrupole …
All-electron configuration interaction theory in the framework of the Dirac-Coulomb Hamiltonian has been applied to the TaN molecule, a promising candidate in the search for Beyond-Standard-Model physics in both the hadron and the lepton sector of matter. We obtain in the first excited {$^3Δ_1$} state a ${\cal{P,T}}$-odd effective electric field of $36.0 \left[\frac{\rm GV}{\rm cm}\right]$, a scalar-pseudoscalar ${\cal{P,T}}$-odd interaction constant of $32.8$ [kHz], and a nuclear magnetic-quadrupole moment interaction constant of $0.74$ [$\frac{10^{33} {\text{Hz}}}{e\, {\text{cm}}^2}$]. The l…
It is demonstrated that the TaN molecule is the best candidate to search for T,P-violating nuclear magnetic quadrupole moment (MQM), it also looks promising to search for other T,P-odd effects. We report results of coupled-cluster calculations of T,P-odd effects in TaN produced by the Ta nucleus MQM, electron electric dipole moment (EDM), scalar$-$pseudoscalar nucleus$-$electron interactions, also of the molecule-axis hyperfine structure constant and dipole moment. Nuclear calculations of $^{181}$Ta MQM are performed to express the T,P-odd effect in terms of the strength constants of T,P-odd n…
We propose methods and present calculations that can be used to search for evidence of cosmic fields by investigating the parity-violating effects, including parity nonconservation amplitudes and electric dipole moments, that they induce in atoms. The results are used to constrain important fundamental parameters describing the strength of the interaction of various cosmic fields with electrons, protons, and neutrons. Candidates for such fields are dark matter (including axions) and dark energy, as well as several more exotic sources described by standard-model extensions. Existing parity nonc…
Parity (P) and time (T) invariance violating effects in the Ra atom are strongly enhanced due to close states of opposite parity, the large nuclear charge Z and the collective nature of P,T-odd nuclear moments. We have performed calculations of the atomic electric dipole moments (EDM) produced by the electron EDM and the nuclear magnetic quadrupole and Schiff moments. We have also calculated the effects of parity non-conservation produced by the nuclear anapole moment and the weak charge. Our results show that as a rule the values of these effects are much larger than those considered so far i…
The TaO$^+$ molecular ion is proposed as a candidate system for detecting signatures of charge parity (${\cal{CP}}$) violating physics beyond the standard model of elementary particles. The electron electric dipole moment (EDM) effective electric field $E_{\text{eff}} = 20.2 \left[\frac{\rm GV}{\rm cm}\right]$, the nucleon-electron scalar-pseudoscalar (ne-SPS) interaction constant $W_{S} = 17.7$ [kHz] and the nuclear magnetic quadrupole interaction constant $W_M = 0.45$ [$\frac{10^{33} {\text{Hz}}}{e\, {\text{cm}}^2}$] are found to be sizeable ${\cal{P,T}}$-odd enhancements. The ratio of the l…
We propose a method for measuring the magnetic dipole (M1) transition moment of the $6S_{1/2} \big(\mathrm{m}=-1/2\big)\leftrightarrow 5D_{3/2}\big(\mathrm{m}=-1/2\big)$ transition in single trapped Ba$^{+}$ by exploiting different symmetries in the electric quadrupole (E2) and M1 couplings between the states. The technique is adapted from a previously proposed method for measuring atomic parity nonconservation in a single trapped ion [Norval Fortson, Phys. Rev. Lett. \textbf{70}, 17 (1993)]. Knowledge of M1 is crucial for any parity nonconservation measurement in Ba$^{+}$, as laser coupling t…
We employ the $Z$-vector method in the four-component relativistic coupled-cluster framework to calculate the parity (${\mathcal{P}}$) and time-reversal (${\mathcal{T}}$) symmetry violating scalar-pseudoscalar (S-PS) nucleus-electron interaction constant ($W_\text{s}$), the effective electric field ($E_\text{eff}$) experienced by the unpaired electron and the nuclear magnetic quadrupole moment (NMQM)-electron interaction constant ($W_\text{M}$) in the open-shell ground electronic state of HgF. The molecular frame dipole moment and the magnetic hyperfine structure (HFS) constant of the molecule…
