Quark chromo-EDM Research Articles

Enhanced Schiff and magnetic quadrupole moments in deformed nuclei and their connection to the search for axion dark matter

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 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 $T$,$P$-violating nuclear forces. This mixing polarises the orientation of the nuclear axis along the nuclear spin, and all moments existing in the intrinsic frame appear in the laboratory frame (provided the nuclear spin $I$ is sufficiently large to allow such a moment). Such a mechanism produces enhanced $T$,$P$-violating nuclear moments. This enhancement also takes place in nuclei with a soft octupole vibration mode. In this paper we present updated estimates for the enhanced Schiff moment in isotopes of Eu, Sm, Gd, Dy, Er, Fr, Rn, Ac, Ra, Th, Pa, U, Np and Pu in terms of the CP-violating $\pi$-meson--nucleon interaction constants $\bar{g}_{0},\bar{g}_{1}$ and $\bar{g}_{2}$, the QCD parameter $\bar{\theta}$ and the quark chromo-EDMs. The implications of the enhanced $T$,$P$-violating moments to the search for axion dark matter in solid state experiments are also discussed, with potential alternative candidate compounds in which we may expect enhanced effects suggested.

Laser-coolable AcOH+ ion for CP -violation searches

The ${\mathrm{AcOH}}^{+}$ molecular ion is identified as a prospective system to search for $\mathcal{CP}$-violation effects. According to our study ${\mathrm{AcOH}}^{+}$ belongs to the class of laser-coolable polyatomic molecular cations implying a 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 the experimentally measurable $\mathcal{T},\mathcal{P}$-violating energy shift in terms of fundamental quantities such as the nuclear magnetic quadrupole moment (MQM), electron electric dipole moment ($e\mathrm{EDM}$) and dimensionless scalar-pseudoscalar nuclear-electron interaction constant. We further express nuclear MQM in terms of the strength constants of $\mathcal{CP}$-violating nuclear forces: quantum chromodynamics vacuum angle $\overline{\ensuremath{\theta}}$ and quark chromo-EDMs. The equilibrium structure of ${\mathrm{AcOH}}^{+}$ in the ground and the four lowest excited electronic states was found to be linear. The calculated Franck-Condon factors and transition dipole moments indicate that the laser cooling using an optical cycle involving the first excited state is possible for the trapped ${\mathrm{AcOH}}^{+}$ ions with the Doppler limit estimated to be $\ensuremath{\sim}$4 nK. The lifetime of the ($0,{1}^{1},0$) excited vibrational state considered as a working one for MQM and $e\mathrm{EDM}$ search experiments is estimated to be $\ensuremath{\sim}0.4$ s.

Parity and time-reversal invariance violation in neutron-nucleus scattering

Parity violating (PV) as well as parity and time-reversal invariance violating (PTRIV) effects are enhanced a million times in neutron reactions near $p$-wave compound resonances. We present the calculation of such effects using a statistical theory based on the properties of chaotic eigenstates and discuss a possibility to extract the strength constants of PTRIV interactions from the experimental data, including nucleon-nucleon and pion-nucleon CP-violating interactions, the QCD $\ensuremath{\theta}$-term, and the quark chromo-EDM. PV effects have random signs for all target nuclei except for $^{232}\mathrm{Th}$, where PV effects of a positive sign have been observed for ten statistically significant $p$-wave resonances, with energy smaller than 250 eV. This may be an indication of a possible regular (nonchaotic) contribution to PV effects. We link this regular effect to the doublets of opposite parity states in the rotation spectra of nuclei with an octupole deformation and suggest other target nuclei where this hypothesis may be tested. We also discuss a permanent sign contribution produced by doorway states. An estimate of the ratio of PTRIV effects to PV effects is presented. Although a polarized target is not needed for the measurement of PV effects, for the interpretation of the results, it may be convenient to do both PV and PTRIV experiments with a polarized target.

Effects of CP -violating internucleon interactions in paramagnetic molecules

We demonstrate that electron electric dipole moment experiments with molecules in paramagnetic state are sensitive to $P,T$-violating nuclear forces and other $CP$-violating parameters in the hadronic sector. These experiments, in particular, measure the coupling constant $C_{SP}$ of the $CP$-odd contact semileptonic interaction. We establish relations between $C_{SP}$ and different $CP$-violating hadronic parameters including strength constants of the $CP$-odd nuclear potentials, $CP$-odd pion-nucleon interactions, quark-chromo EDM and QCD vacuum angle. These relations allow us to find limits on various $CP$-odd hadronic parameters.

Limits on CP-violating hadronic interactions and proton EDM from paramagnetic molecules

Experiments with paramagnetic ground or metastable excited states of molecules (ThO, HfF+, YbF, YbOH, BaF, PbO, etc.) provide strong constraints on the electron electric dipole moment (EDM) and the coupling constant CSP of contact semileptonic interaction. We compute new contributions to CSP arising from the nucleon EDMs due to the 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, |dp| < 1.1 × 10−23e·cm, the QCD vacuum angle, left|overline{theta}right| < 1.4 × 10−8, as well as the quark chromo-EDMs and the π-meson-nucleon couplings. Our results may also be used to search for the axion dark matter which produces oscillating overline{theta} .

Electric dipole moments of atoms and molecules produced by enhanced nuclear Schiff moments

We perform calculations of the CP-violating atomic and molecular electric dipole moments (EDM) induced by the interaction of the nuclear Schiff moments with electrons. EDM of atoms Eu, Dy, Gd, Ac, Th, Pa, U, Np and Pu are of special interest since they have isotopes with strongly enhanced nuclear Schiff moments caused by the octupole nuclear deformation or soft octupole vibration mode. These atoms have open 4f or 5f shells making the calculations complicated. We use our special version of the configuration interaction method combined with the many-body perturbation theory method adopted for open f-shell case. To validate the method we perform similar calculations for simpler atoms (Xe, Hg, Tl+, Pb, Pb++, Rn, Ra) where some earlier results are available. In addition we present the estimates of the CP-violating nuclear spin - molecular axis interaction constants for molecules which may be of experimental interest including AcF, EuN, ThO, PbO and TlF. We also present updated values of the nuclear Schiff moments and atomic and molecular EDM expressed in terms of the CP-violating pi-meson - nucleon interaction constants g_0,g_1, g_2, QCD parameter theta and quark chromo-EDMs. The results may be used to test CP-violation theories and search for axion dark matter in atomic, molecular and solid state experiments.

Time-reversal invariance violation in neutron-nucleus scattering

Planning and interpretation of the experiments searching for the time reversal (T) and parity (P) violation in neutron reactions require values of the matrix elements of the T,P-violating nuclear forces between nuclear compound states. We calculate the root mean square values and the ratio of the matrix elements of the T,P-violating and P-violating interactions using statistical theory based on the properties of chaotic compound states and present the results in terms of the fundamental parameters in four different forms: in terms of the constants of the contact nuclear interaction, meson exchange constants, QCD theta-term and quark chromo-EDMs. Using current limits on these parameters, we obtain the upper bounds on the ratio of the matrix elements.

Evaluation of CP violation in HfF+

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 $\theta$ and quark chromo-EDMs.

Neutron Electric Dipole Moment on the Lattice

For the neutron to have an electric dipole moment (EDM), the theory of nature must have T, or equivalently CP, violation. Neutron EDM is a very good probe of novel CP violation in beyond the standard model physics. To leverage the connection between measured neutron EDM and novel mechanism of CP violation, one requires the calculation of matrix elements for CP violating operators, for which lattice QCD provides a first principle method. In this paper, we review the status of recent lattice QCD calculations of the contributions of the QCD Θ-term, the quark EDM term, and the quark chromo-EDM term to the neutron EDM.

Lattice calculation of electric dipole moments and form factors of the nucleon

We analyze commonly used expressions for computing the nucleon electric dipole form factors (EDFF) $F_3$ and moments (EDM) on a lattice and find that they lead to spurious contributions from the Pauli form factor $F_2$ due to inadequate definition of these form factors when parity mixing of lattice nucleon fields is involved. Using chirally symmetric domain wall fermions, we calculate the proton and the neutron EDFF induced by the CP-violating quark chromo-EDM interaction using the corrected expression. In addition, we calculate the electric dipole moment of the neutron using background electric field that respects time translation invariance and boundary conditions, and find that it decidedly agrees with the new formula but not the old formula for $F_3$. Finally, we analyze some selected lattice results for the nucleon EDM and observe that after the correction is applied, they either agree with zero or are substantially reduced in magnitude, thus reconciling their difference from phenomenological estimates of the nucleon EDM.

Cancellation mechanism in the predictions of electric dipole moments

The interpretation of the baryon asymmetry of the Universe necessitates the CP violation beyond the Standard Model (SM). We present a general cancellation mechanism in the theoretical predictions of the electron electric dipole moments (EDM), quark chromo-EDMs, and Weinberg operators. A relative large CP violation in the Higgs sector is allowed by the current electron EDM constraint released by the ACME collaboration in 2013, and the recent $^{199}$Hg EDM experiment. The cancellation mechanism can be induced by the mass splitting of heavy Higgs bosons around $\sim\mathcal{O}(0.1-1)$ GeV, and the extent of the mass degeneracy determines the magnitude of the CP-violating phase. We explicate this point by investigating the CP-violating two-Higgs-doublet model and the minimal supersymmetric Standard Model. The cancellation mechanism is general when there are CP violation and mixing in the Higgs sector of new physics models. The CP-violating phases in this scenario can be excluded or detected by the projected $^{225}$Ra EDM experiments with precision reaching $\sim10^{-28}\,e\cdot{\rm cm}$, as well as the future colliders.

Enhanced effect of CP -violating nuclear magnetic quadrupole moment in a HfF+ molecule

The ${\mathrm{HfF}}^{+}$ cation is a very promising system to use in the search for the electron electric dipole moment (EDM), and a corresponding experiment is carried out by JILA group [H. Loh, K. C. Cossel, M. C. Grau, K.-K. Ni, E. R. Meyer, J. L. Bohn, J. Ye, and E. A. Cornell, Science 342, 1220 (2013); K.-K. Ni, H. Loh, M. Grau, K. C. Cossel, J. Ye, and E. A. Cornell, J. Mol. Spectrosc. 300, 12 (2014). Here we theoretically investigate the cation to search for another effect which violates time-reversal ($T$) and spatial parity ($P$) symmetries---the nuclear magnetic quadrupole moment (MQM) interaction with electrons. We report an accurate ab initio relativistic electronic structure calculations of the molecular parameter ${W}_{M}=0.494 \frac{{10}^{33}\phantom{\rule{0.16em}{0ex}}\mathrm{Hz}}{e\phantom{\rule{0.16em}{0ex}}{\mathrm{cm}}^{2}}$ that is required to interpret the experimental data in terms of the MQM of the 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. We discuss interpretation of the MQM effect in terms of the strength constants of $T,P$-odd nuclear forces, proton and neutron EDMs, the QCD parameter $\ensuremath{\theta}$, and quark chromo-EDMs.

The Contribution of Novel CP Violating Operators to the nEDM using Lattice QCD

In this talk, we motivate the calculation of the matrix elements of novel CP violating operators, the quark EDM and the quark chromo EDM operators, within the nucleon state using lattice QCD. These matrix elements, combined with the bound on the neutron EDM, would provide stringent constraints on beyond the standard model physics, especially as the next generation of neutron EDM experiments reduce the current bound. We then present our lattice strategy for the calculation of these matrix elements, in particular we describe the use of the Schr\"odinger source method to reduce the calculation of the 4-point to 3-point functions needed to evaluate the quark chromo EDM contribution. We end with a status report on the quality of the signal obtained in the lattice calculations of the connected contributions to the quark chromo EDM operator and the pseudoscalar operator it mixes with under renormalization.

Relativistic Many-body Calculations for Electric Dipole Moments in \(^{129}\)Xe, \(^{199}\)Hg, \(^{223}\)Rn, \(^{225}\)Ra and \(^{171}\)Yb Atom

We present and compare the results of permanent electric dipole moments (EDMs) of various closed-shell atoms due to the nuclear Schiff moment (NSM) and the tensor-pseudotensor (T-PT) interactions between the atomic nuclei and electrons. In order to highlight the role of electron-correlation effects in obtaining accurate EDM results, we employ a number of relativistic many-body methods including coupled-cluster theory at different degrees of approximation. On combining our results obtained from the relativistic coupled-cluster (RCC) at the levels of singles and doubles excitations (CCSD method) with the available EDM measurements we obtain accurate bounds on the couplings \(S\) and \(C_T\) associated with the respective NSM and T-PT interactions. The most precise EDM measurement on \(^{199}\)Hg in combination with our CC results yield limits on the above couplings as \(S&lt;1.45 \times 10^{-12}|e|\)fm\(^3\) and \(C_T &lt; 2.09 \times 10^{-9}\) respectively. Further combining these bounds with the latest nuclear structure and quantum chromodynamics calculations we infer limits on the strong CP-violating parameter and for the combined up- and down- quark chromo-EDMs as \(|\bar{\theta}| &lt; 1.1 \times 10^{-9}\) and \(|\widetilde{d}_u - \widetilde{d}_d| &lt; 2.8 \times 10^{-26} |e|\)cm, respectively.

TaN molecule as a candidate for the search for aT,P-violating nuclear magnetic quadrupole moment

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 nuclear forces, proton and neutron EDM, QCD parameter $\theta$ and quark chromo-EDM.

Rigorous limits on the hadronic and semileptonicCP-violating coupling constants from the electric dipole moment ofHg199

Relativistic many-body methods at different levels of approximations are employed to gain insights into the passage of the electron correlations from lower to higher levels in the accurate determination of the electric dipole polarizability ($\alpha_d$) and electric dipole moment (EDM) due to the electron-nucleus tensor-pseudotensor (T-PT) and nuclear Schiff moment (NSM) interactions of the $^{199}$Hg ground state. Moreover, plausible reasons for the differences in the previous calculations are pointed out. Comparison between the calculated and experimental results of $\alpha_d$ indicates that our EDM calculations are about 3\% accurate which in combination with the measured value of $^{199}$Hg EDM yield limits for the T-PT coupling constant $C_T < 2.31 \times 10^{-9}$ and the NSM $S<1.55 \times 10^{-12}|e|fm^3$. From this limit of $S$ and the latest nuclear structure calculation, we get improved limits for the strong CP violating parameter $\theta_{QCD} < 2.12 \times 10^{-10}$ and the combined up and down quark chromo-EDMs $(-0.004 \tilde{d}_u + 0.020 \tilde{d}_d) < 1.15 \times 10^{-27} |e| cm$.

Time-reversal symmetry violation in molecules induced by nuclear magnetic quadrupole moments.

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 diamagnetic TlF molecule, where the T, P-odd effects are produced by the nuclear Schiff moment. We argue that measurements in molecules with MQMs may provide improved limits on the strength of T, P-odd nuclear forces, on the proton, neutron, and quark EDMs, on quark chromo-EDMs, and on the QCD θ term and CP-violating quark interactions.

Electric dipole moment of theρmeson

At a hadronic scale the effect of CP-violating interactions that typically appear in extensions of the Standard Model may be described by an effective Lagrangian, in which the operators are expressed in terms of lepton and partonic gluon and quark fields and ordered by their mass dimension, $k\ensuremath{\geqslant}4$. Using a global-symmetry-preserving truncation of QCD Dyson-Schwinger equations, we compute the $\ensuremath{\rho}$-meson's electric dipole moment (EDM), ${d}_{\ensuremath{\rho}}$, as generated by the leading dimension-four and dimension-five CP-violating operators and an example of a dimension-six four-quark operator. The two dimension-five operators, i.e., the quark EDM and quark chromo-EDM, produce contributions to ${d}_{\ensuremath{\rho}}$ whose coefficients are of the same sign and within a factor of 2 in magnitude. Moreover, should a suppression mechanism be verified for the $\ensuremath{\theta}$ term in any beyond-Standard-Model theory, the contribution from a four-quark operator can match the quark EDM and quark chromo-EDM in importance. This study serves as a prototype for the more challenging task of computing the neutron's EDM.

CEDM Constraints on Modified Sfermion Universality and Spontaneous CP Violation

We discuss the supersymmetric CP problem that arises when the sfermion soft mass universality is modified. We place the 3rd generation SU(5) ten-plet sfermion masses in the weak scale in view of the naturalness. The other sfermion masses are assumed to be universal and a TeV scale in order to weaken the flavor changing neutral current processes and electric dipole moment (EDM) constraints. However this modification generically induces too large up quark chromo-EDM (CEDM) via the weak scale stop loop. In order to suppress this CEDM, we propose certain type of flavor structure where the parameters of the up-(s)quark sector are real whereas those of the down-(s)quark and the charged (s)lepton sectors are complex at the GUT scale. It is shown that, in this set up, up quark CEDM can be suppressed within the range where the current and future experiments have their sensitivity. We briefly illustrate the simple realization of these particular forms of the modified sfermion universality with real up-(s)quark sector by spontaneous CP violation in E$_6$ SUSY GUT with SU(2) flavor symmetry.