Reducing the electronic-structure uncertainties in P- and P,T-violating enhancement factors for LaO, LaS, and LuO

The relativistic coupled-cluster calculation of the parity (P) and time-reversal (T) violating enhancement factors $W_{\mathrm{d}}$ , $W_{\mathrm{s}}$ , $W_{\mathrm{M}}$ , and $W_{\mathrm{A}}$ for the LaO, LaS, and LuO molecules (Chamorro et al., 2024) carries total uncertainties of up to about 7%, dominated by basis-set incompleteness and by the neglect of connected triple and higher excitations beyond the perturbative CCSD(T) treatment. The derived energy shifts and matrix elements additionally rest on nuclear-structure inputs - the Nilsson deformed-oscillator nuclear magnetic quadrupole moment (NMQM) of $^{175}$ Lu, the spherical single-particle NMQM of $^{139}$ La, and single-particle anapole-moment estimates for $^{139}$ La and $^{175}$ Lu - which enter, as in the source paper, as error-bar-free model inputs whose accuracy has never been quantified. The problem is to determine these molecular enhancement factors with a defensible uncertainty budget for the electronic-structure contributions, reduced, for each of the twelve enhancement factors, strictly below the per-factor total uncertainty reported by Chamorro et al. (2024) by addressing the dominant residual contributions, so that the accuracy and reliability of these coupling parameters, and knowledge of their uncertainties, do not limit the planning and interpretation of future P- and P,T-violation measurements in these molecules.

Background

Permanent electric dipole moments (EDMs) of particles and nuclei, and related parity (P) and time-reversal (T) violating moments, are among the most sensitive probes of new physics beyond the Standard Model - parity being invariance of the laws of physics under spatial mirror reflection, and time reversal being invariance under reversal of the direction of time. In heavy polar diatomics molecules several such effects are strongly enhanced, because the molecular field fully orients the molecule and because closely spaced levels of opposite parity can be mixed by modest laboratory electric fields. The relevant interactions are: the electron electric dipole moment (eEDM) $d_{e}$ ; the scalar-pseudoscalar (S-PS) nucleon-electron interaction of strength $k_{\mathrm{s}}$ ; the nuclear magnetic quadrupole moment (NMQM) $M$ , a P,T-violating nuclear moment that requires a nucleus with spin $I \geq 1$ and is enhanced in quadrupole-deformed nuclei; and the nuclear anapole moment, a P-violating (but T-conserving) moment generated by weak currents inside the nucleus that dominates nuclear-spin-dependent parity violation (NSD-PV) in heavy systems. The measurable energy shifts are products of these fundamental constants with electronic structure parameters, the enhancement factors $W_{\mathrm{d}}$ , $W_{\mathrm{s}}$ , $W_{\mathrm{M}}$ , and $W_{\mathrm{A}}$ , which cannot be measured and must be supplied by theory; their accuracy directly limits how well a measured shift can be converted into a bound on new physics.

The LaO, LaS, and LuO molecules were proposed as attractive candidates (Flambaum and Khriplovich, 1985) because they contain heavy nuclei ( $Z_{\mathrm{La}}=57$ , $Z_{\mathrm{Lu}}=71$ , with P,T-violating effects growing with powers of $Z$ ), have well-studied $^{2}\Sigma$ ground states (electronic states with one unpaired electron and zero projection of the orbital angular momentum onto the internuclear axis), and - most importantly - exhibit a partial cancellation of hyperfine and rotational splittings that leaves intervals below 1 cm $^{-1}$ between levels of opposite parity, which can be narrowed further by a magnetic field so that the molecules are fully polarized in practicable fields. The two Lu isotopes are quadrupole deformed, so an enhanced collective NMQM contribution is expected, and the ratio $W_{\mathrm{s}}/W_{\mathrm{d}}$ in LuO (3.65) differs considerably from that in the most sensitive current eEDM experiments (about 1.7-2.8), so a competitive measurement in this system would constrain the P,T-violating parameter space in a complementary way.

The state of the art for these molecules is the four-component relativistic coupled-cluster study of Chamorro, Flambaum, Garcia Ruiz, Borschevsky, and Pašteka (2024). Coupled-cluster theory builds the correlated electronic wave function as an exponential of excitation operators; the standard high-accuracy level, CCSD(T), includes single and double excitations iteratively and triple excitations only perturbatively. Using Dyall relativistic basis sets (Gaussian basis families optimized for fully relativistic four-component Dirac calculations) with systematic extension in cardinal number $n$ (the 'zeta' quality of the basis, $n = 2, 3, 4, \ldots$ ) and extrapolation to the complete-basis-set limit, they assigned each enhancement factor a total uncertainty of up to about 7%. The dominant contributions are the residual basis-set incompleteness and the missing higher excitations: the effect of excitations beyond CCSD(T) was estimated from the difference between CCSD(T) and CCSD results (up to 4.2% in LaS and LaO and 5.8% in LuO), not from any explicit higher-level calculation.

The nuclear inputs suffer a second, unquantified systematic uncertainty. The NMQM of the spherical $^{139}$ La nucleus is obtained from single-particle considerations (Flambaum, DeMille, and Kozlov, 2014), while for the deformed $^{175}$ Lu nucleus it is built from Nilsson-model single-particle matrix elements (Lackenby and Flambaum, 2018; Maison et al., 2020), the Nilsson model being a deformed-oscillator shell model in which nucleons occupy orbitals in an axially deformed potential. The anapole moments of $^{139}$ La and $^{175}$ Lu rest on single-particle estimates of the Flambaum-Khriplovich type. None of these nuclear models currently carries an error bar, even though the derived NMQM energy shifts and NSD-PV matrix elements in these molecules depend on them linearly, and only one anapole moment (in atomic cesium) has ever been measured anywhere.

Problem Statement

Determine, with a quantified uncertainty, the molecular enhancement factors $W_{\mathrm{d}}$ (electron electric dipole moment), $W_{\mathrm{s}}$ (scalar-pseudoscalar nucleon-electron interaction), $W_{\mathrm{M}}$ (nuclear magnetic quadrupole moment), and $W_{\mathrm{A}}$ (anapole, nuclear-spin-dependent parity violation) of the LaO, LaS, and LuO molecules, with the electronic-structure uncertainty demonstrably reduced for every one of the twelve enhancement factors (the four quantities $W_{\mathrm{d}}$ , $W_{\mathrm{s}}$ , $W_{\mathrm{M}}$ , and $W_{\mathrm{A}}$ for each of LaO, LaS, and LuO), strictly below the per-factor total uncertainty reported for that same factor by Chamorro et al. (2024) - $W_{\mathrm{d}}$ , $W_{\mathrm{s}}$ , $W_{\mathrm{M}}$ , and $W_{\mathrm{A}}$ equal to 7.1%, 6.3%, 6.9%, and 6.2% for LaS, and about 4.9% and about 5.8% for every one of the four factors of LaO and LuO respectively - by addressing the dominant residual contributions identified in that work: basis-set incompleteness in the extrapolation to the complete-basis-set limit and the neglect of higher excitations beyond the CCSD(T) treatment.

As in the source paper, the nuclear moments - the $^{139}$ La and $^{175}$ Lu NMQMs and the single-particle anapole-moment estimates - enter the derived energy shifts and transition matrix elements as inputs from the existing nuclear-structure models, without error bars. These nuclear-model moments are held fixed as error-bar-free inputs and are therefore not part of the requested uncertainty budget, which covers the electronic-structure contributions to the four enhancement factors.

The objective is an internally consistent uncertainty budget for the electronic-structure contributions to these enhancement factors, so that the accuracy and reliability of the coupling parameters, and knowledge of their uncertainties, do not limit the conversion of a measured signal into limits on $d_{e}$ , $k_{\mathrm{s}}$ , and the hadronic P,T-violating constants (such as the QCD vacuum angle $\bar{\theta}$ - the parameter that sets parity and time-reversal violation in the strong interaction - and the electric and chromo-electric dipole moments of the quarks).

References

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