Nuclear spin-dependent parity-violating effects in light polyatomic molecules

Measurements of nuclear spin-dependent parity-violating (NSD-PV) effects provide an excellent opportunity to test nuclear models and to search for physics beyond the Standard Model. Molecules possess closely spaced states with opposite parity which may be easily tuned to degeneracy to greatly enhance the observed parity-violating effects. A high-sensitivity measurement of NSD-PV effects using light triatomic molecules is in preparation [E. B. Norrgard et al., Commun. Phys. 2, 77 (2019)]. Importantly, by comparing these measurements in light nuclei with prior and ongoing measurements in heavier systems, the contribution to NSD-PV from $Z^0$-boson exchange between the electrons and the nuclei may be separated from the contribution of the nuclear anapole moment. Furthermore, light triatomic molecules offer the possibility to search for new particles, such as the postulated $Z^{\prime }$ boson. In this work, we detail a sensitive measurement scheme and present high-accuracy molecular and nuclear calculations needed for interpretation of NSD-PV experiments on triatomic molecules composed of light elements, Be, Mg, N, and C. The ab initio nuclear structure calculations, performed within the no-core shell model provide a reliable prediction of the magnitude of different contributions to the NSD-PV effects in the four nuclei. These results differ significantly from the predictions of the standard single-particle model and highlight the importance of including many-body effects in such calculations. In order to extract the NSD-PV contributions from measurements, a parity-violating interaction parameter $W_{\text{PV}}$, which depends on the molecular structure, needs to be known with a high accuracy. We have calculated these parameters for the triatomic molecules of interest using the relativistic coupled-cluster approach. In order to facilitate the interpretation of future experiments we provide uncertainties on the calculated parameters. A scheme for measurement using laser-cooled polyatomic molecules in a molecular fountain is presented, along with an estimate of the expected sensitivity of such an experiment. This experimental scheme, combined with the presented state-of-the-art calculations, opens exciting prospects for a measurement of the anapole moment and the PV effects due to the electron-nucleon interactions with unprecedented accuracy and for a new path towards detection of signatures of physics beyond the Standard Model.

Figure 1

Dependence of the anapole-moment coupling constant ${\kappa }_{\text{A}}$ for ${}^9\mathrm{Be}$ on the size of the NCSM basis characterized by $N_{\text{max}}$. The dashed line represents ${\kappa }_{\text{A}}$ obtained in the single-particle model.

Figure 2

Theoretical uncertainties in the calculated $W_{\mathrm{PV}}$ parameters of Be, N, and C in BeNC, given as percentages with respect to the final result. The lower panel presents a detailed breakdown of various sources of uncertainty.

Figure 3

Examples of parity-violation-sensitive pair crossings in linear molecules. Energy of MgNC $|X^2\mathrm{\Sigma };\left(v_1{v}_2^{\ell }{v}_3\right)=\left({01}^{1}0\right),N=1\rangle$ spin-rotation and hyperfine Zeeman sublevels as a function of the magnetic field $\mathcal{B}$. Solid (dashed) lines have parity $P=+1$ ($P=-1$). Crossings of parity-violation-sensitive pairs are circled. Gray lines correspond to states which do not have a parity-violation-sensitive crossing in this range of magnetic field. Other colors correspond to states with a particular value of the total angular momentum project $m_F$ (see the legend). Note that $|N=1,G=3,F=3,4,P\rangle$ are nearly degenerate when $\mathcal{B}=0$.

Figure 4

Potential nuclear spin-dependent parity-violation measurement scheme. Left: Laser-cooled triatomic molecules are prepared in the first bending mode to access the $\ell$-doublet structure and are launched upward into an interaction region to form a molecule fountain. The oscillating electric field $\mathcal{E}$ drives electric dipole transitions between states of opposite parity. The magnetic field $\mathcal{B}$ tunes to degeneracy a particular pair of opposite-parity states $|{\psi }^{\pm }\rangle$ to enhance their interaction via the effective parity-violating Hamiltonian $H_{\text{NSD-PV}}^{\text{eff}}$. Population transfer from the initial state to the degenerate opposite-parity state is read out by laser spectroscopy after molecules fall back out of the interaction region. Right: Stark interference: State transfer (orange) is parity dependent due to the combined NSD-PV interactions (wavy line) and electric dipole interaction interfering constructively or destructively depending on the relative orientations of the electron spin, nuclear spin, and molecule axis.