$\mathcal{CP}$-violation sensitivity of closed-shell radium-containing polyatomic molecular ions

Closed-shell atoms and molecules such as Hg and TlF provide some of the best low-energy tests of hadronic $\mathcal{CP}$ violation beyond the standard model of particle physics, which is considered to be a necessary ingredient to explain the observed excess of matter over antimatter in our universe. $\mathcal{CP}$ violation is expected to be strongly enhanced in octupole-deformed nuclei such as ${}^{225}\mathrm{Ra}$. Recently, closed-shell radium-containing symmetric-top molecular ions were cooled sympathetically in a Coulomb crystal [Fan et al., Phys. Rev. Lett. 126, 023002 (2021)] and shown to be well suited for precision spectroscopy in the search for fundamental physics [Yu and Hutzler, Phys. Rev. Lett. 126, 023003 (2021)]. In closed-shell molecules hadronic $\mathcal{CP}$ violation contributes to a net electric dipole moment (EDM) that violates parity and time-reversal symmetries ($\mathcal{P},\mathcal{T}$), which is the target of measurements. To interpret experiments, it is indispensable to know the electronic-structure-enhancement parameters for the various sources of $\mathcal{P},\mathcal{T}$ violation which contribute to the net $\mathcal{P},\mathcal{T}$-odd EDM. We employ relativistic density-functional-theory calculations to determine relevant parameters for interpretation of possible EDM measurements in ${\mathrm{RaOCH}}_3{}^{+}, {\mathrm{RaSH}}^{+}, {\mathrm{RaCH}}_3^{+}, {\mathrm{RaCN}}^{+}$, and ${\mathrm{RaNC}}^{+}$ and perform accurate relativistic coupled-cluster calculations of the Schiff-moment enhancement in ${\mathrm{RaSH}}^{+}$ to gauge the quality of the density-functional-theory approach. Finally, we project to bounds on various $\mathcal{P},\mathcal{T}$-odd parameters that could be achievable from an experiment with ${\mathrm{RaOCH}}_3{}^{+}$ in the near future and assess its complementarity to experiments with Hg and TlF.

Figure 1

Restriction of two-dimensional subspaces including parameters $d_{\mathrm{sr},\mathrm{n}}, k_{\mathrm{T}}, k_{\mathrm{p}}, g_{\pi }^{\left(0\right)}, g_{\pi }^{\left(1\right)}$, and $g_{\pi }^{\left(2\right)}$ of the considered nine-dimensional space of $P,T$-odd parameters in experiments with Hg and TlF and the proposed experiment with ${\mathrm{RaOCH}}_3{}^{+}$. Coverage regions are computed with electronic-structure parameters of ${\mathrm{RaOCH}}_3{}^{+}$ provided in this work, electronic-structure parameters of $\mathrm{TlF}$ and $\mathrm{Hg}$ from Ref. [36], and nuclear-structure parameters from Ref. [38]. The experimental uncertainty of the EDM of TlF ${\sigma }_d=2.9\times {10}^{-23}e\mathrm{cm}$ with an external electric field for a polarization of strength $\mathcal{E}=16000\mathrm{V}\mathrm{c}{\mathrm{m}}^{-1}$ is taken from Ref. [81], and that of the EDM of Hg ${\sigma }_d=3.1\times {10}^{-30}e\mathrm{cm}$ is taken from Ref. [5]. The expected uncertainty of an experiment with a single ${\mathrm{RaOCH}}_3{}^{+}$ molecule $\delta \nu \approx 6.5\times {10}^{-5}\mathrm{Hz}$ is used as proposed in Ref. [29]. All bounds are computed with Gaussian probability distributions of 95% confidence level as described in Ref. [36]. The two-dimensional subspaces that include the parameters $d_{\mathrm{e}}, d_{\mathrm{sr},\mathrm{p}}$, and $k_{\mathrm{s}}$ are shown in the Supplemental Material.