Comprehensive Inclusion of Higher-order Ca$^+$ Isotope Shifts in the King's Plot Yields an Order Improvement on the $e^-$-$n$ Coupling Limit

Abstract

By critically evaluating higher-order nonlinear effects to the isotope shifts (ISs) in the low-lying transition frequencies of the singly charged calcium ion, stringent constraint on the electron-neutron coupling due to a hypothetical boson describing physics beyond the Standard Model is inferred. It shows an order magnitude difference compared to the previously reported limit demonstrating importance of higher-order effects in the analysis of nonlinearity in the King's plot. The first-order IS parameters and enhancement factor ($D$) were evaluated using two complementary approaches in the relativistic coupled-cluster theory framework: namely finite-field (FF) and analytical response (AR) approaches. Extraction of the second-order IS parameters in the FF approach show numerical instabilities, so they are determined in the AR approach. Comparison of these factors with previous calculation shows substantial differences in the magnitudes. However, $D$ values from both the FF and AR approaches display excellent agreement. We also show explicitly roles of electron correlation effects in the evaluation of $D$ values accurately.

Figures (2)

• Figure 1: Plot demonstrating $D$ values (in Hz) for the (a) $4S\rightarrow 3D_{3/2}$ and (b) $4S_{1/2}\rightarrow3D_{5/2}$ transitions in Ca$^+$ with respect to $m_{\phi}$ (in eV/c$^2$) from the DHF, RPA and RCC methods of the AR and FF approaches.
• Figure 2: Exclusion plot demonstrating constraints (2$\sigma$ upper bound) on $|\alpha_{NP}|$ with respect to $m_{\phi}$. The red and green solid line represent limits imposed using the FF results through Case I and Case II analyses from the present work, respectively, black gebert2015precision, cyan solaro2020improved and purple chang2024systematic solid lines are the constraints from the previous IS studies, while constraints from other complementary studies such as Casimir effects bordag2009advances, $(g-2)$ factor of the electron fan2023measurementmorel2020determination and neutron scattering nesvizhevsky2008neutron and star cooling in a supernova raffelt2012limits are also shown.