Universal behaviour of diatomic halo states and the mass sensitivity of their properties

The scattering and spectroscopic properties of molecular halo states can serve as sensitive probes of the constancy of the electron-to-proton mass ratio $\beta=m_e/m_p$. Since halo states are formed by resonant $s$-wave interactions, their properties exhibit universal correlations that are fairly independent of the interactions at short distances. For diatomic molecules, these properties depend on a single parameter only, and so this `universality' means that all the characteristics of a diatomic halo state can be determined with high precision if only one parameter is accurately known. Furthermore, this knowledge can be used to establish the respective property mass sensitivities for investigating the stability of $\beta$. Here, we show for the halo states of the helium dimers that the relationship between the probed properties and their mass sensitivity can be derived from numerically exact solutions of suitable radial Schr\"{o}dinger equations for a set of effective potential energy curves. The resulting relations exhibit a weak dependence on the short-range part of the used potentials and a near-negligible dependence on the `higher-order' nonadiabatic, relativistic, QED and residual retardation effects. The presented approach is thus a robust alternative to other literature approaches, particularly in cases where a lack of experimental data prevents an accurate interaction potential from being determined.