Re-evaluating Lyman $\alpha$ wing opacities and the low mass-problem in cool white dwarfs

Gaia observations have reignited interest in the optical and ultraviolet (UV) opacity problems of cool white dwarfs ($T_{\rm eff} \leq 6000$ K), which were thought to be resolved nearly two decades ago through the inclusion of Lyman $\alpha$ red wing opacity arising from H-H$_2$ collisions in atmospheric models. Recent studies have revealed that their masses derived from Gaia optical photometry are 0.1$-$0.2 M$_{\odot}$ lower than expected from single-star evolution. Since the Ly $\alpha$ H-H$_2$ wing opacity significantly affects the blue end of their optical spectra, it may contribute to the mass discrepancy. To investigate this hypothesis, we revisited the Ly $\alpha$ opacity calculations in the quasi-static single and multi-perturber approximations by explicitly using the ab initio potential energy data of H$_3$ while fully accounting for the H-H$_2$ collision angle. We find that the opacity is slightly smaller than the standard models at the shortest wavelengths ($\leq5000$ angstrom), but larger at longer wavelengths. Comparing synthetic magnitudes (GALEX, Gaia, WISE) to the observations of the 40 pc white dwarf sample, we note that the revised models tentatively reproduce the observed $NUV-G$ colours for stars cooler than 6000 K, but still fail to match $G_{\rm BP} - G_{\rm RP}$ colours, resulting in similarly low inferred masses ($\leq 0.5$ M$_{\odot}$) as obtained with the standard Ly $\alpha$ opacity. Exploring other dominant opacity sources, we discover that decreasing the strength of the bound-free H$^-$ opacity in existing models better reproduces the optical and infrared colours, while collision-induced absorption (CIA) opacity is ineffective in resolving the low-mass problem. We highlight the need for improved opacities and multi-wavelength observations in future studies.