Ruling out dominant electron scattering in Little Red Dots' Rosetta Stone using multiple hydrogen lines

The majority of Little Red Dots (LRDs) hosting Active Galactic Nuclei (AGN) exhibits broad H$\alpha$ emission, which recent studies propose originates from scattering off free electrons within an ionized and dense medium embedding the Broad Line Region (BLR), rather than directly from the BLR itself. This model suggests that the observed broad lines may be intrinsically narrower than observed, which would lead to black hole masses that are up to two orders of magnitude smaller than what inferred when assuming that the whole broad line comes from the BLR. To test this model, we present a joint analysis of multiple hydrogen recombination lines in the ''Rosetta Stone''AGN, the brightest known LRD at $z$=2.26. We show that H$\alpha$, H$\beta$ and Pa$\beta$ have different spectral profiles, which is inconsistent with the predictions of the simple electron scattering scenario. Additionally, we test a variety of exponential models and show that none of them can simultaneously reproduce all three line profiles with physically plausible parameters. The inadequacy of these models for the Rosetta Stone implies that the scenario of electron scattering by an ionized medium surrounding the BLR is not universally applicable to LRDs and AGN, and therefore provides a counterexample to the claim of a universal and systematic overestimation of black hole masses.