BL Lac host galaxies: how to systematically characterise them in optical-NIR spectroscopy

Host galaxies of Active Galactic Nuclei give crucial information on the interaction between accreting Supermassive Black Holes and their surroundings, and on their common evolution. Their study in the case of aligned jetted AGN - BL Lacertae objects in particular - is complicated by the non-thermal jet component, whose bright and multi-frequency emission easily dominates over the whole electromagnetic spectrum. BL Lac host galaxies have thus been sparsely studied, and their elliptical nature is currently a hypothesis supported by few observations. With the broad aim of a systematic analysis of these sources, and in light of the many optical and NIR spectroscopic facilities that are now available, we implement an easily applicable method to determine whether a BL Lac is hosted in an elliptical or spiral galaxy. Building on the only systematic study currently available on BL Lac hosts, we worked on a sample of realistic BL Lac synthetic spectra. We analysed them and characterized their statistics using QSFit, a publicly available spectroscopy software. If BL Lac host galaxies were both elliptical and spiral, our method would be able to discriminate between them, provided that BL Lac jets are fainter than $L_γ\sim10^{46}$erg/s. Just two runs of QSFit for each BL Lac spectrum would return a single parameter, that would allow for a first broad distinction between the two classes. We finally discuss the two galaxy types that introduce some uncertainty in their classification, that might lead to possible classification biases.