Population synthesis of Thorne-\.Zytkow objects: Rejuvenated donors and unexplored progenitors in the common envelope formation channel

Context. Common envelope evolution of a massive star and a neutron star companion has two possible outcomes: formation of a short-period binary (a potential gravitational wave source progenitor) or a merger of the massive star with the neutron star. If the binary merges, a structure with a neutron star core surrounded by a large diffuse envelope, a so-called Thorne-\.Zytkow object (T\.ZO), may form. The predicted appearance of this hypothetical class of star is very similar to red supergiants, making observational identification difficult. Aims. Our objective is to understand the properties of systems that are potential T\.ZO progenitors, e.g., binary systems that enter a common envelope phase with a neutron star companion. We also aim to distinguish those that have been through a previous stable mass transfer phase, which can rejuvenate the accretor. We estimate the number of T\.ZOs in the Milky Way and assess the impact of uncertainties in their formation. Methods. We use the rapid population synthesis code COMPAS at Solar metallicity and with common envelope efficiency parameter set to unity to determine the population demographics of T\.ZOs. We use one-dimensional evolutionary T\.ZO models from the literature to determine a fit for T\.ZO lifetime in order to estimate the current number of T\.ZOs in the Galaxy as well as to assess core disruption during the merger. Results. We explore the progenitors in the Hertzsprung-Russell diagram, calculate formation rates, and investigate kinematics of the progenitor stars. We find that the vast majority ($\approx 92\%$) of T\.ZO progenitors in our population have experienced mass transfer and become rejuvenated before their formation event. Using a constant star formation rate we estimate $\approx 2\times 10 ^{-4}$ T\.ZOs per $M_\odot$ in our Galaxy, corresponding to $\approx 5\pm 1$ T\.ZOs in the Milky Way at present.