Neutron Star - White Dwarf Binaries: Probing Formation Pathways and Natal Kicks with LISA

Neutron star-white dwarf (NS+WD) binaries offer a unique opportunity for studying NS-specific phenomena with gravitational waves. In this paper, we employ the binary population synthesis technique to study the Galactic population of NS+WDs with the future Laser Interferometer Space Antenna (LISA). We anticipate approximately $\mathcal{O}(10^2)$ detectable NS+WDs by LISA, encompassing both circular and eccentric binaries formed via different pathways. Despite the challenge of distinguishing NS+WDs from more prevalent double white dwarfs in the LISA data (especially at frequencies below 2 mHz), we show that their eccentricity and chirp mass distributions may provide avenues to explore the NS natal kicks and common envelope evolution. Additionally, we investigate the spatial distribution of detectable NS+WDs relative to the Galactic plane and discuss prospects for identifying electromagnetic counterparts at radio wavelengths. Our results emphasise LISA's capability to detect and characterise NS+WDs and to offer insights into the properties of the underlying population. Our conclusions carry significant implications for shaping LISA data analysis strategies and future data interpretation.