  Be X-ray binaries (Be-XRBs) are crucial in understanding high-mass X-ray
binaries, featuring a rapidly rotating Be star and a neutron star companion in
an eccentric orbit, intermittently accreting material from the Be star's
decretion disk. Originating from binary stellar evolution, Be-XRBs are of
significant interest to binary population synthesis (BPS) studies,
encapsulating the physics of supernovae, common envelope, and mass transfer
(MT). Using the POSYDON BPS code, employing pre-computed grids of detailed
binary stellar evolution models, we investigate the Galactic Be-XRB population.
POSYDON incorporates stellar rotation self-consistently during MT phases,
enabling a detailed examination of the rotational distribution of Be stars. Our
fiducial BPS and Be-XRB model align well with the orbital properties of
Galactic Be-XRBs, emphasizing the role of rotational constraints. Our modeling
reveals a bimodal rotational distribution of Be-XRB-like systems, in excellent
agreement with literature values. All Be-XRBs undergo an MT phase before the
first compact object forms, with over half experiencing a second MT phase from
a stripped helium companion (Case BB). Computing rotationally-limited MT
efficiencies and applying them to our population, we find that the majority of
Be-XRBs have undergone highly non-conservative MT (beta ~ 0.15). Our study
underscores the importance of detailed angular momentum modeling during MT in
interpreting Be-XRB populations, emphasizing this population as a key probe for
the stability and efficiency of MT in interacting binaries.
