  The ever-expanding observational sample of X-ray binaries (XRBs) makes them
excellent laboratories for constraining binary evolution theory. Such
constraints can be obtained by studying the effects of various physical
assumptions on synthetic X-ray luminosity functions (XLFs) and comparing to
observed XLFs. In this work, we focus on high-mass XRBs (HMXBs) and study the
effects on the XLF of various, poorly-constrained assumptions regarding
physical processes such as the common-envelope phase, the core-collapse, and
wind-fed accretion. We use the new binary population synthesis code POSYDON,
which employs extensive pre-computed grids of detailed stellar structure and
binary evolution models, to simulate the evolution of binaries. We generate 96
synthetic XRB populations corresponding to different combinations of model
assumptions. The generated HMXB XLFs are feature-rich, deviating from the
commonly assumed single-power law. We find a break in our synthetic XLF at
luminosity $\sim 10^{38}$ erg s$^{-1}$, similar to observed XLFs. However, we
find also a general overabundance of XRBs (up to a factor of $\sim$10 for
certain model parameter combinations) driven primarily by XRBs with black hole
accretors. Assumptions about the transient behavior of Be-XRBs, asymmetric
supernova kicks, and common-envelope physics can significantly affect the shape
and normalization of our synthetic XLFs. We find that less well-studied
assumptions regarding the circularization of the orbit at the onset of
Roche-lobe overflow and criteria for the formation of an X-ray emitting
accretion disk around wind-accreting black holes can also impact our synthetic
XLFs. Our study reveals the importance of large-scale parameter studies,
highlighting the power of XRBs in constraining binary evolution theory.
