  Currently, there are 24 black hole (BH) X-ray binary systems that have been
dynamically confirmed in the Galaxy. Most of them are low-mass X-ray binaries
(LMXBs) comprised of a stellar-mass BH and a low-mass donor star. Although the
formation of these systems has been extensively investigated, some crucial
issues remain unresolved. The most noticeable one is that, the low-mass
companion has difficulties in ejecting the tightly bound envelope of the
massive primary during the spiral-in process. While initially intermediate-mass
binaries are more likely to survive the common envelope (CE) evolution, the
resultant BH LMXBs mismatch the observations. In this paper, we use both
stellar evolution and binary population synthesis to study the evolutionary
history of BH LMXBs. We test various assumptions and prescriptions for the
supernova mechanisms that produce BHs, the binding energy parameter, the CE
efficiency, and the initial mass distributions of the companion stars. We
obtain the birthrate and the distributions of the donor mass, effective
temperature and orbital period for the BH LMXBs in each case. By comparing the
calculated results with the observations, we put useful constraints on the
aforementioned parameters. In particular, we show that it is possible to form
BH LMXBs with the standard CE scenario if most BHs are born through failed
supernovae.
