  We study the formation of low-mass X-ray binaries (LMXBs) through helium star
supernovae in binary systems that have each emerged from a common-envelope
phase. LMXB progenitors must satisfy a large number of evolutionary and
structural constraints, which imposed under the assumption of a symmetric
supernova explosion, prohibit the formation of short-period LMXBs transferring
mass at sub-Eddington rates through any channel in which the intermediate
progenitor of the neutron star is not completely degenerate. Barring
accretion-induced collapse, the existence of such systems therefore requires
that natal kicks be imparted to neutron stars. We use an analytical method to
synthesize the distribution of nascent LMXBs over donor masses and orbital
periods, and evaluate their birth rate and systemic velocity dispersion. Within
the limitations imposed by observational incompleteness and selection effects,
and our neglect of secular evolution in the LMXB state, we compare our results
with observations. However, our principal objective is to evaluate how basic
model parameters influence these results. We conclude that the characteristics
of newborn LMXBs are primarily determined by age and stability constraints and
the efficiency of magnetic braking, and are largely independent of the
primordial binary population and the evolutionary history of LMXB progenitors
(except for extreme values of the average kick magnitude or of the
common-envelope ejection efficiency). Theoretical estimates of total LMXB birth
rates are not credible, since they strongly depend on the observationally
indeterminate frequency of primordial binaries with extreme mass ratios in
long-period orbits.
