  We consider the formation of low-mass X-ray binaries containing accreting
neutron stars via the helium-star supernova channel. The predicted relative
number of short-period transients provides a sensitive test of the input
physics in this process. We investigate the effect of varying mean kick
velocities, orbital angular momentum loss efficiencies, and common envelope
ejection efficiencies on the subpopulation of short-period systems, both
transient and persistent. Guided by the thermal-viscous disk instability model
in irradiation-dominated disks, we posit that short-period transients have
donors close to the end of core-hydrogen burning. We find that with increasing
mean kick velocity the overall short-period fraction, s, grows, while the
fraction, r, of systems with evolved donors among short-period systems drops.
This effect, acting in opposite directions on these two fractions, allows us to
constrain models of LMXB formation through comparison with observational
estimates of s and r. Without fine tuning or extreme assumptions about
evolutionary parameters, consistency between models and current observations is
achieved for a regime of intermediate average kick magnitudes of about 100-200
km/s, provided that (i) orbital braking for systems with donor masses in the
range 1-1.5 solar masses is weak, i.e., much less effective than a simple
extrapolation of standard magnetic braking beyond 1.0 solar mass would suggest,
and (ii) the efficiency of common envelope ejection is low.
