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.
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