  Ultra-compact X-ray binaries (UCXBs) are low-mass X-ray binaries with
hydrogen-deficient mass-donors and ultra-short orbital periods. They have been
suggested to be the potential Laser Interferometer Space Antenna (LISA) sources
in the low-frequency region. Several channels for the formation of UCXBs have
been proposed so far. In this article, we carried out a systematic study on the
He star donor channel, in which a neutron star (NS) accretes matter from a He
main-sequence star through Roche-lobe overflow, where the mass-transfer is
driven by gravitational wave radiation. Firstly, we followed the long-term
evolution of the NS+He main-sequence star binaries by employing the stellar
evolution code Modules for Experiments in Stellar Astrophysics, and thereby
obtained the initial parameter spaces for the production of UCXBs. We then used
these results to perform a detailed binary population synthesis approach to
obtain the Galactic rates of UCXBs through this channel. We estimate the
Galactic rates of UCXBs appearing as LISA sources to be $\sim3.1-11.9\, \rm
Myr^{-1}$ through this channel, and the number of such UCXB-LISA sources in the
Galaxy can reach about $1-26$ calibrated by observations. The present work
indicates that the He star donor channel may contribute significantly to the
Galactic UCXB formation rate. We found that the evolutionary tracks of UCXBs
through this channel can account for the location of the five transient sources
with relatively long orbital periods quite well. We also found that such UCXBs
can be identified by their locations in the mass-transfer rate versus the
orbital period diagram.
