  A large number of low-mass (< 0.20 M_sun) helium white dwarfs (He WDs) have
recently been discovered. The majority of these are orbiting another WD or a
millisecond pulsar (MSP) in a close binary system; a few examples are found to
show pulsations or to have a main-sequence star companion. There appear to be
discrepancies between the current theoretical modelling of such low-mass He WDs
and a number of key observed cases, indicating that their formation scenario
remains to be fully understood. Here we investigate the formation of detached
proto-He WDs in close-orbit low-mass X-ray binaries (LMXBs). Our prime focus is
to examine the thermal evolution and the contraction phase towards the WD
cooling track and investigate how this evolution depends on the WD mass. Our
calculations are then compared to the most recent observational data. Numerical
calculations with a detailed stellar evolution code were used to trace the
mass-transfer phase in a large number of close-orbit LMXBs. Subsequently, we
followed the evolution of the detached low-mass proto-He WDs, including stages
with residual shell hydrogen burning and vigorous flashes caused by unstable
CNO burning. We find that the time between Roche-lobe detachment until the
low-mass proto-He WD reaches the WD cooling track is typically Delta_t_proto =
0.5 - 2 Gyr, depending systematically on the WD mass and therefore on its
luminosity. The lowest WD mass for developing shell flashes is ~0.21 M_sun for
progenitor stars of mass M2 <= 1.5 M_sun (and ~0.18 M_sun for M2 = 1.6 M_sun).
The long timescale of low-mass proto-He WD evolution can explain a number of
recent observations, including some MSP systems hosting He WD companions with
very low surface gravities and high effective temperatures. We find no evidence
for Delta_t_proto to depend on the occurrence of flashes and thus question the
suggested dichotomy in thermal evolution of proto-WDs.
