  Justham, Rappaport & Podsiadlowski (2006) recently suggested that black-hole
low-mass X-ray binaries (BHLMXBs) with short orbital periods may have evolved
from black-hole intermediate-mass X-ray binaries (BHIMXBs). In their model the
secondaries in BHIMXBs are assumed to possess anomalously high magnetic fields,
so that magnetic braking can lead to substantial loss of angular momentum. In
this paper we propose an alternative mechanism for orbital angular momentum
loss in BHIMXBs. We assume that a small fraction $\delta$ of the transferred
mass from the donor star form a circumbinary disc surrounding the binary
system. The tidal torques exerted by the disc can effectively drain orbital
angular momentum from the binary. We have numerically calculated the
evolutionary sequences of BHIMXBs, to examine the influence of the circumbinary
disc on the binary evolution. Our results indicate when $\delta\la 0.01-0.1$
(depending on the initial orbital periods), the circumbinary disc can cause
secular orbital shrinking, leading to the formation of compact
  BHLMXBs, otherwise the orbits always expand during the evolution. This
scenario also suggests the possible existence of luminous, persistent BHLMXBs,
but it suffers the same problem as in Justham, Rappaport & Podsiadlowski (2006)
that, the predicted effective temperatures of the donor stars are significantly
higher than those of the observed donor stars in BHLMXBs.
