  Recent observational studies of $\omega$ Centauri by {\it Hubble Space
Telescope} have discovered a double main sequence in the color magnitude
diagrams (CMDs) of its stellar populations. The stellar population with the
blue main sequence (bMS) is observationally suggested to have a helium
abundance much larger, by $\Delta Y\sim 0.12$, than that of the red main
sequence (rMS). By using somewhat idealized models in which stars of the bMS
are formed from gas ejected from those of the rMS, we quantitatively
investigate whether the helium overabundance of the bMS can result from
self-enrichment from massive AGB stars, from mass loss of very massive young
stars, or from type II supernovae within $\omega$ Cen. We show that as long as
the helium enrichment is due to ejecta from the rMS formed earlier than the
bMS, none of the above three enrichment scenarios can explain the observed
properties of the bMS self-consistently for reasonable IMFs. The common,
serious problem in all cases is that the observed number fraction of the bMS
can not be explained without assuming unusually top-heavy IMFs. This failure of
the self-enrichment scenarios implies that most of the helium-enriched gas
necessary for the formation of the bMS originated from other external sources.
We thus suggest a new scenario that most of the second generation of stars
(i.e., the bMS) in $\omega$ Cen could be formed from gas ejected from field
stellar populations that surrounded $\omega$ Cen when it was a nucleus of an
ancient dwarf galaxy.
