  We have calculated the evolution of low metallicity red giant stars under the
assumption of deep mixing between the convective envelope and the hydrogen
burning shell. We find that the extent of the observed abundance anomalies, and
in particular the universal O-Na anticorrelation, can be totally explained by
mixing which does not lead to significant helium enrichment of the envelope. On
the other hand, models with extremely deep mixing and strong helium enrichment
predict anomalies of sodium and oxygen, which are much larger than the observed
ones. This latter result depends solely on the nucleosynthesis inside the
hydrogen burning shell, but not on the details of the mixing descriptions.
These, however, influence the evolution of surface abundances with brightness,
which we compare with the limited observational material available. Our models
allow, nevertheless, to infer details on the depth and speed of the mixing
process in several clusters. Models with strong helium enrichment evolve to
high luminosities and show an increased mass loss. However, under peculiar
assumptions, red giants reach very high luminosities even without extreme
helium mixing. Due to the consequently increased mass loss, such models could
be candidates for blue horizontal branch stars, and, at the same time, would be
consistent with the observed abundance anomalies.
