  Recent Hubble Space Telescope observations have found that the horizontal
branches (HB's) in the metal-rich globular clusters NGC 6388 and NGC 6441 slope
upward with decreasing B-V. Such a slope is not predicted by canonical HB
models and cannot be produced by either a greater cluster age or enhanced mass
loss along the red-giant branch (RGB). The peculiar HB morphology in these
clusters may provide an important clue for understanding the second-parameter
effect.
  We have carried out extensive evolutionary calculations and numerical
simulations in order to explore three non-canonical scenarios for explaining
the sloped HB's in NGC 6388 and NGC 6441: i) A high cluster helium abundance
scenario, where the HB evolution is characterized by long blue loops; ii) A
rotation scenario, where internal rotation during the RGB phase increases the
HB core mass; iii) A helium-mixing scenario, where deep mixing on the RGB
enhances the envelope helium abundance. All three of these scenarios predict
sloped HB's with anomalously bright RR Lyrae variables. We compare this
prediction with the properties of the two known RR Lyrae variables in NGC 6388.
Other possible observational tests are suggested.
