Red Clump Stars as a Tracer of Microlensing Optical Depth

Zaritsky and Lin have recently suggested that the color magnitude diagram of the Large Magellanic Cloud (LMC) contains evidence of foreground red clump stars. They interpret this as evidence of tidal debris or a dwarf galaxy at a distance of $\sim 35$ kpc which may be responsible for the large gravitational microlensing optical depth observed by the MACHO Collaboration. I derive a relationship between the microlensing optical depth of such a foreground population and the observed density of foreground red clump stars. Recent observational determinations for Pop I and Pop II stellar mass functions are used to show that the surface density of foreground red clump stars claimed by Zaritsky and Lin implies a microlensing optical depth in the range $\tau_{fg} = 0.8-3.6\times 10^{-8}$ which is only 3-13% of $\tau_{LMC}$ as determined by the MACHO Collaboration. If the foreground population has a similar star formation history to the LMC, then the implied $\tau_{fg}$ is only 3-4% of $\tau_{LMC}$.