Pulsational Pair-instability and the Mass Gap of Population III Black Holes: Effects of Overshooting

Since the discovery of GW190521, several proposals have been put forward to explain the formation of a black hole in the mass gap caused by (pulsational) pair-instability, $M = 65-130 M_\odot$. We calculate the mass ejection of Pop III stars by the pulsational pair-instability (PPI) process using a stellar evolution and hydrodynamical code. If a relatively small, but reasonable value is adopted for the overshooting parameter, the stars do not become red super giants during the PPI phase. We show that in this case most of the hydrogen envelope remains after the mass ejection by PPI. We find that the BH mass could be at most $M = 109 M_\odot$ below the mass range of pair-instability supernovae.