Can very massive Population III stars produce a super-collapsar?

A fraction of the first generation of stars in the early Universe may be very massive ($\gtrsim 300~\mathrm{M_\odot}$) as they form in metal-free environments. Formation of black holes from these stars can be accompanied by supermassive collapsars to produce long gamma-ray bursts of a unique type having a very high total energy ($\sim 10^{54}~\mathrm{erg}$) as recently suggested by several authors. We present new stellar evolution models of very massive Population III stars including the effect of rotation to provide theoretical constraints on super-collapsar progenitors. We find that the angular momentum condition for super-collapsar can be fulfilled if magnetic torques are ignored, in which case Eddington-Sweet circulations play the dominant role for the transport of angular momentum. We further find that the initial mass range for super-collapsar progenitors would be limited to $300~\mathrm{M_\odot} \lesssim M \lesssim 700~\mathrm{M_\odot}$. However, all of our very massive star models of this mass range end their lives as red supergiants rather than blue supergiants, in good agreement with most of the previous studies. The predicted final fate of these stars is either a jet-powered type IIP supernova or an ultra-long, relatively faint gamma-ray transient, depending on the initial amount of angular momentum.