Formation of SMBH seeds in Pop III star clusters through collisions :
  the importance of mass loss
  Runaway collisions in dense clusters may lead to the formation of
supermassive black hole (SMBH) seeds, and this process can be further enhanced
by accretion, as recent models of SMBH seed formation in Population III star
clusters have shown. This may explain the presence of supermassive black holes
already at high redshift, $z>6$. However, in this context, mass loss during
collisions was not considered and could play an important role for the
formation of the SMBH seed. Here, we study the effect of mass loss, due to
collisions of protostars, in the formation and evolution of a massive object in
a dense primordial cluster. We consider both constant mass loss fractions as
well as analytic models based on the stellar structure of the collision
components. Our calculations indicate that mass loss can significantly affect
the final mass of the possible SMBH seed. Considering a constant mass loss of
5% for every collision, we can lose between 60-80% of the total mass that is
obtained if mass loss were not considered. Using instead analytical
prescriptions for mass loss, the mass of the final object is reduced by 15-40%,
depending on the accretion model for the cluster we study. Altogether, we
obtain masses of the order of $10^4M_{\odot}$, which are still massive enough
to be SMBH seeds.
