The Hubble Space Telescope UV Legacy Survey of Galactic Globular
  Clusters. V. Constraints on Formation Scenarios
  We build on the evidence provided by our Legacy Survey of Galactic globular
clusters (GC) to submit to a crucial test four scenarios currently entertained
for the formation of multiple stellar generations in GCs. The observational
constraints on multiple generations to be fulfilled are manifold, including GC
specificity, ubiquity, variety, predominance, discreteness, supernova
avoidance, p-capture processing, helium enrichment and mass budget. We argue
that scenarios appealing to supermassive stars, fast rotating massive stars and
massive interactive binaries violate in an irreparable fashion two or more
among such constraints. Also the scenario appealing to AGB stars as producers
of the material for next generation stars encounters severe difficulties,
specifically concerning the mass budget problem and the detailed chemical
composition of second generation stars. We qualitatively explore ways possibly
allowing one to save the AGB scenario, specifically appealing to a possible
revision of the cross section of a critical reaction rate destroying sodium, or
alternatively by a more extensive exploration of the vast parameter space
controlling the evolutionary behavior of AGB stellar models. Still, we cannot
ensure success for these efforts and totally new scenarios may have to be
invented to understand how GCs formed in the early Universe.
