ranges from $\sim$0.15 to 0.80 solar masses. Moreover, the range of \[O/Fe\] needed to reproduce the color broadening of stars at the bottom of the MS is similar to that inferred from spectroscopy of more-massive RGB stars. Noticeably, Milone et al.(2014) obtained similar conclusion for M4, where the relative numbers of stars in the two stellar populations of this cluster and the oxygen abundances inferred from M-dwarfs are similar to the corresponding quantities measured among RGB and more-massive MS stars (Marino et al.2008, 2011, 2017; Nardiello et al.2015; Milone et al.2017). These facts suggest that the properties of the multiple populations of GCs do not depend significantly on the stellar mass in the $\sim$0.15-0.80 $\mathcal{M}_{\odot}$ range, and provide new constraints to the scenarios for the formation of multiple populations.

According to some scenarios, GCs have experienced multiple episodes of star formation, where the material ejected by relatively massive first-generation (1G) stars produces a cooling flow, rapidly collects in the innermost regions of the cluster and forms centrally-concentrated second stellar generations (2G, e.g.Ventura et al.2001; Decressin et al.2007; De Mink et al.2009; D’Ercole et al.2010; Denissenkov et al.2014; D’Antona et al.2016). It has been suggested that the proto GCs should have 