Are there any first-generation stars in globular clusters today?

Context. Several models compete to explain the abundance properties of stellar populations in globular clusters. One of the main constraints is the present-day ratio of first- and second-generation stars that are currently identified based on their sodium content. Aims. We propose an alternative interpretation of the observed sodium distribution, and suggest that stars with low sodium abundance that are counted as members of the first stellar generation could actually be second-generation stars. Methods. We compute the number ratio of second-generation stars along the Na distribution following the fast rotating massive star model using the same constraints from the well-documented case of NGC 6752 as in our previous developments. Results. We reproduce the typical percentage of low-sodium stars usually classified as first-generation stars by invoking only secondary star formation from material ejected by massive stars and mixed with original globular cluster material in proportions that account for the Li-Na anti-correlation in this cluster. Conclusions. Globular clusters could be totally devoid of first-generation low-mass stars today. This can be tested with the determination of the carbon isotopic ratio and nitrogen abundance in turn-off globular cluster stars. Consequences and related issues are briefly discussed