Evolution of long-lived globular cluster stars III. Effect of the initial helium spread on the position of stars in a synthetic Hertzsprung-Russell diagram

Context. Globular clusters host multiple populations of long-lived low-mass stars whose origin remains an open question. Several scenarios have been proposed to explain the associated photometric and spectroscopic peculiarities. They differ, for instance, in the maximum helium enrichment they predict for stars of the second population, which these stars can inherit at birth as the result of the internal pollution of the cluster by different types of stars of the first population. Aims. We present the distribution of helium-rich stars in present-day globular clusters as it is expected in the original framework of the fast-rotating massive stars scenario (FRMS) as first-population polluters. We focus on NGC 6752. Methods. We completed a grid of 330 stellar evolution models for globular cluster low-mass stars computed with different initial chemical compositions corresponding to the predictions of the original FRMS scenario for [Fe/H] = 1.75. Starting from the initial helium-sodium relation that allows reproducing the currently observed distribution of sodium in NGC 6752, we deduce the helium distribution expected in that cluster at ages equal to 9 and 13 Gyr. We distinguish the stars that are moderately enriched in helium from those that are very helium-rich (initial helium mass fraction below and above 0.4, respectively), and compare the predictions of the FRMS framework with other scenarios for globular cluster enrichment. Results. The effect of helium enrichment on the stellar lifetime and evolution reduces the total number of very helium-rich stars that remain in the cluster at 9 and 13 Gyr to only 12% and 10%, respectively, from an initial fraction of 21%. Within this age range, most of the stars still burn their hydrogen in their core, which widens the MS band significantly in effective temperature. The fraction of very helium-rich stars drops in the more advanced evolution phases, where the associated spread in effective temperature strongly decreases. These stars even disappear from the horizontal branch and the asymptotic giant branch at 13 Gyr. Conclusions. The helium constraint is no suitable criterion for clearly distinguishing between the scenarios for GC self-enrichment because only few very helium-rich stars are predicted in the investigated framework and because it is difficult to derive the helium content of GC stars observationally. However, the helium constraint indicates some difficulties of the original FRMS scenario that require the exploration of alternatives.