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 .