The Gaia-ESO Survey: No sign of multiple stellar populations in open clusters from their sodium and oxygen abundances

Context. The light element (anti-)correlations shown by globular clusters (GCs) are the main spectroscopic signature of multiple stellar populations. These internal abundance variations provide us with fundamental constraints on the formation mechanism of stellar clusters. Aims. Using Gaia-ESO, the largest and most homogeneous survey of open clusters (OCs), we intend to check whether these stellar aggregates display the same patterns. Based on previous studies of many GCs, several young and massive clusters in the Magellanic Clouds, as well as a few OCs, we do not expect to find any anti-correlation, given the low mass of Milky Way OCs. Methods. We used the results based on UVES spectra of stars in Gaia-ESO to derive the distribution of Na and O abundances and see whether they show an unexplained dispersion or whether they are anti-correlated. By selecting only high-probability members with high-precision stellar parameters, we ended up with more than 700 stars in 74 OCs. We examined the O-Na distribution in 28 OCs with at least 4 stars available as well as the Na distribution in 24 OCs, with at least 10 stars available. Results. We find that the distribution of Na abundances is compatible with a single-value population, within the errors. The few apparent exceptions can be explained by differences in the evolutionary phase (main sequence and giant post first dredge-up episode) or by difficulties in analysing low gravity giants. We did not find any indication of an Na-O anti-correlation in any of the clusters for which O has been derived. Conclusions. Based on the very small spread we find, OCs maintain the status of single stellar populations. However, a definitive answer requires studying more elements and larger samples covering different evolutionary phases. This will be possible with the next generation of large surveys.