APOGEE chemical tagging constraint on the maximum star cluster mass in the $\alpha$-enhanced Galactic disk

Stars born from the same molecular cloud should be nearly homogeneous in their element abundances. The concept of chemical tagging is to identify members of disrupted clusters by their clustering in element abundance space. Chemical tagging requires large samples of stars with precise abundances for many individual elements. With uncertainties of $\sigma_{[X/{\rm Fe}]}$ and $\sigma_{\rm [Fe/H]} \simeq 0.05$ for 10 elements measured for $>10^4$ stars, the APOGEE DR12 spectra may be the first well-suited data set to put this idea into practice. We find that even APOGEE data offer only $\sim 500$ independent volume elements in the 10-dimensional abundance space, when we focus on the $\alpha$-enhanced Galactic disk. We develop and apply a new algorithm to search for chemically homogeneous sets of stars against a dominant background. By injecting star clusters into the APOGEE data set we show that chemically homogeneous clusters with masses $\gtrsim 3 \times 10^7 \, {\rm M}_\odot$ would be easily detectable and yet no such signal is seen in the data. By generalizing this approach, we put a first abundance-based constraint on the cluster mass function for the old disk stars in the Milky Way.