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.
