  The precise mechanism for production of quarkonium at hadron colliders is
still an open question. Within non-relativistic quantum chromodynamics,
quarkonium production cross sections can be factorized into short-distance,
perturbative contributions and universal, non-perturbative, long-distance
matrix elements, and then summed over quantum numbers of the heavy quark pair.
In principle, at short-distances, the heavy quark pair can be either in a
color-singlet or color-octet state, and it is desirable to establish the
relative contributions to compare with data and to make predictions in
different experimental environments. From the explicit form of the lowest-order
perturbative matrix elements for color-singlet and color-octet production, we
show that the structure of the optimal observable for discrimination on phase
space, the likelihood ratio, has strong dependence on the angular momentum
state of the heavy quark pair. This presents an obstruction for construction of
a single, robust discrimination observable that can be applied to production of
an arbitrary quarkonium state.
