  In this paper we discuss the systematics of quarkonium production at the LHC.
In particular, we focus on the necessity to sum logs of the form log(Q/p_perp)
and log(p_perp/m_Q). We show that the former contributions are power
suppressed, while the latter, whose contribution in fragmentation is well
known, also arise in the short distance (i.e., non-fragmentation) production
mechanisms. Though these contributions are suppressed by powers of m_Q/p_perp,
they can be enhanced by inverse powers of v, the relative velocity between
heavy quarks in the quarkonium. In the limit p_perp >> m_Q short distance
production can be thought of as the fragmentation of a pair of partons (i.e.,
the heavy quark and anti-quark) into the final state quarkonium. We derive an
all order factorization theorem for this process in terms of double parton
fragmentation functions (DPFF) and calculate the one-loop anomalous dimension
matrix for the DPFF.
