Using our measurement of the ratio between and and their known radiative decay branching ratio we can calculate the ratio between plus and direct cross sections. This is plotted in Figure 8 and shows that , and are produced with about equal cross sections. To obtain the direct cross section we subtract from the inclusive prompt cross section the contribution from and decays. The contribution from is obtained from the measured fraction of from . These fractions, together with the fraction of from , are shown in Figure 9 .
Direct production is the largest production mechanism of prompt .
While the normalization of the from cross section is in reasonable agreement with the theoretical calculation the direct data is a factor above the prediction. We conclude that the color singlet charmonium production model underestimates direct production for both and by the same factor of .