Kinematic distributions of the ${\eta }_c$ photoproduction in $ep$ collisions within the nonrelativistic QCD framework

We study the ${\eta }_c$ photoproduction in $ep$ collisions in this paper. The short-distance coefficients for $c\overline{c}({}^1{S}_0^{[1]})$, $c\overline{c}({}^1{S}_0^{[8]})$, $c\overline{c}({}^3{S}_1^{[8]})$, and $c\overline{c}({}^1{P}_1^{[8]})$ photoproductions via both direct and resolved photons are evaluated at leading order in ${\alpha }_s$ expansion, where the color-singlet contribution is achieved for the first time. The dependence of the cross sections on the renormalization and factorization scales, and the charm quark mass is studied in this paper. As an interesting result, the color-singlet cross section is dominated by the resolved photoproduction, which is almost two orders of magnitude larger than the contributions from the direct photons. The resolved photoproduction also enhances the color-octet cross sections considerably. By including the contributions from direct and resolved photons, we find the color-singlet channel dominates the integrated cross section. Making use of different sets of long-distance matrix elements, we observe some apparently distinguishable predictions, which can be utilized to scrutinize the validity of these matrix elements. Moreover, we carefully analyze different kinematic distributions of the cross sections and find that, in high $p_t$ regions, the color-singlet contribution is considerably suppressed comparing with the color-octet parts for most sets of long-distance matrix elements. This feature renders the ${\eta }_c$ photoproduction process another laboratory to test the color-octet mechanism in nonrelativistic QCD.

Figure 1

The renormalization scale dependence of the cross sections. The upper and mid plots display the contributions from the direct and resolved photons, respectively. Their summation is given in the lower plot.

Figure 2

The factorization scale dependence of the cross sections. The upper and mid plots display the contributions from the direct and resolved photons, respectively. Their summation is given in the lower plot.

Figure 3

The charm mass dependence of the cross sections. The upper and mid plots display the contributions from the direct and resolved photons, respectively. Their summation is given in the lower plot.

Figure 4

The $p_t$ (upper), $W$ (mid), and $z$ (lower) distribution of ${\eta }_c$ photoproduction employing different sets of LDMEs.

Figure 5

The $p_t$ (upper), $W$ (mid), and $z$ (lower) distribution of ${\eta }_c$ photoproduction for each channel.