Associated production of ${\chi }_c$ pairs with a gluon in the collinear-factorization approach

We calculate cross section for production of ${\chi }_c$ pairs in proton-proton collisions. The cross section for the $gg\to {\chi }_{c{J}_1}{\chi }_{c{J}_2}$ is considerably smaller (especially for the ${\chi }_{c1}{\chi }_{c1}$ final state) than that obtained recently in the $k_T$-factorization approach. We calculate therefore next-to-leading order (NLO) contributions with the ${\chi }_c$ pair and one extra associated (mini-)jet. We find these contributions to be much larger than those for the $2\to 2$ contribution. Especially the emission of a leading gluon (carrying a large momentum fraction of one of the incoming gluons) is important. These emissions in the $k_T$-factorization approach are absorbed into the initial state unintegrated gluon distributions. A smaller contribution to the cross section comes from the production of central gluons emitted with rapidities between the ${\chi }_c$ mesons. They do lead, however, to an enhancement of the ${\chi }_c$-pair production at large rapidity distance between the mesons. Our present study explains the size of the cross section for the ${\chi }_c$-pair production obtained previously in the $k_T$-factorization approach. Several differential distributions are presented.

Figure 1

A diagrammatic representation of the leading order mechanisms for the $pp\to {\chi }_{c{J}_1}{\chi }_{c{J}_2}$ reaction.

Figure 2

The lowest-order mechanisms for the ${\chi }_{c{J}_1}{\chi }_{c{J}_2}g$ production in the high-energy kinematics described in the text.

Figure 3

Differential cross section for the processes from Figs. 1 and 2 (C) at the parton level, where the energy in the center of mass of two gluons was fixed for $W=50\mathrm{GeV}$. The left panel is for extra rapidity veto and the right panel without the extra condition.

Figure 4

Differential cross section for the processes from Figs. 1 and 2 (C) at the parton level, where the energy in the center of mass of two gluons was fixed for $W=50\mathrm{GeV}$. The left panel is for extra rapidity veto and the right panel without the extra condition.

Figure 5

Transverse momenta distributions of one of the ${\chi }_c$ for the $pp\to {\chi }_{cJ}{\chi }_{cJ}$ and $pp\to {\chi }_{cJ}{\chi }_{cJ}g$ reaction for $\sqrt{s}=8\mathrm{TeV}$.

Figure 6

Rapidity distributions of ${\chi }_c$ mesons from $2\to 2$ and $2\to 3$ processes shown in Fig. 2 for ${\chi }_{c0}{\chi }_{c0}$ (left), ${\chi }_{c1}{\chi }_{c1}$ (middle) and ${\chi }_{c2}{\chi }_{c2}$ (right). Here ${\mu }_f^2=\widehat{s}$ was used.

Figure 7

Rapidity distributions of ${\chi }_c$ mesons and gluons from the $gg\to \chi g\chi$ processes with a central gluon. Here ${\mu }_f^2=\widehat{s}$ was used.

Figure 8

Rapidity distributions of ${\chi }_c$ mesons and gluons from the $gg\to g{\chi }_c{\chi }_c$ processes with a leading gluon. Here ${\mu }_f^2=\widehat{s}$ was used.

Figure 9

Distribution in the difference of rapidities of ${\chi }_{cJ}$ mesons from processes shown in diagrams [2]. The most external lines are from the process where the gluon is emitted among the two ${\chi }_{cJ}$ mesons.

Figure 10

Distribution in the difference of rapidities between ${\chi }_{cJ}$ meson and gluon from diagrams presented in Fig. 2. The most external lines in the plot are for difference between the external gluon and the external ${\chi }_{cJ}$ meson [diagrams (A) and (B) from Fig. 2].

Figure 11

Two-dimensional distributions ($p_{T1}\times p_{T2}$) for ${\chi }_{c0}{\chi }_{c0}$ production. Here ${\mu }_f^2=\widehat{s}$.

Figure 12

Two-dimensional distributions ($p_{T1}\times p_{T2}$) for ${\chi }_{c1}{\chi }_{c1}$ production. Here ${\mu }_f^2=\widehat{s}$.

Figure 13

Two-dimensional distributions ($p_{T1}\times p_{T2}$) for ${\chi }_{c2}{\chi }_{c2}$ production. Here ${\mu }_f^2=\widehat{s}$.

Figure 14

Distribution in ${\overrightarrow{p}}_{T,\mathrm{sum}}$, where ${\overrightarrow{p}}_{T,\mathrm{sum}}={\overrightarrow{p}}_{1T}+{\overrightarrow{p}}_{2T}$ for $pp\to g[{\chi }_{cJ}{\chi }_{cJ}]$ (the left panel) and $pp\to {\chi }_{cJ}g{\chi }_{cJ}$ (the right panel) processes for $\sqrt{s}=8\mathrm{TeV}$. Here ${\mu }_f^2=\widehat{s}$.