Inclusive photoproduction of charmonia-bottomonia pairs

In this article we analyze the inclusive photoproduction of heavy charmonia-bottomonia pairs in the color glass condensate framework and demonstrate that the cross section of the process is sensitive to dipole and quadrupole forward scattering amplitudes (2- and 4-point correlators of Wilson lines). Using the phenomenological parametrizations of these amplitudes, we estimate numerically the production cross sections in the kinematics of the forthcoming Electron Ion Collider and the ultraperipheral collisions at the LHC. We found that the contribution controlled by the quadrupole amplitude is dominant, and for this reason, the suggested channel can be used as a gateway for studies of this nonperturbative object.

Figure 1

Diagrams which describe the inclusive heavy quark pair production in the CGC picture in leading order over ${\alpha }_s$. The subscript numbers 1, 2 enumerate the heavy quark lines in our convention. The subscript letter $z$ is the coordinate of the interaction vertex in the configuration space. The red block represents the interaction with the target (shockwave).

Figure 2

Diagrams which describe the inclusive charmonia-bottomonia pairs production in the CGC picture in the leading order over ${\alpha }_s$. The diagrams in the right column differ from diagrams in the left column just by inversion of the quark line in the upper loop (charge conjugation). The dominant color singlet contribution comes from the diagrams in the last row (diagrams 5 and 6 respectively). The subscript numbers 1–4 enumerate the heavy quark lines in our convention. The subscript letters $z_1\dots z_3$ stand for the coordinates of the interaction vertices in the configuration space. The red block represents the interaction with the target (shockwave).

Figure 3

Additional diagrams which in general should be taken into account for inclusive quarkonia pair production in the CGC picture in the leading order over ${\alpha }_s$ do not contribute to charmonia-bottomonia pairs. The subscript numbers 1–4 enumerate the heavy quark lines in our convention. The subscript letters $z_1\dots z_3$ stand for the coordinates of the interaction vertices in the configuration space. The red block represents the interaction with the target (shockwave).

Figure 4

The cross sections of inclusive production of different quarkonia pairs with opposite $C$ parities. For the sake of definiteness we considered that both quarkonia are produced with the same absolute value of the transverse momenta $p_1^{\perp }=p_2^{\perp }=p_T$. Upper and lower rows correspond to kinematics of ultraperipheral collisions and the Electron-Ion Collider, respectively. The left column corresponds to cross sections of the photon-proton subprocess, the right column includes predictions for the cross section of the full process. For ultraperipheral kinematics at the LHC we consider collisions of protons with lead (${}_{82}{}^{208}\mathrm{Pb}$) ions, and the cross sections are given per nucleon. For the EIC we considered the electron-proton collisions only; for heavy nuclei these results should be understood as cross sections per nucleon. For the sake of definiteness we considered production at central rapidities ($y_1=y_2=0$) in the lab frame. The shape of the $p_T$ dependence has a very mild dependence on rapidity and azimuthal angle $\varphi$ between the produced quarkonia.

Figure 5

Upper row: dependence of the normalized ratio $R(\varphi )$, defined in (54), on the angle $\varphi$ (azimuthal angle between transverse momenta of quarkonia). Left plot corresponds to the kinematics of ultraperipheral collisions at the LHC; right plot is for the kinematics of the Electron-Ion Collider. The cross section has almost the same angular dependence for all rapidities and transverse momenta $p_T$ (see the text for more detailed explanation). Lower row: comparison of angular dependence found with parametrization (41) (curve with label “Iancu et al.”) and a trivial parametrization (56).

Figure 6

The ratio of the cross sections (57) found with different parametrizations of the quadrupole amplitude. The deviation of this ratio from unity demonstrates the dominance of the quadrupole contribution in the parametrization (41).

Figure 7

Rapidity dependence of the cross section in the kinematics of the ultraperipheral collisions at the LHC. Plots in the upper row correspond to a configuration with equal rapidities of the produced quarkonia, $y_1=y_2$, whereas the lower row corresponds to rapidities which differ by a sign in lab frame, $y_1=-y_2$. In both rows the left plot corresponds to the cross section of the photoproduction subprocess, and the right column shows predictions for the cross section of the full process $pA\to A+M_1{M}_{2}X$, assuming proton-lead collisions, as defined in (2).

Figure 8

Rapidity dependence of the cross section in the kinematics of the future Electron-Ion Collider. The left column corresponds to the cross section of the photoproduction subprocess, and the right column shows predictions for the cross section of the full electroproduction process $eA\to eA{M}_1{M}_{2}X$. Due to the heavy mass of the final state, the CGC approach is not applicable for $y\lesssim -1$, and for $y\gtrsim 1$ the flux of the equivalent photons is suppressed due to leptonic factor (energy of the emitted virtual photon approaches from below the energy of the electron).

Figure 9

Single-differential cross-section $d\sigma /d{\mathcal{M}}_{12}$ as a function of invariant mass ${\mathcal{M}}_{12}$ of the produced heavy quarkonia pair. For $pA$ collisions the cross section is given on a per nucleon basis, for proton-lead (${}_{82}{}^{208}\mathrm{Pb}$) collisions. In order to have comparable magnitudes in different channels, the cross section of $ep\to eJ/\psi {\eta }_{b}X$ is multiplied by a common factor 200 in both plots.

Figure 10

Single-differential cross sections at the EIC (upper row) and in ultraperipheral kinematics at the LHC (lower row). Left column: rapidity dependence of the cross-section $d\sigma /d{y}_1$. Solid line corresponds to the cross section with constraint (58); without it the cross sections would follow the dashed line. Right plot: transverse momentum dependence of the cross-section $d\sigma /d{p}_1^2$. In all plots the scale on the right-hand side shows the quarkonia pair production rates per unit of time (hour) per unit of rapidity ($d{y}_1$) or transverse momentum ($d{p}_1^2$), assuming the instantaneous luminosities given in Table 1. For the $pA$ collisions, the cross sections are given on a per nucleon basis.

Figure 11

Leading order contribution to the wave function ${\psi }_{\overline{Q}Q\overline{Q}Q}^{(\gamma )}$ defined in the text. The momenta $k_i$ shown in the right-hand side are Fourier conjugates of the coordinates $x_i$. It is implied that both diagrams should be supplemented by all possible permutations of final state quarks (see the text for more details).