The exclusive photoproduction of $χ_{c}γ$ pairs in the small-$x$ kinematics

TL;DR

This work investigates exclusive photoproduction of $χ_{c} γ$ pairs in small-$x$ kinematics using the Color Glass Condensate framework, showing that the amplitude factorizes into process-dependent impact factors convolved with the universal forward dipole amplitude $\mathcal{N}(x,\boldsymbol{r},\boldsymbol{b})$ and incorporating both real-part and skewedness corrections for exclusive processes. By employing light-cone wave functions for the $χ_c$ states and NRQCD-inspired connections to LDMEs, the authors derive explicit helicity amplitudes and perform numerical estimates with the bCGC dipole model, predicting cross sections of order tens to hundreds of picobarns for $χ_c$ states in $\gamma p$ collisions at $W \sim 100$ GeV and extending to LHC and EIC kinematics. They find that the $χ_{c} γ$ channel can provide large event rates and represents a potentially dominant background to odderon-mediated $χ_{c}$ photoproduction at small momentum transfer, with backgrounds from $ψ(2S)\to χ_{c} γ$ decays remaining significant at moderate $|t|$. The results underscore the importance of kinematic cuts on $t$, $t'$, and $M_{γ χ_c}$ for disentangling direct odderon signals from non-resonant $χ_{c} γ$ production and radiative decays, illustrating the utility of CGC-based exclusive processes as probes of dipole correlators and HQSS in heavy quarkonia.

Abstract

In this manuscript we analyze the exclusive photoproduction of the $χ_{c}γ$ pairs. We focus on the small-$x$ kinematics and evaluate the cross-sections in the Color Glass Condensate framework. We found that in the leading order in the strong coupling $α_{s}$, this process is sensitive only to the forward color dipole scattering amplitude. We estimated numerically the cross-sections for different polarizations of $χ_{c}$ mesons in the kinematics of the ultraperipheral collisions at LHC and the future Electron Ion Collider. We also analyzed the role of this process as a potential background to exclusive $χ_{c}$ photoproduction, which has been recently suggested as an alternative channel for studies of odderons. According to our estimates, the cross-section of $χ_{c}γ$ with undetected photons is comparable to that of odderons at small momentum transfer $|t|\lesssim1$ GeV$^{2}$, but becomes less relevant at larger $|t|$. We also found that a dominant background to odderon-mediated production of $χ_{c}$ comes from the radiative decays of $ψ(2S)$ mesons, which potentially can present a challenge for odderon studies via $χ_{c}$ photoproduction.

Figures (13)

• Figure 1: (Color online) The definition of the angle $\varphi$ between leptonic and hadronic planes for the $ep\to e'p'\chi_{c}\gamma$ electroproduction channel, as seen from the target rest frame. The lepton scattering plane is formed by the three-momenta of incoming and scattered electrons (marked with labels $e,e'$), and the hadron scattering plane may be defined as a plane that includes the three-momenta of the recoiled proton $P_{{\rm out}}$ and the total three-momentum $p_{\chi_{c}}+k_{\gamma}$ of the produced $\gamma\chi_{c}$ pair. Due to momentum conservation, this plane also includes the 3-momentum of the virtual photon $\gamma^{*}$. In the photon-proton collision frame, the incoming proton propagates in direction anticollinear to the incoming virtual photon $\gamma^{*}$, so the momentum of the recoil proton $P_{{\rm out}}$ is almost anticollinear to the momentum of the virtual photon $\gamma^{*}$.
• Figure 2: The dominant mechanism of $\chi_{c}\gamma$ pair photoproduction in the CGC framework in the leading order in $\alpha_{s}$. The red block stands for a shock wave. The diagrams in the first and the second row are related by charge conjugation (inversion of the quark line). In the left column we disregard interaction of the emitted photon with a shock wave as tiny $\mathcal{O}\left(\alpha_{{\rm em}}\right)$-correction, and to emphasize this we don't put “ dot” on a photon line which crosses the shock wave.
• Figure 3: The definitions of the parton momenta $k_{0},k_{1},k_{0}',k_{1}'$ before and after interaction with the shock wave. The variables $\boldsymbol{x}_{0},\boldsymbol{x}_{1}$ stand for the transverse coordinates of the quark and antiquark when they pass through the shock wave.
• Figure 4: The diagrams which correspond to the $\gamma\to\bar{Q}Q\gamma$ amplitude in the leading order in strong coupling $\alpha_{s}$. The contributions of the two diagrams in the first row correspond to expressions in the second lines of (\ref{['eq:a-1']}) and (\ref{['eq:a-1-1']}). Similarly, the contributions of the diagrams in the last row are given by the expressions in the third lines of the same Equations (\ref{['eq:a-1']}) and (\ref{['eq:a-1-1']}). The vertical fermion line with a short horizontal dash in the diagrams of the last row denotes instantaneous part of the quark propagator. The colored vertical dashed lines stand for the light-cone energy denominators of the canonical light-cone perturbation theory Brodsky:1997deLepage:1980fj.
• Figure 5: The leading order diagrams which contribute to the amplitude of the subprocess $Q\bar{Q}\to\gamma\chi_{c}$. The vertical dashed lines denote the energy denominators of the light-cone perturbation theoryLepage:1980fjBrodsky:1997de.