Exclusive quarkonium photoproduction: predictions with the Balitsky-Kovchegov equation including the full impact-parameter dependence

TL;DR

The paper addresses exclusive diffractive photoproduction of quarkonia as a probe of small-$x$ gluons through ultra-peripheral collisions. It employs the Balitsky-Kovchegov equation with full impact-parameter dependence to evolve the dipole amplitude $N(r,b;\eta)$ in target rapidity, couples it to photon–vector-meson overlap in a dipole framework, and computes cross-sections as functions of $W_{\gamma p}$ and Mandelstam-$t$, including nonlinear saturation effects controlled by a tunable parameter $\kappa$. Vector-meson wave functions are modeled with boosted Gaussian forms, with parameters extracted from PDG data, enabling predictions for ground and excited states ${\rm J}/\psi$, ${\Psi(2S)}$, and ${\Upsilon(nS)}$ and their ratios. The key findings are that nonlinear BK terms slow the evolution (more prominently at larger $|t|$) and that excited-to-ground-state cross-section ratios exhibit a steeper energy rise under nonlinear dynamics, yielding testable predictions for LHC UPCs and future Electron-Ion Collider measurements to constrain saturation effects in exclusive quarkonium production.

Abstract

The ongoing Run 3 at the Large Hadron Collider (LHC) is substantially increasing the luminosity delivered to the experiments during Run 1 and Run 2. The advent of the high-luminosity upgrade of the LHC (Run 4 to 6), as well as the improvements to all detectors, will allow for the collection of an unprecedented amount of data in the next decade. This opens the possibility of performing measurements which have been limited by the smallness of the available data samples. This is the case of multi-differential studies of $J/ψ$, as well as of $Υ$ excited states, in exclusive diffractive photon-induced interactions. Here, we present predictions for the cross-sections of these processes utilising the dipole amplitude from the Balitsky-Kovchegov (BK) equation solved in the target rapidity and including the full impact-parameter dependence. Cross-sections are computed as a function of the photon--proton centre-of-mass energy as well as a function of Mandelstam-$t$. Ratios of cross-sections for different states and for the same state at different Mandelstam-$t$ values are also presented. The contribution to these observables of the non-linear terms in the BK equation is discussed.

Figures (6)

• Figure 1: The $J/\psi$ production data used for extraction of the model parameters. Measurements by H1 H1:2013okq were used for the differential cross section (left panel) and a combination of measurements by ZEUS, H1, ALICE, and LHCb ZEUS:2002wfjH1:2005dtpH1:2013okqALICE:2014eofLHCb:2018rcm for the total cross section (right panel).
• Figure 2: Dipole amplitude at the initial condition and after evolution to rapidity 10 for the case of parallel dipole-size and impact-parameter vectors.
• Figure 3: Scalar part of charmonium (left panel) and bottomonium (right panel) wave functions according to the boosted Gaussian model for transverse polarisation.
• Figure 4: Energy dependence of exclusive diffractive quarkonium photoproduction. The upper panels show the comparison of the predictions with data of the H1 collaboration H1:2005dtp for photo- (left) and electroproduction (right). The lower panels show the prediction for the energy dependence of the $\Psi(2S)$ (left) and $\Upsilon(nS)$ (right) cross-sections integrated over Mandelstam-$t$ in the range $|t|\in (0,1.2)$ GeV$^2$. The predictions are compared to $\Psi(2S)$ data from the LHCb collaboration LHCb:2018rcm and to $\Upsilon(1S)$ data from the H1 and ZEUS collaborations at HERA and the LHCb and CMS collaborations at the LHC Breitweg:1998kiH1:2000kisChekanov:2009zzLHCb:2015wlxCMS:2018bbk
• Figure 5: Left: Ratio of the cross-sections for ${\rm J}/\psi$ exclusive diffractive photoproduction considering only the linear part of the BK evolutions, obtained by setting $\kappa=0$, to the full evolution, including non-linear terms, corresponding to use $\kappa=1$, see Eq. (\ref{['eq:BK']}). The dependence on the value of the Mandelstam-$t$ variable is shown with the different lines. Right: Ratio of the cross-section for quarkonium exclusive diffractive photoproduction for $|t|=0.84$ GeV$^2$ to $|t|=0.03$ GeV$^2$. Solid lines represent the case $\kappa=1$ and long-dashed lines $\kappa=0$.