Searching for gluon saturation effects in the momentum transfer dependence of coherent charmonium electroproduction off nuclei

TL;DR

The paper develops a rigorous Green function–based color-dipole framework for coherent charmonium electroproduction on nuclei, incorporating color transparency, reduced coherence, and leading-twist gluon shadowing through $|Q\bar{Q}G\rangle$ Fock components. By using a realistic, $x$-dependent dipole amplitude with dipole–orientation correlations and Lorentz-boosted quarkonium wave functions (including Melosh rotations) and solving a two-dimensional Schrödinger equation for the $Q\bar{Q}$ pair in the nuclear medium, the authors generate predictions for the $t$-dependent cross sections and nuclear modification factors across RHIC/LHC and LHeC kinematics. They find non-monotonic $W$-dependence of $d\sigma/dt$ at large $t$ as a potential signal of gluon saturation, while also highlighting strong CT and GS effects at LHeC energies and notable reduced coherence effects at EIC energies. They discuss several caveats—wave-function modeling, $\vec{r}$–$\vec{b}$ correlations, BK kernel choices, and higher-order shadowing—requiring careful consideration before interpreting a saturation signal, especially for radially excited states such as $\psi'(2S)$.

Abstract

We study for the first time the transverse momentum transfer distributions $dσ/dt$ in coherent production of charmonia in nuclear ultra-peripheral and electron-ion collisions within the QCD color dipole approach based on a rigorous Green function formalism. This allows us to treat properly the color transparency effects, as well as the higher and leading-twist shadowing corrections associated with the $|Q\bar Q\rangle$ and $|Q\bar QnG\rangle$ Fock components of the photon. While the multi-gluon photon fluctuations represent the dominant source of nuclear shadowing at kinematic regions related to the recent LHC and its future upgrade to LHeC, the upcoming electron-ion collider at RHIC will additionally require the proper incorporation of reduced quark shadowing. The latter effect leads to a significant decrease in the differential cross sections $dσ/dt$ compared to standard calculations based on the eikonal form for the dipole-nucleus amplitude. The leading-twist shadowing corrections, corresponding to a non-linear QCD evolution of a partial dipole-nucleus amplitude, reduce substantially charmonium $t$-distributions in the LHeC energy range. We predict a non-monotonic energy dependence of $dσ/dt$ suggesting so possible gluon saturation effects with increased onset at larger $t$-values. In addition to shadowing corrections, we study how the color transparency effects affect the shape of $t$-dependent nuclear modification factor. We also briefly discuss several aspects that can modify the charmonium production rate and thus may have a large impact on the search for gluon saturation effects.

Figures (5)

• Figure 1: Gluon shadowing factor $R_G$ for photoproduction of $J\!/\!\psi$ on the gold target as function of impact parameter $b_A$ at several fixed values of $x$ (left), and as function of $x$ at fixed values of the photon virtuality $Q^2 = 0$ and $50\,\,\hbox{GeV}^2$ (right).
• Figure 2: Model calculations of $d\sigma^{\gamma(\gamma^{\ast}) p\to J\!/\!\psi p}(t)/dt$ at $Q^2=0.05\,\,\hbox{GeV}^2$ (upper left panel) and at $Q^2 = 8.9\,\,\hbox{GeV}^2$ (upper right panel) as a function of photon energy $W$ at several fixed values of $t$ vs. data from the H1 collaboration H1:2005dtp. Results for the $t$-integrated differential cross section in the photoproduction of $\psi^{\,\prime}(2S)$ are shown in the lower left panel and compared with data from the LHCb LHCb:2018rcm collaboration. The lower right panel shows the model calculations of $d\sigma^{\gamma Pb\to J\!/\!\psi Pb}(t)/dt$ in comparison with data from the ALICE experiment Acharya:2021bnz.
• Figure 3: The $t$-dependent nuclear transparency, Eq. (\ref{['tr-t']}), for the coherent electroproduction of $J\!/\!\psi(1S)$ (left panel) and $\psi^{\,\prime}(2S)$ (right panel) in the limit $l_c\gg R_A$ and at two fixed values of $Q^2 =0$ and $50\,\,\hbox{GeV}^2$.
• Figure 4: Model predictions for the energy behavior of the $t$-dependent differential cross section $d\sigma/dt$ for coherent photoproduction of $J\!/\!\psi(1S)$ (upper left panel), $\psi^{\,\prime}(2S)$ (upper middle panel) and for the $\psi^{\,\prime}(2S)$-to-$J\!/\!\psi(1S)$ ratio of $d\sigma/dt$ (upper right panel) at several fixed values of $t=0.001, 0.004, 0.008$ and $0.012\,\,\hbox{GeV}^2$. The lower panels show the electroproduction process at $Q^2 = 50\,\,\hbox{GeV}^2$. In all panels the Green function formalism with gluon shadowing corrections (solid lines) is compared with the standard eikonal approximation without gluon shadowing (dashed lines).
• Figure 5: (Color online) The comparison of the CCV and KKN model predictions for the ratio $R_{\perp/\parallel}(b)$ as a function of the impact parameter of a collision $b$. Here the solid, dashed, and dotted lines correspond to the production of $\Upsilon$, $J\!/\!\psi$, and $\rho^0$ mesons, respectively.