Prompt hadroproduction of $\eta_c(1S,2S)$ in the $k_T$-factorization approach

In this work, we present a thorough analysis of $\eta_c(1S,2S)$ quarkonia hadroproduction in $k_\perp$-factorisation in the framework of the light-front potential approach for the quarkonium wave function. The off-shell matrix elements for the $g^* g^* \eta_c(1S,2S)$ vertices are derived. We discuss the importance of taking into account the gluon virtualities. %We discuss the role of relativistic corrections, the Melosh spin-rotation and theoretical uncertainties by comparing our results with the standard NRQCD approach. We present the transverse momentum distributions of $\eta_c$ for several models of the unintegrated gluon distributions. Our calculations are performed for four distinct parameterisations for the $c\bar c$ interaction potential consistent with the meson spectra. We compare our results for $\eta_c(1S)$ to measurements by the LHCb collaboration and present predictions for $\eta_c(2S)$ production.