We cure the perturbative instability of the total-inclusive-photoproduction cross sections of vector $S$-wave quarkonia observed at high photon-proton-collision energies ($\sqrt{ s_{ \gamma p}}$) in Next-to-Leading Order (NLO) Collinear-Factorisation (CF) computations. This is achieved using High-Energy Factorisation (HEF) in the Doubly-Logarithmic Approximation (DLA), which is a subset of the Leading-Logarithmic Approximation (LLA) of HEF which resums higher-order QCD corrections proportional to $\alpha_s^n \ln^{ n-1} (\hat{ s}/M^2)$ in the Regge limit $\hat{ s}\gg M^2$ with $M^2$ being the quarkonium mass and $\hat{ s}$ is the squared partonic-center-of-mass energy. Such a DLA is strictly consistent with the NLO and NNLO DGLAP evolutions of the Parton Distribution Functions. By improving the treatment of the large-$\hat{ s}$ asymptotics of the CF coefficient function, the resummation cures the unphysical results of the NLO CF calculation. The matching is directly performed in $\hat{ s}$ space using the Inverse-Error Weighting matching procedure which avoids any possible double counting. The obtained cross sections are in good agreement with data. In addition, the scale-variation uncertainty of the matched result is significantly reduced compared to the LO results. Our calculations also yield closed-form analytic limits for $\hat{ s}\gg M^2$ of the NLO partonic CF and numerical limits for contributions to those at NNLO scaling like $\alpha_s^2 \ln(\hat{ s}/M^2)$.
Categories: hep-ph hep-ex