Curing the high-energy perturbative instability of vector-quarkonium-photoproduction cross sections at order $\alpha \alpha_s^3$ with high-energy factorisation

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 $\alpha_s^2 \ln(\hat{s}/M^2)$.